A Basic Introduction to Surgery Theory

Wolfgang Lück^{1}

Fachbereich Mathematik und Informatik Westfälische Wilhelms-Universität Münster

Einsteinstr. 62 48149 Münster

Germany

October 2, 2002

1email: lueck@math.uni-muenster.de www: http://www.math.uni-muenster.de/u/lueck/ FAX: 49 251 8338370

II

Preface

This manuscript contains extended notes of the lectures presented by the author at the summer school “High-dimensional Manifold Theory” in Trieste in May/June 2001. It is written not for experts but for talented and well educated graduate students or Ph.D. students who have some background in algebraic and differential topology. Surgery theory has been and is a very successful and well established theory. It was initiated and developed by Browder, Kervaire, Milnor, Novikov, Sullivan, Wall and others and is still a very active research area. The idea of these notes is to give young mathematicians the possibility to get access to the field and to see at least a small part of the results which have grown out of surgery theory. Of course there are other good text books and survey articles about surgery theory, some of them are listed in the references.

The Chapters 1 and 2 contain interesting and beautiful results such as the s-Cobordism Theorem and the classification of lens spaces including their illuminating proofs. If one wants to start with the surgery machinery immediately, one may skip these chapters and pass directly to Chapters 3, 4 and 5. As an application we present the classification of homotopy spheres in Chapter 6. Chapters 7 and 8 contain material which is directly related to the main topic of the summer school.

I thank the members of the topology group at Münster and the participants of the summer school who made very useful comments and suggestions, in particular C.L. Douglas, J. Verrel, T. A. Kro, R. Sauer and M. Szymik. I thank the ICTP for its hospitality and financial support for the school and the conference.

Münster, October 2, 2002 Wolfgang Lück IV Preface

Contents

1 The s-Cobordism Theorem 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Handlebody Decompositions . . . . . . . . . . . . . . . . . . . . . 4 1.2 Handlebody Decompositions and CW -Structures . . . . . . . . . 9 1.3 Reducing the Handlebody Decomposition . . . . . . . . . . . . . 12 1.4 Handlebody Decompositions and Whitehead Groups . . . . . . . 18 1.5 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2 Whitehead Torsion 23 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.1 Whitehead Groups . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.2 Algebraic Approach to Whitehead Torsion . . . . . . . . . . . . . 27 2.3 The Geometric Approach to Whitehead Torsion . . . . . . . . . . 34 2.4 Reidemeister Torsion and Lens Spaces . . . . . . . . . . . . . . . 38 2.5 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3 Normal Maps and the Surgery Problem 49 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.1 Poincaré Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.2 The Spivak Normal Fibration . . . . . . . . . . . . . . . . . . . . 56

3.2.1 Pontrjagin-Thom Construction . . . . . . . . . . . . . . . 57 3.2.2 Spherical Fibrations . . . . . . . . . . . . . . . . . . . . . 60 3.2.3 The Existence and Uniqueness of the Spivak Normal Fi-

bration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.3 Normal Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.4 The Surgery Step . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.4.1 Motivation for the Surgery Step . . . . . . . . . . . . . . 69 3.4.2 Immersions and Embeddings . . . . . . . . . . . . . . . . 73 3.4.3 The Surgery Step . . . . . . . . . . . . . . . . . . . . . . . 75

3.5 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4 The Algebraic Surgery Obstruction 79 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.1 Intersection and Selfintersection Pairings . . . . . . . . . . . . . . 80 4.1.1 Intersections of Immersions . . . . . . . . . . . . . . . . . 80 4.1.2 Selfintersections of Immersions . . . . . . . . . . . . . . . 82

4.2 Kernels and Forms . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.2.1 Symmetric Forms and Surgery Kernels . . . . . . . . . . . 85 4.2.2 Quadratic Forms and Surgery Kernels . . . . . . . . . . . 89

4.3 Even Dimensional L-Groups . . . . . . . . . . . . . . . . . . . . . 93 4.4 The Surgery Obstruction in Even Dimensions . . . . . . . . . . . 95 4.5 Formations and Odd Dimensional L-Groups . . . . . . . . . . . . 99 4.6 The Surgery Obstruction in Odd Dimensions . . . . . . . . . . . 102 4.7 Variations of the Surgery Obstruction and the L-Groups . . . . . 104

4.7.1 Surgery Obstructions for Manifolds with Boundary …. 105 4.7.2 Surgery Obstructions and Whitehead Torsion . . . . . . . 107

4.8 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

5 The Surgery Exact Sequence 113 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 5.1 The Structure Set . . . . . . . . . . . . . . . . . . . . . . . . . . 113 5.2 Realizability of Surgery Obstructions . . . . . . . . . . . . . . . . 114 5.3 The Surgery Exact Sequence . . . . . . . . . . . . . . . . . . . . 117 5.4 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

6 Homotopy Spheres 123 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 6.1 The Group of Homotopy Spheres . . . . . . . . . . . . . . . . . . 124 6.2 The Surgery Sequence for Homotopy Spheres . . . . . . . . . . . 127 6.3 The J-Homomorphism and Stably Framed Bordism . . . . . . . 130 6.4 Computation of bP^{n+1} . . . . . . . . . . . . . . . . . . . . . . . . 133 6.5 Computation of Θ^{n}/bP^{n+1} . . . . . . . . . . . . . . . . . . . . . 137 6.6 The Kervaire-Milnor Braid . . . . . . . . . . . . . . . . . . . . . 137 6.7 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

7 The Farrell-Jones and Baum-Connes Conjectures 143 7.1 G-Homology Theories . . . . . . . . . . . . . . . . . . . . . . . . 143 7.2 Isomorphisms Conjectures and Assembly Maps . . . . . . . . . . 148

7.2.1 Classifying Spaces of Families . . . . . . . . . . . . . . . . 148 7.2.2 The Formulation of the Isomorphism Conjectures . . . . . 149 7.2.3 Conclusions from the Isomorphism Conjectures . . . . . . 151

7.3 Characterizing Assembly Maps . . . . . . . . . . . . . . . . . . . 153 7.4 The Choice of the Families . . . . . . . . . . . . . . . . . . . . . 155 7.5 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

8 Computations of K- and L-Groups for Infinite Groups 161 8.1 Equivariant Chern Characters . . . . . . . . . . . . . . . . . . . . 161 8.2 Rational Computations . . . . . . . . . . . . . . . . . . . . . . . 167 8.3 Some Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . . 170 CONTENTS VII References 174 Notation 185 Index 187

Chapter 1

The s-Cobordism Theorem

Introduction

In this chapter we want to discuss and prove the following result

Theorem 1.1 (s-Cobordism Theorem) Let M_{0} be a closed connected oriented manifold of dimension n ≥ 5 with fundamental group π = π_{1}(M_{0}). Then

1. Let (W; M_{0}, f_{0}, M_{1}, f_{1}) be an h-cobordism over M_{0}. Then W is trivial over M_{0} if and only if its Whitehead torsion τ(W, M_{0}) ∈ Wh(π) vanishes;

2. For any x ∈ Wh(π) there is an h-cobordism (W; M_{0}, f_{0}, M_{1}, f_{1}) over M_{0} with τ(W, M_{0}) = x ∈ Wh(π);

3. The function assigning to an h-cobordism (W; M_{0}, f_{0}, M_{1}, f_{1}) over M_{0} its Whitehead torsion yields a bijection from the diffeomorphism classes relative M_{0} of h-cobordisms over M_{0} to the Whitehead group Wh(π).

2 CHAPTER 1. THE S-COBORDISM THEOREM

Here are some explanations. An n-dimensional cobordism (sometimes also called just bordism) (W; M_{0}, f_{0}, M_{1}, f_{1}) consists of a compact oriented n-dimen-sional manifold W, closed (n − 1)-dimensional manifolds M_{0} and M_{1}, a disjoint decomposition ∂W = ∂_{0}W ∐ ∂_{1}W of the boundary ∂W of W and orientation preserving diffeomorphisms f_{0} : M_{0} → ∂W_{0} and f_{1} : M_{1}^{−} → ∂W_{1}. Here and in the sequel we denote by M_{1}^{−} the manifold M_{1} with the reversed orientation and we use on ∂W the orientation with respect to the decomposition T_{x}W = T_{x}∂W ⊕ℝ coming from an inward normal field for the boundary. If we equip D^{2} with the standard orientation coming from the standard orientation on ℝ^{2}, the induced orientation on S^{1} = ∂D^{2} corresponds to the anti-clockwise orientation on S^{1}. If we want to specify M_{0}, we say that W is a cobordism over M_{0}. If ∂_{0}W = M_{0}, ∂_{1}W = M_{1}^{−} and f_{0} and f_{1} are given by the identity or if f_{0} and f_{1} are obvious from the context, we briefly write (W; ∂_{0}W, ∂_{1}W). Two cobordisms (W, M_{0}, f_{0}, M_{1}, f_{1}) and (W′, M_{0}, f′_{0}, M′_{1}, f′_{1}) over M_{0} are diffeomorphic relative M_{0} if there is an orientation preserving diffeomorphism F: W → W′ with F ∘ f_{0} = f′_{0}. We call an h-cobordism over M_{0} trivial, if it is diffeomorphic relative M_{0} to the trivial h-cobordism (M_{0} × [0, 1]; M_{0} × {0}, (M_{0} × {1})^{−}). Notice that the choice of the diffeomorphisms f_{i} do play a role although they are often suppressed in the notation. We call a cobordism (W; M_{0}, f_{0}, M_{1}, f_{1}) an h-cobordism, if the inclusions ∂_{i}W → W for i = 0, 1 are homotopy equivalences.

We will later see that the Whitehead group of the trivial group vanishes. Thus the s-Cobordism Theorem 1.1 implies

Theorem 1.2 (h-Cobordism Theorem) Any h-cobordism (W; M_{0}, f_{0}, M_{1}, f_{1}) over a simply connected closed n-dimensional manifold M_{0} with dim(W) ≥ 6 is trivial.

Theorem 1.3 (Poincaré Conjecture) The Poincaré Conjecture is true for a closed n-dimensional manifold M with dim(M) ≥ 5, namely, if M is simply connected and its homology H_{p}(M) is isomorphic to H_{p}(S^{n}) for all p ∈ ℤ, then M is homeomorphic to S^{n}.

\underline{\underline{Proof}} : We only give the proof for dim(M) ≥ 6. Since M is simply connected and H_{∗}(M) ≅ H_{∗}(S^{n}), one can conclude from the Hurewicz Theorem and Whitehead Theorem [121, Theorem IV.7.13 on page 181 and Theorem IV.7.17 on page 182] that there is a homotopy equivalence f: M → S^{n}. Let D_{i}^{n} ⊂ M for i = 0, 1 be two embedded disjoint disks. Put W = M −(int(D_{0}^{n}) ∐ int(D_{1}^{n})). Then W turns out to be a simply connected h-cobordism. Hence we can find a diffeomorphism F: (∂D_{0}^{n} × [0, 1], ∂D_{0}^{n} × {0}, ∂D_{0}^{n} × {1}) → (W, ∂D_{0}^{n}, ∂D_{1}^{n}) which is the identity on ∂D_{0}^{n} = ∂D_{0}^{n} × {0} and induces some (unknown) diffeomorphism f_{1} : ∂D_{0}^{n} × {1} → ∂D_{1}^{n}. By the Alexander trick one can extend f_{1} : ∂D_{0}^{n} = ∂D_{0}^{n}×\{1\} → ∂D_{1}^{n} to a homeomorphism \overline{f_{1}} : D_{0}^{n} → D_{1}^{n}. Namely, any homeomorphism f: S^{n−1} → S^{n−1} extends to a homeomorphism \overline{f}: D^{n} → D^{n} by sending t · x for t ∈ [0, 1] and x ∈ S^{n−1} to t · f(x). Now define a homeomorphism h: D_{0}^{n} ×{0}∪_{i_{0}} ∂D_{0}^{n} ×[0, 1]∪_{i_{1}} D_{0}^{n} ×{1} → M for the canonical inclusions i_{k}: ∂D_{0}^{n} × {k} → ∂D_{0}^{n} × [0, 1] for k = 0, 1 by h|_{D_{0}^{n}×{0}} = id, h|_{∂D_{0}^{n}×[0,1]} = F and h|_{D_{0}^{n}×\{1\}} = \overline{f_{1}}. Since the source of h is obviously homeomorphic to S^{n}, Theorem 1.3 follows.

In the case dim(M) = 5 one uses the fact that M is the boundary of a contractible 6-dimensional manifold W and applies the s-cobordism theorem to W with an embedded disc removed.

Remark 1.4 Notice that the proof of the Poincaré Conjecture in Theorem 1.3 works only in the topological category but not in the smooth category. In other words, we cannot conclude the existence of a diffeomorphism h: S^{n} → M. The proof in the smooth case breaks down when we apply the Alexander trick. The construction of \overline{f} given by coning f yields only a homeomorphism \overline{f} and not a diffeomorphism even if we start with a diffeomorphism f. The map \overline{f} is smooth outside the origin of D^{n} but not necessarily at the origin. We will see that not every diffeomorphism f: S^{n−1} → S^{n−1} can be extended to a diffeomorphism D^{n} → D^{n} and that there exist so called exotic spheres, i.e. closed manifolds which are homeomorphic to S^{n} but not diffeomorphic to S^{n}. The classification of these exotic spheres is one of the early very important achievements of surgery theory and one motivation for its further development (see Chapter 6).

Remark 1.5 In some sense the s-Cobordism Theorem 1.1 is one of the first theorems, where diffeomorphism classes of certain manifolds are determined by an algebraic invariant, namely the Whitehead torsion. Moreover, the Whitehead group Wh(π) depends only on the fundamental group π = π_{1}(M_{0}), whereas the diffeomorphism classes of h-cobordisms over M_{0} a priori depend on M_{0} itself. The s-Cobordism Theorem 1.1 is one step in a program to decide whether two closed manifolds M and N are diffeomorphic what is in general a very hard question. The idea is to construct an h-cobordism (W; M, f, N, g) with vanishing Whitehead torsion. Then W is diffeomorphic to the trivial h-cobordism over M what implies that M and N are diffeomorphic. So the surgery program would be:

1. Construct a homotopy equivalence f: M → N;

2. Construct a cobordism (W; M, N) and a map (F, f, id): (W; M, N) → (N × [0, 1], N × {0}, N × {1});

3. Modify W and F relative boundary by so called surgery such that F becomes a homotopy equivalence and thus W becomes an h-cobordism. During these processes one should make certain that the Whitehead torsion of the resulting h-cobordism is trivial.

The advantage of this approach will be that it can be reduced to problems in homotopy theory and algebra which can sometimes be handled by well-known techniques. In particular one will get sometimes computable obstructions for two homotopy equivalent manifolds to be diffeomorphic. Often surgery theory has proved to be very useful when one wants to distinguish two closed manifolds which have very similar properties. The classification of homotopy spheres (see Chapter 6) is one example. Moreover, surgery techniques can be applied to problems which are of different nature than of diffeomorphism or homeomorphism classifications.

In this chapter we want to present the proof of the s-Cobordism Theorem and explain why the notion of Whitehead torsion comes in. We will encounter a typical situation in mathematics. We will consider an h-cobordism and try to prove that it is trivial. We will introduce modifications which we can apply to a handlebody decomposition without changing the diffeomorphism type and which are designed to reduce the number of handles. If we could get rid of all handles, the h-cobordism would be trivial. When attempting to cancel all handles, we run into an algebraic difficulty. A priori this difficulty could be a lack of a good idea or technique. But it will turn out to be the principal obstruction and lead us to the definition of the Whitehead torsion and Whitehead group.

The rest of this Chapter is devoted to the proof of the s-cobordism Theorem 1.1. Its proof is interesting and illuminating and it motivates the definition of Whitehead torsion. But we mention that it is not necessary to go through it in order to understand the following chapters.

1.1 Handlebody Decompositions

In this section we explain basic facts about handles and handlebody decompositions.

Definition 1.6 The n-dimensional handle of index q or briefly q-handle is D^{q}× D^{n−q}. Its core is D^{q} ×{0}. The boundary of the core is S^{q−1} ×{0}. Its cocore is {0} × D^{n−q} and its transverse sphere is {0} × S^{n−q−1}.

Let (M, ∂M) be an n-dimensional manifold with boundary ∂M. If ϕ^{q}: S^{q−1}× D^{n−q} → ∂M is an embedding, then we say that the manifold M + (ϕ^{q}) defined by M ∪_{ϕ^{q}} D^{q} × D^{n−q} is obtained from M by attaching a handle of index q by ϕ^{q}.

Obviously M + (ϕ^{q}) carries the structure of a topological manifold. To get a smooth structure, one has to use the technique of straigthening the angle to get rid of the corners at the place, where the handle is glued to M. The boundary ∂(M + (ϕ^{q})) can be described as follows. Delete from ∂M the interior of the image of ϕ^{q}. We obtain a manifold with boundary together with a diffeomorphism from S^{q−1} × S^{n−q−1} to its boundary induced by ϕ^{q}|_{S^{q−1}×S^{n−q−1}}. If we use this diffeomorphism to glue D^{q} × S^{n−q−1} to it, we obtain a closed manifold, namely, ∂(M + (ϕ^{q})).

Let W be a compact manifold whose boundary ∂W is the disjoint sum ∂_{0}W ∐ ∂_{1}W. Then we want to construct W from ∂_{0}W × [0, 1] by attaching handles as follows. Notice that the following construction will not change ∂_{0}W = ∂_{0}W × {0}. If ϕ^{q}: S^{q−1} × D^{n−q} → ∂_{1}W is an embedding, we get by attaching a handle the compact manifold W_{1} = ∂_{0}W × [0, 1] + (ϕ^{q}) which is given by W ∪_{ϕ^{q}} D^{q} × D^{n−q}. Its boundary is a disjoint sum ∂_{0}W_{1} ∐ ∂_{1}W_{1}, where ∂_{0}W_{1} is the same as ∂_{0}W. Now we can iterate this process, where we attach a handle to ∂_{1}W_{1}. Thus we obtain a compact manifold with boundary

W = ∂_{0}W × [0, 1] + (ϕ_{1}^{q_{1}}) + (ϕ_{2}^{q_{2}}) + … + (ϕ_{r}^{q_{r}}),

whose boundary is the disjoint union ∂_{0}W ∐ ∂_{1}W, where ∂_{0}W is just ∂_{0}W ×{0}. We call such a description of W as above a handlebody decomposition of W relative ∂_{0}W. We get from Morse theory [54, Chapter 6], [82, part I].

Lemma 1.7 Let W be a compact manifold whose boundary ∂W is the disjoint sum ∂_{0}W ∐ ∂_{1}W. Then W possesses a handlebody decomposition relative ∂_{0}W, i.e. W is up to diffeomorphism relative ∂_{0}W = ∂_{0}W × {0} of the form

W = ∂_{0}W × [0, 1] + (ϕ_{1}^{q_{1}}) + (ϕ_{2}^{q_{2}}) + … + (ϕ_{r}^{q_{r}}).

If we want to show that W is diffeomorphic to ∂_{0}W × [0, 1] relative ∂_{0}W = ∂_{0}W ×{0}, we must get rid of the handles. For this purpose we have to find possible modifications of the handlebody decomposition which reduce the number of handles without changing the diffeomorphism type of W relative ∂_{0}W.

Lemma 1.8 (Isotopy lemma) Let W be an n-dimensional compact manifold whose boundary ∂W is the disjoint sum ∂_{0}W ∐ ∂_{1}W. If ϕ^{q}, ψ^{q}: S^{q−1} ×D^{n−q} → ∂_{1}W are isotopic embeddings, then there is a diffeomorphism W + (ϕ^{p}) → W + (ψ^{q}) relative ∂_{0}W.

\underline{\underline{Proof}} : Let i: S^{q−1} × D^{n−q} × [0, 1] → ∂_{1}W be an isotopy from ϕ^{q} to ψ^{q}. Then one can find a diffeotopy H: W × [0, 1] → W with H_{0} = id_{W} such that the composition of H with ϕ^{q} ×id_{[0,1]} is i and H is stationary on ∂_{0}W [54, Theorem 1.3 in Chapter 8 on page 184]. Thus H_{1} : W → W is a diffeomorphism relative ∂_{0}W and satisfies H_{1} ∘ ϕ^{q} = ψ^{q}. It induces a diffeomorphism W + (ϕ^{p}) → W + (ψ^{q}) relative ∂_{0}W.

Lemma 1.9 (Diffeomorphism lemma) Let W resp. W′ be a compact manifold whose boundary ∂W is the disjoint sum ∂_{0}W ∐ ∂_{1}W resp. ∂_{0}W′ ∐ ∂_{1}W′. Let F: W → W′ be a diffeomorphism which induces a diffeomorphism f_{0} : ∂_{0}W → ∂_{0}W′. Let ϕ^{q}: S^{q−1} × D^{n−q} → ∂_{1}W be an embedding. Then there is an embedding \overline{ϕ}^{q}: S^{q−1} ×D^{n−q} → ∂_{1}W′ and a diffeomorphism F′: W +(ϕ^{q}) → W′ +(\overline{ϕ}^{q}) which induces f_{0} on ∂_{0}W.

\underline{\underline{Proof}} : Put \overline{ϕ}^{q} = F ∘ ϕ^{q}.

Lemma 1.10 Let W be an n-dimensional compact manifold whose boundary ∂W is the disjoint sum ∂_{0}W ∐ ∂_{1}W. Suppose that V = W + (ψ^{r}) + (ϕ^{q}) for q ≤ r. Then V is diffeomorphic relative ∂_{0}W to V′ = W + (\overline{ϕ}^{q}) + (ψ^{r}) for an appropriate \overline{ϕ}^{q}.

\underline{\underline{Proof}} : By transversality and the assumption (q − 1) + (n − 1 − r) < n − 1 we can show that the embedding ϕ^{q}|_{S^{q−1}×{0}}: S^{q−1} × {0} → ∂_{1}(W + (ψ^{r})) is isotopic to an embedding which does not meet the transverse sphere of the handle (ψ^{r}) attached by ψ^{r} [54, Theorem 2.3 in Chapter 3 on page 78]. This isotopy can be embedded in a diffeotopy on ∂_{1}(W + (ψ^{r})). Thus the embedding ϕ^{q}: S^{q−1} × D^{n−q} → ∂_{1}(W + (ψ^{r})) is isotopic to an embedding whose restriction to S^{q−1} × {0} does not meet the transverse sphere of the handle (ψ^{r}). Since we can isotope an embedding S^{q−1}×D^{n−q} → W +(ψ^{r}) such that its image becomes arbitrary close to the image of S^{q−1} × {0}, we can isotope ϕ^{q}: S^{q−1} × D^{n−q} → ∂_{1}(W + (ψ^{r})) to an embedding which does not meet a closed neighborhood U ⊂ ∂_{1}(W + (ψ^{r})) of the transverse sphere of the handle (ψ^{r}). There is an obvious diffeotopy on ∂_{1}(W +(ψ^{r})) which is stationary on the transverse sphere of (ψ^{r}) and moves any point on ∂_{1}(W + (ψ^{r})) which belongs to the handle (ψ^{r}) but not to U to a point outside the handle (ψ^{r}). Thus we can find an isotopy of ϕ^{q} to an embedding \overline{ϕ}^{p} which does not meet the handle (ψ^{r}) at all. Obviously W + (ψ^{r}) + (\overline{ϕ}^{q}) and W + (\overline{ϕ}^{q}) + (ψ^{r}) agree. By the Isotopy Lemma 1.8 there is a diffeomorphism relative ∂_{0}W from W + (ψ^{r}) + (\overline{ϕ}^{q}) to W + (ψ^{r}) + (ϕ^{q}).

Example 1.11 Here is a standard situation, where attaching first a q-handle and then a (q + 1)-handle does not change the diffeomorphism type of an ndimensional compact manifold W with the disjoint union ∂_{0}W ∐ ∂_{1}W as boundary ∂W. Let 0 ≤ q ≤ n − 1. Consider an embedding

μ: S^{q−1} × D^{n−q} ∪_{S^{q−1}×S_{+}^{n−1−q}} D^{q} × S_{+}^{n−1−q} → ∂_{1}W,

where S_{+}^{n−1−q} is the upper hemisphere in S^{n−1−q} = ∂D^{n−q}. Notice that the source of μ is diffeomorphic to D^{n−1}. Let ϕ^{q}: S^{q−1} × D^{n−q} → ∂_{1}W be its restriction to S^{q−1} × D^{n−q}. Let ϕ_{+}^{q+1} : S_{+}^{q} × S_{+}^{n−q−1} → ∂_{1}(W + (ϕ^{q})) be the embedding which is given by

S_{+}^{q} × S_{+}^{n−q−1} = D^{q} × S_{+}^{n−q−1} ⊂ D^{q} × S^{n−q−1} = ∂(ϕ^{q}) ⊂ ∂_{1}(W + (ϕ^{q})).

It does not meet the interior of W. Let ϕ_{−}^{q+1} : S_{−}^{q} × S_{+}^{n−1−q} → ∂_{1}(W ∪ (ϕ^{q})) be the embedding obtained from μ by restriction to S_{−}^{q} × S_{+}^{n−1−q} = D^{q} × S_{+}^{n−1−q}. Then ϕ_{−}^{q+1} and ϕ_{+}^{q+1} fit together to yield an embedding ψ^{q+1} : S^{q} × D^{n−q−1} = S_{−}^{q} × S_{+}^{n−q−1} ∪_{S^{q−1}×S_{+}^{n−q−1}} S_{+}^{q} × S_{+}^{n−q−1} → ∂_{1}(W + (ϕ^{q})). Then it is not difficult to check that W + (ϕ^{q}) + (ψ^{q+1}) is diffeomorphic relative ∂_{0}W to W since up to diffeomorphism W + (ϕ^{q}) + (ψ^{q+1}) is obtained from W by taking the boundary connected sum of W and D^{n} along the embedding μ of D^{n−1} = S_{+}^{n−1} = S^{q−1} × D^{n−q} ∪_{S^{q−1}×S_{+}^{n−1−q}} D^{q} × S_{+}^{n−1−q} into ∂_{1}W.

This cancellation of two handles of consecutive index can be generalized as follows.

Lemma 1.12 (Cancellation Lemma) Let W be an n-dimensional compact manifold whose boundary ∂W is the disjoint sum ∂_{0}W ∐ ∂_{1}W. Let ϕ^{q}: S^{q−1} × D^{n−q} → ∂_{1}W be an embedding. Let ψ^{q+1} : S^{q} × D^{n−1−q} → ∂_{1}(W + (ϕ^{q})) be an embedding. Suppose that ψ^{q+1}(S^{q} × {0}) is transversal to the transverse sphere of the handle (ϕ^{q}) and meets the transverse sphere in exactly one point. Then there is a diffeomorphism relative ∂_{0}W from W to W + (ϕ^{q}) + (ψ^{q+1}).

\underline{\underline{Proof}} : Given any neighborhood U ⊂ ∂(ϕ^{q}) of the tranverse sphere of (ϕ^{q}), there is an obvious diffeotopy on ∂_{1}(W + (ϕ^{q})) which is stationary on the transverse sphere of (ϕ^{q}) and moves any point on ∂_{1}(W + (ϕ^{q})) which belongs to the handle (ϕ^{q}) but not to U to a point outside the handle (ϕ^{q}). Thus we can achieve that ψ^{q+1} maps the lower hemisphere S_{−}^{q} × {0} to points outside (ϕ^{q}) and is on the upper hemisphere S_{+}^{q} × {0} given by the obvious inclusion D^{q} × {x} → D^{p} × D^{n−q} = (ϕ^{q}) for some x ∈ S^{n−q−1} and the obvious identification of S_{+}^{q}×{0} with D^{q}×{x}. Now it is not hard to construct an diffeomorphism relative ∂_{0}W from W + (ϕ^{q}) + (ψ^{q+1}) to W modelling the standard situation of Example 1.11.

The Cancellation Lemma 1.12 will be our only tool to reduce the number of handles. Notice that one can never get rid of one handle alone, there must always be involved at least two handles simultaneously. The reason is that the Euler characteristic χ(W, ∂_{0}W) is independent of the handle decomposition and can be computed by ∑_{q≥0}(−1)^{q} · p_{q}, where p_{q} is the number of q-handles (see Section 1.2).

We call an embedding S^{q} × D^{n−q} → M for q < n into an n-dimensional manifold trivial if it can be written as the composition of an embedding D^{n} → W and a fixed standard embedding S^{q} × D^{n−q} → D^{n}. We call an embedding S^{q} → M for q < n trivial if it can be extended to a trivial embedding S^{q} × D^{n−q} → M. We conclude from the Cancellation Lemma 1.12

Lemma 1.13 Let ϕ^{q}: S^{q−1} × D^{n−q} → ∂_{1}W be a trivial embedding. Then there is an embedding ϕ^{q+1} : S^{q} × D^{n−1−q} → ∂_{1}(W + (ϕ^{q})) such that W and W + (ϕ^{q}) + (ϕ^{q+1}) are diffeomorphic relative ∂_{0}W.

Consider a compact n-dimensional manifold W whose boundary is the disjoint union ∂_{0}W ∐ ∂_{1}W. In view of Lemma 1.7 and Lemma 1.10 we can write it

W ≅ ∂_{0}W × [0, 1] + ∑_{i=1}^{p_{0}}(ϕ_{i}^{0}) + ∑_{i=1}^{p_{1}}(ϕ_{i}^{1}) + … + ∑_{i=1}^{p_{n}}(ϕ_{i}^{n}),  (1.14)

where ≅ means diffeomorphic relative ∂_{0}W.

Notation 1.15 Put for −1 ≤ q ≤ n

W_{q} := ∂_{0}W × [0, 1] + ∑_{i=1}^{p_{0}}(ϕ_{i}^{0}) + ∑_{i=1}^{p_{1}}(ϕ_{i}^{1}) + … + ∑_{i=1}^{p_{q}}(ϕ_{i}^{q});
∂_{1}W_{q} := ∂W_{q} − ∂_{0}W × {0};
∂_{1}^{∘}W_{q} := ∂_{1}W_{q} − ∐_{i=1}^{p_{q+1}} ϕ_{i}^{q+1}(S^{q} × int(D^{n−1−q})).

Notice for the sequel that ∂_{1}^{∘}W_{q} ⊂ ∂_{1}W_{q+1}.

Lemma 1.16 (Elimination Lemma) Fix an integer q with 1 ≤ q ≤ n − 3. Suppose that p_{j} = 0 for j < q, i.e. W looks like

W = ∂_{0}W × [0, 1] + ∑_{i=1}^{p_{q}}(ϕ_{i}^{q}) + ∑_{i=1}^{p_{q+1}}(ϕ_{i}^{q+1}) + … + ∑_{i=1}^{p_{n}}(ϕ_{i}^{n}).

Fix an integer i_{0} with 1 ≤ i_{0} ≤ p_{q}. Suppose that there is an embedding ψ^{q+1} : S^{q} × D^{n−1−q} → ∂_{1}^{∘}W_{q} with the following properties:

1. ψ^{q+1}|_{S^{q}×{0}} is isotopic in ∂_{1}W_{q} to an embedding ψ_{1}^{q+1} : S^{q} × {0} → ∂_{1}W_{q} which meets the transverse sphere of the handle (ϕ_{i_{0}}^{q}) transversally and in exactly one point and is disjoint from to the transverse sphere of ϕ_{i}^{q} for i ≠ i_{0};

2. ψ^{q+1}|_{S^{q}×{0}} is isotopic in ∂_{1}W_{q+1} to a trivial embedding ψ_{2}^{q+1} : S^{q}×{0} → ∂_{1}W_{q+1}.

Then W is diffeomorphic relative ∂_{0}W to a manifold of the shape

∂_{0}W ×[0, 1]+ ∑_{i=1,2,...,p_{q},i≠i_{0}}(ϕ_{i}^{q})+ ∑_{i=1}^{p_{q+1}}(\overline{ϕ}_{i}^{q+1})+(ψ^{q+2})+ ∑_{i=1}^{p_{q+2}}(\overline{ϕ}_{i}^{q+2})+…+∑_{i=1}^{p_{n}}(\overline{ϕ}_{i}^{n}).

\underline{\underline{Proof}} : Since ψ^{q+1}|_{S^{q}×\{0\}} is isotopic to ψ_{1}^{q+1} and ψ_{2}^{q+1} is trivial, we can extend ψ_{1}^{q+1} and ψ_{2}^{q+1} to embeddings denoted in the same way ψ_{1}^{q+1} : S^{q} × D^{n−q−1} → ∂_{1}W_{q} and ψ_{2}^{q+1} : S^{q} × D^{n−1−q} → ∂_{1}^{∘}W_{q+1} with the following properties [54, Theorem 1.5 in Chapter 8 on page 180]: ψ^{q+1} is isotopic to ψ_{1}^{q+1} in ∂_{1}W_{q}, ψ_{1}^{q+1} does not meet the transverse spheres of the handles (ϕ_{i}^{q}) for i ≠ i_{0}, ψ_{1}^{q+1}|_{S^{q}×{0}} meets the transverse sphere of the handle (ϕ_{i_{0}}^{q}) transversally and in exactly one point, ψ^{q+1} is isotopic to ψ_{2}^{q+1} within ∂_{1}W_{q+1} and ψ_{2}^{q+1} is trivial. Because of the Diffeomorphism Lemma 1.9 we can assume without loss of generality that there are no handles of index ≥ q + 2, i.e. p_{q+2} = p_{q+3} = … = p_{n} = 0. It suffices to show for appropriate embeddings \overline{ϕ}_{i}^{q+1} and ψ^{q+2} that

∂_{0}W × [0, 1] + ∑_{i=1}^{p_{q}}(ϕ_{i}^{q}) + ∑_{i=1}^{p_{q+1}}(ϕ_{i}^{q+1})

≅ ∂_{0}W × [0, 1] + ∑_{i=1,2,...,p_{q},i≠i_{0}}(ϕ_{i}^{q}) + ∑_{i=1}^{p_{q+1}}(\overline{ϕ}_{i}^{q+1}) + (ψ^{q+2}),

where ≅ means diffeomorphic relative ∂_{0}W. Because of Lemma 1.13 there is an embedding (ψ^{q+2}) satisfying

∂_{0}W × [0, 1] + ∑_{i=1}^{p_{q}}(ϕ_{i}^{q}) + ∑_{i=1}^{p_{q+1}}(ϕ_{i}^{q+1})

≅ ∂_{0}W × [0, 1] + ∑_{i=1}^{p_{q}}(ϕ_{i}^{q}) + ∑_{i=1}^{p_{q+1}}(ϕ_{i}^{q+1}) + (ψ_{2}^{q+1}) + (ψ^{q+2}).

We conclude from the Isotopy Lemma 1.8 and the Diffeomorphism Lemma 1.9 for appropriate embeddings ψ_{k}^{q+1} for k = 1, 2

∂_{0}W × [0, 1] + ∑_{i=1}^{p_{q}}(ϕ_{i}^{q}) + ∑_{i=1}^{p_{q+1}}(ϕ_{i}^{q+1}) + (ψ_{2}^{q+1}) + (ψ^{q+2})

≅ ∂_{0}W × [0, 1] + ∑_{i=1}^{p_{q}}(ϕ_{i}^{q}) + ∑_{i=1}^{p_{q+1}}(ϕ_{i}^{q+1}) + (ψ^{q+1}) + (ψ_{1}^{q+2})

≅ ∂_{0}W × [0, 1] + ∑_{i=1,2,...,p_{q},i≠i_{0}}^{p_{q}}(ϕ_{i}^{q}) + (ϕ_{i_{0}}^{q}) + (ψ^{q+1}) + ∑_{i=1}^{p_{q+1}}(ϕ_{i}^{q+1}) + (ψ_{2}^{q+2}).

1.2. HANDLEBODY DECOMPOSITIONS AND CW -STRUCTURES 9

We get from the Diffeomorphism Lemma 1.9 and the Cancellation Lemma 1.12 for appropriate embeddings \overline{ϕ}_{i}^{q+1} and ψ_{3}^{q+2}

∂_{0}W × [0, 1] + ∑_{i=1,2,...,p_{q},i≠i_{0}}^{p_{q}}(ϕ_{i}^{q}) + (ϕ_{i_{0}}^{q}) + (ψ^{q+1}) + ∑_{i=1}^{p_{q+1}}(ϕ_{i}^{q+1}) + (ψ_{2}^{q+2})

≅ ∂_{0}W × [0, 1] + ∑_{i=1,2,...,p_{q},i≠i_{0}}^{p_{q}}(ϕ_{i}^{q}) + ∑_{i=1}^{p_{q+1}}(\overline{ϕ}_{i}^{q+1}) + (ψ_{3}^{q+2}).

This finishes the proof of the Elimination Lemma 1.16.

1.2 Handlebody Decompositions and CW -Struc-tures

Next we explain how we can associate to a handlebody decomposition (1.14) a CW -pair (X, ∂_{0}W) such that there is a bijective correspondence between the q-handles of the handlebody decomposition and the q-cells of (X, ∂_{0}W). The key ingredient is the elementary fact that the projection (D^{q} × D^{n−q}, S^{q−1} × D^{n−q}) → (D^{q}, S^{q−1}) is a homotopy equivalence and actually – as we will explain later – a simple homotopy equivalence.

Recall that a (relative) CW -complex (X, A) consists of a pair of topological spaces (X, A) together with a filtration

X_{−1} = A ⊂ X_{0} ⊂ X_{1} ⊂ … ⊂ X_{q} ⊂ X_{q+1} ⊂ … ⊂ ∪_{q≥0}X_{q} = X

such that X carries the colimit topology with respect to this filtration and for any q ≥ 0 there exists a pushout of spaces

∐_{i∈I_{q}} S^{q−1} \underrightarrow{∐_{i∈Iq} ϕ_{i}^{q}} X_{q−1}
↓  ↓
∐_{i∈I_{q}} D^{q} \overrightarrow{∐_{i∈Iq} Φ_{i}^{q}} X_{q}

The map ϕ_{i}^{q} is called the attaching map and the map (Φ_{i}^{q}, ϕ_{i}^{q}) is called the characteristic map of the q-cell belonging to i ∈ I_{q}. The pushouts above are not part of the structure, only their existence is required. Only the filtration {X_{q} | q ≥ −1} is part of the structure. The path components of X_{q} − X_{q−1} are called the open cells. The open cells coincide with the sets Φ_{i}^{q}(D^{q} − S^{q−1}). The closure of an open cell Φ_{i}^{q}(D^{q} − S^{q−1}) is called closed cell and turns out to be Φ_{i}^{q}(D^{q}).

Suppose that X is connected with fundamental group π. Let p: X̃ → X be the universal covering of X. Put X̃_{q} = p^{−1}(X_{q}) and Ã = p^{−1}(A). Then (X̃, Ã) inherits a CW -structure from (X, A) by the filtration {X̃_{q} | q ≥ −1}. The cellular ℤπ-chain complex C_{∗}(X̃, Ã) has as q-th ℤπ-chain module the singular homology H_{q}(X̃_{q}, { \atop X_{q−1}}) with ℤ-coefficients and the π-action coming from the deck transformations. The q-th differential d_{q} is given by the composition

H_{q}(X̃_{q}, { \atop X_{q−1}}) ⟶^{∂_{q}} H_{q−1}({ \atop X_{q−1}}) ⟶^{i_{q}} H_{q−1}({ \atop X_{q−1}}, { \atop X_{q−2}}),

where ∂_{q} is the boundary operator of the long exact sequence of the pair (X̃_{q}, { \atop X_{q−1}}) and i_{q} is induced by the inclusion. If we choose for each i ∈ I_{q} a lift (Φ̃_{i}^{q}, ϕ̃_{i}^{q}): (D^{q}, S^{q−1}) → (X̃_{q}, { \atop X_{q−1}}) of the characteristic map (Φ_{i}^{q}, ϕ_{i}^{q}), we obtain a ℤπ-basis {b_{i} | i ∈ I_{n}} for C_{n}(X̃, Ã), if we define b_{i} as the image of a generator in H_{q}(D^{q}, S^{q−1}) ≅ ℤ under the map H_{q}(Φ̃_{i}^{q}, ϕ̃_{i}^{q}): H_{q}(D^{q}, S^{q−1}) → H_{q}(X̃_{q}, { \atop X_{q−1}}) = C_{q}(X̃, Ã). We call \{b_{i} | i ∈ I_{n}\} the cellular basis. Notice that we have made several choices in defining the cellular basis. We call two ℤπ-bases {α_{j} | j ∈ J} and {β_{k} | k ∈ K} for C_{q}(X̃, Ã) equivalent if there is a bijection ϕ: J → K and elements ϵ_{j} ∈ {±1} and γ_{j} ∈ π for j ∈ J such that ϵ_{j} · γ_{j} · α_{j} = β_{ϕ(j)}. The equivalence class of the basis {b_{i} | i ∈ I_{n}} constructed above does only depend on the CW -structure on (X, A) and is independent of all further choices such as (Φ_{i}^{q}, ϕ_{i}^{q}), its lift (Φ̃_{i}^{q}, ϕ̃_{i}^{q}) and the generator of H_{n}(D^{n}, S^{n−1}).

Now suppose we are given a handlebody decomposition (1.14). We construct by induction over q = −1, 0, 1, …, n a sequence of spaces X_{−1} = ∂_{0}W ⊂ X_{0} ⊂ X_{1} ⊂ X_{2} ⊂ … ⊂ X_{n} together with homotopy equivalences f_{q}: W_{q} → X_{q} such that f_{q}|_{W_{q−1}} = f_{q−1} and (X, ∂_{0}W) is a CW -complex with respect to the filtration {X_{q} | q = −1, 0, 1, …, n}. The induction beginning f_{1} : W_{−1} = ∂_{0}W × [0, 1] → X_{−1} = ∂_{0}W is given by the projection. The induction step from (q −1) to q is done as follows. We attach for each handle (ϕ_{i}^{q}) for i = 1, 2, …, p_{q} a cell D^{q} to X_{q−1} by the attaching map f_{q−1} ∘ ϕ_{i}^{q}|_{S^{q−1}×{0}}. In other words, we define X_{q} by the pushout

∐_{i=1}^{p_{q}} S^{q−1} \underrightarrow{∐_{i=1}^{pq} f_{q−1}∘ϕ_{i}^{q}|_{S^{q−1}×\{0\}}} X_{q−1}
↓  ↓
∐_{i=1}^{p_{q}} D^{q}  ⟶  X_{q}

Recall that W_{q} is the pushout

∐_{i=1}^{p_{q}} S^{q−1} × D^{n−q} \underrightarrow{∐_{i=1}^{pq} ϕ_{i}^{q}} W_{q−1}
↓  ↓
∐_{i=1}^{p_{q}} D^{q} × D^{n−q} ⟶ W_{q}

Define a space Y_{q} by the pushout

∐_{i=1}^{p_{q}} S^{q−1} \underrightarrow{∐_{i=1}^{pq} ϕ_{i}^{q}|_{S^{q−1}×\{0\}}} W_{q−1}
↓  ↓
∐_{i=1}^{p_{q}} D^{q}  ⟶  Y_{q}

Define (g_{q}, f_{q−1}): (Y_{q}, W_{q−1}) → (X_{q}, X_{q−1}) by the pushout property applied to homotopy equivalences given by f_{q−1} : W_{q−1} → X_{q−1} and the identity maps on S^{q−1} and D^{q}. Define (h_{q}, id): (Y_{q}, W_{q−1}) → (W_{q}, W_{q−1}) by the pushout property applied to homotopy equivalences given by the obvious inclusions S^{q−1} → S^{q−1} × D^{n−q} and D^{q} → D^{q} × D^{n−q} and the identity on W_{q−1}. The resulting maps are homotopy equivalences of pairs since the upper horizontal arrows in the three pushouts above are cofibrations (see [18, page 249]). Choose a homotopy inverse (h_{q}^{−1}, id): (W_{q}, W_{q−1}) → (Y_{q}, W_{q−1}). Define f_{q} by the composition g_{q} ∘ h_{q}^{−1}.

In particular we see that the inclusions W_{q} → W are q-connected since the inclusion of the q-skeleton X_{q} → X is always q-connected for a CW -complex X.

Denote by p: W̃ → W the universal covering with π = π_{1}(W) as group of deck transformations. Let W̃_{q} be the preimage of W_{q} under p. Notice that this is the universal covering for q ≥ 2 since each inclusion W_{q} → W induces an isomorphism on the fundamental groups. Let C_{∗}(W̃, { \atop ∂_{0}W}) be the ℤπ-chain complex whose q-th chain group is H_{q}(W̃_{q}, { \atop W_{q−1}}) and whose q-th differential is given by the composition

H_{q}(W̃_{q}, { \atop W_{q−1}}) ⟶^{∂_{p}} H_{q}({ \atop W_{q−1}}) ⟶^{i_{q}} H_{q−1}({ \atop W_{q−1}}, { \atop W_{q−2}}),

where ∂_{q} is the boundary operator of the long homology sequence associated to the pair (W̃_{p}, { \atop W_{p−1}}) and i_{q} is induced by the inclusion. The map f: W → X induces an isomorphism of ℤπ-chain complexes

C_{∗}(f̃): C_{∗}(W̃, { \atop ∂_{0}W}) ⟶^{≅} C_{∗}(X̃, { \atop ∂_{0}W}).  (1.17)

Each handle (ϕ_{i}^{q}) determines an element

[ϕ_{i}^{q}] ∈ C_{q}(W̃, { \atop ∂_{0}W})  (1.18)

after choosing a lift (Φ̃_{i}^{q}, ϕ̃_{i}^{q}): (D^{q} × D^{n−q}, S^{q−1} × D^{n−q}) → (W̃_{q}, { \atop W_{q−1}}) of

its characteristic map (Φ_{i}^{q}, ϕ_{i}^{q}): (D^{q} × D^{n−q}, S^{q−1} × D^{n−q}) → (W_{q}, W_{q−1}), namely, the image of the preferred generator in H_{q}(D^{q} ×D^{n−q}, S^{q−1} ×D^{n−q}) ≅ H_{0}({∗}) = ℤ under the map H_{q}(Φ̃_{i}^{q}, ϕ̃_{i}^{q}). This element is only well-defined up to multiplication with an element γ ∈ π. The elements {[ϕ_{i}^{q}] | i = 1, 2, …, p_{q}} form a ℤπ-basis for C_{q}(W̃, { \atop ∂_{0}W}). Its image under the isomorphism (1.17) is a cellular ℤπ-basis.

If W has no handles of index ≤ 1, i.e. p_{0} = p_{1} = 0, one can express C_{∗}(W̃, { \atop ∂_{0}W}) also in terms of homotopy groups as follows. Fix a base point z ∈ ∂_{0}W and a lift z̃ ∈ { \atop ∂_{0}W}. All homotopy groups are taken with respect to these base points. Let π_{∗}(W_{∗}, W_{∗−1}) be the ℤπ-chain complex, whose q-th ℤπmodule is π_{q}(W_{q}, W_{q−1}) for q ≥ 2 and zero for q ≤ 1 and whose q-th differential is given by the composition

π_{q}(W_{q}, W_{q−1}) ⟶^{∂_{q}} π_{q−1}(W_{q−1}) ⟶^{π_{q−1}(i)} π_{q−1}(W_{q−1}, W_{q−2}).

The ℤπ-action comes from the canonical π_{1}(Y)-action on the group π_{q}(Y, A) [121, Theorem I.3.1 on page 164]. Notice that π_{q}(Y, A) is abelian for any pair of spaces (Y, A) for q ≥ 3 and is abelian also for q = 2 if A is simply connected or empty. For q ≥ 2 the Hurewicz homomorphism is an isomorphism [121, Corollary IV.7.11 on page 181] π_{q}(W̃_{q}, { \atop W_{q−1}}) → H_{q}(W̃_{q}, { \atop W_{q−1}}) and the projection p: W̃ → W induces isomorphisms π_{q}(W̃_{q}, { \atop W_{q−1}}) → π_{q}(W_{q}, W_{q−1}). Thus we obtain an isomorphism of ℤπ-chain complexes

C_{∗}(W̃, { \atop ∂_{0}W}) ⟶^{≅} π_{∗}(W_{∗}, W_{∗−1}).  (1.19)

Fix a path w_{i} in W from a point in the transverse sphere of (ϕ_{i}^{q}) to the base point z. Then the handle (ϕ_{i}^{q}) determines an element

[ϕ_{i}^{q}] ∈ π_{q}(W_{q}, W_{q−1}).  (1.20)

It is represented by the obvious map (D^{q} × {0}, S^{q−1} × {0}) → (W_{q}, W_{q−1}) together with w_{i}. It agrees with the element [ϕ_{i}^{q}] ∈ C_{q}(W̃, { \atop ∂_{0}W}) defined in (1.18) under the isomorphism (1.19) if we use the lift of the characteristic map determined by the path w_{i}.

1.3 Reducing the Handlebody Decomposition

In the next step we want to get rid of the handles of index zero and one in the handlebody decomposition (1.14).

Lemma 1.21 Let W be an n-dimensional manifold for n ≥ 6 whose boundary is the disjoint union ∂W = ∂_{0}W ∐ ∂_{1}W. Then the following statements are equivalent

1. The inclusion ∂_{0}W → W is 1-connected;

2. We can find a diffeomorphism relative ∂_{0}W

W ≅ ∂_{0}W × [0, 1] + ∑_{i=1}^{p_{2}}(ϕ_{i}^{2}) + ∑_{i=1}^{p_{3}}(\overline{ϕ}_{i}^{3}) + ∑_{i=1}^{p_{n}}(\overline{ϕ}_{i}^{n}).

\underline{\underline{Proof}} : (2) ⇒ (1) has already been proved in Section 1.2. It remains to conclude (2) provided that (1) holds.

We first get rid of all 0-handles in the handlebody decomposition (1.14). It suffices to give a procedure to reduce the number of handles of index 0 by one. Since the inclusion ∂_{0}W → W is 1-connected, the inclusion ∂_{0}W → W_{1} induces a bijection on the set of path components. Given any index i_{0}, there must be an index i_{1} such that the core of the handle ϕ_{i_{1}}^{1} is a path connecting a point in ∂_{0}W ×{1} with a point in (ϕ_{i_{0}}^{0}). We conclude from the Diffeomorphism Lemma 1.9 and the Cancellation Lemma 1.12 that (ϕ_{i_{0}}) and (ϕ_{i_{1}}^{1}) cancel one another, i.e. we have

W ≅ ∂_{0}W × [0, 1] + ∑_{i=1,2,...,p_{0},i≠i_{0}}(ϕ_{i}^{0}) + ∑_{i=1,2,...,p_{1},i≠i_{1}}(ϕ_{i}^{1}) + … + ∑_{i=1}^{p_{n}}(ϕ_{i}^{n}).

Hence we can assume p_{0} = 0 in (1.14).

Next we want to get rid of the 1-handles assuming that the inclusion ∂_{0}W → W is 1-connected. It suffices to give a procedure to reduce the number of handles of index 1 by one. We want to do this by constructing an embedding ψ^{2} : S^{1} × D^{n−2} → ∂_{1}^{∘}W_{1} which satisfies the two conditions of the Elimination Lemma 1.16 and then applying the Elimination Lemma 1.16. Consider the embedding ψ_{+}^{2} : S_{+}^{1} = D^{1} = D^{1} × {x} ⊂ D^{1} × D^{n−1} = (ϕ_{1}^{1}) for some fixed x ∈ S^{n−2} = ∂D^{n−1}. The inclusion ∂_{1}^{∘}W_{0} → ∂_{1}W_{0} = ∂_{0}W × {1} induces an isomorphism on the fundamental group since ∂_{1}^{∘}W_{0} is obtained from ∂_{1}W_{0} = ∂_{0}W × {1} by removing the interior of a finite number of embedded (n − 1)- dimensional disks. Since by assumption the inclusion ∂_{0}W → W is 1-connected, the inclusion ∂_{1}^{∘}W_{0} → W induces an epimorphism on the fundamental groups. Therefore we can find an embedding ψ_{−}^{2} : S_{−}^{1} → ∂_{1}^{∘}W_{0} with ψ_{−}^{2}|_{S^{0}} = ψ_{+}^{2}|_{S^{0}} such that the map ψ_{0}^{2} : S^{1} = S_{+}^{1} ∪_{S^{0}} S_{−}^{1} → ∂_{1}W_{1} given by ψ_{+}^{2} ∪ ψ_{−}^{2} is nullhomotopic in W. One can isotope the attaching maps ϕ_{i}^{2} : S^{1} × D^{n−2} → ∂_{1}W_{1} of the 2-handles (ϕ_{i}^{2}) such that they do not meet the image of ψ_{0}^{2} because the sum of the dimension of the source of ψ_{0}^{2} and of S^{1} × {0} ⊂ S^{1} × D^{n−2} is less than the dimension (n − 1) of ∂_{1}W_{1} and one can always shrink inside D^{n−2}. Thus we can assume without loss of generality by the Isotopy Lemma 1.8 and the Diffeomorphism Lemma 1.9 that the image of ψ_{0}^{2} lies in ∂_{1}^{∘}W_{1}. The inclusion ∂_{1}W_{2} → W is 2-connected. Hence ψ_{0}^{2} is nullhomotopic in ∂_{1}W_{2}. Let h: D^{2} → ∂_{1}W_{2} be a nullhomotopy for ψ_{0}^{2}. Since 2 · dim(D^{2}) < dim(∂_{1}W_{2}), we can change h relative to S^{1} into an embedding. (Here we need for the first time the assumption n ≥ 6.) Since D^{2} is contractible the normal bundle of h and thus of ψ_{0}^{2} = ψ_{+}^{2} ∪ ψ_{−}^{2} are trivial. Therefore we can extend ψ_{0}^{2} to an embedding ψ^{2} : S^{1} × D^{n−1} → ∂_{1}^{∘}W_{1} which is isotopic to a trivial embedding in ∂_{1}W_{2} and meets the transverse sphere of the handle (ϕ_{1}^{1}) transversally and in exactly one point and does not meet the transverse spheres of the handles (ϕ_{i}^{1}) for 2 ≤ i ≤ p_{1}. Now Lemma 1.21 follows from the Elimination Lemma 1.16.

Now consider an h-cobordism (W; ∂_{0}W, ∂_{1}W). Because of Lemma 1.21 we can write it as

W ≅ ∂_{0}W × [0, 1] + ∑_{i=1}^{p_{2}}(ϕ_{i}^{2}) + ∑_{i=1}^{p_{3}}(\overline{ϕ}_{i}^{3}) + ….

Lemma 1.22 (Homology Lemma) Suppose n ≥ 6. Fix 2 ≤ q ≤ n − 3 and i_{0} ∈ {1, 2, …, p_{q}}. Let f: S^{q} → ∂_{1}W_{q} be an embedding. Then the following statements are equivalent

1. f is isotopic to an embedding g: S^{q} → ∂_{1}W_{q} such that g meets the transverse sphere of (ϕ_{i_{0}}^{q}) transversally and in exactly one point and is disjoint from transverse spheres of the handles (ϕ_{i}^{q}) for i ≠ i_{0};

2. Let f̃: S^{q} → W̃_{q} be a lift of f under p|W̃_{q}: W̃_{q} → W_{q}. Let [f̃] be the image

of the class represented by f̃ under the obvious composition

π_{q}(W̃_{q}) → π_{q}(W̃_{q}, { \atop W_{q−1}}) → H_{q}(W̃_{q}, { \atop W_{q−1}}) = C_{q}(W̃).

Then there is γ ∈ π with

[f̃] = ±γ · [ϕ_{i_{0}}^{q}].

\underline{\underline{Proof}} : (1) ⇒ (2) We can isotop f such that f|_{S_{+}^{q}}: S_{+}^{q} → ∂_{1}W_{q} looks like the canonical embedding S_{+}^{q} = D^{q} × {x} ⊂ D^{q} × S^{n−1−q} = ∂(ϕ_{i_{0}}^{q}) for some x ∈ S^{n−1−q} and f(S_{−}^{q}) does not meet any of the handles (ϕ_{i}^{q}) for i = 1, 2, …, p_{q}. One easily checks that then (2) is true.

(2) ⇒ (1) We can isotop f such that it is transversal to the transverse spheres of the handles (ϕ_{i}^{q}) for i = 1, 2, …, p_{q}. Since the sum of the dimension of the source of f and of the dimension of the transverse spheres is the dimension of ∂_{1}W_{q}, the intersection of the image of f with the transverse sphere of the handle (ϕ_{i}^{q}) consists of finitely many points x_{i,1}, x_{i,2}, …, x_{i,r_{i}} for i = 1, 2, …, p_{q}. Fix a base point y ∈ S^{q}. It yields a base point z = f(y) ∈ W. Fix for each handle (ϕ_{i}^{q}) a path w_{i} in W from a point in its transverse sphere to z. Let u_{i,j} be a path in S^{q} with the property that u_{i,j}(0) = y and f(u_{i,j}(1)) = x_{i,j} for 1 ≤ j ≤ r_{i} and 1 ≤ i ≤ p_{q}. Let v_{i,j} be any path in the transverse sphere of (ϕ_{i}^{q}) from x_{i,j} to w_{i}(0). Then the composition f(u_{i,j}) ∗ v_{i,j} ∗ w_{i} is a loop in W with base point z and thus represents an element denoted by γ_{i,j} in π = π_{1}(W, z). It is independent of the choice of u_{i,j} and v_{i,j} since S^{q} and the transverse sphere of each handle (ϕ_{i}^{q}) are simply connected. The tangent space T_{x_{i,j}}∂_{1}W_{q} is the direct sum of T_{f^{−1}(x_{i,j})}S^{p} and the tangent space of the transverse sphere {0} × S^{n−1−q} of the handle (ϕ_{i}^{q}) at x_{i,j}. All these three tangent spaces come with preferred orientations. We define elements ϵ_{i,j} ∈ {±1} by requiring that it is 1 if these orientations fit together and −1 otherwise. Now one easily checks that

[f̃] = ∑_{i=1}^{p_{q}} ∑_{j=1}^{r_{i}} ϵ_{i,j} · γ_{i,j} · [ϕ_{i}^{q}],

where [ϕ_{i}^{q}] is the element associated to the handle (ϕ_{i}^{q}) after the choice of the path w_{i} (see (1.18) and (1.20)). We have by assumption [f̃] = ± · γ · [ϕ_{i_{0}}^{q}] for some γ ∈ π. We want to isotop f such that f does not meet the transverse spheres of the handles (ϕ_{i}^{q}) for i ≠ i_{0} and does meet the transverse sphere of (ϕ_{i_{0}}^{q}) transversally and in exactly one point. Therefore it suffices to show in the case that the number ∑_{i=1}^{p_{q}} r_{i} of all intersection points of f with the transverse spheres of the handles (ϕ_{i}^{q}) for i = 1, 2, …, p_{i} is bigger than one that we can change f by an isotopy such that this number becomes smaller. We have

±γ · [ϕ_{i_{0}}^{q}] = ∑_{i=1}^{p_{q}} ∑_{j=1}^{r_{i}} ϵ_{i,j} · γ_{i,j} · [ϕ_{i}^{q}].

Recall that the elements [ϕ_{i}^{q}] for i = 1, 2, …, p_{q} form a ℤπ-basis. Hence we can find an index i ∈ {1, 2, …, p_{q}} and two different indices j_{1}, j_{2} ∈ {1, 2, …, r_{i}} such that the composition of the paths f(u_{i,j_{1}}) ∗ v_{i,j_{1}} ∗ v_{i,j_{2}}^{−} ∗ f(u_{i,j_{2}}^{−}) is nullhomotopic in W and hence in ∂_{1}W_{q} and the signs ϵ_{i,j_{1}} and ϵ_{i,j_{2}} are different. Now by the Whitney trick (see [83, Theorem 6.6 on page 71], [123]) we can change f by an isotopy such that the two intersection points x_{i,j_{1}} and x_{i,j_{2}} disappear, the other intersection points of f with transverse spheres of the handles (ϕ_{i}^{q}) for i ∈ {1, 2, …, p_{q}} remain and no further intersection points are introduced. For the application of the Whitney trick we need the assumption n − 1 ≥ 5. This finishes the proof of the Homology Lemma 1.22.

Lemma 1.23 (Modification Lemma) Let f: S^{q} → ∂_{1}^{∘}W_{q} be an embedding and let x_{j} ∈ ℤπ be elements for j = 1, 2 …, p_{q+1}. Then there is an embedding g: S^{q} → ∂_{1}^{∘}W_{q} with the following properties:

1. f and g are isotopic in ∂_{1}W_{q+1};

2. For a given lifting f̃: S^{q} → W̃_{q} of f one can find a lifting g̃: S^{q} → W̃_{q} of g such that we get in C_{q}(W̃)

[g̃] = [f̃] + ∑_{j=1}^{p_{q+1}} x_{j} · d_{q+1}[ϕ_{j}^{q+1}],

where d_{q+1} is the (q + 1)-th differential in C_{∗}(W̃, { \atop ∂_{0}W}).

\underline{\underline{Proof}} : Any element in ℤπ can be written as a sum of elements of the form ±γ for γ ∈ π. Hence it suffices to prove for a fixed number j ∈ {1, 2 …, p_{q}}, a fixed element γ ∈ π and a fixed sign ϵ ∈ {±1} that one can find an embedding g: S^{q} → ∂_{1}^{∘}W_{q} which is isotopic to f in ∂_{1}W_{q+1} and satisfies for an appropriate lifting g̃

[g̃] = [f̃] + ϵ · γ · d_{p+1}[ϕ_{j}^{q+1}].

Consider the embedding t_{j}: S^{q} = S^{q} × {z} ⊂ S^{q} × S^{n−2−q} ⊂ ∂(ϕ_{j}^{q+1}) ⊂ ∂_{1}W_{q} for some point z ∈ S^{n−2−q} = ∂D^{n−1−q}. It is in ∂_{1}W_{q+1} isotopic to a trivial embedding. Choose a path w in ∂_{1}^{∘}W_{q} connecting a point in the image of f with a point in the image of t_{j}. Without loss of generality we can arrange w to be an embedding. Moreover, we can thicken w: [0, 1] → ∂_{1}^{∘}W_{q} to an embedding \overline{w}[0, 1] × D^{q} → ∂_{1}^{∘}W_{q} such that \overline{w}(\{0\} × D^{q}) and \overline{w}(\{1\} × D^{q}) are embedded q-dimensional disks in the images of f and t_{j} and \overline{w}((0, 1) × D^{q}) does not meet the images of f and t_{j}. Now one can form a new embedding, the connected sum g := f♯_{w}t_{j}: S^{q} → ∂_{1}^{∘}W_{q}. It is essentially given by restriction of f and t_{j} to the part of S^{q}, which is not mapped under f and t_{j} to the interior of the disks \overline{w}(\{0\} × D^{q}), \overline{w}(\{1\} × D^{q}), and \overline{w}|_{[0,1]×S^{q−1}}. Since t_{j} is isotopic to a trivial embedding in ∂_{1}W_{q+1}, the embedding g is isotopic in ∂_{1}W_{q+1} to f. Recall that we have fixed a lifting f̃ of f. This determines a unique lifting of g̃, namely, we require that f̃ and g̃ coincide on those points, where f and g already coincide. For an appropriate element γ′ ∈ π one gets [g̃] = [f̃]+γ′ ·d_{q+1}([ϕ_{j}^{q+1}]), since t_{j}: S^{q} → ∂_{1}W_{q} ⊂ W_{q} is homotopic to ϕ_{j}^{q+1}|_{S^{q}×{0}}: S^{q} × {0} = S^{q} → W_{q} in W_{q}. We can change the path w by composing it with a loop representing γ · (γ′)^{−1} ∈ π. Then we get for the new embedding g that

[g̃] = [f̃] + γ · d_{q+1}([ϕ_{j}^{q+1}]).

If we compose t_{j} with a diffeomorphism S^{q} → S^{q} of degree −1, we still get an embedding g which is isotopic to f in ∂_{1}W_{q+1} and satisfies

[g̃] = [f̃] − γ · d_{q+1}([ϕ_{j}^{q+1}]).

This finishes the proof of the Modification Lemma 1.23.

Lemma 1.24 (Normal Form Lemma) Let (W; ∂_{0}W, ∂_{1}W) be an oriented compact h-cobordism of dimension n ≥ 6. Let q be an integer with 2 ≤ q ≤ n−3. Then there is a handlebody decomposition which has only handles of index q and (q + 1), i.e. there is a diffeomorphism relative ∂_{0}W

W ≅ ∂_{0}W × [0, 1] + ∑_{i=1}^{p_{q}}(ϕ_{i}^{q}) + ∑_{i=1}^{p_{q+1}}(ϕ_{i}^{q+1}).

\underline{\underline{Proof}} : In the first step we show that we can arrange W_{−1} = W_{q−1}, i.e. p_{r} = 0 for r ≤ q − 1. We do this by induction over q. The induction beginning q = 2 has already been carried out in Lemma 1.21. In the induction step from q to (q + 1) we must explain how we can decrease the number of q-handles provided that there are no handles of index < q. In order to get rid of the handle (ϕ_{1}^{q}) we want to attach a new (q + 1)-handle and a new (q + 2)-handle such that (ϕ_{1}^{q}) and the new (q + 1)-handle cancel and the new (q + 1)-handle and the new (q+2)-handle cancel each other. The effect will be that the number of q-handles is decreased by one at the cost of increasing the number of (q + 2)-handles by one.

Fix a trivial embedding \overline{ψ}^{q+1} : S^{q} × D^{n−1−q} → ∂_{1}^{∘}W_{q}. Since the inclusion ∂_{0}W → W is a homotopy equivalence, H_{p}(W̃, { \atop ∂_{0}W}) = 0 for all p ≥ 0. Since the p-th homology of C_{∗}(W̃, { \atop ∂_{0}W}) is H_{p}(W̃, { \atop ∂_{0}W}) = 0, the ℤπ-chain complex C_{∗}(W̃, { \atop ∂_{0}W}) is acyclic. Since C_{q−1}(W̃, { \atop ∂_{0}W}) is trivial, the q-th differential of C_{∗}(W̃, { \atop ∂_{0}W}) is zero and hence the (q + 1)-th differential d_{q+1} is surjective. We can choose elements x_{j} ∈ ℤπ such that

[ϕ_{1}^{q}] = ∑_{i=1}^{p_{q+1}} x_{j} · d_{q+1}([ϕ_{i}^{q+1}]).

Since α := \overline{ψ}^{q+1}|_{S^{q}×\{0\}} → ∂_{1}^{∘}W_{q} is nullhomotopic, [α̃] = 0 in H_{q}(W̃_{q}, { \atop W_{q−1}}). Because of the Modification Lemma 1.23 we can find an embedding ψ^{q+1} : S^{q} × D^{n−1−q} → ∂_{1}^{∘}W_{q} such that β := ψ^{q}|_{S^{q}×{0}} is isotopic in ∂_{1}W_{q+1} to α and we get

[β̃] = [α̃] + ∑_{i=1}^{p_{q+1}} x_{j} · d_{q+1}([ϕ_{i}^{q+1}]) = [ϕ_{1}^{q}].

Because of the Homology Lemma 1.22 the embedding β = ψ^{q}|_{S^{q}×{0}} is isotopic in ∂_{1}W_{q} to an embedding γ: S^{q} → ∂_{1}W_{q} which meets the transverse sphere of (ϕ_{1}^{q}) transversally and in exactly one point and is disjoint from the transverse spheres of all other handles of index q. By construction ψ^{q+1} is isotopic in ∂_{1}W_{q+1} to the trivial embedding \overline{ψ}^{q+1}. Now we can apply the Elimination Lemma 1.16. This finishes the proof that we can arrange W_{−1} = W_{q−1}.

Next we explain the dual handlebody decomposition. Suppose that W is obtained from ∂_{0}W × [0, 1] by attaching one q-handle (ϕ^{q}), i.e. W = ∂_{0}W × [0, 1]+(ϕ^{q}). Then we can interchange the role of ∂_{0}W and ∂_{1}W and try to built W from ∂_{1}W by handles. It turns out that W can be written as

W = ∂_{1}W × [0, 1] + (ψ^{n−q})  (1.25)

by the following argument.

Let M be the manifold with boundary S^{q−1} × S^{n−1−q} obtained from ∂_{0}W by removing the interior of ϕ^{q}(S^{q−1} × D^{n−q}). We get

W ≅ ∂_{0}W × [0, 1] ∪_{S^{q−1}×D^{n−q}} D^{q} × D^{n−q}
= M × [0, 1] ∪_{S^{q−1}×S^{n−2−q}×[0,1]}

(S^{q−1} × D^{n−1−q} × [0, 1] ∪_{S^{q−1}×D^{n−q}×{1}} D^{q} × D^{n−q}) .

Inside S^{q−1} × D^{n−1−q} × [0, 1] ∪_{S^{q−1}×D^{n−q}×{1}} D^{q} × D^{n−q} we have the following submanifolds

X := S^{q−1} × 1/2 · D^{n−1−q} × [0, 1] ∪_{S^{q−1}×1/2·D^{n−q}×{1}} D^{q} × 1/2 · D^{n−q};
Y := S^{q−1} × 1/2 · S^{n−1−q} × [0, 1] ∪_{S^{q−1}×1/2·S^{n−q}×{1}} D^{q} × 1/2 · S^{n−q}.

The pair (X, Y) is diffeomorphic to (D^{q} × D^{n−q}, D^{q} × S^{n−1−q}), i.e. it is a handle of index (n − q). Let N be obtained from W by removing the interior of X. Then W is obtained from N by adding a (n − q)-handle, the so called dual handle. One easily checks that N is diffeomorphic to ∂_{1}W × [0, 1] relative ∂_{1}W × {1}. Thus (1.25) follows.

Suppose that W is relatively ∂_{0}W of the shape

W ≅ ∂_{0}W × [0, 1] + ∑_{i=1}^{p_{0}}(ϕ_{i}^{0}) + ∑_{i=1}^{p_{1}}(ϕ_{i}^{1}) + … + ∑_{i=1}^{p_{n}}(ϕ_{i}^{n}),

Then we can conclude inductively using the Diffeomorphism Lemma 1.9 and (1.25) that W is diffeomorphic relative to ∂_{1}W to

W ≅ ∂_{1}W × [0, 1] + ∑_{i=1}^{p_{n}}(\overline{ϕ}_{i}^{0}) + ∑_{i=1}^{p_{n−1}}(\overline{ϕ}_{i}^{1}) + … + ∑_{i=1}^{p_{0}}(\overline{ϕ}_{i}^{n}). (1.26)

This corresponds to replacing a Morse function f by −f. The effect is that the number of q-handles becomes now the number of (n − q)-handles.

Now applying the first step to the dual handlebody decomposition for q replaced by (n−q −1) and then considering the dual handlebody decomposition of the result finishes the proof of the Normal Form Lemma 1.24.

1.4 Handlebody Decompositions and Whitehead Groups

Let (W, ∂_{0}W, ∂_{1}W) be an n-dimensional compact oriented h-cobordism for n ≥ 6. By the Normal Form Lemma 1.24 we can fix a handlebody decomposition for some fixed number 2 ≤ q ≤ n − 3

W ≅ ∂_{0}W × [0, 1] + ∑_{i=1}^{p_{q}}(ϕ_{i}^{q}) + ∑_{i=1}^{p_{q+1}}(ϕ_{i}^{q+1}).

Recall that the ℤπ-chain complex C_{∗}(W̃, { \atop ∂_{0}W}) is acyclic. Hence the only nontrivial differential d_{q+1} : H_{q+1}({ \atop W_{q+1}}, W̃_{q}) → H_{q}(W̃_{q}, { \atop W_{q−1}}) is bijective. Recall

that \{[ϕ_{i}^{q+1}] | i = 1, 2 …, p_{q+1}\} is a ℤπ-basis for H_{q+1}({ \atop W_{q+1}}, W̃_{q}) and \{[ϕ_{i}^{q}] |

i = 1, 2 …, p_{q}\} is a ℤπ-basis for H_{q}(W̃_{q}, { \atop W_{q−1}}). In particular p_{q} = p_{q+1}. The matrix A, which describes the differential d_{q+1} with respect to these basis, is an invertible (p_{q}, p_{q})-matrix over ℤπ. Since we are working with left modules, d_{q+1} sends an element x ∈ (ℤG)^{n} to x · A ∈ ℤG^{n}, or equivalently, d_{q+1}([ϕ_{i}^{q+1}]) = ∑_{j=1}^{n} a_{i,j}[ϕ_{j}^{q}].

Next we define an abelian group Wh(π) as follows. It is the set of equivalence classes of invertible matrices of arbitrary size with entries in ℤπ, where we call an invertible (m, m)-matrix A and an invertible (n, n)-matrix B over ℤπ equivalent, if we can pass from A to B by a sequence of the following operations:

1. B is obtained from A by adding the k-th row multiplied with x from the left to the l-th row for x ∈ ℤπ and k ≠ l; 1.4. HANDLEBODY DECOMPOSITIONS AND WHITEHEAD GROUPS 19

2. B is obtained by taking the direct sum of A and the (1, 1)-matrix I_{1} = (1),

i.e. B looks like the block matrix ( \begin{matrix}A & 0 \\ 0 & 1 \end{matrix} );

3. A is the direct sum of B and I_{1}. This is the inverse operation to (2);

4. B is obtained from A by multiplying the i-th row from the left with a trivial unit , i.e. with an element of the shape ±γ for γ ∈ π;

5. B is obtained from A by interchanging two rows or two columns.

The group structure is given on representatives A and B as follows. By taking the direct sum A ⊕ I_{m} and B ⊕ I_{n} with the identity matrices I_{m} and I_{n} of size m and n for appropriate m and n one can arrange that A ⊕ I_{m} and B ⊕ I_{n} are invertible matrices of the same size and can be multiplied. Define [A] · [B] by [(A ⊕ I_{m}) · (B ⊕ I_{n})]. The zero element 0 ∈ Wh(π) is represented by I_{n} for any positive integer n. The inverse of [A] is given by [A^{−1}]. We will show later in Lemma 2.4 that the multiplication is well-defined and yields an abelian group Wh(π).

Lemma 1.27 1. Let (W, ∂_{0}W, ∂_{1}W) be an n-dimensional compact oriented h-cobordism for n ≥ 6 and A be the matrix defined above. If [A] = 0 in Wh(π), then the h-cobordism W is trivial relative ∂_{0}W;

2. Consider an element u ∈ Wh(π), a closed oriented manifold M of dimension n − 1 ≥ 5 with fundamental group π and an integer q with 2 ≤ q ≤ n − 3. Then we can find an h-cobordism of the shape

W = M × [0, 1] + ∑_{i=1}^{p_{q}}(ϕ_{i}^{q}) + ∑_{i=1}^{p_{q+1}}(ϕ_{i}^{q+1})

such that [A] = u.

\underline{\underline{Proof}} : (1) Let B be a matrix which is obtained from A by applying one of the operations (1), (2), (3), (4) and (5). It suffices to show that we can modify the given handlebody decomposition in normal form of W with associated matrix A such that we get a new handlebody decomposition in normal form whose associated matrix is B.

We begin with (1). Consider W′ = ∂_{0}W ×[0, 1]+∑_{i=1}^{p_{q}}(ϕ_{i}^{q})+∑_{j=1,j≠l}^{p_{q+1}}(ϕ_{j}^{q+1}). Notice that we get from W′ our h-cobordism W if we attach the handle (ϕ_{l}^{q+1}). By the Modification Lemma 1.23 we can find an embedding \overline{ϕ}_{l}^{q+1} : S^{q}×D^{n−1−q} → ∂_{1}W′ such that \overline{ϕ}_{l}^{q+1} is isotopic to ϕ_{l}^{q+1} and we get

\left[{ \atop \overline{ϕ}_{l}^{q+1}|_{S^{q}×\{0\}}}\right] = \left[{ \atop ϕ_{l}^{q+1}|_{S^{q}×\{0\}}}\right] + x · d_{q+1}([ϕ_{k}^{q+1}]).

If we attach to W′ the handle (\overline{ϕ}_{l}^{q+1}), the result is diffeomorphic to W relative ∂_{0}W by the Isotopy Lemma 1.8. One easily checks that the associated invertible matrix B is obtained from A by adding the k-th row multiplied with x from the left to the the l-th row.

The claim for the operations (2) and (3) follow from the Cancellation Lemma 1.12 and the Homology Lemma 1.22. The claim for the operation (4) follows from the observation that we can replace the attaching map of a handle ϕ^{q}: S^{q} × D^{n−1−q} → ∂_{1}W_{q} by its composition with f × id for some diffeomorphism f: S^{q} → S^{q} of degree −1 and that the base element [ϕ_{i}^{q}] can also be changed to γ · [ϕ_{i}^{q}] by choosing a different lift along W̃_{q} → W_{q}. Operation (5) can be realized by interchanging the numeration of the q-handles and (q + 1)- handles.

(2) Fix an invertible matrix A = (a_{i,j}) ∈ GL(n, ℤπ). Choose trivial pairwise disjoint embeddings ϕ_{i}^{2} : S^{1} × D^{n−2} → M_{0} × {1}. Consider

W_{2} = M_{0} × [0, 1] + (ϕ_{1}^{2}) + (ϕ_{2}^{2}) + … + (ϕ_{n}^{2}).

Since the embeddings ϕ_{i}^{2} are trivial, we can construct embeddings ϕ_{i}^{3} : S^{2} ×

D^{n−3} → ∂_{1}W_{2} and lifts ϕ̃_{i}^{3} : S^{2} × D^{n−3} → { \atop ∂_{1}W_{2}} such that in π_{2}(W̃_{2}, { \atop ∂_{0}W})

[ϕ̃_{i}^{3}|_{S^{2}×{0}}] = ∑_{j=1}^{n} a_{i,j} · [ϕ_{j}^{2}].

Put W = W_{2} + (ϕ_{1}^{3}) + (ϕ_{2}^{3}) + … + (ϕ_{n}^{3}). One easily checks that W is an hcobordism over M_{0} with a handlebody decomposition which realizes the matrix A. This finishes the proof Lemma 1.27.

Remark 1.28 If π is trivial, then Wh(π) is trivial. This follows from the fact that any invertible matrix over the integers can be reduced by elementary column operations, permutations of columns and rows and multiplication of a row with ±1 to the identity matrix. This is essentially a consequence of the existence of an Euclidean algorithm for ℤ. Hence Lemma 1.27 (2) implies already the h-Cobordism Theorem 1.2. As soon as we have shown that [A] ∈ Wh(π) agrees with the Whitehead torsion τ(W, M_{0}) of the h-cobordism W over M_{0} and that this invariant depends only on the diffeomorphism type of W relative M_{0}, the s-Cobordism Theorem 1.1 (1) will follow.

Obviously Lemma 1.27 (2) implies the s-Cobordism Theorem 1.1 (2). We will later see that assertion (3) of the s-Cobordism Theorem 1.1 follows from assertions (1) and (2) if we have more information about the Whitehead torsion, namely the sum and the composition formulas.

1.5 Miscellaneous

The s-Cobordism Theorem 1.1 is due to Barden, Mazur, Stallings. Its topological version was proved by Kirby and Siebenmann [59, Essay II]. More information about the s-cobordism theorem can be found for instance in [57], [83] [100, page 87-90]. The s-cobordism theorem is known to be false for n = dim(M_{0}) = 4 in general, by the work of Donaldson [35], but it is true for n = dim(M_{0}) = 4 for so called “good” fundamental groups in the topological category by results of Freedman [46], [47]. The trivial group is an example of a “good” fundamental groups. Counterexamples in the case n = dim(M_{0}) = 3 are constructed by Cappell and Shaneson [22]. The Poincaré Conjecture (see Theorem 1.3) is at the time of writing known in all dimensions except dimension 3.

22 CHAPTER 1. THE S-COBORDISM THEOREM

Whitehead Torsion

Introduction

In this section we will assign to a homotopy equivalence f: X → Y of finite CW -complexes its Whitehead torsion τ(f) in the Whitehead group Wh(π(Y)) associated to Y. The main properties of this invariant are summarized in the following

Theorem 2.1 1. Sum formula

Let the following two diagrams be cellular pushouts of finite CW -complexes

\begin{matrix}X_{0} & ⟶^{i_{1}} & X_{1} \\ ^{i_{2}}↓ & & ^{j_{1}}↓ \\ X_{2} & ⟶^{j_{2}} & X \end{matrix}  \begin{matrix}Y_{0} & ⟶^{k_{1}} & Y_{1} \\ ^{k_{2}}↓ & & ^{l_{1}}↓ \\ Y_{2} & ⟶^{l_{2}} & Y \end{matrix}

Put l_{0} = l_{1} ∘ k_{1} = l_{2} ∘ k_{2} : Y_{0} → Y. Let f_{i}: X_{i} → Y_{i} be homotopy equivalences for i = 0, 1, 2 satisfying f_{1} ∘ i_{1} = k_{1} ∘ f_{0} and f_{2} ∘ i_{2} = k_{2} ∘ f_{0}. Denote by f: X → Y the map induced by f_{0}, f_{1} and f_{2} and the pushout property. Then f is a homotopy equivalence and

τ(f) = (l_{1})_{∗}τ(f_{1}) + (l_{2})_{∗}τ(f_{2}) − (l_{0})_{∗}τ(f_{0});

2. Homotopy invariance

Let f ≃ g: X → Y be homotopic maps of finite CW -complexes. Then the homomorphisms f_{∗}, g_{∗}: Wh(π(X)) → Wh(π(Y)) agree. If additionally f and g are homotopy equivalences, then

τ(g) = τ(f); 3. Composition formula

Let f: X → Y and g: Y → Z be homotopy equivalences of finite CW - complexes. Then

τ(g ∘ f) = g_{∗}τ(f) + τ(g);

4. Product formula

Let f: X′ → X and g: Y′ → Y be homotopy equivalences of connected finite CW -complexes. Then

τ(f × g) = χ(X) · j_{∗}τ(g) + χ(Y) · i_{∗}τ(f),

where χ(X), χ(Y) ∈ ℤ denote the Euler characteristics, j_{∗}: Wh(π(Y)) → Wh(π(X×Y)) is the homomorphism induced by j: Y → X×Y, y ↦ (y, x_{0}) for some base point x_{0} ∈ X and i_{∗} is defined analogously;

5. Topological invariance

Let f: X → Y be a homeomorphism of finite CW -complexes. Then

τ(f) = 0.

Given an h-cobordism (W; M_{0}, f_{0}, M_{1}, f_{1}) over M_{0}, we define its Whitehead torsion τ(W, M_{0}) by the Whitehead torsion of the inclusion ∂_{0}W → W (see (2.14)). This is the invariant appearing in the s-Cobordism Theorem 1.1. We will give some information about the Whitehead group Wh(π(Y)) in Section 2.1. We will present the algebraic definition of Whitehead torsion and the proof of Theorem 2.1 in Section 2.2. A geometric approach to the Whitehead torsion is summarized in Section 2.3. A similar invariant, the Reidemeister torsion, will be treated in Section 2.4. It will be used to classify lens spaces. In order to understand the following chapters, it suffices to comprehend the statements in the s-Cobordism Theorem 1.1 and Theorem 2.1

2.1 Whitehead Groups

In this section we define K_{1}(R) for an associative ring R with unit and the Whitehead group Wh(G) of a group G and relate the definitions of this section with the one of Section 1.4. Furthermore we give some basic information about its computation.

Let R be an associative ring with unit. Denote by GL(n, R) the group of invertible (n, n)-matrices with entries in R. Define the group GL(R) by the colimit of the system indexed by the natural numbers … ⊂ GL(n, R) ⊂ GL(n + 1, R) ⊂ …, where the inclusion GL(n, R) to GL(n + 1, R) is given by stabilization

A ↦ ( \begin{matrix}A & 0 \\ 0 & 1 \end{matrix} ) .

Define K_{1}(R) by the abelianization GL(R)/[GL(R), GL(R)] of GL(R). Let K̃_{1}(R) be the cokernel of the map K_{1}(ℤ) → K_{1}(R) induced by the canonical ring homomorphism ℤ → R. The homomorphism det: K_{1}(ℤ) → {±1}, [A] ↦ det(A) is a bijection, because ℤ is a ring with Euclidian algorithm. Hence K̃_{1}(R) is the same as the quotient of K_{1}(R) by the cyclic subgroup of at most order two generated by the class of the (1, 1)-matrix (−1). Define the Whitehead group Wh(G) of a group G to be the cokernel of the map G × {±1} → K_{1}(ℤG) which sends (g, ±1) to the class of the invertible (1, 1)-matrix (±g). This will be the group, where Whitehead torsion will take its values in.

The Whitehead group Wh(G) is known to be trivial if G is the free abelian group ℤ^{n} of rank n [8] or the free group ∗_{i=1}^{n}ℤ of rank n [109]. There is the conjecture that it vanishes for any torsionfree group. This has been proved by Farrell and Jones [37], [38], [39] for a large class of groups. This class contains any subgroup G ⊂ G′, where G′ is a discrete cocompact subgroup of a Lie group with finitely many path components, and any group G which is the fundamental group of a non-positively curved closed Riemannian manifold or of a complete pinched negatively curved Riemannian manifold. Recall that a G-space X is called cocompact if its orbit space G\X is compact. The Whitehead group satisfies Wh(G ∗ H) = Wh(G) ⊕ Wh(H) [109].

If G is finite, then Wh(G) is very well understood (see [91]). Namely, Wh(G) is finitely generated, its rank as abelian group is the number of conjugacy classes of unordered pairs {g, g^{−1}} in G minus the number of conjugacy classes of cyclic subgroups, and its torsion subgroup is isomorphic to the kernel SK_{1}(G) of the change of coefficient homomorphism K_{1}(ℤG) → K_{1}(ℚG). For a finite cyclic group G the Whitehead group Wh(G) is torsionfree. For instance the Whitehead group Wh(ℤ/p) of a cyclic group of order p for an odd prime p is the free abelian group of rank (p − 3)/2 and Wh(ℤ/2) = 0. The Whitehead group of the symmetric group S_{n} is trivial. The Whitehead group of ℤ^{2} ×ℤ/4 is not finitely generated as abelian group. Next we want to relate the definitions above to the one of Section 1.4. Denote by E_{n}(i, j) for n ≥ 1 and 1 ≤ i, j ≤ n the (n, n)-matrix whose entry at (i, j) is one and is zero elsewhere. Denote by I_{n} the identity matrix of size n. An elementary (n, n)-matrix is a matrix of the form I_{n} + r · E_{n}(i, j) for n ≥ 1, 1 ≤ i, j ≤ n, i ≠ j and r ∈ R. Let A be a (n, n)-matrix. The matrix B = A · (I_{n} + r · E_{n}(i, j)) is obtained from A by adding the i-th column multiplied with r from the right to the j-th column. The matrix C = (I_{n} + r · E_{n}(i, j)) · A is obtained from A by adding the jth row multiplied with r from the left to the i-th row. Let E(R) ⊂ GL(R) be the subgroup generated by all elements in GL(R) which are represented by elementary matrices.

Lemma 2.2 We have E(R) = [GL(R), GL(R)]. In particular E(R) ⊂ GL(R) is a normal subgroup and K_{1}(R) = GL(R)/E(R).

\underline{\underline{Proof}} : For n ≥ 3, pairwise distinct numbers 1 ≤ i, j, k ≤ n and r ∈ R we can write I_{n} + r · E_{n}(i, k) as a commutator in GL(n, R), namely

I_{n} + r · E_{n}(i, k) = (I_{n} + r · E_{n}(i, j)) · (I_{n} + E_{n}(j, k)) ·
(I_{n} + r · E_{n}(i, j))^{−1} · (I_{n} + E_{n}(j, k))^{−1}.

This implies E(R) ⊂ [GL(R), GL(R)].

Let A and B be two elements in GL(n, R). Let [A] and [B] be the elements in GL(R) represented by A and B. Given two elements x and y in GL(R), we write x ∼ y if there are elements e_{1} and e_{2} in E(R) with x = e_{1}ye_{2}, in other words, if the classes of x and y in E(R)\GL(R)/E(R) agree. One easily checks

[AB] ∼ [( \begin{matrix}AB & 0 \\ 0 & I_{n} \end{matrix} )] ∼ [( \begin{matrix}AB & A \\ 0 & I_{n} \end{matrix} )]

∼ [( \begin{matrix}0 & A \\ −B & I_{n} \end{matrix} )] ∼ [( \begin{matrix}0 & A \\ −B & 0 \end{matrix} )] ,

since each step is given by multiplication from the right or left with a block

matrix of the form ( \begin{matrix}I_{n} & 0 \\ C & I_{n} \end{matrix} ) or ( \begin{matrix}I_{n} & C \\ 0 & I_{n} \end{matrix} ) and such a block matrix is

obviously obtained from I_{2n} by a sequence of column and row operations and hence its class in GL(R) belongs to E(R). Analogously we get

[BA] ∼ [( \begin{matrix}0 & B \\ −A & 0 \end{matrix} )] .

Since the element in GL(R) represented by ( \begin{matrix}0 & −I_{n} \\ I_{n} & 0 \end{matrix} ) belongs to E(R), we

conclude

[( \begin{matrix}0 & A \\ −B & 0 \end{matrix} )] ∼ [( \begin{matrix}A & 0 \\ 0 & B \end{matrix} )] ∼ [( \begin{matrix}0 & B \\ −A & 0 \end{matrix} )] .

This shows

[AB] ∼ [BA].  (2.3)

This implies for any element x ∈ GL(R) and e ∈ E(R) that xex^{−1} ∼ ex^{−1}x = e and hence xex^{−1} ∈ E(R). Therefore E(R) is normal. Given a commutator xyx^{−1}y^{−1} for x, y ∈ GL(R), we conclude for appropriate elements e_{1}, e_{2}, e_{3} in E(R)

xyx^{−1}y^{−1} = e_{1}yxe_{2}x^{−1}y^{−1} = e_{1}yxx^{−1}y^{−1}(yx)e_{2}(yx)^{−1} = e_{1}e_{3} ∈ E(R).

This finishes the proof of Lemma 2.2.

Lemma 2.4 The definition of Wh(G) of Section 1.4 makes sense and yields an abelian group which can be identified with the definition of Wh(G) given in this section above.

\underline{\underline{Proof}} : Notice that the operation (4) appearing in the definition of Wh(G) in Section 1.4 corresponds to multiplication with an elementary matrix from the left. Since E(R) is normal by Lemma 2.2, two invertible matrices A and B over ℤG are equivalent under the equivalence relation appearing in the definition of Wh(G) as explained in Section 1.4 if and only their classes [A] and [B] in Wh(G) as defined in this section agree. Now the claim follows from Lemma 2.2.

Remark 2.5 Often K_{1}(R) is defined in a more conceptual way in terms of automorphisms as follows. Namely, K_{1}(R) is defined as the abelian group whose generators [f] are conjugacy classes of automorphisms f: P → P of finitely generated projective R-modules P and which satisfies the following relations. For any commutative diagram of finitely generated projective R-modules with exact rows and automorphisms as vertical arrows

0 ⟶ P_{0} ⟶^{i} P_{1} ⟶^{p} P_{2} ⟶ 0
f_{0}↓≅  ^{f_{1}}↓≅  ^{f_{2}}↓≅
0 ⟶ P_{0} ⟶^{i} P_{1} ⟶^{p} P_{2} ⟶ 0

we get the relation [f_{0}] − [f_{1}] + [f_{2}] = 0. If f, g: P → P are automorphisms of a finitely generated projective R-module P, then [g ∘ f] = [g] + [f]. Using Lemma 2.2 one easily checks that sending the class [A] of an invertible (n, n)- matrix A to the class of the automorphism R_{A}: R^{n} → R^{n}, x ↦ xA defines an isomorphism from GL(R)/[GL(R), GL(R)] to the abelian group defined above.

2.2 Algebraic Approach to Whitehead Torsion

In this section we give the definition and prove the basic properties of the Whitehead torsion using an algebraic approach via chain complexes. This will enable us to finish the proof of the s-Cobordism Theorem 1.1. The idea which underlies the notion of Whitehead torsion will become more transparent in Section 2.3, where we will develop a geometric approach to Whitehead torsion and link it to the strategy of proof of the s-Cobordism Theorem 1.1.

We begin with some input about chain complexes. Let f_{∗}: C_{∗} → D_{∗} be a chain map of R-chain complexes for some ring R. Define cyl_{∗}(f_{∗}) to be the chain complex with p-th differential

C_{p−1} ⊕ C_{p} ⊕ D_{p} \underrightarrow{( \begin{matrix}−c_{p−1} & 0 & 0 \\ − id & c_{p} & 0 \\ f_{p−1} & 0 & d_{p} \end{matrix} )} C_{p−2} ⊕ C_{p−1} ⊕ D_{p−1}.

Define cone_{∗}(f_{∗}) to be the quotient of cyl_{∗}(f_{∗}) by the obvious copy of C_{∗}. Hence the p-th differential of cone_{∗}(f_{∗}) is

C_{p−1} ⊕ D_{p} \underrightarrow{( \begin{matrix}−c_{p−1} & 0 \\ f_{p−1} & d_{p} \end{matrix} )} C_{p−2} ⊕ D_{p−1}.

Given a chain complex C_{∗}, define ΣC_{∗} to be the quotient of cone_{∗}(id_{C_{∗}}) by the obvious copy of C_{∗}, i.e. the chain complex with p-th differential

C_{p−1} ⟶^{−c_{p−1}} C_{p−2}.

Definition 2.6 We call cyl_{∗}(f_{∗}) the mapping cylinder, cone_{∗}(f_{∗}) the mapping cone of the chain map f_{∗} and ΣC_{∗} the suspension of the chain complex C_{∗}.

These algebraic notions of mapping cylinder, mapping cone and suspension are modelled on their geometric counterparts. Namely, the cellular chain complex of a mapping cylinder of a cellular map of CW -complexes is the mapping cylinder of the chain map induced by f. From the geometry it is also clear why one obtains obvious exact sequences such as 0 → C_{∗} → cyl(f_{∗}) → cone(f_{∗}) → 0 and 0 → D_{∗} → cone_{∗}(f_{∗}) → ΣC_{∗} → 0.

A chain contraction γ_{∗} for an R-chain complex C_{∗} is a collection of Rhomomorphisms γ_{p}: C_{p} → C_{p+1} for p ∈ ℤ such that c_{p+1} ∘ γ_{p} + γ_{p−1} ∘ c_{p} = id_{C_{p}} holds for all p ∈ ℤ. We call an R-chain complex C_{∗} finite if there is a number N with C_{p} = 0 for |p| > N and each R-chain module C_{p} is a finitely generated Rmodule. We call an R-chain complex C_{∗} projective resp. free resp. based free if each R-chain module C_{p} is projective resp. free resp. free with a preferred basis. Suppose that C_{∗} is a finite based free R-chain complex which is contractible, i.e. which possesses a chain contraction. Put C_{odd} = ⊕_{p∈ℤ}C_{2p+1} and C_{ev} = ⊕_{p∈ℤ}C_{2p}. Let γ_{∗} and δ_{∗} be two chain contractions. Define R-homomorphisms

(c_{∗} + γ_{∗})_{odd} : C_{odd} → C_{ev};
(c_{∗} + δ_{∗})_{ev} : C_{ev} → C_{odd}.

Let A be the matrix of (c_{∗} + γ_{∗})_{odd} with respect to the given bases. Let B be the matrix of (c_{∗} + δ_{∗})_{ev} with respect to the given bases. Put μ_{n} := (γ_{n+1} − δ_{n+1}) ∘ δ_{n} and ν_{n} := (δ_{n+1} − γ_{n+1}) ∘ γ_{n}. One easily checks that (id +μ_{∗})_{odd}, (id +ν_{∗})_{ev} and both compositions (c_{∗} + γ_{∗})_{odd} ∘ (id +μ_{∗})_{odd} ∘ (c_{∗} + δ_{∗})_{ev} and (c_{∗} + δ_{∗})_{ev} ∘ (id +ν_{∗})_{ev} ∘ (c_{∗} + γ_{∗})_{odd} are given by upper triangular matrices whose diagonal entries are identity maps. Hence A and B are invertible and their classes [A], [B] ∈ K̃_{1}(R) satisfy [A] = −[B]. Since [B] is independent of the choice of γ_{∗}, the same is true for [A]. Thus we can associate to a finite based free contractible R-chain complex C_{∗} an element

τ(C_{∗}) = [A] ∈ K̃_{1}(R).  (2.7)

Let f_{∗}: C_{∗} → D_{∗} be a homotopy equivalence of finite based free R-chain complexes. Its mapping cone cone(f_{∗}) is a contractible finite based free R-chain complex. Define the Whitehead torsion of f_{∗} by

τ(f_{∗}) := τ(cone_{∗}(f_{∗})) ∈ K̃_{1}(R).  (2.8)

We call a sequence of finite based free R-chain complexes 0 → C_{∗} ⟶^{i_{∗}} D_{∗} ⟶^{q_{∗}} E_{∗} → 0 based exact if for any p ∈ ℤ the basis B for D_{p} can be written as a disjoint union B′ ∐ B′′ such that the image of the basis of C_{p} under i_{p} is B′ and the image of B′′ under q_{p} is the basis for E_{p}.

Lemma 2.9 1. Consider a commutative diagram of finite based free R-chain complexes whose rows are based exact.

0 ⟶ C′_{∗} ⟶ D′_{∗} ⟶ E′_{∗} ⟶ 0
f_{∗}↓  ^{g_{∗}}↓  ^{h_{∗}}↓
0 ⟶ C_{∗} ⟶ D_{∗} ⟶ E_{∗} ⟶ 0

Suppose that two of the chain maps f_{∗}, g_{∗} and h_{∗} are R-chain homotopy equivalences. Then all three are R-chain homotopy equivalences and

τ(f_{∗}) − τ(g_{∗}) + τ(h_{∗}) = 0;

2. Let f_{∗} ≃ g_{∗}: C_{∗} → D_{∗} be homotopic R-chain homotopy equivalences of finite based free R-chain complexes. Then

τ(f_{∗}) = τ(g_{∗});

3. Let f_{∗}: C_{∗} → D_{∗} and g_{∗}: D_{∗} → E_{∗} be R-chain homotopy equivalences of based free R-chain complexes. Then

τ(g_{∗} ∘ f_{∗}) = τ(g_{∗}) + τ(f_{∗}).

\underline{\underline{Proof}} : (1) A chain map of projective chain complexes is a homotopy equivalence if and only if it induces an isomorphism on homology. The five-lemma and the long homology sequence of a short exact sequence of chain complexes imply that all three chain maps f_{∗}, h_{∗} and g_{∗} are chain homotopy equivalences if two of them are.

To prove the sum formula, it suffices to show for a based free exact sequence 0 → C_{∗} ⟶^{i_{∗}} D_{∗} ⟶^{q_{∗}} E_{∗} → 0 of contractible finite based free R-chain complexes that

τ(C_{∗}) − τ(D_{∗}) + τ(E_{∗}) = 0.  (2.10)

Let u_{∗}: F_{∗} → G_{∗} be an isomorphism of contractible finite based free R-chain complexes. Since the choice of chain contraction does not affect the values of the Whitehead torsion, we can compute τ(F_{∗}) and τ(G_{∗}) with respect to chain contractions which are compatible with u_{∗}. Then one easily checks in K̃_{1}(R)

τ(G_{∗}) − τ(F_{∗}) = ∑_{p∈ℤ}(−1)^{p} · [u_{p}],  (2.11)

where [u_{p}] is the element represented by the matrix of u_{p} with respect to the given bases.

Let ϵ_{∗} be a chain contraction for E_{∗}. Choose for any p ∈ ℤ an R-homomorphism σ_{p}: E_{p} → D_{p} satisfying p_{q} ∘ σ_{q} = id. Define s_{p}: E_{p} → D_{p} by d_{p+1} ∘ σ_{p+1} ∘ ϵ_{p} + σ_{p}∘ϵ_{p−1}∘e_{p}. One easily checks that the collection of the s_{p}-s defines a chain map s_{∗}: E_{∗} → D_{∗} with q_{∗} ∘ s_{∗} = id. Thus we obtain an isomorphism of contractible based free R-chain complexes

i_{∗} ⊕ q_{∗}: C_{∗} ⊕ E_{∗} → D_{∗}.

Since the matrix of i_{p} ⊕ s_{p} with respect to the given basis is a block matrix

of the shape ( \begin{matrix}I_{m} & ∗ \\ 0 & I_{n} \end{matrix} ) we get [i_{p} ⊕ s_{p}] = 0 in K̃_{1}(R). Now (2.11) implies

τ(C_{∗} ⊕ D_{∗}) = τ(E_{∗}). Since obviously τ(C_{∗} ⊕ D_{∗}) = τ(C_{∗}) + τ(D_{∗}), (2.10) and thus assertion (1) follows.

(2) If h_{∗}: f_{∗} ≃ g_{∗} is a chain homotopy, we obtain an isomorphism of based free R-chain complexes

( \begin{matrix}id & 0 \\ h_{∗−1} & id \end{matrix} ) : cone_{∗}(f_{∗}) = C_{∗−1} ⊕ D_{∗} → cone_{∗}(g_{∗}) = C_{∗−1} ⊕ D_{∗}.

We conclude from (2.11)

τ(g_{∗}) − τ(f_{∗}) = ∑_{p∈ℤ}(−1)^{p} · [( \begin{matrix}id & 0 \\ h_{∗−1} & id \end{matrix} )] = 0.

(3) Define a chain map h_{∗}: Σ^{−1} cone_{∗}(g_{∗}) → cone_{∗}(f_{∗}) by

( \begin{matrix}0 & 0 \\ − id & 0 \end{matrix} ) : D_{p} ⊕ E_{p+1} → C_{p−1} ⊕ D_{p}.

There is an obvious based exact sequence of contractible finite based free Rchain complexes 0 → cone_{∗}(f_{∗}) → cone(h_{∗}) → cone(g_{∗}) → 0. There is also a based exact sequence of contractible finite based free R-chain complexes 0 → cone_{∗}(g_{∗} ∘ f_{∗}) ⟶^{i_{∗}} cone_{∗}(h_{∗}) → cone_{∗}(id: D_{∗} → D_{∗}) → 0, where i_{p} is given by

( \begin{matrix}f_{p−1} & 0 \\ 0 & id \\ id & 0 \\ 0 & 0 \end{matrix} ) : C_{p−1} ⊕ E_{p} → D_{p−1} ⊕ E_{p} ⊕ C_{p−1} ⊕ D_{p}.

We conclude from assertion (1)

τ(h_{∗}) = τ(f_{∗}) + τ(g_{∗});
τ(h_{∗}) = τ(g_{∗} ∘ f_{∗}) + τ(id_{∗}: D_{∗} → D_{∗});
τ(id_{∗}: D_{∗} → D_{∗}) = 0.

This finishes the proof of Lemma 2.9.

Now we can pass to CW -complexes. Let f: X → Y be a homotopy equivalence of connected finite CW -complexes. Let p_{X}: X̃ → X and p_{Y} : Ỹ → Y be the universal coverings. Fix base points x̃ ∈ X̃ and ỹ ∈ Ỹ such that f maps x = p_{X}(x̃) to y = p_{Y} (ỹ). Let f̃: X̃ → Ỹ be the unique lift of f satisfying f̃(x̃) = ỹ. We abbreviate π = π_{1}(Y, y) and identify π_{1}(X, x) in the sequel with π by π_{1}(f, x). After the choice of the base points x̃ and ỹ we get unique operations of π on X̃ and Ỹ. The lift f̃ is π-equivariant. It induces a ℤπ-chain homotopy equivalence C_{∗}(f̃): C_{∗}(X̃) → C_{∗}(Ỹ). We can apply (2.8) to it and thus obtain an element

τ(f) ∈ Wh(π_{1}(Y, y)).  (2.12)

So far this definition depends on the various choices of base points. We can get rid of these choices as follows. If y′ is a second base point, we can choose path w from y to y′ in Y. Conjugation with w yields a homomorphism c_{w}: π_{1}(Y, y) → π_{1}(Y, y′) which induces (c_{w})_{∗}: Wh(π_{1}(Y, y)) → Wh(π_{1}(Y, y′)). If v is a different path from y to y′, then c_{w} and c_{v} differ by an inner automorphism of π_{1}(Y, y). Since an inner automorphism of π_{1}(Y, y) induces the identity on Wh(π_{1}(Y, y)), we conclude that (c_{w})_{∗} and (c_{v})_{∗} agree. Hence we get a unique isomorphism t(y, y′): Wh(π_{1}(Y, y)) → Wh(π_{1}(Y, y′)) depending only on y and y′. Moreover t(y, y) = id and t(y, y′′) = t(y′, y′′) ∘ t(y, y′). Therefore we can define Wh(π(Y)) independently of a choice of a base point by ∐_{y∈Y} Wh(π_{1}(Y, y))/ ∼, where ∼ is the obvious equivalence relation generated by a ∼ b ⇔ t(y, y′)(a) = b for a ∈ Wh(π_{1}(Y, y)) and b ∈ Wh(π_{1}(Y, y′)). Define τ(f) ∈ Wh(π_{1}(Y)) by the element represented by the element introduced in (2.12). Notice that Wh(π(Y)) is isomorphic to Wh(π_{1}(Y, y)) for any base point y ∈ Y. It is not hard to check using Lemma 2.9 that τ(f) depends only on f: X → Y and not on the choice of the universal coverings and base points. Finally we want to drop the assumption that Y is connected. Notice that f induces a bijection π_{0}(f): π_{0}(X) → π_{0}(Y).

Definition 2.13 Let f: X → Y be a homotopy equivalence of finite CW - complexes. Define the Whitehead group Wh(π(Y)) of Y and the Whitehead torsion τ(f) ∈ Wh(π(Y)) by

Wh(π(Y)) = ⊕_{C∈π_{0}(Y)} Wh(π_{1}(C));
τ(f) = ⊕_{C∈π_{0}(Y)}τ (f|_{π_{0}(f)^{−1}(C)} : π_{0}(f)^{−1}(C) → C) .

In the notation Wh(π(Y)) one should think of π(Y) as the fundamental groupoid of Y. Notice that a map f: X → Y induces a homomorphism f_{∗}: Wh(π(X)) → Wh(π(Y)) such that id_{∗} = id, (g ∘ f)_{∗} = g_{∗} ∘ f_{∗} and f ≃ g ⇒ f_{∗} = g_{∗}.

Suppose that the following diagram is a pushout

A ⟶^{f} B
i↓  ↓^{j}
X ⟶_{g} Y

the map i is an inclusion of CW -complexes and f is a cellular map of CW - complexes, i.e. respects the filtration given by the CW -structures. Then Y inherits a CW -structure by defining Y_{n} as the union of j(B_{n}) and g(X_{n}). If we equip Y with this CW -structure, we call the pushout above a cellular pushout.

Next we give the proof of Theorem 2.1.

(1), (2) and (3) follow from Lemma 2.9.

(4) Because of assertion (3) we have

τ(f × g) = τ(f × id_{Y}) + (f × id_{Y})_{∗}τ(id_{X} ×g).

Hence it suffices to treat the case g = id_{Y}. Now one proceeds by induction over the cells of Y using assertions (1), (2) and (3).

(5) This (in comparision with the other assertions much deeper result) is due to Chapman [25], [26]. This finishes the proof of Theorem 2.1.

We define the Whitehead torsion of an h-cobordism (W; M_{0}, M_{1}, f_{0}, f_{1})

τ(W, M_{0}) ∈ Wh(π(M_{0})).  (2.14)

by τ(W, M_{0}) := (i_{0} ∘ f_{0})_{∗}^{−1} (τ(i_{0} ∘ f_{0} : M_{0} → W)), where we equip W and M_{0} with some CW -structure, for instance one coming from a smooth triangulation. This is independent of the choice of CW -structure by Theorem 2.1 (5). Because of Theorem 2.1 (5) two h-cobordisms over M_{0} which are diffeomorphic relative M_{0} have the same Whitehead torsion.

Let R be a ring with involution i: R → R, r → \overline{r}, i.e. a map satisfying \overline{r + s} = \overline{r} + \overline{s}, \overline{r · s} = \overline{s} · \overline{r} and \overline{1} = 1. Given a (m, n)-matrix A = (a_{i,j}) define the (n, m)-matrix A^{∗} by (\overline{a_{j,i}}). We obtain an involution

∗: K_{1}(R) → K_{1}(R), [A] ↦ [A^{∗}].  (2.15)

Let P be a left R-module. Define the dual R-module P^{∗} to be the left Rmodule whose underlying abelian group is P^{∗} = hom(P, R) and whose left R-module structure is given by (rf)(x) := f(x)\overline{r} for f ∈ P^{∗} and x ∈ P. Then the induced involution on K_{1}(R) corresponds to [f: P → P] ↦ [f^{∗}: P^{∗} → P^{∗}] if one defines K_{1}(R) as in Remark 2.5. We equip ℤG with the involution \overline{∑_{g∈G} λ_{g} · g} = ∑_{g∈G} λ_{g} · g^{−1}. Thus we get an involution on K_{1}(ℤG) which induces an involution ∗: Wh(G) → Wh(G).

Lemma 2.16 1. Let (W; M_{0}, f_{0}, M_{1}, f_{1}) and (W′; M′_{0}, f′_{0}, M′_{1}, f′_{1}) be h-co-bordisms over M_{0} and M′_{0} and let g: M_{1} → M′_{0} be a diffeomorphism. Let W ∪ W′ be the h-cobordism over M_{0} obtained from W and W′ by glueing with the diffeomorphism f′_{0} ∘ g ∘ f_{1}^{−1} : ∂_{1}W → ∂_{0}W′. Let u: Wh(M′_{0}) → Wh(M_{0}) be the isomorphism given by the composition (f_{0})_{∗}^{−1} ∘ (i_{0})_{∗}^{−1} ∘ (i_{1})_{∗} ∘ (f_{1})_{∗} ∘ (g_{∗})^{−1}, where i_{k}: ∂_{k}W → W is the inclusion for k = 0, 1. Then

τ(W ∪ W′, M_{0}) = τ(W, M_{0}) + u (τ(W′, M′_{0})) .

2. Let (W; M_{0}, f_{0}, M_{1}, f_{1}) be an h-cobordism over M_{0}. Let v: Wh(M_{1}) → Wh(M_{0}) be the isomorphism given by the composition (f_{0})_{∗}^{−1} ∘ (i_{0})_{∗}^{−1} ∘ (i_{1})_{∗} ∘ (f_{1})_{∗}. Then

∗(τ(W, M_{0})) = (−1)^{dim(M_{0})} · v(τ(W, M_{1})).

\underline{\underline{Proof}} : (1) follows from Theorem 2.1.

(2) Let C_{∗}(W̃, { \atop ∂_{k}W}) be the cellular ℤπ-chain complex with respect to a triangulation of W of the universal covering W̃ → W for π = π_{1}(W) = π_{1}(∂_{0}W) = π_{1}(∂_{1}W). Put n = dim(W). The dual ℤπ-chain complex C^{n−∗}(W̃, { \atop ∂_{1}W}) has as p-th chain module the dual module of C_{n−p}(W̃, { \atop ∂_{1}W}) and its p-th differential is the dual of c_{n−p+1} : C_{n−p+1}(W̃, { \atop ∂_{1}W}) → C_{n−p}(W̃, { \atop ∂_{1}W}). Poincaŕe duality yields a ℤπ-chain homotopy equivalence for n = dim(W)

∩[W, ∂W]: C^{n−∗}(W̃, { \atop ∂_{1}W}) → C_{∗}(W̃, { \atop ∂_{0}W}).

Inspecting the proof of Poincaré duality using dual cells [67] shows that this is a base preserving chain map if one passes to subdivisions. This implies that this ℤπ-chain homotopy equivalence has trivial Whitehead torsion. We conclude from Lemma 2.9

τ(C_{∗}(W̃, { \atop ∂_{0}W})) = τ(C^{n−∗}(W̃, { \atop ∂_{1}W})) = (−1)^{n−1} · ∗ (τ(C_{∗}(W̃, { \atop ∂_{1}W}))) .

This finishes the proof of Lemma 2.16

Next we can finish the proof of the s-Cobordism Theorem 1.1. As mentioned already in Remark 1.28, Theorem 1.1 (1) follows if we can show that the class of the matrix [A] ∈ Wh(π) agrees with τ(W, M_{0}) and that τ(W, M_{0}) depends only on the diffeomorphism type of W relative M_{0} (see Lemma 1.27 (1)). Using Theorem 2.1 one can show that the homotopy equivalence f: W → X of Section 1.2 satisfies τ(f) = 0 and hence τ(W, M_{0}) = τ(C_{∗}(X̃, { \atop ∂_{0}W})). We conclude [A] = τ(C_{∗}(X̃, { \atop ∂_{0}W})) from the existence of the base preserving isomorphism (1.17). In view of Remark 1.28 and Theorem 1.1 assertions (1) and (2) of Theorem 1.1 follow. In order to prove Theorem 1.1 (3), we must show for two h-cobordisms (W; M_{0}, M_{1}, f_{0}, f_{1}) and (W′; M_{0}, M′_{1}, f′_{0}, f′_{1}) over M_{0} with τ(W, M_{0}) = τ(W′, M_{0}) that they are diffeomorphic relative M_{0}. Choose an h-cobordism (W′′; M_{1}, M′_{2}, f′_{1}, f′_{2}) over M_{1} such that τ(W′′; M_{1}, M_{2}, f′_{1}, f_{2}) ∈ Wh(π(M_{1})) is the image of τ(W, M_{0}) under the isomorphism (f_{1} ∘ i_{1})_{∗}^{−1} ∘ (f_{0} ∘ i_{0})_{∗}: Wh(π(M_{0})) → Wh(π(M_{1})), where i_{k}: ∂_{k}W → W for k = 0, 1 is the inclusion. We can glue W and W′′ along M_{1} to get an h-cobordism W ∪_{M_{1}} W′′ over M_{0}. From Lemma 2.16 (1) we get τ(W ∪_{M_{1}} W′′, M_{0}) = 0. Hence there is a diffeomorphism G: W ∪_{M_{1}} W′′ → M × [0, 1] which induces the identity on M_{0} = M_{0} × {0} and a diffeomorphism g_{1} : M_{2} → M_{0} × {1} = M_{0}. Now we can form the h-cobordism W ∪_{M_{1}} W′′ ∪_{g_{1}} W′. Using G we can construct a diffeomorphism relative M_{0} from W ∪_{M_{1}} W′′ ∪_{g_{1}} W′ to W′. Similarly one can show that W′′∪_{g_{1}} W′ is diffeomorphic relative M_{1} to the trivial h-cobordism over M_{1}. Hence there is also a diffeomorphism relative M_{0} from W ∪_{M_{1}} W′′ ∪_{g_{1}} W′ to W. Hence W and W′ are diffeomorphic relative M_{0}. This finishes the proof of the s-Cobordism Theorem 1.1.

2.3 The Geometric Approach to Whitehead Torsion

In this section we introduce the concept of a simple homotopy equivalence f: X → Y of finite CW -complexes geometrically. We will show that the obstruction for a homotopy equivalence f: X → Y of finite CW -complexes to be simple is the Whitehead torsion. This proof is a CW -version or homotopy version of the proof of the s-Cobordism Theorem 1.1.

We have the inclusion of spaces S^{n−2} ⊂ S_{+}^{n−1} ⊂ S^{n−1} ⊂ D^{n}, where S_{+}^{n−1} ⊂ S^{n−1} is the upper hemisphere. The pair (D^{n}, S_{+}^{n−1}) carries an obvious relative CW -structure. Namely, attach an (n − 1)-cell to S_{+}^{n−1} by the attaching map id: S^{n−2} → S^{n−2} to obtain S^{n−1}. Then we attach to S^{n−1} an n-cell by the attaching map id: S^{n−1} → S^{n−1} to obtain D^{n}. Let X be a CW -complex. Let q: S_{+}^{n−1} → X be a map satisfying q(S^{n−2}) ⊂ X_{n−2} and q(S_{+}^{n−1}) ⊂ X_{n−1}. Let Y be the space D^{n} ∪_{q} X, i.e. the pushout

S_{+}^{n−1} ⟶^{q} X
i↓  ↓^{j}
D^{n} ⟶_{g} Y

where i is the inclusion. Then Y inherits a CW -structure by putting Y_{k} = j(X_{k}) for k ≤ n − 2, Y_{n−1} = j(X_{n−1}) ∪ g(S^{n−1}) and Y_{k} = j(X_{k}) ∪ g(D^{n}) for k ≥ n. Notice that Y is obtained from X by attaching one (n − 1)-cell and one ncell. Since the map i: S_{+}^{n−1} → D^{n} is a homotopy equivalence and cofibration, the map j: X → Y is a homotopy equivalence and cofibration. We call j an elementary expansion and say that Y is obtained from X by an elementary expansion. There is a map r: Y → X with r ∘ j = id_{X}. This map is unique up to homotopy relative j(X). We call any such map an elementary collaps and say that X is obtained from Y by an elementary collaps.

An elementary expansion is the CW -version or homotopy version of the construction in Example 1.11, where we have added a q-handle and a (q + 1)- handle to W without changing the diffeomorphism type of W. This corresponds to an elementary expansion for the CW -complex X which we have assigned to W in Section 1.2.

Definition 2.17 Let f: X → Y be a map of finite CW -complexes. We call it a simple homotopy equivalence if there is a sequence of maps

X = X[0] ⟶^{f_{0}} X[1] ⟶^{f_{1}} X[2] … ⟶^{f_{n−1}} X[n] = Y

such that each f_{i} is an elementary expansion or elementary collaps and f is homotopic to the composition of the maps f_{i}.

The idea of the definition of a simple homotopy equivalence is that such a map can be written as a composition of elementary maps which are obviously homotopy equivalences. This is similar to the idea in knot theory that two knots are equivalent if one can pass from one knot to the other by a sequence of elementary moves, the so called Reidemeister moves. A Reidemeister move obviously does not change the equivalence class of a knot and, indeed, it turns out that one can pass from one knot to a second knot by a sequence of Reidemeister moves if the two knots are equivalent. The analogous statement is not true for homotopy equivalences f: X → Y of finite CW -complexes because there is an obstruction for f to be simple, namely its Whitehead torsion.

Lemma 2.18 1. Let f: X → Y be a simple homotopy equivalence. Then its Whitehead torsion τ(f) ∈ Wh(Y) vanishes;

2. Let X be a finite CW -complex. Then for any element x ∈ Wh(π(X)) there is an inclusion i: X → Y of finite CW -complexes such that i is a homotopy equivalence and i_{∗}^{−1}(τ(i)) = x.

\underline{\underline{Proof}} : (1) Because of Theorem 2.1 it suffices to prove for an elementary expansion j: X → Y that its Whitehead torsion τ(j) ∈ Wh(Y) vanishes. We can assume without loss of generality that Y is connected. In the sequel we write π = π_{1}(Y) and identify π = π_{1}(X) by π_{1}(f). The following diagram of based free finite ℤπ-chain complexes

0 ⟶ C_{∗}(X̃) ⟶^{C_{∗}(j̃)} C_{∗}(Ỹ) ⟶^{pr_{∗}} C_{∗}(Ỹ, X̃) ⟶ 0
id_{∗}↑  ^{C_{∗}(j̃)}↑  ^{0_{∗}}↑
0 ⟶ C_{∗}(X̃) ⟶^{id_{∗}} C_{∗}(X̃) ⟶^{pr_{∗}}  0  ⟶ 0

has based exact rows and ℤπ-chain homotopy equivalences as vertical arrows. We conclude from Lemma 2.9 (1)

τ (C_{∗}(j̃)) = τ (id_{∗}: C_{∗}(X̃) → C_{∗}(X̃)) + τ (0_{∗}: 0 → C_{∗}(Ỹ, X̃))

= τ (C_{∗}(Ỹ, X̃)) .

The ℤπ-chain complex C_{∗}(Ỹ, X̃) is concentrated in two consecutive dimensions and its only non-trivial differential is id: ℤπ → ℤπ if we identify the two non-trivial ℤπ-chain modules with ℤπ using the cellular basis. This implies τ(C_{∗}(Ỹ, X̃)) = 0 and hence τ(j) := τ(C_{∗}(j̃)) = 0.

(2) We can assume without loss of generality that X is connected. Put π = π_{1}(X). Choose an element A ∈ GL(n, ℤπ) representing x ∈ Wh(π). Choose n ≥ 2. In the sequel we fix a zero-cell in X as base point. Put X′ = X ∨∨_{j=1}^{n}S^{n}. Let b_{j} ∈ π_{n}(X′) be the element represented by the inclusion of the j-th copy of S^{n} into X for j = 1, 2 …, n. Recall that π_{n}(X′) is a ℤπ-module. Choose for i = 1, 2 …, n a map f_{i}: S^{n} → X′_{n} such that [f_{i}] = ∑_{j=1}^{n} a_{i,j} · b_{j} holds in π_{n}(X′). Attach to X′ for each i ∈ {1, 2 …, n} an (n + 1)-cell by f_{i}: S^{n} → X′_{n}. Let Y be the resulting CW -complex and i: X → Y be the inclusion. Then i is an inclusion of finite CW -complexes and induces an isomorphism on the fundamental groups. In the sequel we identify π and π_{1}(Y) by π_{1}(i). The cellular ℤπ-chain complex C_{∗}(Ỹ, X̃) is concentrated in dimensions n and (n + 1) and its (n + 1)-differential is given by the matrix A with respect to the cellular basis. Hence C_{∗}(Ỹ, X̃) is a contractible finite based free ℤπ-chain complex with τ(C_{∗}(Ỹ, X̃)) = [A] in Wh(π). This implies that i: X → Y is a homotopy equivalence with i_{∗}^{−1}(τ(i)) = x. This finishes the proof of Lemma 2.18.

Notice that Lemma 2.18 (2) is the CW -analog of Theorem 1.1 (2).

Recall that the mapping cylinder cyl(f) of a map f: X → Y is defined by the pushout

X × {0} ⟶^{f} Y
↓  ↓
X × [0, 1] ⟶ cyl(f)

There are natural inclusions i_{X}: X = X × {1} → cyl(f) and i_{Y} : Y → cyl(f) and a natural projection p: cyl(f) → Y. Notice that i_{X} is a cofibration and p ∘ i_{X} = f and p_{Y} ∘ Y = id_{Y}. Define the mapping cone cone(f) by the quotient cyl(f)/i_{X}(X).

Lemma 2.19 Let f: X → Y be a cellular map of finite CW -complexes and A ⊂ X be a CW -subcomplex. Then the inclusion cyl(f|_{A}) → cyl(f) and (in the case A = ∅) i_{Y} : Y → cyl(f) is a composition of elementary expansions and hence a simple homotopy equivalence.

It suffices to treat the case, where X is obtained from A by attaching an n-cell by an attaching map q: S^{n−1} → X. Then there is an obvious pushout

S^{n−1} × [0, 1] ∪_{S^{n−1}×{0}} D^{n} × {0} ⟶ cyl(f|_{A})
↓  ↓
D^{n} × [0, 1]  ⟶ cyl(f)

and an obvious homeomorphism (D^{n}×[0, 1], S^{n−1}×[0, 1]∪_{S^{n−1}×{0}}D^{n}×{0}) → (D^{n+1}, S_{+}^{n}).

Lemma 2.20 A map f: X → Y of finite CW -complexes is a simple homotopy equivalence if and only if i_{X}: X → cyl(f) is a simple homotopy equivalence.

\underline{\underline{Proof}} : follows from Lemma 2.19 since a composition of simple homotopy equivalence and a homotopy inverse of a simple homotopy equivalence is again a simple homotopy equivalence.

Fix a finite CW -complex X. Consider to pairs of finite CW -complexes (Y, X) and (Z, X) such that the inclusions of X into Y and Z are homotopy equivalent. We call them equivalent, if there is a chain of pairs of finite CW - complexes

(Y, X) = (Y [0], X), (Y [1], X), (Y [2], X), …, (Y [n], X) = (Z, X),

such that for each k ∈ {1, 2, …, n} either Y [k] is obtained from Y [k − 1] by an elementary expansion or Y [k − 1] is obtained from Y [k] by an elementary expansion. Denote by Wh^{geo}(X) the equivalence classes [Y, X] of such pairs (Y, X). This becomes an abelian group under the addition [Y, X] + [Z, X] := [Y ∪_{X} Z, X]. The zero element is given by [X, X]. The inverse of [Y, X] is constructed as follows. Choose a map r: Y → X with r_{X} = id. Let p: X × [0, 1] → X be the projection. Then [(cyl(r) ∪_{p} X) ∪_{r} X, X] + [Y, X] = 0. A map g: X → X′ induces a homomorphism g_{∗}: Wh^{geo}(X) → Wh^{geo}(X′) by sending [Y, X] to [Y ∪_{g}X′, X′]. We obviously have id_{∗} = id and (g∘h)_{∗} = g_{∗}∘h_{∗}. In other words, we obtain a covariant functor on the category of finite CW -complexes with values in abelian groups.

The next result may be viewed as the homotopy theoretic analog of the s-Cobordism Theorem 1.1 (3), where Wh^{geo}(X) plays the role of the set of the diffeomorphism classes relative M_{0} of h-cobordism over M_{0}

Theorem 2.21 1. Let X be a finite CW -complex. The map

τ: Wh^{geo}(X) → Wh(X)

sending [Y, X] to i_{∗}^{−1}τ(i) for the inclusion i: X → Y is a natural isomorphism;

2. A homotopy equivalence f: X → Y is a simple homotopy equivalence if and only if τ(f) ∈ Wh(Y) vanishes.

\underline{\underline{Proof}} : (1) The map τ is a well-defined homomorphism by Theorem 2.1 and Lemma 2.18 (1). It is surjective by Lemma 2.18 (2).

We give only a sketch of the proof of injectivity which is similar but much easier than the proof of s-Cobordism Theorem 1.1 (1). Consider an element [Y, X] in Wh^{geo}(X) with i_{∗}^{−1}τ(i) = 0 for the inclusion i: X → Y. We want to show that [Y, X] = [X, X] by reducing the number of cells, which must be attached to X to obtain Y, to zero without changing the class [Y, X] ∈ Wh^{geo}(X). This corresponds in the proof of the s-Cobordism Theorem 1.1 (1) to reducing the number of handles in the handlebody decomposition to zero without changing the diffeomorphism type of the s-cobordism.

In the first step one arranges that Y is obtained from X by attaching only cells in two dimensions r and (r + 1) for some integer r. This is analogous to, but much easier to achieve than in the case of the s-Cobordism Theorem 1.1 (1) (see Normal Form Lemma 1.24). Details of this construction for CW -complexes can be found in [28, page 25-26].

Let A ∈ GL(n, ℤπ) be the matrix describing the (r + 1)-differential in the ℤπ-chain complex C_{∗}(Ỹ, X̃). As in the proof of the s-Cobordism Theorem 1.1 (1) (see also [28, chapter II, Section §8]) one shows that we can modify (Y, X) without changing its class [Y, X] ∈ Wh^{geo}(X) such that the new matrix B is obtained from A by applying one of the operations (1), (2), (3), (4) and (5) introduced in Section 1.4. Since one can reduce A by a sequence of these operation to the trivial matrix if and only if its class [A] ∈ Wh(X) vanishes and this class [A] is i_{∗}^{−1}τ(i), the map τ is injective. Hence τ is a natural isomorphism of abelian groups.

(2) follows from Lemma 2.18 (1), Lemma 2.20 and the obvious fact that i: X → Y is a simple homotopy equivalence if [Y, X] = 0 in Wh^{geo}(X). This finishes the proof of Theorem 2.21.

2.4 Reidemeister Torsion and Lens Spaces

In this section we deal with Reidemeister torsion which was defined earlier than Whitehead torsion and motivated the definition of Whitehead torsion. Reidemeister torsion was the first invariant in algebraic topology which could distinguish between spaces which are homotopy equivalent but not homeomorphic. Namely, it can be used to classify lens spaces up to homeomorphism.

Let X be a finite CW -complex with fundamental group π. Let U be an orthogonal finite-dimensional π-representation. Denote by H_{p}(X; U) the homology of X with coefficients in U, i.e. the homology of the ℝ-chain complex U ⊗_{ℤπ} C_{∗}(X̃). Suppose that X is U-acyclic, i.e. H_{p}(X; U) = 0 for all p ≥ 0. If we fix a cellular basis for C_{∗}(X̃) and some orthogonal ℝ-basis for U, then U ⊗_{ℤπ} C_{∗}(X̃) is a contractible based free finite ℝ-chain complex and defines an element τ(U ⊗_{ℤπ} C_{∗}(X̃)) ∈ K̃_{1}(ℝ) (see (2.7)). Define the Reidemeister torsion

ρ(X; U) ∈ ℝ^{>0}  (2.22)

to be the image of τ(U ⊗_{ℤπ}C_{∗}(X̃)) ∈ K̃_{1}(ℝ) under the homomorphism K̃_{1}(ℝ) → ℝ^{>0} sending the class [A] of A ∈ GL(n, ℝ) to det(A)^{2}. Notice that for any trivial unit ±γ the automorphism of U given by multiplication with ±γ is orthogonal and that the square of the determinant of any orthogonal automorphism of U is 1. Therefore ρ(X; U) ∈ ℝ^{>0} is independent of the choice of cellular basis for C_{∗}(X̃) and the orthogonal basis for U and hence is an invariant of the CW - complex X and U.

Lemma 2.23 Let f: X → Y be a homotopy equivalence of connected finite CW -complexes and let U be an orthogonal finite-dimensional π = π_{1}(Y)-repre-sentation. Suppose that Y is U-acyclic. Let f^{∗}U be the orthogonal π_{1}(X)- representation obtained from U by restriction with the isomorphism π_{1}(f). Let det_{U}: Wh(π(Y)) → ℝ^{>0} be the map sending the class [A] of A ∈ GL(n, ℤπ_{1}(Y)) to det(id_{U} ⊗_{ℤπ}R_{A}: U ⊗_{ℤπ} ℤπ^{n} → U ⊗_{ℤπ} ℤπ^{n})^{2}, where R_{A}: ℤπ^{n} → ℤπ^{n} is the ℤπ-automorphism induced by A. Then

\frac{ρ(Y, U)}{ρ(X, f^{∗}U)} = det_{U}(τ(f)).

\underline{\underline{Proof}} : This follows from Lemma 2.9 (1) applied to the based exact sequence of contractible based free finite ℝ-chain complexes 0 → U ⊗_{ℤπ} C_{∗}(Ỹ) → U ⊗_{ℤπ} cone_{∗}(C_{∗}(f̃)) → Σ (U ⊗_{ℤπ_{1}(f)} C_{∗}(X̃)) → 0.

Next we introduce the family of spaces which we want to classify completely using Reidemeister torsion. Let G be a cyclic group of finite order |G|. Let V be a unitary finite-dimensional G-representation. Define its unit sphere SV and its unit disk DV to be the G-subspaces SV = {v ∈ V | ||v|| = 1} and DV = {v ∈ V | ||u|| ≤ 1} of V. Notice that a complex finite-dimensional vector space has a preferred orientation as real vector space, namely the one given by the ℝ-basis {b_{1}, ib_{1}, b_{2}, ib_{2}, …, b_{n}, ib_{n}} for any ℂ-basis {b_{1}, b_{2}, …, b_{n}}. Any ℂlinear automorphism of a complex finite-dimensional vector space preserves this orientation. Thus SV and DV are compact oriented Riemannian manifolds with isometric orientation preserving G-action. We call a unitary G-representation V free if the induced G-action on its unit sphere SV = {v ∈ V | ||u|| = 1} is free. Then SV → G\SV is a covering and the quotient space L(V) := G\SV inherits from SV the structure of an oriented closed Riemannian manifold.

Definition 2.24 We call the closed oriented Riemannian manifold L(V) the lens space associated to the finite-dimensional unitary representation V of the finite cyclic group G.

One can specify these lens spaces also by numbers as follows.

Notation 2.25 Let ℤ/t be the cyclic group of order t ≥ 2. The 1-dimensional unitary representation V_{k} for k ∈ ℤ/t has as underlying vector space ℂ and l ∈ ℤ/t acts on it by multiplication with exp(2πikl/t). Notice that V_{k} is free if and only if k ∈ ℤ/t^{∗}, and is trivial if and only if k = 0 in ℤ/t. Define the lens space L(t; k_{1}, …, k_{c}) for an integer c ≥ 1 and elements k_{1}, …, k_{c} in ℤ/t^{∗} by L(⊕_{i=1}^{c}V_{k_{i}}).

These lens spaces form a very interesting family of manifolds which can be completely classified as we will see. Two lens spaces L(V) and L(W) of the same dimension n ≥ 3 have the same homotopy groups, namely their fundamental group is G and their p-th homotopy group is isomorphic to π_{p}(S^{n}). They also have the same homology with integral coefficients, namely H_{p}(L(V)) ≅ ℤ for p = 0, 2n − 1, H_{p}(L(V)) ≅ G for p odd and 1 ≤ p < n and H_{p}(L(V)) = 0 for all other values of p. Also their cohomology groups agree. Nevertheless not all of them are homotopic. Moreover, there are homotopic lens spaces which are not diffeomorphic (see Example 2.41).

Suppose that dim_{ℂ}(V) ≥ 2. We want to give an explicit identification

π_{1}(L(V), x) = G.  (2.26)

Given a point x ∈ L(V), we obtain an isomorphism s(x): π_{1}(L(V), x) ⟶^{≅} G by sending the class of a loop w in L(V) with base point x to the element g ∈ G for which there is a lift w̃ in SV of w with w̃(1) = g·w̃(0). One easily checks using elementary covering theory that this is a well-defined isomorphism. If y is another base point, we obtain a homomorphism t(x, y): π_{1}(L(V), x) → π_{1}(L(V), y) by conjugation with any path v in L(V) from x to y. Since π_{1}(L(V), x) is abelian, t(x, y) is independent of the choice of v. One easily checks t(x, x) = id, t(y, z) ∘ t(x, y) = t(x, z) and s(y) ∘ t(x, y) = s(x). Hence we can in the sequel identify π_{1}(L(V), x) with G and ignore the choice of the base point x ∈ L(V).

Let p: EG → BG be a model for the universal principal G-bundle. It has the property that for any principal G-bundle q: E → B there is a map f: B → BG called classifying map of q which is up to homotopy uniquely determined by the property that the pull back of p with f is isomorphic over B to q. Equivalently p can be characterized by the property that BG is a CW -complex and EG is contractible. The space BG is called the classifying space for G. Let f(V): L(V) → BG be the classifying map of the principal G-bundle SV → L(V). Put n = dim(L(V)) = 2 · dim_{ℂ}(V) − 1. Define the element

l(V) ∈ H_{n}(BG)  (2.27)

by the image of the fundamental class [L(V)] ∈ H_{n}(L(V)) associated to the preferred orientation of L(V) under the map H_{n}(f(V)): H_{n}(L(V)) → H_{n}(BG) induced by f(V) on homology with integer coefficients. The map f(V): L(V) → BG is n-connected since its lift SV → EG is n-connected. Hence H_{n}(f(V)) is surjective. As H_{n}(L(V)) is infinite cyclic with [L(V)] as generator, l(V) generates H_{n}(BG). Notice that n is odd and that H_{n}(BG) is isomorphic to ℤ/|G| for a cyclic group G of finite order |G|. A map f: L(V) → L(W) of lens spaces of the same dimension n = dim(L(V)) = dim(L(W)) for two free unitary G-representations V and W induces a homomorphism π_{1}(f, x): π_{1}(L(V)) → π_{1}(L(W)). Under the identification (2.26) this is an endomorphism π_{1}(f) of G. Define

e(f) ∈ ℤ/|G|  (2.28)

to be the element for which π_{1}(f) sends g ∈ G to g^{e(f)}. Notice that e(f) depends only on the homotopy class of f and satisfies e(g ∘f) = e(g)·e(f) and e(id) = 1. In particular e(f) ∈ ℤ/|G|^{∗} for a homotopy equivalence f: L(V) → L(W). Define the degree

deg(f) ∈ ℤ  (2.29)

of f to be the integer for which H_{n}(f) sends [L(V)] ∈ H_{n}(L(V)) to deg(f) · [L(W)] ∈ H_{n}(L(W)).

Given two spaces X and Y, define their join X ∗ Y by the pushout

X × Y  ⟶ X × cone(Y)
↓  ↓
cone(X) × Y ⟶  X ∗ Y

If X and Y are G-spaces, X ∗ Y inherits a G-operation by the diagonal operation. Given two free finite-dimensional unitary G-representations, there is a G-homeomorphism

S(V ⊕ W) ≅ SV ∗ SW.  (2.30)

Theorem 2.31 (Homotopy Classification of Lens Spaces) Let L(V) and L(W) be two lens spaces of the same dimension n ≥ 3. Then

1. The map

e × deg: [L(V), L(W)] → ℤ/|G| × ℤ, [f] ↦ (e(f), deg(f))

is injective, where [L(V), L(W)] is the set of homotopy classes of maps from L(V) to L(W);

2. An element (u, d) ∈ ℤ/|G| × ℤ is in the image of e × deg if and only if we

get in H_{n}(BG)

d · l(W) = u^{dim_{ℂ}(V)} · l(V);

3. L(V) and L(W) are homotopy equivalent if and only if there is an element e ∈ ℤ/|G|^{∗} satisfying in H_{n}(BG)

±l(W) = e^{dim_{ℂ}(V)} · l(V).

L(V) and L(W) are oriented homotopy equivalent if and only if there is an element e ∈ ℤ/|G|^{∗} satisfying in H_{n}(BG)

l(W) = e^{dim_{ℂ}(V)} · l(V);

4. The lens spaces L(t; k_{1}, …, k_{c}) and L(t; l_{1}, …, l_{c}) are homotopy equivalent if and only if there is e ∈ ℤ/t^{∗} such that we get ∏_{i=1}^{c} k_{i} = ±e^{c} · ∏_{i=1}^{c} l_{i} in ℤ/t^{∗}.

The lens spaces L(t; k_{1}, …, k_{c}) and L(t; l_{1}, …, l_{c}) are oriented homotopy equivalent if and only if there is e ∈ ℤ/t^{∗} such that we get ∏_{i=1}^{c} k_{i} = e^{c} · ∏_{i=1}^{c} l_{i} in ℤ/t^{∗}.

\underline{\underline{Proof}} : (1) Obviously e(f) and deg(f) depend only on the homotopy type of f. Consider two maps f_{0}, f_{1} : : L(V) → L(W) with e(f_{0}) = e(f_{1}) and deg(f_{0}) = deg(f_{1}). Choose lifts f̃_{0}, f̃_{1} : SV → SW. Let α: G → G be the automorphism sending g to g^{e(f_{0})} = g^{e(f_{1})}. Then both lifts f̃_{k} are α-equivariant. Since G acts orientation preserving and freely on SV and SW, the projection induces a map H_{n}(SV) → H_{n}(L(V)) resp. H_{n}(SW) → H_{n}(L(W)) which send the fundamental class [SV] resp. [SW] to |G| · [L(V)] resp. |G| · [L(W)]. Hence deg(f_{k}) = deg(f̃_{k}) for k = 0, 1. Thus deg(f_{0}) = deg(f_{1}) implies deg(f̃_{0}) = deg(f̃_{1}). Let α^{∗}SW be the G-space obtained from SW by restricting the group action with α. Then f̃_{0} and f̃_{1} : SV → α^{∗}SW are G-maps with deg(f̃_{0}) = deg(f̃_{1}). It suffices to show that they are G-homotopic.

We outline an elementary proof of this fact. There is a G-CW -structure on SV such that there is exactly one equivariant cell G × D^{k} in each dimension. It induces a G-CW -structure on SV × [0, 1] using the standard CW - structure on [0, 1]. We want to define inductively for k = −1, 0, …, n Gmaps h_{k}: SV × {0, 1} ∪ SV_{k} × [0, 1] → α^{∗}SW such that h_{k} extends h_{k−1} and h_{−1} = f̃_{0} ∐ f̃_{1} : SV ×{0, 1} → α^{∗}SW. Then h_{n} will be the desired G-homotopy between f_{0} and f_{1}. Notice that SV × {0, 1} ∪ SV_{k} × [0, 1] is obtained from SV × {0, 1} ∪ SV_{k−1} × [0, 1] by attaching one equivariant cell G × D^{k+1} with some attaching G-map q_{k}: G × S^{k} → SV. In the induction step we must show that the composition h_{k−1} ∘ q_{k}: G × S^{k} → SW can be extended to a G-map G × D^{k+1} → SW. This is possible if and only if its restriction to S^{k} = {1} × S^{k} can be extended to a (non-equivariant) map D^{k+1} → SW. Since SW is nconnected, this can be done for any map S^{k} → SW for k < n. In the final step we run into the obstruction that a map S^{n} → SW can be extended to a map D^{n+1} → SW if and only if its degree is zero. Now one has to check that the degree of the map (h_{n−1} ∘ q_{n})|_{{1}×S^{n}}: {1} × S^{n} = S^{n} → SW is exactly the difference deg(f_{0}) − deg(f_{1}) and hence zero.

(2) Given a map f: LV → LW, let π_{1}(f): G → G be the map induced on the fundamental groups under the identification (2.26). Then the following diagram commutes

H_{n}(LV) ⟶^{H_{n}(f)} H_{n}(LW)
H_{n}(f(V))↓  ↓^{H_{n}(f(W))}
H_{n}(BG) ⟶_{H_{n}(Bπ_{1}(f))} H_{n}(BG)

This implies

deg(f) · l(W) = H_{n}(Bπ_{1}(f))(l(V)).  (2.32)

Given e ∈ ℤ, let m_{e}: ℤ/t → ℤ/t be multiplication with e. Let e, k_{1}, …, k_{c}, l_{1}, …, l_{c} be integers which are prime to t. Fix integers k′_{1}, …, k′_{c} such that we get in ℤ/t the equation k_{i} · k′_{i} = 1. Define a m_{e}-equivariant map d_{i}: V_{k_{i}} → V_{l_{i}} by z ↦ z^{k′_{i}l_{i}e}. It has degree k′_{i}l_{i}e. The m_{e}-equivariant map ∗_{i=1}^{c}d_{i}: ∗_{i=1}^{c} SV_{k_{i}} → ∗_{i=1}^{c}SV_{l_{i}} yields under the identification (2.30) a m_{e}- equivariant map f̃: S(⊕_{i=1}^{c}V_{k_{i}}) → S(⊕_{i=1}^{c}V_{l_{i}}) of degree e^{c} · ∏_{i=1}^{c} k′_{i}l_{i}. By taking the quotient under the G-action yields a map f: L(t; k_{1}, …, k_{c}) → L(t; l_{1}, …, l_{c}) of n = (2c−1)-dimensional lens spaces with deg(f) = e^{c}·∏_{i=1}^{c} k′_{i}l_{i} and e(f) = e. We conclude from (2.32) in the special case k_{i} = l_{i} for i = 1, …, c that H_{n}(Bm_{e}): H_{n}(Bℤ/t) → H_{n}(Bℤ/t) is multiplication with e^{c} since l(t; k_{1} …, k_{c}) := l(⊕_{i=1}^{c}V_{k_{i}}) is always a generator of H_{n}(BG). Thus (2.32) becomes

deg(f) · l(W) = e(f)^{dim_{ℂ}(V)} · l(V).  (2.33)

We conclude from (2.33) in the special case e = 1 that we get in H_{n}(B(ℤ/t)) the equation

∏_{i=1}^{c} l_{i} · l(t; l_{1}, …, l_{c}) = ∏_{i=1}^{c} k_{i} · l(t; k_{1}, …, k_{c}).  (2.34)

Next we show for a map f: L(V) → L(W) and m ∈ ℤ that we can find another map f′: LV → LW with e(f) = e(f′) and deg(f′) = deg(f) + m · |G|.

Fix some embedded disk D^{n} ⊂ SV such that g · D^{n} ∩ D^{n} ≠ ∅ implies g = 1. Let f̃: SV → SW be a lift of f. Recall that f is m_{e}-equivariant for m_{e}: G → G, g ↦ g^{e}. Define a new m_{e}-equivariant map f̃′: SV → SW as follows. Outside G·D^{n} the maps f̃and f̃′ agree. Let \frac{1}{2}D^{n} be \{x ∈ D^{n} | ||x|| ≤ 1/2\}. Define f̃′ on G · (D^{n} − \frac{1}{2}D^{n}) by sending (g, t · x) for g ∈ G, t ∈ [1/2, 1] and x ∈ S^{n−1} = ∂D^{n} to g^{e} · f̃((2t − 1)x). Let c: (\frac{1}{2}D^{n}, ∂ \frac{1}{2}D^{n}) → (SW, f̃(0)) for 0 ∈ D^{n} the origin be a map such that the induced map (\frac{1}{2}D^{n})/∂(\frac{1}{2}D^{n}) → SW has degree m. Define f′|_{G· \frac{1}{2} D^{n}}: G · \frac{1}{2}D^{n} → SW by sending (g, x) to g^{e} · c(x). One easily checks that

f̃′ has degree deg(f̃) + m · |G|. Then G\f̃′ is the desired map f′.

Notice that there is at least one map f: L(V) → L(W) with e(f) = e for any given e ∈ ℤ/|G|. This follows by an argument similar to the one above, since G acts freely on SV and SW is (dim(SV) − 1)-connected. Now assertion (2) follows.

(3) Let f: SV → SW be a map. Then f is a homotopy equivalence if and only if it induces isomorphisms on all homotopy groups. This is the case if and only if π_{1}(f) is an automorphism and f̃: SV → SW induces an isomorphism on all homology groups. This follows from the Hurewicz Theorem [121, Theorem IV.7.13 on page 181] and the fact that a covering induces an isomorphism on π_{n} for n ≥ 2. Hence f is a homotopy equivalence if and only if e(f) ∈ ℤ/|G|^{∗} and deg(f) = deg(f̃) = ±1. Recall that f is an oriented homotopy equivalence if and only if f is a homotopy equivalence and deg(f) = 1. Now the claim follows from assertion (2).

(4) follows from (2.34) and assertion (3). This finishes the proof of Theorem 2.31.

Lemma 2.35 Let G be a cyclic group of finite order |G|.

1. Let V be a free unitary finite-dimensional G-representation. Let U be an orthogonal finite-dimensional G-representation with U^{G} = 0. Then L(V) is U-acyclic and the Reidemeister torsion ρ(L(V); U) ∈ ℝ^{>0} is defined;

2. Let V,V_{1}, V_{2} be free unitary finite-dimensional G-representations. Let U, U_{1} and U_{2} be orthogonal finite-dimensional G-representations with U^{G} = U_{1}^{G} = U_{2}^{G} = 0. Then

ρ(L(V_{1} ⊕ V_{2}), U) = ρ(L(V_{1}); U) · ρ(L(V_{2}); U);
ρ(L(V), U_{1} ⊕ U_{2}) = ρ(L(V); U_{1}) · ρ(L(V); U_{2}).

\underline{\underline{Proof}} : (1) Let X be a finite CW -complex with fundamental group G. We show that X is U-acyclic if G acts trivial on H_{p}(X̃) and U is an orthogonal finite-dimensional G-representation with U^{G} = 0. We have to show that H_{p}(C_{∗}(X̃) ⊗_{ℤG} U) ≅ H_{p}(X̃) ⊗_{ℤ} ℝ ⊗_{ℝG} U vanishes. By assumption H_{p}(X̃) ⊗_{ℤ} ℝ is a direct sum of copies of the trivial G-representation ℝ and U is a direct sum of non-trivial irreducible representations. Since for any non-trivial irreducible G-representation W the tensor product ℝ ⊗_{ℝG} W is trivial, the claim follows.

(2) The claim for the second equation is obvious, it remains to prove the first. Since G-acts trivially on the homology of SV and SW and hence on the homology of SV × SW, SV × DV and DV × SW, we conclude from Theorem 2.1 (1), Lemma 2.9 (1), Lemma 2.23 and (2.30)

ρ(L(V ⊕ W); U)

= ρ(G\(DV × SW); U) · ρ(G\(SV × DW); U) · (ρ(G\(SV × SW); U))^{−1}.

Hence it remains to show

ρ(G\(DV × SW); U) = ρ(L(W); U);
ρ(G\(SV × DW); U) = ρ(L(V); U);
ρ(G\(SV × SW); U) = 1.

These equations will follow from the following slightly more general formula (2.36) below. Let D_{∗} be a finite ℝG-chain complexes such that G-acts trivially on its homology. Assume that D_{∗} comes with a ℝ-basis. Then C_{∗}(SV) ⊗_{ℤ} D_{∗} with the diagonal G-action is a finite ℝG-chain complex such that G acts trivially on its homology and there is a preferred ℝG-basis. Let χ_{ℝ}(D_{∗}) be the Euler characteristic of D_{∗}, i.e.

χ_{ℝ}(D_{∗}) := ∑_{p∈ℤ}(−1)^{p} · dim_{ℝ}(D_{p}) = ∑_{p∈ℤ}(−1)^{p} · dim_{ℝ}(H_{p}(D_{∗})).

Then

ρ(C_{∗}(SV) ⊗_{ℤ} D_{∗} ⊗_{ℝG} U) = ρ(L(V); U)^{χ_{ℝ}(D_{∗})},  (2.36)

where ρ(C_{∗}(SV) ⊗_{ℤ} D_{∗} ⊗_{ℝG} U) ∈ ℝ^{>0} is the Reidemeister torsion of the acyclic based free finite ℝ-chain complex C_{∗}(SV) ⊗_{ℤ} D_{∗} ⊗_{ℝG} U which is defined to be the image of the Whitehead torsion τ(C_{∗}(SV)⊗_{ℤ}D_{∗}⊗_{ℝG}U) ∈ K̃_{1}(ℝ) (see (2.7)) under the homomorphism K̃_{1}(ℝ) → ℝ^{>0}, [A] ↦ det(A)^{2}.

It remains to prove (2.36). Since ℝG is semi-simple, there is a ℝG-chain homotopy equivalence p_{∗}: D_{∗} → H_{∗}(D_{∗}), where we consider H_{∗}(D_{∗}) as ℝGchain complex with the trivial differential.

Equip H_{∗}(D_{∗}) with an ℝ-basis. Then we get in K̃_{1}(ℝG) from Lemma 2.9 (1) and χ(L(V)) = 0

τ (id_{C_{∗}(SV)} ⊗p_{∗}: C_{∗}(SV) ⊗_{ℤ} D_{∗} → C_{∗}(SV) ⊗_{ℤ} H_{∗}(D_{∗}))

= ∑_{q∈ℤ}(−1)^{q} · τ (id_{C_{q}(SV)} ⊗p_{∗}: C_{q}(SV) ⊗_{ℤ} D_{∗} → C_{q}(SV) ⊗_{ℤ} H_{∗}(D_{∗}))
= ∑_{q∈ℤ}(−1)^{q} · dim_{ℤG}(C_{p}(SV)) · τ (id_{ℤG} ⊗p_{∗}: ℤG ⊗_{ℤ} D_{∗} → ℤG ⊗_{ℤ} H_{∗}(D_{∗}))
= χ(L(V)) · τ (id_{ℤG} ⊗p_{∗}: ℤG ⊗_{ℤ} D_{∗} → ℤG ⊗_{ℤ} H_{∗}(D_{∗}))
= 0.

The chain complex version of Lemma 2.23 shows

ρ(C_{∗}(SV) ⊗_{ℤ} D_{∗} ⊗_{ℝG} U) = ρ(C_{∗}(SV) ⊗_{ℤ} H_{∗}(D_{∗}) ⊗_{ℝG} U).

Obviously

ρ(C_{∗}(SV)⊗_{ℤ} H_{∗}(D_{∗})⊗_{ℝG} U) = ρ(C_{∗}(SV)⊗_{ℤG} U)^{χ_{ℝ}(D_{∗})} = ρ(L(V); U)^{χ_{ℝ}(D_{∗})}.

This finishes the proof of Lemma 2.35.

Theorem 2.37 (Diffeomorphism Classification of Lens Spaces) Let L(V) and L(W) be two lens spaces of the same dimension n ≥ 3. Then the following statements are equivalent.

1. There is an automorphism α: G → G such that V and α^{∗}W are isomorphic as orthogonal G-representations;

2. There is an isometric diffeomorphism L(V) → L(W);

3. There is a diffeomorphism L(V) → L(W);

4. There is homeomorphism L(V) → L(W);

5. There is simple homotopy equivalence L(V) → L(W);

6. There is an automorphism α: G → G such that for any orthogonal finitedimensional representation U with U^{G} = 0

ρ(L(W), U) = ρ(L(V), α^{∗}U)

holds.

7. There is an automorphism α: G → G such that for any non-trivial 1dimensional unitary G-representation U

ρ(L(W), res U) = ρ(L(V), α^{∗} res U)

holds, where the orthogonal representation res U is obtained from U by restricting the scalar multiplication from ℂ to ℝ.

\underline{\underline{Proof}} : The implications (1) ⇒ (2) ⇒ (3) ⇒ (4) ⇒ (5) ⇒ (6) ⇒ (7) are obvious or follow directly from Theorem 2.1 (5) and Lemma 2.23. Hence it remains to prove the implication (7) ⇒ (1).

Fix a generator g ∈ G, or equivalently, an identification G = ℤ/|G|. Choose e ∈ ℤ/|G|^{∗} such that α: G → G sends g to g^{e}. Recall that V_{k} denotes the 1-dimensional unitary G-representation for which g acts by multiplication with exp(2πik/|G|). The based ℤG-chain complex of SV_{k} is concentrated in dimension 0 and 1 and its first differential is g^{k} − 1: ℤG → ℤG. If g acts on U by multiplication with the |G|-th root of unity ζ, then we conclude

ρ(LV_{k}; U) = ||ζ^{k} − 1||^{2};
ρ(LV_{k}; α^{∗}U) = ||ζ^{ek} − 1||^{2}.

We can write for appropriate numbers c ∈ ℤ, c ≥ 1, k_{i}, l_{j} ∈ ℤ/G^{∗}

V = ⊕_{i=1}^{c}V_{k_{i}};
W = ⊕_{j=1}^{c}V_{l_{j}}.

We conclude from Lemma 2.35

ρ(LV; U) = ∏_{i=1}^{c}(ζ^{k_{i}} − 1) · (ζ^{−k_{i}} − 1);

ρ(LW; α^{∗}U) = ∏_{i=1}^{c}(ζ^{el_{j}} − 1) · (ζ^{−el_{j}} − 1).

This implies that for any |G|-th root ζ of unity with ζ ≠ 1 the following equation holds

∏_{i=1}^{c}(ζ^{k_{i}} − 1) · (ζ^{−k_{i}} − 1) = ∏_{i=1}^{c}(ζ^{el_{j}} − 1) · (ζ^{−el_{j}} − 1).  (2.38)

We will need the following number theoretic result due to Franz whose proof can be found for instance in [33].

Lemma 2.39 (Franz’ Independence Lemma) Let t ≥ 2 be an integer and S = {j ∈ ℤ | 0 < j < t, (j, t) = 1}. Let (a_{j})_{j∈S} be a sequence of integers indexed by S such that ∑_{j∈S} a_{j} = 0, a_{j} = a_{t−j} for j ∈ S and ∏_{j∈S}(ζ^{j} − 1)^{a_{j}} = 1 holds for every t-th root of unity ζ ≠ 1. Then a_{j} = 0 for j ∈ S.

Put t = |G|. For j ∈ S = {j ∈ ℤ | 0 < j < |G|, (j, |G|) = 1} let x_{j} be the number of elements in the sequence k_{1}, −k_{1}, k_{2}, −k_{2}, …, k_{c}, −k_{c}, which are congruent j modulo |G|. Each of the elements k_{i} is prime to |G| and hence ±x_{j} is congruent mod |G| to some j ∈ S. This implies ∑_{j∈S} x_{j} = 2c. Obviously x_{j} = x_{|G|−j} for j ∈ S. Define analogously a sequence (y_{j})_{j∈S} for the sequence el_{1}, −el_{1}, el_{2}, −el_{2}, …, el_{c}, −el_{c}. Put a_{j} = x_{j} −y_{j} for j ∈ S. Then ∑_{∈S} a_{j} = 0, a_{j} = a_{|G|−j} for j ∈ S and we conclude from (2.38) that ∏_{j∈S}(ζ^{j} − 1)^{a_{j}} = 1 for any |G|-th root of unity ζ ≠ 1. We conclude from Franz’ Independence Lemma 2.39 that a_{j} = 0 and hence x_{j} = y_{j} holds for j ∈ S. This implies that there is a permutation σ ∈ Σ_{c} together with signs ϵ_{i} ∈ {±1} for i = 1, 2, …, c such that k_{i} and ϵ_{i} · l_{σ(i)} are congruent modulo |G|. But this implies that V_{i} and α^{∗}W_{σ(i)} are isomorphic as orthogonal G-representations. Hence V and α^{∗}W are isomorphic as orthogonal representations. This finishes the proof of Theorem 2.37.

Corollary 2.40 Two lens spaces L(t; k_{1}, …, k_{c}) and L(t; l_{1}, …, l_{c}) are homeomorphic if and only there are e ∈ ℤ/t^{∗}, signs ϵ_{i} ∈ {±1} and a permutation σ ∈ Σ_{c} such that k_{i} = ϵ_{i} · e · l_{σ(i)} holds in ℤ/t^{∗} for i = 1, 2, …, c.

Notice that ⊕_{i=1}^{c}V_{k_{i}} and ⊕_{i=1}^{c}V_{l_{i}} are isomorphic as orthogonal representations if and only if there are signs ϵ_{i} ∈ {±1} and a permutation σ ∈ Σ_{c} such that k_{i} = ϵ_{i} · l_{σ(i)} holds in ℤ/t^{∗} for i = 1, 2, …, c. If m_{e}: ℤ/t → ℤ/t is multiplication with e ∈ ℤ/t^{∗}, then the restriction m_{e}^{∗} (⊕_{i=1}^{c}V_{l_{i}}) is ⊕_{i=1}^{c}V_{el_{i}}. Now apply Theorem 2.37.

Example 2.41 We conclude from Theorem 2.31 and Corollary 2.40 the following facts:

1. Any homotopy equivalence L(7; k_{1}, k_{2}) → L(7; k_{1}, k_{2}) has degree 1. Thus L(7; k_{1}, k_{2}) possesses no orientation reversing selfdiffeomorphism;

2. L(5; 1, 1) and L(5; 2, 1) have the same homotopy groups, homology groups and cohomology groups, but they are not homotopy equivalent;

3. L(7; 1, 1) and L(7; 2, 1) are homotopy equivalent, but not homeomorphic.

2.5 Miscellaneous

We mention that lens spaces are the only closed manifolds M which carry a Riemannian metric with sectional curvature which is constant 1, provided π_{1}(M) is cyclic. Reidemeister torsion can be used to classify all such manifolds without any assumption on π_{1}(M) and to show that two finite-dimensional (not necessarily free) orthogonal representations V and W have G-diffeomorphic unit spheres SV and SW if and only if they are isomorphic as orthogonal representations (see [32]). The corresponding statement is false if one replaces G-diffeomorphic by G-homeomorphic (see [21],[23], [49]).

Let (W, L, L′) be an h-cobordism of lens spaces which is compatible with the orientations and the identifications of π_{1}(L) and π_{1}(L′) with G. Then W is diffeomorphic relative L to L × [0, 1] and L and L′ are diffeomorphic [84, Corollary 12.13 on page 410].

We refer to [28] and [84] for more information about Whitehead torsion and lens spaces.

48 CHAPTER 2. WHITEHEAD TORSION

Normal Maps and the Surgery Problem

Introduction

In this chapter we want to take the first step for the solution of the following problem

Problem 3.1 Let X be a topological space. When is X homotopy equivalent to a closed manifold?

We will begin with discussing Poincaré duality, which is the first obstruction from algebraic topology for a positive answer to Problem 3.1 above, in Section 3.1. In Section 3.2 we will deal with the Spivak normal fibration of a finite Poincaré complex X which is a reminiscence of the normal bundle of the embedding of a closed manifold into some high-dimensional Euclidean space. We will explain and motivate the notion of a normal map of degree one in Section 3.3. This is a map of degree one f: M → X from a closed manifold M to a finite Poincaré complex X covered by some bundle data. The surgery problem is to change it to a homotopy equivalence leaving the target fixed and changing the source without loosing the structure of a closed manifold. In Section 3.4 we will introduce the surgery step. This is the manifold analog of the process in the world of CW -complexes which is given by adding a cell in order to kill an element in the homotopy group. We will show that we can make a normal map f: M → X highly connected by carrying out a finite number of surgery steps. The surgery obstructions will later arise as obstructions to make f connected in the middle dimension. If f is also connected in the middle dimension, then Poincaré duality implies that f is a homotopy equivalence from a closed manifold to X.

We mention that Problem 3.1 is an important problem but the main success of surgery comes from its contribution to the question whether two closed manifolds are diffeomorphic (see the surgery program in Remark 1.5) and the construction of exotic structures, i.e. different smooth structures on the same topological manifold. We will restrict our attention for some time to Problem 3.1 because it is a good motivation for certain techniques such as the surgery step, for certain notions such as normal maps of degree one and for surgery obstructions and surgery obstruction groups such as the signature and the L-groups. After we have developed the machinery which allows us to solve Problem 3.1 in principle, we will develop it further in order to carry out the surgery program.

3.1 Poincaré Duality

In order to state Poincaré duality on the level we will need we have to introduce some algebra. Recall that a ring with involution R is an associative ring R with unit together with an involution of rings : R → R, r ↦ \overline{r}, i.e. a map satisfying \overline{\overline{r}} = r, \overline{r + s} = \overline{r} + \overline{s}, \overline{rs} = \overline{s} \overline{r} and \overline{1} = 1 for r, s ∈ R. Our main example will be the group ring AG for some commutative associative ring A with unit and a group G together with a homomorphism w: G → {±1}. The so called w-twisted involution sends ∑_{g∈G} a_{g} · g to ∑_{g∈G} w(g) · a_{g} · g^{−1}. Let M be a left R-module. Then M^{∗} := hom_{R}(M, R) carries a canonical right R-module structure given by (fr)(m) = f(m) · r for a homomorphism of left R-modules f: M → R and m ∈ M. The involution allows us to view M^{∗} = hom_{R}(M; R) as a left R-module, namely define rf for r ∈ R and f ∈ M^{∗} by (rf)(m) := f(m)·\overline{r} for m ∈ M. Given an R-chain complex of left R-modules C_{∗} and n ∈ ℤ, we define its dual chain complex C^{n−∗} to be the chain complex of left R-modules whose p-th chain module is hom_{R}(C_{n−p}, R) and whose p-th differential is given by

hom_{R}(c_{n−p+1}, id): (C^{n−∗})_{p} = hom_{R}(C_{n−p}, R)

→ (C^{n−∗})_{p−1} = hom_{R}(C_{n−p+1}, R).

Consider a connected finite CW -complex X with fundamental group π and a group homomorphism w: π → {±1}. In the sequel we use the w-twisted involution. Denote by C_{∗}(X̃) the cellular ℤπ-chain complex of the universal covering. Recall that this is a free ℤπ-chain complex and the cellular structure on X determines a cellular ℤπ-basis on it such that each basis element corresponds to a cell in X. This basis is not quite unique but its equivalence class depends only on the CW -structure of X (see Section 1.2). The product X̃ ×X̃ equipped with the diagonal π-action is again a π-CW -complex. The diagonal map D: X̃ → X̃ ×X̃ sending x̃ to (x̃, x̃) is π-equivariant but not cellular. By the equivariant cellular Approximation Theorem (see for instance [69, Theorem 2.1 on page 32]) there is up to cellular π-homotopy precisely one cellular π-map \overline{D}: X̃ → X̃ ×X̃ which is π-homotopic to D. It induces a ℤπ-chain map unique up to ℤπ-chain homotopy

C_{∗}(\overline{D}): C_{∗}(X̃) → C_{∗}(X̃ × X̃).  (3.2)

There is a natural isomorphism of based free ℤπ-chain complexes

i_{∗}: C_{∗}(X̃) ⊗_{ℤ} C_{∗}(X̃) ⟶^{≅} C_{∗}(X̃ × X̃).  (3.3)

Denote by ℤ^{w} the ℤπ-module whose underlying abelian group is ℤ and on which g ∈ G acts by w(g) · id. Given two projective ℤπ-chain complexes C_{∗} and D_{∗} we obtain a natural ℤ-chain map unique up to ℤ-chain homotopy

s: ℤ^{w} ⊗_{ℤπ} (C_{∗} ⊗_{ℤ} D_{∗}) → hom_{ℤπ}(C^{−∗}, D_{∗})  (3.4)

by sending 1 ⊗ x ⊗ y ∈ ℤ ⊗ C_{p} ⊗ D_{q} to

s(1 ⊗ x ⊗ y): hom_{ℤπ}(C_{p}, ℤπ) → D_{q}, (ϕ: C_{p} → ℤπ) ↦ \overline{ϕ(x)} · y.

The composite of the chain map (3.4) for C_{∗} = D_{∗} = C_{∗}(X̃), the inverse of the chain map (3.3) and the chain map (3.2) yields a ℤ-chain map

ℤ^{w} ⊗_{ℤπ} C_{∗}(X̃) → hom_{ℤπ}(C^{−∗}(X̃), C_{∗}(X̃)).

Notice that the n-homology of hom_{ℤπ}(C^{−∗}(X̃), C_{∗}(X̃)) is the set of ℤπ-chain homotopy classes [C^{n−∗}(X̃), C_{∗}(X̃)]_{ℤπ} of ℤπ-chain maps from C^{n−∗}(X̃) to C_{∗}(X̃). Define H_{n}(X; ℤ^{w}) := H_{n}(ℤ^{w} ⊗_{ℤπ} C_{∗}(X̃)). Taking the n-th homology group yields a well-defined ℤ-homomorphism

∩: H_{n}(X; ℤ^{w}) → [C^{n−∗}(X̃), C_{∗}(X̃)]_{ℤπ}  (3.5)

which sends a class x ∈ H_{n}(X; ℤ^{w}) = H_{n}(ℤ^{w} ⊗_{ℤπ} C_{∗}(X̃)) to the ℤπ-chain homotopy class of a ℤπ-chain map denoted by ? ∩ x: C^{n−∗}(X̃) → C_{∗}(X̃).

Definition 3.6 A connected finite n-dimensional Poincaré complex is a connected finite CW -complex of dimension n together with a group homomorphism w = w_{1}(X): π_{1}(X) → {±1} called orientation homomorphism and an element [X] ∈ H_{n}(X; ℤ^{w}) called fundamental class such that the ℤπ-chain map ? ∩ [X]: C^{n−∗}(X̃) → C_{∗}(X̃) is a ℤπ-chain homotopy equivalence. We will call it the Poincaré ℤπ-chain homotopy equivalence.

The orientation homomorphism w: π_{1}(X) → {±1} is uniquely determined by the homotopy type of X by the following argument. Denote by C^{n−∗}(X̃)_{untw} the ℤπ-chain complex which is analogously defined as C^{n−∗}(X̃), but now with respect to the untwisted involution. Its n-th homology H_{n}(C^{n−∗}(X̃)_{untw}) depends only on the homotopy type of X. If X carries the structure of a Poincaré complex with respect to w: π_{1}(X) → {±1}, then the Poincaré ℤπ-chain homotopy equivalence induces a ℤπ-isomorphism H_{n}(C^{n−∗}(X̃)_{untw}) ≅ ℤ^{w}. Thus we rediscover w from H_{n}(C^{n−∗}(X̃)_{untw}). Obviously there are two possible choices for [X], since it has to be a generator of the infinite cyclic group H_{n}(X, ℤ^{w}) ≅ H^{0}(X; ℤ) ≅ ℤ. A choice of [X] will be part of the Poincaré structure.

The connected finite n-dimensional Poincaré complex X is called simple if the Whitehead torsion (see (2.8)) of the ℤπ-chain homotopy equivalence of finite based free ℤπ-chain complexes ? ∩ [X]: C^{n−∗}(X̃) → C_{∗}(X̃) vanishes.

Theorem 3.7 Let M be a connected closed manifold of dimension n. Then M carries the structure of a simple connected finite n-dimensional Poincaré complex.

For a proof we refer for instance to [119, Theorem 2.1 on page 23]. We explain at least the idea and give some evidence for it.

Any closed manifold admits a smooth triangulation h: K → M, i.e. a finite simplicial complex together with a homeomorphism h: K → M such that h restricted to a simplex is a smooth C^{∞}-embedding. Two such smooth triangulations have common subdivisions. In particular M is homeomorphic to a finite CW -complex. Fix such a triangulation K. Denote by K′ its barycentric subdivision. The vertices of K′ are the barycenters σ̂^{r} of simplices σ^{r} in K. A p-simplex in K′ is given by a sequence σ̂^{i_{0}} σ̂^{i_{1}} … σ̂^{i_{p}}, where σ^{i_{j}} is a face of σ^{i_{j+1}}. Next we define the dual CW -complex K^{∗} as follows. It is not a simplicial complex but shares the property of a simplicial complex that all attaching maps are embeddings. Each p-simplex σ in K determines a (n − p)-dimensional cell σ^{∗} of K^{∗} which is the union of all simplices in K′ which begin with σ̂^{p}. So K has as many p-simplices as K^{∗} has (n − p)-cells. The cap product with the fundamental cycle, which is given by the sum of the n-dimensional simplices, yields an isomorphism of ℤπ-chain complexes C^{n−∗}(K̃^{∗}) → C_{∗}(K̃). It preserves the cellular ℤπ-bases and in particular its Whitehead torsion is trivial. Since K′ is a common subdivision of K and K^{∗}, there are canonical ℤπ-chain homotopy equivalences C_{∗}(K̃′) → C_{∗}(K̃) and C_{∗}(K̃′) → C_{∗}(K̃^{∗}) which have trivial Whitehead torsion. Thus we can write the ℤπ-chain map ?∩[M]: C^{n−∗}(K̃′) → C_{∗}(K̃′) as a composite of three simple ℤπ-chain homotopy equivalences. Hence it is a simple ℤπ-chain homotopy equivalence.

Remark 3.8 Theorem 3.7 gives us the first obstruction for a topological space X to be homotopy equivalent to a connected closed n-dimensional manifold (see Problem 3.1). Namely, X must be homotopy equivalent to a connected finite simple n-dimensional Poincaré complex.

Remark 3.9 Suppose that X is a Poincaré complex with respect to the trivial orientation homomorphism. Definition 3.6 of Poincaré duality implies that Poincaré duality holds for any G-covering \overline{X} → X and yields Poincaré duality for all possible coefficient systems. In particular we get a ℤ-chain homotopy equivalence

ℤ ⊗_{ℤπ} (? ∩ [X]): ℤ ⊗_{ℤπ} C^{n−∗}(X̃) = C^{n−∗}(X) → ℤ ⊗_{ℤπ} C_{∗}(X̃) = C_{∗}(X),

which induces for any commutative ring R an R-isomorphism on (co-)homology with R-coefficients

? ∩ [X]: H^{n−∗}(X; R) ⟶^{≅} H_{∗}(X; R).

Remark 3.10 From a Morse theoretic point of view Poincaré duality corresponds to the dual handlebody decomposition of a manifold which we have already described in (1.26).

Remark 3.11 From the analytic point of view Poincaré duality can be explained as follows. Let M be a connected closed oriented Riemannian manifold. Let (Ω^{∗}(M), d^{∗}) be the de Rham complex of smooth p-forms on M. The p-th Laplacian is defined by Δ_{p} = (d^{p})^{∗}d^{p} + d^{p−1}(d^{p−1})^{∗}: Ω^{p}(M) → Ω^{p}(M), where (d^{p})^{∗} is the adjoint of the p-th differential d^{p}. The kernel of the p-th Laplacian is the space H^{p}(M) of harmonic p-forms on M. The Hodge-de Rham Theorem yields an isomorphism

A^{p}: H^{p}(M) ⟶^{≅} H^{p}(M; ℝ)  (3.12)

from the space of harmonic p-forms to the singular cohomology of M with coefficients in ℝ. Let [M]_{ℝ} ∈ H^{n}(M; ℝ) be the fundamental cohomology class with ℝ-coefficients which is characterized by the property ⟨[M]_{ℝ}, i_{∗}([M])⟩ = 1 for ⟨ , ⟩ the Kronecker product and i_{∗}: H_{n}(M; ℤ) → H_{n}(M; ℝ) the change of rings homomorphism. Then A^{n} sends the volume form dvol to the class \frac{1}{vol(M)} · [M]_{ℝ}. The Hodge-star operator ∗: Ω^{n−p}(M) → Ω^{p}(M) induces isomorphisms

∗: H^{n−p}(M) ⟶^{≅} H^{p}(M).  (3.13)

We obtain from (3.12) and (3.13) isomorphisms

H^{n−p}(M; ℝ) ⟶^{≅} H^{p}(M; ℝ).

This is the analytic version of Poincaré duality. It is equivalent to the claim that the bilinear pairing

P^{p}: H^{p}(M) ⊗_{ℝ} H^{n−p}(M) → ℝ,  (ω, η) ↦ ∫_{M} ω ∧ η  (3.14)

is non-degenerate. Recall that for any commutative ring R with unit we have the intersection pairing

I^{p}: H^{p}(M; R) ⊗_{R} H^{n−p}(M; R) → R, (x, y) ↦ ⟨x ∪ y, i_{∗}[M]⟩, (3.15)

where i_{∗} is the change of coefficients map associated to ℤ → R. The fact that the intersection pairing is non-degenerate is for R a field equivalent to the bijectivity of the homomorphism ? ∩ [X]: H^{n−∗}(X; R) → H_{∗}(X; R) appearing in Remark 3.9. If we take R = ℝ, then the pairings (3.14) and (3.15) agree under the Hodge-de Rham isomorphism (3.12).

One basic invariant of a finite CW -complex X is its Euler characteristic χ(X). It is defined by χ(X) := ∑_{p}(−1)^{p} · n_{p}, where n_{p} is the number of p-cells. Equivalently it can be defined in terms of its homology by χ(X) = ∑_{p}(−1)^{p} · dim_{ℚ} H_{p}(X; ℚ). A basic invariant for Poincaré complexes is defined next.

Consider a symmetric bilinear non-degenerate pairing s: V ⊗_{ℝ} V → ℝ for a finite-dimensional real vector space V. Choose a basis for V and let A be the square matrix describing s with respect to this basis. Since s is symmetric and non-degenerate, A is symmetric and invertible. Hence A can be diagonalized by an orthogonal matrix U to a diagonal matrix whose entries on the diagonal are non-zero real numbers. Let n_{+} be the number of positive entries and n_{−} be the number of negative entries on the diagonal. These two numbers are independent of the choice of the basis and the orthogonal matrix U. Namely n_{+} is the maximum of the dimensions of subvector spaces W ⊂ V on which s is positive-definite, and analogous for n_{−}. Obviously n_{+} + n_{−} = dim_{ℝ}(V). Define the signature of s to be the integer n_{+} − n_{−}.

Definition 3.16 Let X be a finite connected Poincaré complex. Suppose that X is orientable, i.e. w_{1}(X): π_{1}(X) → {±1} is trivial and that its dimension n = 4k is divisible by four. Define its intersection pairing to be the symmetric bilinear non-degenerate pairing

I: H^{2k}(X; ℝ) ⊗_{ℝ} H^{2k}(X; ℝ) ⟶^{∪} H^{n}(X; ℝ) ⟶^{⟨−,[X]_{ℝ}⟩} ℝ.

Define the signature sign(X) to be the signature of the intersection pairing.

Remark 3.17 The notion of a Poincaré complex can be extended to pairs as follows. Let X be a connected finite n-dimensional CW -complex with fundamental group π together with a subcomplex A ⊂ X of dimension (n − 1). Denote by Ã ⊂ X̃ the preimage of A under the universal covering X̃ → X. We call (X, A) a finite n-dimensional Poincaré pair with respect to the orientation homomorphism w: π_{1}(X) → {±1} if there is a fundamental class [X, A] ∈ H_{n}(X, A; ℤ^{w}) such that the ℤπ-chain maps ? ∩ [X, A]: C^{n−∗}(X̃, Ã) → C_{∗}(X̃) and ?∩[X, A]: C^{n−∗}(X̃) → C_{∗}(X̃, Ã) are ℤπ-chain equivalences. We call (X, A) simple if the Whitehead torsion of these ℤπ-chain maps vanish.

Each component C of the space A inherits from (X, A) the structure of a finite (n − 1)-dimensional Poincaré complex. Its orientation homomorphisms w_{1}(C) is obtained from w_{1}(X) by restriction with the homomorphism π_{1}(C) → π_{1}(X) induced by the inclusion. The fundamental classes [C] of the components C ∈ π_{0}(A) are given by the image of the fundamental class [X, A] under the boundary map H_{n}(X, A, ℤ^{w}) → H_{n−1}(A, ℤ^{w_{1}(A)}) ≅ ⊕_{C∈π_{0}(A)}H_{n−1}(C, ℤ^{w_{1}(C)}). If M is a compact connected manifold of dimension n with boundary ∂M, then (M, ∂M) is a simple finite n-dimensional Poincaré pair.

The signature will be the first and the most elementary surgery obstruction which we will encounter. This is due to the following lemma.

Lemma 3.18 Let (X, A) be a (4k + 1)-dimensional oriented finite Poincaré pair. Then

∑_{C∈π_{0}(A)} sign(C) = 0.

For its proof and later purposes we need the following lemma.

Lemma 3.19 Let s: V ⊗_{ℝ} V → ℝ be a symmetric bilinear non-degenerate pairing for a finite-dimensional real vector space V. Then sign(s) = 0 if and only if there exists a subvector space L ⊂ V such that dim_{ℝ}(V) = 2 · dim_{ℝ}(L) and s(a, b) = 0 for a, b ∈ L.

\underline{\underline{Proof}} : Suppose that sign(s) = 0. Then one can find an orthonormal (with respect to s) basis {b_{1}, b_{2}, …, b_{n_{+}}, c_{1}, c_{2}, …, c_{n_{−}}} such that s(b_{i}, b_{i}) = 1 and s(c_{j}, c_{j}) = −1 holds. Since 0 = sign(s) = n_{+} − n_{−}, we can define L to be the subvector space generated by {b_{i} + c_{i} | i = 1, 2, …, n_{+}}. One easily checks that L has the desired properties.

Suppose such an L ⊂ V exists. Choose subvector spaces V_{+} and V_{−} of V such that s is positive-definite on V_{+} and negative-definite on V_{−} and V_{+} and V_{−} are maximal with respect to this property. Then V_{+} ∩ V_{−} = {0} and V = V_{+} ⊕ V_{−}. Obviously V_{+} ∩ L = V_{−} ∩ L = {0}. From

dim_{ℝ}(V_{±}) + dim_{ℝ}(L) − dim_{ℝ}(V_{±} ∩ L) ≤ dim_{ℝ}(V)

we conclude dim_{ℝ}(V_{±}) ≤ dim_{ℝ}(V) − dim_{ℝ}(L). Since 2 · dim_{ℝ}(L) = dim_{ℝ}(V) = dim_{ℝ}(V_{+}) + dim_{ℝ}(V_{−}) holds, we get dim_{ℝ}(V_{±}) = dim_{ℝ}(L). This implies

sign(s) = dim_{ℝ}(V_{+}) − dim_{ℝ}(V_{−}) = dim_{ℝ}(L) − dim_{ℝ}(L) = 0.

Now we can give the proof of Lemma 3.18. \underline{\underline{Proof}} : Let i: A → M be the inclusion. Then the following diagram commutes for n = 4k.

H^{2k}(X; ℝ) ⟶^{H^{2k}(i)} H^{2k}(A; ℝ) ⟶^{δ^{2k}} H^{2k+1}(X, A; ℝ)
?∩[X,A]↓≅  ^{?∩∂_{4k+1}([X,A])}↓≅  ^{?∩[X,A]}↓≅
H_{2k+1}(X, A; ℝ) ⟶^{∂_{2k+1}} H_{2k}(A; ℝ) ⟶^{H_{2k}(i)} H_{2k}(X; ℝ)

This implies dim_{ℝ}(ker(H_{2k}(i))) = dim_{ℝ}(im(H^{2k}(i))). Since ℝ is a field, we get from the Kronecker pairing an isomorphism H^{2k}(X; ℝ) ≅ (H_{2k}(X; ℝ))^{∗} and analogously for A. Under these identifications H^{2k}(i) becomes (H_{2k}(i))^{∗}. Hence dim_{ℝ}(im(H_{2k}(i))) = dim_{ℝ}(im(H^{2k}(i))). From

dim_{ℝ}(H_{2k}(A; ℝ)) = dim_{ℝ}(ker(H_{2k}(i))) + dim_{ℝ}(im(H_{2k}(i)))

we conclude

dim_{ℝ}(H^{2k}(A; ℝ)) = 2 · dim_{ℝ}(im(H^{2k}(i))).

We have for x, y ∈ H^{2k}(M; ℝ)

⟨H^{2k}(i)(x) ∪ H^{2k}(i)(y), ∂_{4k+1}([X, A])⟩ = ⟨H^{2k}(i)(x ∪ y), ∂_{4k+1}([X, A])⟩
= ⟨x ∪ y, H_{2k}(i) ∘ ∂_{4k+1}([X, A])⟩
= ⟨x ∪ y, 0⟩ = 0.

If we apply Lemma 3.19 to the non-degenerate symmetric bilinear pairing

H^{2k}(A; ℝ) ⊗_{ℝ} H^{2k}(A; ℝ) ⟶^{∪} H^{4k}(A; ℝ) ⟶^{⟨?,∂_{4k+1}([X,A])⟩} H^{0}(A; ℝ) ≅ ⊕_{π_{0}(A)}ℝ

\underrightarrow{∑} ℝ

with L the image of H^{2k}(i): H^{2k}(X; ℝ) → H^{2k}(A; ℝ), we see that the signature of this pairing is zero. One easily checks that its signature is the sum of the signatures of the components of A.

The signature has the following further properties.

Lemma 3.20 1. Let M and N be compact oriented manifolds and f: ∂M → ∂N be an orientation reversing diffeomorphism. Then M ∪_{f} N inherits an orientation from M and N and

sign(M ∪_{f} N) = sign(M) + sign(N);

2. Let p: \overline{M} → M be a finite covering with d sheets of closed oriented mani-

folds. Then

sign(\overline{M}) = d · sign(M).

\underline{\underline{Proof}} : (1) is due to Novikov. For a proof see for instance [5, Proposition 7.1 on page 588].

(2) For a smooth manifold M this follows from Atiyah’s L^{2}-index theorem [4, (1.1)]. Topological closed manifolds are treated in [102, Theorem 8].

Example 3.21 Wall has constructed a finite connected Poincaré space X together with a finite covering with d sheets \overline{X} → X such that the signature does not satisfy sign(\overline{X}) = d · sign(X) (see [98, Example 22.28], [118, Corollary 5.4.1])). Hence X cannot be homotopy equivalent to a closed manifold by Lemma 3.20.

3.2 The Spivak Normal Fibration

In this section we introduce the Spivak normal fibration which is the homotopy theoretic analog of the normal fiber bundle of an embedding of a manifold into some Euclidean space. In order to motivate the construction we briefly recall the Pontrjagin-Thom construction which we will need later anyway.

3.2.1 Pontrjagin-Thom Construction

Let (M, i) be an embedding i: M^{n} → ℝ^{n+k} of a closed n-dimensional manifold M into ℝ^{n+k}. Notice that T ℝ^{n+k} comes with an explicit trivialization ℝ^{n+k} × ℝ^{n+k} ⟶^{≅} T ℝ^{n+k} and the standard Euclidean inner product induces a Riemannian metric on T ℝ^{n+k}. Denote by ν(M) = ν(M, i) the normal bundle of i which is the orthogonal complement of T M in i^{∗}T ℝ^{n+k}. For a vector bundle ξ: E → X with Riemannian metric define its disk bundle p_{DE}: DE → X by DE = {v ∈ E | ||v|| ≤ 1} and its sphere bundle p_{SE}: SE → X by SE = {v ∈ E | ||v|| = 1}, where p_{DE} and p_{SE} are the restrictions of p. Its Thom space Th(ξ) is defined by DE/SE. It has a preferred base point ∞ = SE/SE. The Thom space can be defined without choice of a Riemannian metric as follows. Put Th(ξ) = E ∪ {∞} for some extra point ∞. Equip Th(ξ) with the topology for which E ⊂ Th(E) is an open subset and a basis of open neighborhoods for ∞ is given by the complements of closed subsets A ⊂ E for which A∩E_{x} is compact for each fiber E_{x}. If X is compact, E is locally compact and Th(ξ) is the one-point-compactification of E. The advantage of this definition is that any bundle map (\overline{f}, f): ξ_{0} → ξ_{1} of vector bundles ξ_{0} : E_{0} → X_{0} and ξ_{1} : E_{1} → X_{1} induces canonically a map Th(\overline{f}): Th(ξ_{0}) → Th(ξ_{1}). Notice that we require that \overline{f} induces a bijective map on each fiber. Denote by \underline{ℝ^{k}} the trivial vector bundle with fiber ℝ^{k}. We mention that there are homeomorphisms

Th(ξ × η) ≅ Th(ξ) ∧ Th(η);  (3.22)
Th(ξ ⊕ \underline{ℝ^{k}}) ≅ Σ^{k} Th(ξ),  (3.23)

where ∧ stands for the smash product of pointed spaces

X ∧ Y := (X × Y) / (X × {y} ∪ {x} × Y)  (3.24)

and Σ^{k}Y = S^{k} ∧ Y is the (reduced) suspension. Let (N(M), ∂N(M)) be a tubular neighborhood of M. Recall that there is a diffeomorphism

u: (Dν(M), Sν(M)) → (N(M), ∂N(M))

which is up to isotopy relative M uniquely determined by the property that its restriction to M is i and its differential at M is ϵ · id for small ϵ > 0 under the canonical identification T (Dν(M))|_{M} = T M ⊕ ν(M) = i^{∗}T ℝ^{n+k}. The collapse map

c: S^{n+k} = ℝ^{n+k} ∐{∞} → Th(ν(M))  (3.25)

is the pointed map which is given by the diffeomorphism u^{−1} on the interior of N(M) and sends the complement of the interior of N(M) to the preferred base point ∞. The homology group H_{n+k}(Th(T M)) ≅ H_{n+k}(N(M), ∂N(M)) is infinite cyclic, since N(M) is a compact orientable (n+k)-dimensional manifold with boundary ∂N(M). The Hurewicz homomorphism h: π_{n+k}(T h(T M)) → H_{n+k}(Th(T M)) sends the class [c] of c to a generator. This follows from the fact that any point in the interior of N(M) is a regular value of c and has precisely one point in his preimage.

Before we deal with the Spivak normal fibration, we apply this construction to bordism. Fix a space X together with a k-dimensional vector bundle ξ: E → X. Let us recall the definition of the bordism set Ω_{n}(ξ) . An element in it is represented by a quadruple (M, i, f, \overline{f}) which consists of a closed n-dimensional manifold M, an embedding i: M → ℝ^{n+k}, a map f: M → X and a bundle map \overline{f}: ν(M) → ξ covering f. We briefly explain what a bordism (W, I, F, \overline{F}) from one such quadruple (M_{0}, i_{0}, f_{0}, \overline{f_{0}}) to another quadruple (M_{1}, i_{1}, f_{1}, \overline{f_{1}}) is. We need a compact (n + 1)-dimensional manifold W together with a map F: W → X × [0, 1]. Its boundary ∂W is written as a disjoint sum ∂_{0}W ∐ ∂_{1}W such that F maps ∂_{0}W to X ×{0} and ∂_{1}W to X ×{1}. There is an embedding I: W → ℝ^{n+k} × [0, 1] such that I^{−1}(ℝ^{n+k} × {j}) = ∂_{j}W holds for j = 0, 1 and W meets ℝ^{n+k} × {j} for j = 0, 1 transversally. We require a bundle map (\overline{F} , F): ν(W) → ξ × [0, 1]. Moreover for j = 0, 1 there is a diffeomorphism U_{j}: ℝ^{n+k} → ℝ^{n+k} ×{j} which maps M_{j} to ∂_{j}W. It satisfies F ∘U_{j}|_{M_{j}×{j}} = f_{j}. Notice that U_{j} induces a bundle map ν(U_{j}): ν(M_{j}) → ν(W) covering U_{j}|_{M_{j}}. The composition of \overline{F} with ν(U_{j}) is required to be \overline{f_{j}}.

Theorem 3.26 (Pontrjagin-Thom Construction) Let ξ: E → X be a kdimensional vector bundle over a CW -complex X. Then the map

P_{n}(ξ): Ω_{n}(ξ) ⟶^{≅} π_{n+k}(Th(ξ)),

which sends the class of (M, i, f, \overline{f}) to the class of the composite S^{n+k} ⟶^{c}

Th(ν(M)) ⟶^{Th(\overline{f})} Th(ξ) is a well-defined bijection and natural in ξ.

\underline{\underline{Proof}} : The details can be found in [15, Satz 3.1 on page 28, Satz 4.9 on page 35,]. The basic idea becomes clear after we have explained the construction of the inverse for a finite CW -complex X. Consider a pointed map (S^{n+k}, ∞) → (Th(ξ), ∞). We can change f up to homotopy relative {∞} such that f becomes transverse to X. Notice that transversality makes sense although X is not a manifold, one needs only the fact that X is the zero-section in a vector bundle. Put M = f^{−1}(X). The transversality construction yields a bundle map \overline{f}: ν(M) → ξ covering f|_{M}. Let i: M → ℝ^{n+k} = S^{n+k} − \{∞\} be the inclusion. Then the inverse of P_{n}(ξ) sends the class of f to the class of (M, i, f|_{M}, \overline{f}).

Let Ω_{n}(X) be the bordism group of pairs (M, f) of oriented closed ndimensional manifolds M together with reference maps g: M → X. Let ξ_{k}: E_{k} → BSO(k) be the universal oriented k-dimensional vector bundle. In the sequel we will denote for a finite-dimensional vector space V by \underline{V} the trivial bundle with fiber V. Let \overline{j_{k}}: ξ_{k} ⊕ \underline{ℝ} → ξ_{k+1} be a bundle map covering a map j_{k}: BSO(k) → BSO(k + 1). Up to homotopy of bundle maps this map is unique. Denote by γ_{k} the bundle X × E_{k} → X × BSO(k) and by (\overline{i_{k}}, i_{k}): γ_{k} ⊕ \underline{ℝ} → γ_{k+1} the bundle map id_{X} ×(\overline{j_{k}}, j_{k}). The bundle map (\overline{i_{k}}, i_{k}) is unique up to homotopy of bundle maps and hence induces a well-defined map Ω_{n}(\overline{i_{k}}): Ω_{n}(γ_{k}) → Ω_{n}(γ_{k+1}), which sends the class of (M, i, f, \overline{f}) to the class of the quadruple which comes from the embedding j: M ⟶^{i} ℝ^{n+k} ⊂ ℝ^{n+k+1} and the canonical isomorphism ν(i) ⊕ \underline{ℝ} = ν(j). Consider the homomorphism

V_{k}: Ω_{n}(γ_{k}) → Ω_{n}(X)

which sends the class of (M, i, f, \overline{f}) to (M, pr_{X} ∘f), where we equip M with the orientation determined by \overline{f}. Let colim_{k→∞} Ω_{n}(γ_{k}) be the colimit of the directed system indexed by k ≥ 0

… ⟶^{Ω_{n}(\overline{i_{k−1}})} Ω_{n}(γ_{k}) ⟶^{Ω_{n}(\overline{i_{k}})} Ω_{n}(γ_{k+1}) ⟶^{Ω_{n}(\overline{i_{k+1}})} …

Since V_{k+1} ∘ i_{k} = V_{k} holds for all k ≥ 0, we obtain a map

V : colim_{k→∞} Ω_{n}(γ_{k}) ⟶^{≅} Ω_{n}(X).  (3.27)

This map is bijective because of the classifying property of γ_{k} and the facts that for k > n+1 any closed manifold M of dimension n can be embedded into ℝ^{n+k} and two such embeddings are isotopic.

We see a sequence of spaces Th(γ_{k}) together with maps

Th(\overline{i_{k}}): Σ Th(γ_{k}) = Th(γ_{k} ⊕ \underline{ℝ}) → Th(γ_{k+1}).

They induce homomorphisms

s_{k}: π_{n+k}(Th(γ_{k})) → π_{n+k+1}(Σ Th(γ_{k})) ⟶^{π_{n+k}(Th(\overline{i_{k}}))} π_{n+k+1}(Th(γ_{k+1})),

where the first map is the suspension homomorphism. Let colim_{k→∞} π_{n+k}(Th(γ_{k})) be the colimit of the directed system

… ⟶^{s_{k−1}} π_{n+k}(Th(γ_{k})) ⟶^{s_{k}} π_{n+k+1}(Th(γ_{k+1})) ⟶^{s_{k+1}} ….

We get from Theorem 3.26 a bijection

P: colim_{k→∞} Ω_{n}(γ_{k}) ⟶^{≅} colim_{k→∞} π_{n+k}(Th(γ_{k})).

This implies

Theorem 3.28 (Pontrjagin Thom Construction and Oriented Bordism) There is an isomorphism of abelian groups natural in X

P: Ω_{n}(X) ⟶^{≅} colim_{k→∞} π_{n+k}(Th(γ_{k})).

Remark 3.29 Notice that this is the beginning of the theory of spectra and stable homotopy theory. A spectrum E consists of a sequence of spaces (E_{k})_{k∈ℤ} together with so called structure maps σ_{k}: ΣE_{k} → E_{k+1}. The n-th stable homotopy group π_{n}(E) is the defined as the colimit colim_{k→∞} π_{n+k}(E_{k}) with respect to the directed system given by the composites

π_{n+k}(E_{k}) → π_{n+k+1}(ΣE_{k}) ⟶^{π_{n+k}(σ_{k})} π_{n+k+1}(E_{k+1}).

Theorem 3.28 is a kind of mile stone in homotopy theory since it is the prototype of a result, where the computation of geometrically defined objects are translated into a computation of (stable) homotopy groups. It applies to all other kind of bordism groups, where one puts additional structures on the manifolds, for instance a Spin-structure. The bijection is always of the same type, but the sequence of bundles ξ_{k} depends on the additional structure. If we want to deal with the unoriented bordism ring we have to replace the bundle ξ_{k} → BSO(k) by the universal k-dimensional vector bundle over BO(k).

3.2.2 Spherical Fibrations

A spherical (k − 1)-fibration p: E → X is a fibration, i.e. a map having the homotopy lifting property, whose typical fiber is homotopy equivalent to S^{k−1}. Define its associated disk fibration Dp: DE → X by DE = cyl(p), where cyl(p) is the mapping cylinder of p and Dp the obvious map. Define its Thom space Th(p) to be the pointed space cone(p), where cone(p) is the mapping cone of p with its canonical base point. Notice that Th(p) = DE/E.

If ξ: E → X is a k-dimensional vector bundle with Riemannian metric, its sphere bundle p: SE → X is an example of a (k−1)-spherical fibration. The role of the disk bundle Dξ is now played by DE = cyl(p). Notice that the canonical inclusion of X in cyl(p) is a homotopy equivalence analogous to the fact that the inclusion of the zero-section of ξ into E is a homotopy equivalence. The canonical inclusion of E into cyl(p) corresponds to the inclusion of SE ⊂ DE. Hence it is clear that Th(p) = DE/E = cone(p) for a (k − 1)-spherical fibration corresponds to Th(ξ) = DE/SE for a k-dimensional vector bundle ξ: E → X.

If one has two vector bundles ξ_{0} : E_{0} → X and ξ_{1} : E_{1} → X, one can form the Whitney sum ξ_{0} ⊕ ξ_{1}. Notice that S(E_{0} ⊕ E_{1})_{x} is homeomorphic to the join S(E_{0})_{x} ∗ S(E_{1})_{x}. This allows us to rediscover S(ξ_{1} ⊕ ξ_{2}) as a spherical fibration from Sξ_{0} and Sξ_{1} by the fiberwise join construction. Namely, given a (k − 1)-spherical fibration p_{0} : E_{0} → X and an (l − 1)-spherical fibration p_{1} : E_{1} → X, define the (k + l − 1)-spherical fibration p_{0} ∗ p_{1} : E_{0} ∗ E_{1} → X called fiberwise join as follows. The total space E_{0} ∗ E_{1} is the quotient of the space {(e_{0}, e_{1}, t) ∈ E_{0} × E_{1} × [0, 1] | p_{0}(e_{0}) = p_{1}(e_{1})} under the equivalence relation generated by (e_{0}, e_{1}, 1) ∼ (e_{0}, e′_{1}, 1) and (e_{0}, e_{1}, 0) ∼ (e′_{0}, e_{1}, 0). The projection p_{0} ∗ p_{1} sends the class of (e_{0}, e_{1}, t) to p_{0}(e_{0}) = p_{1}(e_{1}). Given a spherical (k − 1)-fibration p: E → X, its l-fold suspension is the spherical (k + l − 1)-fibration given by the fiberwise join of p and the trivial (l − 1)- spherical fibration pr: X × S^{l−1} → X.

There is natural homeomorphism for a spherical fibrations ξ

Th(ξ ∗ \underline{S^{k−1}}) ≅ Σ^{k} Th(ξ).  (3.30)

Given two fibrations p_{0} : E_{0} → X and p_{1} : E_{1} → X, a fiber map (\overline{f}, f): p_{0} → p_{1} consists of maps \overline{f}: E_{0} → E_{1} and f: X_{0} → X_{1} satisfying p_{1}∘\overline{f} = f ∘p_{0}. There is an obvious notion of fiber homotopy (\overline{h}, h). A fiber homotopy is called a strong fiber homotopy if h_{t}: X_{0} → X_{1} is stationary for t ∈ [0, 1]. Two fibrations p_{0} and p_{1} over the same base space are called strongly fiber homotopy equivalent if there are fiber maps (\overline{f}, id): p_{0} → p_{1} and (\overline{g}, id): p_{1} → p_{0} such that both composition are strongly fiber homotopy equivalent to the identity. Given a topological space F, let G(F) be the monoid of selfhomotopy equivalences of F. One can associate to such a monoid a classifying space BG(F) together with a fibration p_{F} : EG(F) → BG(F) with typical fiber F such that the pullback construction yields a bijection between the homotopy classes of maps from X to BG(F) and the set of strong fiber homotopy classes of fibrations over X with typical fiber F [110]. We will abbreviate in the sequel G(k) := G(S^{k−1}).

Given a fibration p: E → X, and x ∈ X, the fiber transport along loops at x defines a map of monoids t_{x}: π_{1}(X, x) → [p^{−1}(x), p^{−1}(x)]. We call p orientable if t_{x} is trivial for any base point x ∈ X. In the case of (k − 1)-spherical fibration this fiber transport is the same as a homomorphism w: π_{1}(X) → {±1} called the orientation homomorphism since the degree defines a bijection [S^{k−1}, S^{k−1}] → ℤ for k ≥ 2. An orientation for an orientable spherical (k − 1)-fibration p is the choice of fundamental class [p^{−1}(x)] ∈ H_{k−1}(p^{−1}(x)) ≅ ℤ for any fiber p^{−1}(x) for x ∈ X such that for any path w in X the fiber transport along w yields a homotopy equivalence p^{−1}(w(0)) → p^{−1}(w(1)) for which the induced map on H_{k−1} sends [p^{−1}(w(0))] to [p^{−1}(w(1))].

Theorem 3.31 (Thom Isomorphism) Let p: E → X be a (k − 1)-spherical fibration over a connected CW -complex X. Then there exists a group homomorphism w: π_{1}(X) → {±1} and a so called Thom class U_{p} ∈ H^{k}(DE, E; ℤ^{w}) such that the composites

H_{p}(DE, E; ℤ) ⟶^{U_{p}∩?} H_{p−k}(DE; ℤ^{w}) ⟶^{H_{p−k}(p)} H_{p−k}(X; ℤ^{w}); (3.32)
H_{p}(DE, E; ℤ^{w}) ⟶^{U_{p}∩?} H_{p−k}(DE; ℤ) ⟶^{H_{p−k}(p)} H_{p−k}(X; ℤ);  (3.33)
H^{p}(X; ℤ) ⟶^{H^{p}(p)} H^{p}(DE; ℤ) ⟶^{?∪U_{p}} H^{p+k}(DE, SE; ℤ^{w});  (3.34)
H^{p}(X; ℤ^{w}) ⟶^{H^{p}(p)} H^{p}(DE; ℤ^{w}) ⟶^{?∪U_{p}} H^{p+k}(DE, SE; ℤ);  (3.35)

are bijective. These maps are called Thom isomorphisms.

There are precisely two possible choices for U_{p} because H^{k}(DE, E; ℤ^{w}) ≅ H^{0}(X; ℤ) ≅ ℤ is infinite cyclic. Moreover, w is uniquely determined by the spherical (k − 1)-fibration. The proof is analogous to the one for Poincaré complexes.

If the spherical fibration p is orientable, then U_{p} is uniquely determined by the property that the composite

H^{k}(DE, E) → H^{k}(p_{DE}^{−1}(x), p^{−1}(x); ℤ) ⟶^{(δ^{k−1})^{−1}} H^{k−1}(p^{−1}(x))

sends U_{p} to [p^{−1}(x)]. Moreover, a choice of orientation for p is the same as a choice of Thom class.

3.2.3 The Existence and Uniqueness of the Spivak Normal Fibration

Definition 3.36 A Spivak normal fibration for an n-dimensional connected finite Poincaré complex X is a (k − 1)-spherical fibration p = p_{X}: E → X together with a pointed map c = c_{X}: S^{n+k} → Th(p) such that X and p have the same orientation homomorphism w: π_{1}(X) → {±1} and for some choice of Thom class U_{p} ∈ H^{k}(DE, E; ℤ^{w}) the fundamental class [X] ∈ H_{n}(X; ℤ^{w}) and the image h(c) ∈ H_{n+k}(Th(p)) ≅ H_{n+k}(DE, E; ℤ) of [c] under the Hurewicz homomorphism h: π_{n+k}(Th(p)) → H_{n+k}(Th(p), ℤ) are related by the formula

[X] = H_{n}(p)(U_{p} ∩ h(c)).

Remark 3.37 A closed manifold M of dimension n admits a Spivak normal fibration. Namely, choose an embedding i: M → ℝ^{n+k} and take p to be the sphere bundle Sν(i) → M and c to be the collaps map defined in (3.25).

Theorem 3.38 (Existence and Uniqueness of the Spivak Normal Fibration) Let X be a connected finite n-dimensional Poincaré complex. Then:

1. For k > n there exists a Spivak normal (k − 1)-fibration for X;

2. Let p_{i}: E_{i} → X together with c_{i}: S^{n+k_{i}} → Th(p_{i}) be a Spivak normal (k_{i} − 1)-fibration of X for i = 0, 1. Let k be an integer satisfying k ≥ k_{0}, k_{1}. Then there is up to strong fiber homotopy precisely one strong fiber homotopy equivalence \overline{f}: Σ^{k−k_{0}}p_{0} → Σ^{k−k_{1}}p_{1} for which

π_{n+k}(Th(\overline{f})): π_{n+k}(Th(p_{0})) → π_{n+k}(Th(p_{0}))

sends [c_{0}] to [c_{1}].

At least we want to give the idea of the proof of assertion (1) of Theorem 3.38 provided that the orientation homomorphism is trivial. For a detailed proof we refer for instance to [17, I.4], [108].

Consider a connected finite n-dimensional CW -complex X. One can always find an embedding X ⊂ ℝ^{n+k} for k > n together with a regular neighborhood (N(X), ∂N(X)). The regular neighboorhood is a compact manifold N(X) with boundary ∂N(K) such that X ⊂ N(K) is a strong deformation retraction [100, Chapter 3]. This regular neighborhood corresponds in the case, where X is a closed manifold M, to a tubular neighborhood (N(M), ∂N(M)) ≅ (Dν(M), Sν(M)). Let [N(X), ∂N(X)] ∈ H_{n+k}(N(X), ∂N(X)) be a fundamental class of the compact manifold N(X) with boundary ∂N(X) which corresponds to the orientation inherited from the standard orientation on ℝ^{n+k}. Let i: X → N(X) be the inclusion which is a homotopy equivalence. Then the following diagram commutes for any class u ∈ H^{k}(N(X), ∂N(X))

H^{n−p}(X)  ⟶^{?∩(H_{n}(i)^{−1}(u∩[N(X),∂N(X)]))} H_{p}(X)
(u∪?)∘H^{n−p}(i)^{−1}↓  ↓^{H_{p}(i)}
H^{n+k−p}(N(X), ∂N(X))  ⟶_{?∩[N(X),∂N(X)]}  H_{p}(N(X))

The lower horizontal arrow and the right vertical arrow are bijective by Poincaré duality and homotopy invariance. Hence the upper horizontal arrow is bijective if and only if the left vertical arrow is bijective. Notice that bijectivity of the upper horizontal arrow corresponds to Poincaré duality for X with H_{n}(i)^{−1}(u ∩ [N(X), ∂N(X)]) as fundamental class and the bijectivity of the left vertical arrow corresponds to the Thom isomorphism with u as Thom class.

Suppose that X is a connected finite Poincaré complex with fundamental class [X] ∈ H_{n}(X). Let u ∈ H^{k}(N(X), ∂N(X)) be the class uniquely determined by the property that H_{n}(i)^{−1}(u ∩ [N(X), ∂N(X)]) = [X]. Then the map

H^{n−p}(X) → H^{n+k−p}(N(X), ∂N(X), ℤ), v ↦ u ∪ H^{n−p}(i)^{−1}(v)

is bijective. Our first approximation of the Spivak normal (k − 1)-fibration

is the composition f: ∂N(X) ⟶^{j} N(X) ⟶^{i^{−1}} X, where j is the inclusion and i^{−1} a homotopy inverse of i. Of course f is not a fibration but possess a candidate for a Thom class, namely u. By a general construction we can turn f into a fibration. More precisely, for any map g: X → Y there is a functorial construction which yields a fibration p_{g}: E_{g} → Y together with a homotopy equivalence i_{g}: X → E_{g} satisfying p_{g} ∘ i_{g} = g. Namely, define E_{g} = {(x, w) ∈ X × map([0, 1], Y) | f(x) = w(0)}, p_{g}(x, w) = w(1) and i_{g}(x) = (x, c_{f(x)}) for c_{f(x)} the constant path at f(x) (see [121, Theorem 7.30 on page 42]). We apply this to f: ∂N(X) → X and obtain a fibration p_{f}: E_{f} → X together with a homotopy equivalence i_{f}: ∂N(X) → E_{f} satisfying p_{f} ∘ i_{f} = f. Since j: ∂N(X) → N(X) is a cofibration and i^{−1} : N(X) → X a homotopy equivalence, we can find an extension of i_{f} to a homotopy equivalence of pairs (I_{f}, i_{f}): (N(X), ∂N(X)) → (DE_{f}, E_{f}) for DE_{f} = cyl(p_{f}). This extension is unique up to homotopy relative ∂N(X). Let U_{p_{f}} ∈ H^{k}(DE_{f}, E_{f}) be the preimage of u ∈ H^{k}(N(X), ∂N(X)) under the isomorphism induces by (I_{f}, i_{f}). One easily checks that the map

H^{p}(X) ⟶^{H^{p}(p_{f})} H^{p}(DE_{f}) ⟶^{?∪U_{p_{f}}} H^{p+k}(DE_{f}, E)

is bijective. So U_{p_{f}} looks like a Thom class. We have already seen that a spherical fibration has a Thom class and it turns out that this does characterize the homotopy fiber of a fibration. Hence p_{f}: ∂N(X) → X is a spherical (k −1)- fibration. The collaps map c: S^{n+k} → N(X)/∂N(X) can be composed with the map N(X)/∂N(X) → Th(p_{f}) induced by (I_{g}, i_{g}) and yields a pointed map c_{p_{f}} : S^{n+k} → Th(p_{f}). It has the desired property [X] = H_{n}(p_{f})(U_{p_{f}} ∩ h(c_{p_{f}})). Hence p_{f} and U_{p_{f}} yield a normal Spivak fibration for X.

Remark 3.39 Recall that we want to address Problem 3.1, whether a space X is homotopy equivalent to a connected closed n-dimensional manifold. We have already seen in Remark 3.8 that we only have to consider connected finite n-dimensional Poincaré complexes X. From Theorem 3.38 and Remark 3.37 we get the following new necessary condition. Namely, we must be able to find for k > n a k-dimensional vector bundle ξ: E → X such that the associated sphere bundle SE → X is strongly fiber homotopy equivalent to the Spivak normal (k − 1)-fibration of X.

Lemma 3.40 Let p_{i}: E_{i} → X_{i} be a spherical (k − 1)-fibration over a connected finite n-dimensional Poincaré complexes X_{i} for i = 0, 1. Let (\overline{f}, f): p_{0} → p_{1} be a fiber map which is fiberwise a homotopy equivalence. Then we get for the orientation homomorphisms w(p_{1}) ∘ π_{1}(f) = w(p_{0}). Consider a pointed map c_{0} : S^{n+k} → Th(p_{0}). Let c_{1} : S^{n+k} → Th(p_{0}) be the composition Th(\overline{f}) ∘ c_{0}. Then

1. Suppose that the degree of f is ± − 1. Then (p_{0}, c_{0}) is the Spivak normal fibration of X_{0} if and only if (p_{1}, c_{1}) is the Spivak normal fibration for X_{1};

2. Suppose that (p_{i}, c_{i}) is the Spivak normal (k−1)-fibration of X_{i} for i = 0, 1. Then the degree of f is ±1.

\underline{\underline{Proof}} : The claim about the orientation homomorphisms follows from the fact that the fiber transport of p_{0} and p_{1} are compatible with π_{1}(f). We can choose the Thom classes of p_{0} and p_{1} such that the map

H^{k}(Th(\overline{f})): H^{n+k}(DE_{1}, E_{1}; ℤ^{w(p_{1})}) → H^{k}(DE_{0}, E_{0}; ℤ^{w(p_{0})})

sends U_{p_{1}} to U_{p_{0}}. Thus we get

H_{n}(f) (H_{n}(p_{0}) (U_{p_{0}} ∩ h(c_{0}))) = H_{n}(p_{1}) (U_{p_{1}} ∩ h(c_{1})) .

Now the claim follows from the definitions.

3.3 Normal Maps

Motivated by Remark 3.39 we define

Definition 3.41 Let X be a connected finite n-dimensional Poincaré complex. A normal k-invariant (ξ, c) consists of a k-dimensional vector bundle ξ: E → X together with an element c ∈ π_{n+k}(Th(ξ)) such that for some choice of Thom class U_{p} ∈ H^{k}(DE, SE; ℤ^{w}) the equation [X] = H_{n}(p)(U_{p} ∩ h(c)) holds. We call a normal k-invariant (ξ_{0}, c_{0}) and a normal k-invariant (ξ_{1}, c_{1}) equivalent if there is a bundle isomorphism (\overline{f}, id): ξ_{0} ⟶^{≅} ξ_{1} such that

π_{n+k}(Th(\overline{f})): π_{n+k}(Th(E_{0})) ⟶^{≅} π_{n+k}(Th(ξ_{1}))

maps c_{0} to c_{1}. The set of normal k-invariants T_{n}(X, k) is the set of equivalence classes of normal k-invariants of X.

Given a normal k-invariant (ξ, c), we obtain a normal (k + 1)-invariant (ξ ⊕ \underline{ℝ}, Σc), where Σ: π_{k}(Th(ξ)) → π_{k+1}(Σ Th(ξ)) = π_{k+1}(Th(ξ ⊕ \underline{ℝ})) is the suspension homomorphism. Define

Definition 3.42 Let X be a connected finite n-dimensional Poincaré complex X. Define the set of normal invariants T_{n}(X) of X to be colim_{k→∞} T_{n}(X, k).

Let J_{k}: BO(k) → BG(k) be the classifying map for the universal k-dimen-sional vector bundle ξ_{k}: E_{k} → BO(k) viewed as a spherical fibration. Taking the Whitney sum with \underline{ℝ} or the fiberwise join with S\underline{ℝ} yields stabilization maps BO(k) → BO(k +1) and BG(k) → BG(k +1). This corresponds to the obvious stabilization maps O(k) → O(k + 1) and G(k) → G(k + 1) given by direct sum with the identity map ℝ → ℝ and by suspending a selfhomotopy equivalence of S^{k}. One can arrange that these stabilization maps are cofibrations by a mapping cone construction and are compatible with the various maps J_{k} by a cofibration argument. Put

O = colim_{k→∞} O(k);
G = colim_{k→∞} G(k);
BO = colim_{k→∞} BO(k);
BG = colim_{k→∞} BG(k).

Define J: BO → BG by colim_{k→∞} J_{k}.

From Theorem 3.38 we get for a connected finite n-dimensional Poincaré complex X a map s_{X}: X → BG which is given by the classifying map of the Spivak normal (k − 1)-bundle for large k. It is unique up to homotopy. We obtain from the universal properties of the classifying spaces

Theorem 3.43 Let X be a connected finite n-dimensional Poincaré complex. Then T_{n}(X) is non-empty if and only there is a map S: X → BO such that J ∘ S is homotopic to s_{X}.

Remark 3.44 In view of Remark 3.40 we see that a necessary condition for a connected finite n-dimensional Poincaré complex to be homotopy equivalent to a closed manifold is that the classifying map s: X ⟶^{s_{X}} BG(k) lifts along J: BO → BG. There is a fibration BO → BG → BG/O. Hence this condition is equivalent to the statement that the composition X ⟶^{s_{X}} BG → BG/O is homotopic to the constant map. There exists a finite Poincaré complex X for which composition X ⟶^{s_{X}} BG → BG/O is not nullhomotopic (see [78, page 32 f]). In particular X cannot be homotopy equivalent to a closed manifold.

Let G/O be the homotopy fiber of J: BO → BG. This is the fiber of the fibration Ĵ: E_{J} → BG associated to J. Then the following holds

Theorem 3.45 Let X be a connected finite n-dimensional Poincaré complex. Suppose that T_{n}(X) is non-empty. Then there is a canonical group structure on the set [X, G/O] of homotopy classes of maps from X to G/O and a transitive free operation of this group on T_{n}(X).

\underline{\underline{Proof}} : Define an abelian group G/O(X) as follows. We consider pairs (ξ, t) consisting of a k-dimensional vector bundle ξ over X for some k ≥ 1 and a strong fiber homotopy equivalence t: Sξ → \underline{S^{k−1}} from the associated spherical fibration given by the sphere bundle Sξ and the trivial spherical (k−1)-fibration over X. We call two such pairs (ξ_{0}, t_{0}) and (ξ_{1}, t_{1}) equivalent, if for some k which is greater or equal to both k_{0} = dim(ξ_{0}) and k_{1} = dim(ξ_{1}) there is a bundle isomorphism \overline{f}: ξ_{0} ⊕ \underline{ℝ^{k−k_{0}}} ⟶^{≅} ξ_{1} ⊕ \underline{ℝ^{k−k_{1}}} covering the identity on X such that Σ^{k−k_{1}}t_{1} ∘ S\overline{f} is fiber homotopic to Σ^{k−k_{0}}t_{0}. Let G/O(X) be the set of equivalence classes [ξ, t] of such pairs (ξ, t). Notice that [ξ, t] for a pair (ξ, t) depends only on ξ and the fiber homotopy class of t. Addition is given by the Whitney sum. The neutral element is represented by (\underline{ℝ^{k}}, id) for some k ≥ 1. The existence of inverses follows from the fact that for a vector bundle ξ over X we can find another vector bundle η such that ξ ⊕ η is trivial and for a map f: X → G(k) we can find l and a map f′: X → G(l) such that the map given by the join f ∗ f′: X → G(k + l) is homotopic to the identity.

Next we describe an action ρ: G/O(X) × T_{n}(X) → T_{n}(X). Consider a pair (ξ, t) representing an element [ξ, t] ∈ G/O(X) and a normal k_{1}-invariant (η, c) representing an element [η, c] ∈ T_{n}(X). If k_{0} is the dimension of ξ, then ξ ⊕ η is a (k_{0} + k_{1})-dimensional vector bundle. Consider the composition

Th(ξ ⊕ η) ⟶^{≅} Th(Sξ ∗ Sη) ⟶^{t∗id} Th(\underline{S^{k_{0}−1}} ∗ Sη) ⟶^{≅} Σ^{k_{0}} Th(η).

This together with the suspension Σ^{k_{0}}c yields an element d ∈ π_{n+k_{0}+k_{1}}(Th(ξ ⊕ η)). One easily checks that (ξ ⊕ η, d) is a normal (k_{0} + k_{1})-invariant for X. Define

ρ([ξ, t], [η, c]) := [ξ ⊕ η, d].

We leave it to the reader to check that this is a well-defined group action. It follows from Theorem 3.38 that this operation is transitive and free. Hence it remains to construct a bijection

μ: [X, G/O] ⟶^{≅} G/O(X).

Recall that G/O is the fiber of the fibration Ĵ: E_{J} → BG associated to J: BO → BG over some point z ∈ BG. By definition this is the space {(y, w) ∈ BO × map([0, 1], BG) | J(y) = w(0), w(1) = z}. Hence a map F: X → G/O is the same as a pair (f, h) consisting of a map f: X → BO and a homotopy h: X × [0, 1] → BG such that h_{0} = J ∘ f and h_{1} is the constant map c_{z}. Since X is compact, the can find k such that the image of f and h lie in BO(k) and BG(k). Recall that J_{k}: BO(k) → BG(k) is covered by a fiber map \overline{J_{k}}: Sξ_{k} → η_{k} which is fiberwise a homotopy equivalence, where ξ_{k} is the universal k-dimensional bundle over BO(k) and η_{k} the universal spherical (k − 1)-fibration. By the pullback construction applied to f and ξ_{k} and h and η_{k}, we get a vector bundle ξ over X and a spherical (k − 1)-fibration η over X × [0, 1] together with fiber homotopy equivalences (u, id): Sξ → η|_{X×{0}} and (v, id): η|_{X×{1}} → X × S^{k−1}. Up to fiber homotopy there is precisely one fiber homotopy equivalence (\overline{g}, pr): η → X × [0, 1] × S^{k−1} whose restriction to X × {1} = X is v [113, Proposition 15.11 on page 342]. Thus we obtain a fiber homotopy equivalence v|_{X×{0}}: η_{X×{0}} → X × S^{k−1} covering the identity X × {0} → X which is unique up to fiber homotopy. Composing v|_{X×{0}} and u yields a fiber homotopy equivalence unique up to fiber homotopy (w, id): Sξ → \underline{S^{k−1}}. Thus we can assign to map F: X → G/O an element [ξ, w] ∈ G/O(X). We leave it to the reader to check that this induces the desired bijection μ: [X, G/O] ⟶^{≅} G/O(X). This finishes the proof of Theorem 3.45.

Notice that Theorem 3.45 yields after a choice of an element in T_{n}(X) a bijection of sets [X, G/O] ⟶^{≅} T_{n}(X).

Definition 3.46 Let X be a connected finite n-dimensional Poincaré complex together with a k-dimensional vector bundle ξ: E → X. A normal k-map (M, i, f, \overline{f}) consists of a closed manifold M of dimension n together with an embedding i: M → ℝ^{n+k} and a bundle map (\overline{f}, f): ν(M) → ξ. A normal map of degree one is a normal map such that the degree of f: M → X is one.

Notice that this definition is the same as the definition of an element representing a class in Ω_{n}(ξ) except that we additionally require the map of degree 1. Analogously one requires in the definition of a bordism that the map F: (W, ∂W) → (X × [0, 1], X × ∂[0, 1]) has degree one. Denote by N_{n}(X, k) the set of normal bordism classes of normal k-maps of degree one to X. Define N_{n}(X, k) → N_{n}(X, k + 1) by sending the class of (M, i, f, \overline{f}) to the class of (M, i′, f, \overline{f}′), where i′: M → ℝ^{n+k+1} is the composition of i with the standard

inclusion ℝ^{n+k} → ℝ^{n+k+1} and \overline{f}′ is ν(i′) = ν(i) ⊕ \underline{ℝ} ⟶^{ν(f)⊕id} ξ ⊕ \underline{ℝ}.

Definition 3.47 Let X be a connected finite n-dimensional Poincaré complex. Define the set of normal maps to X

N_{n}(X) := colim_{k→∞} N_{n}(X, k).

Notice that in the definition of N_{n}(X, k) and N_{n}(X) the bundle ξ over X may vary, we only fix X. The proof of the next result is similar to the one of Theorem 3.26.

Theorem 3.48 Let X be a connected finite n-dimensional Poincaré complex. Then the Pontrjagin-Thom construction yields for each k ≥ 1 a bijection

P_{k}(X): N_{n}(X, k) ⟶^{≅} T_{n}(X, k).

This induces a bijection

P (X): N_{n}(X) ⟶^{≅} T_{n}(X).

Remark 3.49 In view of the Pontrjagin Thom construction it is convenient to work with the normal bundle. On the other hand one always needs an embedding and one would prefer an intrinsic definition. This is possible if one defines the normal map in terms of the tangent bundle which we will do below. Both approaches are equivalent. We will use in the sequel the one which is adequate for the concrete purpose. We mention that for a generalization to the equivariant setting the approach using the tangent bundle is more useful [73], [74].

Definition 3.50 Let X be a connected finite n-dimensional Poincaré complex together with a vector bundle ξ: E → X. A normal map with respect to the tangent bundle (\overline{f}, f): M → X consists of an oriented closed manifold M of dimension n together with a bundle map (\overline{f}, f): T M ⊕ \underline{ℝ^{a}} → ξ for some integer a ≥ 0. If f has degree one, we call (\overline{f}, f) a normal map of degree one with respect to the tangent bundle.

We define also a bordism relation as follows. Consider two normal maps of degree one with respect to the tangent bundle (\overline{f}_{i}, f_{i}): T M ⊕ \underline{ℝ^{a_{i}}} → ξ_{i} covering f_{i}: M_{i} → X. A normal bordism is a normal map of degree one (\overline{F} , F): T W ⊕ \underline{ℝ}^{b} → η covering F: W → X such that ∂W is a disjoint union ∂_{0}W ∐ ∂_{1}W and we have the following data for i = 0, 1. We require diffeomorphisms u_{i}: M_{i} → ∂_{i}W with F ∘ u_{i} = f_{i}. Moreover, we require bundle isomorphisms v_{i}: ξ_{i} ⊕ \underline{ℝ}^{b−a_{i}+1} → η covering the identity on X such that following diagram commutes

T M_{i} ⊕ \underline{ℝ} ⊕ \underline{ℝ^{b}} ⟶^{\overline{f}_{i}⊕id\underline{ℝ^{b−a_{i}+1}}} ξ_{i} ⊕ \underline{ℝ^{b−a_{i}+1}}
T u_{i}⊕n_{i}⊕id\underline{ℝ^{b}} ↓  ↓^{v_{i}}
T W |_{∂_{i}W} ⊕ \underline{ℝ^{b}}  ⟶_{F |_{∂_{i}W}}  η|_{∂_{i}W}

Here T u_{i}: T M_{i} → T W is given by the differential and n_{i}: \underline{ℝ} → T W is the bundle monomorphism given by an inward normal field of T W |_{∂_{i}W}. Denote by N_{n}^{T} (X) the set of bordism classes of normal maps of degree one with respect to the tangent bundle.

Lemma 3.51 Let X be a connected n-dimensional finite Poincaré complex. There is a natural bijection

N_{n}(X) ≅ N_{n}^{T} (X).

\underline{\underline{Proof}} : We define a map ϕ_{n}(k): N_{n}(X, k) → N_{n}^{T} (X) as follows. Consider a normal k-map \overline{f}: ν(M) → ξ covering the map f: M → X of degree one for some closed oriented n-dimensional manifold M with an embedding M → ℝ^{n+k}. Since X is compact, we can find a bundle η together with an isomorphism u: η ⊕ ξ ⟶^{≅} \underline{ℝ^{a}}. There is an explicit isomorphism v: ν(M) ⊕ T M ≅ \underline{ℝ^{n+k}}. We get from \overline{f}, u and v an isomorphism of bundles covering the identity on M

f^{∗}η ⊕ \underline{ℝ^{n+k}} ≅ f^{∗}η ⊕ T M ⊕ ν(M) ≅ f^{∗}η ⊕ T M ⊕ f^{∗}ξ ≅ T M ⊕ \underline{ℝ^{a}}.

The inverse of this isomorphism is the same as a bundle map \overline{g}: T M ⊕ \underline{ℝ^{a}} → η ⊕ \underline{ℝ^{n+k}} covering f. Define the image under ϕ_{n}(k) of the class [\overline{f}, f] to be the class [\overline{g}, f]. One easily checks that this is well-defined and that the ϕ_{n}(k)-s fit together to a map ϕ_{n}: N_{n}(X) → N_{n}^{T} (X). By the analogous construction one gets an inverse.

Theorem 3.52 (Normal Maps and G/O) Let M be a closed n-dimensional manifolds. Then the sets T_{n}(M), N_{n}(M) and N_{n}^{T} (M) are non-empty and come with preferred base points and there are canonical bijections

[M, G/O] ≅ T_{n}(M) ≅ N_{n}(M) ≅ N_{n}^{T} (M).

\underline{\underline{Proof}} : follows from Theorem 3.45, Theorem 3.48 and Lemma 3.51.

Remark 3.53 Let X be a finite Poincaré complex of dimension n. Then there exists a closed manifold M, which is homotopy equivalent to X, only if there exists a normal map of degree one with target X. Namely, suppose f: M → X is a homotopy equivalence with a closed manifold as source. Choose a homotopy inverse f^{−1} : X → M. Put ξ = (f^{−1})^{∗}T M. Then we can cover f by a bundle map \overline{f}: T M → ξ. Thus we get a normal map (\overline{f}, f) of degree one with M as source and X as target. In order to solve Problem 3.1 we have to address the following problem.

Problem 3.54 Suppose we have some normal map (\overline{f}, f) from a closed manifold M to a finite Poincaré complex X. Can we change M and f leaving X fixed to get a normal map (\overline{g}, g) such that g is a homotopy equivalence?

3.4 The Surgery Step

In this section we explain the surgery step. We begin with a motivation.

3.4.1 Motivation for the Surgery Step

We first consider the CW -complex version of Problem 3.54.

Let f: Y → X be a map of CW -complexes. We want to find a procedure which changes Y and f leaving X fixed to map f′: Y′ → X which is a homotopy equivalence. Of course this procedure should have the potential to carry over to the case, where Y is a manifold and the resulting source Y′ is also a manifold. The Whitehead Theorem [121, Theorem V.3.1 and Theorem V.3.5 on page 230] says that f is a homotopy equivalence if and only if it is k-connected for all k ≥ 0. Recall that k-connected means that π_{j}(f, y): π_{j}(Y, y) → π_{j}(X, f(y)) is bijective for j < k and surjective for j = k for all base points y in Y. Hence we would expect from the procedure to get π_{k}(f) to be closer and closer to be trivial for all k. It is reasonable to try to work out an inductive procedure, where f is already k-connected and we would like to make it (k + 1)-connected. Recall that there is a long exact homotopy sequence of a map f: Y → X

… → π_{l+1}(Y) → π_{l+1}(X) → π_{l+1}(f) → π_{l}(Y) → π_{l}(X) → …,

where π_{l+1}(f) ≅ π_{l+1}(cyl(f), Y) consists of homotopy classes of commutative squares with j the inclusion

S^{l} ⟶^{q} Y
j↓  ↓^{f}
D^{l+1} ⟶_{Q} X

The map f is k-connected, if and only if π_{l}(f) = 0 for l ≤ k. Suppose that f is k-connected. In order to achieve that f is (k + 1)-connected, we must arrange that π_{k+1}(f) = 0 without changing π_{l}(f) for l ≤ k. Consider an element ω in π_{k+1}(f) given by square as above. Define Y′ to be the pushout

S^{k} ⟶^{q} Y
j↓  ↓^{j′}
D^{k+1} ⟶_{\overline{q}} Y′

Then the universal property of the pushout gives a map f′: Y′ → X. We will say that f′ is obtained from f by attaching a cell.

One easily checks that π_{l}(f′) = π_{l}(f) for l ≤ k and that there is a natural map π_{k+1}(f) → π_{k+1}(f′) which is surjective and whose kernel contains ω ∈ π_{l}(f). Recall that in general π_{l}(f) is an abelian group for l ≥ 3 and a group for l = 2 but carries no group structure for l = 0, 1. Moreover π_{1}(X) acts on π_{l}(f). Hence π_{l}(f) is a ℤπ_{1}(X)-module for l ≥ 3. For k ≥ 3 the kernel of the epimorphism π_{k+1}(f) → π_{k+1}(f′) is the ℤπ_{1}(Y)-submodule generated by ω [121, Section V.1]. We see that we can achieve by applying this construction that π_{l+1}(f) becomes zero. Suppose that Y and X are finite CW -complexes. One can achieve π_{k+1}(f) = 0 in a finite number of steps by the following result.

Lemma 3.55 Let f: Y → X be a map of finite CW -complexes. Suppose that f is (k − 1)-connected for some integer k ≥ 0.

1. Suppose that X is connected, k ≥ 2 and π_{1}(f): π_{1}(Y) → π_{1}(X) is bijective. Then π_{k}(f) is a finitely generated ℤπ_{1}(X)-module;

2. One can make f k-connected by attaching finitely many cells.

\underline{\underline{Proof}} : (1) Notice that both Y and X are connected. We will identify π := π_{1}(Y) = π_{1}(X). The map f lifts to π-map f̃: Ỹ → X̃ between the universal coverings. The obvious map π_{k}(f̃) → π_{k}(f) is bijective and compatible with the π-operations which are given by the π-actions on the universal covering and the operation of the fundamental group on the homotopy groups. The Hurewicz homomorphism induces an isomorphism of ℤπ-modules π_{k}(f̃) ⟶^{≅} H_{k}(f̃) since Ỹ and X̃ are simply-connected [121, Corollary IV.7.10 on page 181]. In particular we see that π_{k}(f) is indeed an abelian group what is true in general only for k ≥ 3. Since f and hence f̃ is (k − 1)-connected the ℤπ-chain complex C_{∗}(f̃) which is defined to be the mapping cone D_{∗} := cone_{∗}(C_{∗}(f̃)) of the ℤπ-chain map C_{∗}(f̃): C_{∗}(Ỹ) → C_{∗}(X̃) is (k − 1)-connected. Thus C_{∗}(f̃) yields an exact sequence of finitely generated free ℤπ-modules

0 → ker(d_{k}) → D_{k} ⟶^{d_{k}} D_{k−1} ⟶^{d_{k−1}} … ⟶^{d_{1}} D_{0} → 0.

Hence ker(d_{k}) is a finitely generated free projective ℤπ-module which is stably free, i.e. after adding a finitely generated free ℤπ-module it becomes free. Since H_{k}(f̃) is a quotient of ker(d_{k}), it is finitely generated. Hence π_{k}(f) is finitely generated free as ℤπ-module.

(2) We begin with k = 0. Attaching zero cells means taking the disjoint union of Y with finitely many points. Obviously one can achieve in this way that π_{0}(f): π_{0}(Y) → π_{0}(X) is surjective. In the sequel we assume that X is connected, otherwise we treat each component separately.

Next we treat the case k = 1. Since Y is by assumption a finite CW - complex, π_{0}(Y) is finite. If two path components of Y are mapped to the same path component in X, one can attach a 1-cell in the obvious manner to connect these components. Thus we can achieve that π_{0}(f) is bijective. Since X is finite, π_{1}(X) is finitely generated. By attaching 1-cells trivially to Y which is the same as taking the one-point union of Y with S^{1}, we can achieve that π_{1}(f) is an epimorphism.

Next we consider k = 2. Since both Y and X are finite, π_{1}(f): π_{1}(Y) → π_{1}(X) is an epimorphism of a finitely generated group onto a finitely presented group. One easily checks that the kernel of such a group homomorphism is always finitely generated. For any element in a finite set of generators we can attach 2-cells to kill these elements. The resulting map induces an isomorphism on π_{1}(Y). Now we can apply assertion (1). The cases k ≥ 3 follow directly from assertion (1). This finishes the proof of Lemma 3.55.

Actually one can achieve in the world of CW -complexes the desired homotopy equivalence f′: Y′ → X directly by the following construction. Namely consider the projection pr: cyl(f) → X of the mapping cylinder of f to X. Obviously the mapping cylinder is in general no manifold even if f: Y → X is a smooth map of closed manifolds. Neither there is a chance that the space Y′ obtained from Y by attaching a cell is a manifold even if Y is a closed manifold and f smooth. But this single step of attaching a cell can be modified so that it applies to manifolds as source such that the resulting map has a manifold as source. This will be explained next.

Suppose that M is a compact manifold of dimension n and X is a CW - complex. Suppose that f: M → X is a k-connected map. Consider an element ω ∈ π_{k+1}(f) represented by a diagram

S^{k} ⟶^{q} M
j↓  ↓^{f}
D^{k+1} ⟶_{Q} X

We cannot attach a single cell to M without distroying the manifold structure. But one can glue two manifolds together along a common boundary such that the result is a manifold. Suppose that the map q: S^{k} → M extends to an embedding \overline{q}: S^{k} × D^{n−k} → M. Let int(im(\overline{q})) be the interior of the image of \overline{q}. Then M − int(im(\overline{q})) is a manifold with boundary im(\overline{q}|_{S^{k}×S^{n−k−1}}). We can get rid of the boundary by attaching D^{k+1} × S^{n−k−1} along im(\overline{q}|_{S^{k}×S^{n−k−1}}). Call the result

M′ := D^{k+1} × S^{n−k−1} ∪_{im(\overline{q}|_{S^{k}×S^{n−k−1}} )} (M − int(im(\overline{q}))) .

Here and elsewhere we apply without further mentioning the technique of straightening the angle in order to get a well-defined smooth structure (see [14, Definition 13.11 on page 145 and (13.12) on page 148] and [54, Chapter 8, Section 2]). Choose a map \overline{Q}: D^{k+1} × D^{n−k} → X which extends Q and \overline{q}. The restriction of f to M − int(im(\overline{q})) extends to a map f′: M′ → X using \overline{Q}|_{D^{k+1}×S^{n−k}}. Notice that the inclusion M − int(im(\overline{q})) → M is (n − k − 1)-connected since S^{k} × S^{n−k−1} → S^{k} × D^{n−k} is (n − k − 1)-connected. So the passage from M to M − int(im(\overline{q})) will not affect π_{j}(f) for j < n − k − 1. All in all we see that π_{l}(f) = π_{l}(f′) for l ≤ k and that there is an epimorphism π_{k+1}(f) → π_{k+1}(f′) whose kernel contains ω, provided that 2(k+1) ≤ n. The condition 2(k+1) ≤ n can be viewed as a consequence of Poincaré duality. Roughly speaking, if we change something in a manifold in dimension l, Poincaré duality forces also a change in dimension (n − l). This phenomenon will cause surgery obstructions to appear.

It is important to notice that f: M → X and f′: M′ → X are bordant. The relevant bordism is given by W = D^{k+1} × D^{n−k} ∪\overline{q} M × [0, 1], where we think of \overline{q} as an embedding S^{k} × D^{n−k} → M × \{1\}. In other words, W is obtained from M × [0, 1] by attaching a handle D^{k+1} × D^{n−k} to M × {1}. Then M appears in W as M × {0} and M′ as other part of the boundary of W. Define F: W → X by f × id_{[0,1]} and \overline{Q}. Then F restricted to M and M′ is f and f′.

But before we come to surgery obstructions, we must figure out, whether we can arrange that q is an embedding and extends to an embedding \overline{q}. If 2k < n, we can change q up to homotopy such that it becomes an embedding. In the case 2k = n we can change q up to homotopy to an immersion and the surgery obstruction will actually be the obstruction to change it up to homotopy to an embedding. Let us assume that q is an embedding. Then the existence of the extension \overline{q} is equivalent to the triviality of the normal bundle of the embedding q: S^{k} → M.

A priori there is no reason why this normal bundle should be trivial. At this point the bundle data attached to a normal map become useful. So far we have only considered them since it was a necessary condition to be able to solve Problem 3.1 although in Problem 3.1 no bundle maps appear. But now it will pay off that we have this bundle data available. Namely, if we want to kill an element ω ∈ π_{k+1}(f) represented by a diagram

S^{k} ⟶^{q} M
j↓  ↓^{f}
D^{k+1} ⟶_{Q} X

then q^{∗}T M ⊕\underline{ℝ^{a}} is isomorphic to q^{∗}f^{∗}ξ = j^{∗}Q^{∗}ξ and hence is trivial since Q^{∗}ξ is a bundle over the contractible space D^{k+1}. Since ν(S^{k}, M) ⊕ T S^{k} is isomorphic to q^{∗}T M, the bundle ν(S^{k}, M)⊕\underline{ℝ}^{a} is trivial for some a ≥ 0. Suppose 2k ≤ n−1. Then the natural map BO(n − k) → BO(n − k + a) is (k + 1)-connected. Hence ν(S^{k}, M) itself is trivial. So we are able to carry out one step. But we have to ensure that we can repeat this process. So we must arrange that the bundle data are also available for the resulting map f′: M′ → X. Therefore we must be more careful with the choice of embedding (resp. immersion) which is homotopy equivalent to g. The given bundle data should tell us which embedding we should choose. For this we need some information about embeddings and immersions which we will give next.

3.4.2 Immersions and Embeddings

Given two vector bundles ξ: E → M and η: F → N, we have so far only considered bundle maps (\overline{f}, f): ξ → η which are fiberwise isomorphisms. We need to consider now more generally bundle monomorphisms, i.e. we only will require that the map is fiberwise injective. Consider two bundle monomorphism (\overline{f}_{0}, f_{0}), (\overline{f}_{1}, f_{1}): ξ → η. Let ξ × [0, 1] be the vector bundle ξ × id: E × [0, 1] → M ×[0, 1]. A homotopy of bundle monomorphisms (\overline{h}, h) from (\overline{f}_{0}, f_{0}) to (\overline{f}_{1}, f_{1}) is a bundle monomorphism (\overline{h}, h): ξ × [0, 1] → η whose restriction to X × \{j\} is (\overline{f}_{j}, f_{j}) for j = 0, 1. Denote by π_{0}(Mono(ξ, η)) the set of homotopy classes of bundle monomorphisms.

An immersion f: M → N is a map whose differential T f: T M → T N is a bundle monomorphism. An immersion is locally an embedding but it is not isotopic to an embedding in general. A regular homotopy h: M × [0, 1] → N from an immersion f_{0} : M → N to an immersion f_{1} : M → N is a homotopy h such that h_{0} = f_{0}, h_{1} = f_{1} and h_{t}: M → N is an immersion for each t ∈ [0, 1]. Denote by π_{0}(Imm(M, N)) the set of regular homotopy classes of immersions from M to N. The next result is due to Whitney [122], [123].

Theorem 3.56 Let M and N be closed manifolds of dimensions m and n. Then any map f: M → N is arbitrarily close to an immersion provided that 2m ≤ n and arbitrarily close to an embedding provided that 2m < n.

For a proof of the following result we refer to Haefliger-Poenaru [48], Hirsch [53] and Smale [107].

Theorem 3.57 (Immersions and Bundle Monomorphisms) Let M be a m-dimensional and N be a n-dimensional closed manifold.

1. Suppose that 1 ≤ m < n. Then taking the differential of an immersion yields a bijection

T: π_{0}(Imm(M, N)) ⟶^{≅} π_{0}(Mono(T M, T N));

2. Suppose that 1 ≤ m ≤ n and that M has a handlebody decomposition consisting of q-handles for q ≤ n − 2. Then taking the differential of an immersion yields a bijection

T: π_{0}(Imm(M, N)) ⟶^{≅} colim_{a→∞} π_{0}(Mono(T M ⊕ \underline{ℝ^{a}}, T N ⊕ \underline{ℝ^{a}})),

where the colimit is given by stabilization.

Example 3.58 Theorem 3.57 (1) has the following remarkable consequence. We claim that π_{0}(Imm(S^{2}, ℝ^{3})) consists of one element. We have to show that π_{0}(Mono(T S^{2}, T ℝ^{3})) consists of one element. Consider bundle monomorphisms (\overline{f}_{i}, f_{i}): T S^{2} → T ℝ^{3} for i = 0, 1. Since T_{x}f_{i}(T_{x}S^{2}) ⊂ T_{f_{i}(x)}ℝ^{3} is an oriented 2-dimensional subspace of the oriented Euclidean vector space T_{f(x)}ℝ^{3}, there is precisely one vector v_{i}(x) ∈ T_{f_{i}(x)}ℝ^{3} whose norm is one and for which the orientation on T_{x}f_{i}(T_{x}S^{2}) ⊕ ℝv(x) = T_{f_{i}(x)}ℝ^{3} induced by the one of T_{x}S^{2} and v(x) and the standard one on T_{f_{i}(x)}ℝ^{3} agree. Hence we can find an orientation preserving bundle isomorphism (\overline{g_{i}}, f_{i}): T S^{2} ⊕ \underline{ℝ} → T ℝ^{3} covering f_{i} and extending \overline{f}_{i}. Since f_{i} is homotopic to the constant map c with value 0, we can find a homotopy of bundle maps which are fiberwise orientation preserving isomorphisms from (\overline{g}_{i}, f_{i}) to (\overline{c}_{i}, c) for i = 0, 1. It suffices to show that (\overline{c}_{0}, c) and (\overline{c}_{1}, c) are strongly fiber homotopic as bundle maps (which are fiberwise isomorphisms) because then we get by restriction a homotopy of bundle monomorphism between (\overline{f}_{0}, f_{0}) and (\overline{f}_{1}, f_{1}). Now (\overline{c}_{0}, c) and (\overline{c}_{1}, c) differ by an orientation preserving bundle automorphism covering the identity (\overline{u}, id): S^{2} × T_{0}ℝ^{3} → S^{2} × T_{0}ℝ^{3}. This is the same a map \overline{u}: S^{2} → GL(3, ℝ)^{+}, where we identify T_{0}ℝ^{3} = ℝ^{3} and GL(3, ℝ)^{+} is the Lie group of orientation preserving linear automorphism of ℝ^{3}. The inclusion SO(3) → GL(3, ℝ)^{+} is a homotopy equivalence by the polar decomposition. Since π_{2}(SO(3)) is known to be zero, there is a strong homotopy of bundle maps which are fiberwise isomorphisms and cover the identity from (\overline{u}, id) to (id, id). This proves that π_{0}(Imm(S^{2}, ℝ^{3})) consists of precisely one element.

Let f_{0} : S^{2} → ℝ^{3} be the standard embedding. Let f_{1} : S^{2} → ℝ^{3} be its composition with the involution i: ℝ^{3} − \{0\} → ℝ^{3} − \{0\} which sends x to \frac{x}{||x||^{2}} . By the argument above f_{0} and f_{1} are regular homotopic. Notice that i is the identity on im(f_{0}). Its differential at a point x ∈ im(f_{0}) sends the normal vector v pointing to the origin in ℝ^{3} to −v. Therefore a regular homotopy from f_{0} to f_{1} will turn the inside of the standard sphere f_{0} : S^{2} → ℝ^{3} to the outside.

3.4.3 The Surgery Step

Now we can carry out the surgery step.

Theorem 3.59 (The Surgery Step) Consider a normal map (\overline{f}, f): T M ⊕ \underline{ℝ^{a}} → ξ covering f: M → X and an element ω ∈ π_{k+1}(f) for k ≤ n − 2 for n = dim(M). Let T j ⊕ n: T (S^{k} × D^{n−k}) ⊕ \underline{ℝ} → T (D^{k+1} × D^{n−k}) be the bundle map covering the inclusion j which is given by the differential T j of j and the inward normal field of the boundary of D^{k+1} × D^{n−k}. Then

1. We can find a commutative diagram of vector bundles

T (S^{k} × D^{n−k}) ⊕ \underline{ℝ^{a+b}}  ⟶^{\overline{q}} T M ⊕ \underline{ℝ^{a+b}}
T j⊕n⊕id\underline{ℝ^{a+b−1}} ↓  \sqrt{f}
T (D^{k+1} × D^{n−k}) ⊕ \underline{ℝ^{a+b−1}} ⟶_{\overline{Q}} ξ ⊕ \underline{ℝ^{b}}

covering a commutative diagram

S^{k} × D^{n−k} ⟶^{q} M
j↓  ↓^{f}
D^{k+1} × D^{n−k} ⟶_{Q} X

such that the restriction of the last diagram to D^{k+1} × {0} represents ω and q: S^{k} × D^{n−k} → M is an immersion;

2. The regular homotopy class of the immersion q appearing in assertion (1) is uniquely determined by the properties above and depends only on ω and (\overline{f}, f);

3. Suppose that the regular homotopy class of the immersion q appearing in (1) contains an embedding. Then one can arrange q in assertion (1) to be an embedding. If 2k < n, one can always find an embedding in the regular homotopy class of q;

4. Suppose that the map q appearing in assertion (1) is an embedding.

Let W be the manifold obtained from M × [0, 1] by attaching a handle D^{k+1} × D^{n−k} by q: S^{k} × D^{n−k} → M = M × {1}. Let F: W → X be the map induced by M × [0, 1] ⟶^{pr} M ⟶^{f} X and Q: D^{k} × D^{k+1} → X. After possibly stabilizing \overline{f} the bundle maps \overline{f} and \overline{Q} induce a bundle map \overline{F}: T W ⊕\underline{ℝ^{a+b}} → ξ ⊕\underline{ℝ^{b}} covering F: W → X. Thus we get a normal map (\overline{F} , F): T W ⊕ \underline{ℝ^{a+b}} → ξ ⊕ \underline{ℝ^{b}} which extends (\overline{f} ⊕ (f × id_{ℝ^{b}}), f): T M ⊕ \underline{ℝ^{a+b}} → ξ ⊕ \underline{ℝ^{b}}. The normal map (\overline{f}′, f′): T M′ ⊕ \underline{ℝ^{a+b}} → ξ ⊕ \underline{ℝ^{b}} obtained by restricting (\overline{F} , F) to ∂W − M × \{0\} =: M′ is a normal map of degree one which is normally bordant to (\overline{f}, f) and has as underlying manifold M′ = M − int(q(S^{k} × D^{n−k})) ∪_{q} D^{k} × S^{n−k−1}.

\underline{\underline{Proof}} : (1) Choose a commutative diagram of smooth maps

S^{k} ⟶^{q′} M
j′↓  ↓^{f}
D^{k+1} ⟶_{Q′} X

representing ω. Of course it can be extended to a diagram

S^{k} × D^{n−k} ⟶^{q} M
j↓  ↓^{f}
D^{k+1} × D^{n−k} ⟶_{Q} X

Since D^{k+1} × D^{n−k+1} is contractible, we can find a bundle map \overline{Q}: T D^{k+1} × D^{n−k} ⊕ \underline{ℝ^{a−1}} → ξ covering Q. There is precisely one bundle map \overline{q} covering q such that the following diagram commutes

T (S^{k} × D^{n−k}) ⊕ \underline{ℝ}^{a}  ⟶^{\overline{q}} T M ⊕ \underline{ℝ^{a}}
T j⊕n⊕id\underline{ℝ^{a−1}} ↓  \sqrt{f}
T (D^{k+1} × D^{n−k+1}) ⊕ \underline{ℝ^{a−1}} ⟶_{\overline{Q}}  ξ

Suppose that k ≤ n − 2. From Theorem 3.57 (2) we get an immersion q_{0} : S^{k} × D^{n−k} → M, such that (T q_{0}, q_{0}): T (S^{k} × D^{n−k}) → T M and (\overline{q}, q) define the same element in

colim_{c→∞} π_{0}(Mono(T S^{k} × D^{n−k} ⊕ \underline{ℝ^{c}}, T M ⊕ \underline{ℝ^{s}})).

Now after possibly stabilization \overline{f} and thus enlarging a to a + b we can achieve the desired diagram by a cofibration argument.

(2) The stable homotopy class of T \overline{q} is uniquely determined by the commutativity of the diagram of vector bundles

T (S^{k} × D^{n−k}) ⊕ \underline{ℝ^{a+b}}  ⟶^{T \overline{q}⊕(q×id_{ℝ^{a+b}})} T M ⊕ \underline{ℝ^{a+b}}
T j⊕n⊕id\underline{ℝ^{a+b−1}} ↓  \sqrt{f}
T (D^{k+1} × D^{n−k}) ⊕ \underline{ℝ^{a+b−1}}  ⟶_{\overline{Q}}  ξ

since D^{k+1} × D^{n+k} is contractible and hence (\overline{Q}, Q) is unique up to fiber homotopy. Now apply Theorem 3.57 (2).

(3) By a cofibration argument we can use a regular homotopy from q to an embedding q_{0} to change all the diagrams for q up to homotopy to diagrams for q_{0}. If 2k < n, we can find an embedding q_{0} which is arbitrary close to q by Theorem 3.56 and hence regular homotopic to q since the condition being an immersion is an open condition.

(4) We leave it to the reader to check that the construction of (\overline{F} , F) makes sense.

Definition 3.60 Consider a normal map (\overline{f}, f): M → N and an element ω ∈ π_{k+1}(f) for k ≤ n−2 for n = dim(M). We call the normal map (\overline{f′}, f′): T M′ ⊕ \underline{ℝ^{a+b}} → ξ ⊕ \underline{ℝ^{b}} appearing in Theorem 3.59 (4) the result of surgery on (\overline{f}, f) and ω if it exists. Sometimes we call the step from (\overline{f}, f) to (\overline{f′}, f′) a surgery step.

We conclude from Lemma 3.55, from the discussion of the effect of surgery after Lemma 3.55 and from Theorem 3.59

Theorem 3.61 Let X be a connected finite n-dimensional Poincaré complex. Let \overline{f}: T M ⊕\underline{ℝ^{a}} → ξ be a normal map of degree one covering f: M → X. Then we can carry out a finite sequence of surgery steps to obtain a normal map of degree one \overline{g}: T N ⊕ \underline{ℝ^{a+b}} → ξ ⊕ \underline{ℝ^{b}} covering g: N → X such that (\overline{f}, f) and (\overline{g}, g) are normally bordant and g is k-connected, where n = 2k or n = 2k + 1.

Recall that we want to address Problem 3.54. The strategy we have developed so far is to do surgery to change our normal map into a homotopy equivalence. Theorem 3.61 gives us some hope to carry out this program successfully, at least we can get a highly connected map. So we can give the final version of the surgery problem.

Problem 3.62 (Surgery problem) Suppose we have some normal map (\overline{f}, f) from a closed manifold M to a finite Poincaré complex X. Can we change M and f leaving X fixed by finitely many surgery steps to get a normal map (\overline{g}, g) from a closed manifold N to X such that g is a homotopy equivalence?

Remark 3.63 Suppose that X appearing in Problem 3.62 is orientable and of dimension n = 4k. Notice that the surgery step does not change the normal bordism class. In particular the manifolds M and N are oriented bordant. Hence Lemma 3.18 implies that M and N have the same signature. This implies that sign(M) − sign(X) is an invariant of the normal bordism class and is not changed by a surgery step, where for non-connected M we mean by sign(M) the sum of the signatures of the components of M. Since the signature is also an oriented homotopy invariant, we see an obstruction to solve the Surgery Problem 3.62, namely sign(M) − sign(X) must be zero. We will see that this is the only obtruction if X is a simply connected orientable 4k-dimensional Poincaré complex for k ≥ 2. If X is not simply connected, the vanishing of sign(M) − sign(X) will not be sufficient, more complicated surgery obstructions will occur.

3.5 Miscellaneous

We give more information about the space G/O in Section 6.6.

The Algebraic Surgery Obstruction

Introduction

In this chapter we want to give the solution to the surgery Problem 3.62, whether we can change a normal map (\overline{f}, f) from a closed manifold M to a finite Poincaré complex X by finitely many surgery steps to get a normal map (\overline{f′}, f′) from a closed manifold N to X such that f′ is a homotopy equivalence. We have already seen in Theorem 3.61 that we can make f k-connected if n = 2k or n = 2k + 1. So it remains to achieve that f is (k + 1)-connected because then f is a homotopy equivalence by Poincaré duality. Of course we want to do further surgery on elements in π_{k+1}(f) to make f (k + 1)-connected. It will turn out that this is not possible in general. We will encounter an obstruction, the so called surgery obstruction. It takes values in the so called L-groups which are defined in terms of forms and formations.

Let us start with the case n = 2k for k ≥ 3. Then the problem will be that we cannot do surgery on each element ω in π_{k+1}(f). The main obstacle is that an immersion f: S^{k} → M associated to ω may not be regularly homotopic to an embedding. This assumption appears in Theorem 3.59 (3). If we put f in general position, we may encounter double points. We have to figure out whether we can get rid of these double points. The main tool will be the Whitney trick which allows to get rid of two of the double points under certain algebraic conditions. In Section 4.1 we introduce intersection numbers and selfintersection numbers for immersions S^{k} → M and show that the selfintersection number of f is trivial if and only f is regularly homotopic to an embedding provided that k ≥ 3 (see Theorem 4.8).

We will explain in Section 4.2 that the intersection pairing λ and the selfintersection numbers μ(f) for pointed immersions f: S^{k} → M are linked to one another. They together define the structure of a non-degenerate (−1)^{k}- quadratic form on the surgery kernel K_{k}(M̃) which is ℤπ_{1}(M)-isomorphic to π_{k+1}(f). We will show that we are able to kill the surgery kernel K_{k}(M̃) by finitely many surgery steps if and only if this non-degenerate (−1)^{k}-quadratic form is isomorphic to a standard (−1)^{k}-hyperbolic form H_{(−1)^{k}}(R^{b}) for some b after adding a standard (−1)^{k}-hyperbolic form H_{(−1)^{k}}(R^{a}) for some a (see Theorem 4.27). This leads in a natural way to the definition of the even-dimensional L-groups in Section 4.3 and of the surgery obstruction in even dimensions in Section 4.4.

In the odd-dimensional case n = 2k + 1 the embedding question has always a positive answer because of 2k < n. Hence one can always do surgery on an element in π_{k+1}(f). But it is not clear that one can find appropriate elements so that after doing surgery on them the surgery kernel is trivial. The problem is that surgery on such an element simultaneously affects K_{k}(M̃) and K_{k+1}(M̃) since these are related by Poincaré duality. We will explain the definition of the relevant odd-dimensional L-groups in Section 4.5 and we will only sketch the definition and the proof of the main properties of the surgery obstruction in Section 4.6.

Our main concern is not the surgery Problem 3.62 but the question whether two closed manifolds are diffeomorphic. We have explained the surgery program in Remark 1.5. We will explain in Section 4.7 why this forces us to consider also normal maps whose underlying map (f, ∂f): (M, ∂M) → (X, ∂X) is a map from a compact manifold with boundary to a Poincaré pair such that ∂f is already a homotopy equivalence. In this situation the aim of surgery is to change f into a homotopy equivalence without changing ∂f. The relevant modification of the surgery obstruction will be introduced. Because of the appearance of the Whitehead torsion in the s-Cobordism Theorem 1.1 we are also forced to take Whitehead torsion into account. So we want to achieve that f is a simple homotopy equivalence provided that (X, ∂X) is a simple pair and ∂f is a simple homotopy equivalence. This will lead to the definition of simple L-groups and the simple surgery obstruction.

4.1 Intersection and Selfintersection Pairings

4.1.1 Intersections of Immersions

We are facing the problem to decide whether we can change an immersion f: S^{k} → M within its regular homotopy class to an embedding, where M is a closed manifold of dimension n = 2k. This problem occurs when we want to carry out a surgery step in the middle dimension (see Theorem 3.59). We first deal with the necessary algebraic obstructions and then address the question whether their vanishing is also sufficient.

We fix base points s ∈ S^{k} and b ∈ M and assume that M is connected and k ≥ 2. We will consider pointed immersions (f, w), i.e. an immersion f: S^{k} → M together with a path w from b to f(s). A pointed regular homotopy from (f_{0}, w_{0}) to (f_{1}, w_{1}) is a regular homotopy h: S^{k} × [0, 1] → M from h_{0} = f_{0} to h_{1} = f_{1} such that w_{0} ∗h(s,?) and w_{1} are homotopic paths relative end points.

Here h(s,?) is the path from f_{0}(s) to f_{1}(s) given by restricting h to {s} × [0, 1]. Denote by I_{k}(M) the set of pointed homotopy classes of pointed immersions from S^{k} to M. We need the paths to define the structure of an abelian group on I_{k}(M). The sum of [(f_{0}, w_{0})] and [(f_{1}, w_{1})] is given by the connected sum along the path w_{0}^{−} ∗ w_{1} from f_{0}(s) to f_{1}(s). The zero element is given by the composition of the standard embedding S^{k} → ℝ^{k+1} ⊂ ℝ^{k+1} × ℝ^{k−1} = ℝ^{n} with some embedding ℝ^{n} ⊂ M and any path w from b to the image of s. The inverse of the class of (f, w) is the class of (f ∘ a, w) for any base point preserving diffeomorphism a: S^{k} → S^{k} of degree −1.

The fundamental group π = π_{1}(M, b) operates on I_{k}(M) by composing the path w with a loop at b. Thus I_{k}(M) inherits the structure of a ℤπ-module.

Next we want to define the intersection pairing

λ: I_{k}(M) × I_{k}(M) → ℤπ.  (4.1)

For this purpose we will have to fix an orientation of T_{b}M at b. Consider α_{0} = [(f_{0}, w_{0})] and α_{1} = [(f_{1}, w_{1})] in I_{k}(M). Choose representatives (f_{0}, w_{0}) and (f_{1}, w_{1}). We can arrange without changing the pointed regular homotopy class that D = im(f_{0}) ∩ im(f_{1}) is finite, for any y ∈ D both the preimage f_{0}^{−1}(y) and the preimage f_{1}^{−1}(y) consists of precisely one point and for any two points x_{0} and x_{1} in S^{k} with f_{0}(x_{0}) = f_{1}(x_{1}) we have T_{x_{0}}f_{0}(T_{x_{0}}S^{k}) + T_{x_{1}}f_{1}(T_{x_{1}}S^{k}) = T_{f_{0}(x_{0})}M. Consider d ∈ D. Let x_{0} and x_{1} in S^{k} be the points uniquely determined by f_{0}(x_{0}) = f_{1}(x_{1}) = d. Let u_{i} be a path in S^{k} from s to x_{i}. Then we obtain an element g(d) ∈ π by the loop at b given by the composition w_{1} ∗f_{1}(u_{1})∗f_{0}(u_{0})^{−} ∗w_{0}^{−}. Recall that we have fixed an orientation of T_{b}M. The fiber transport along the path w_{0} ∗ f(u_{0}) yields an isomorphism T_{b}M ≅ T_{d}M which is unique up to isotopy. Hence T_{d}M inherits an orientation from the given orientation of T_{b}M. The standard orientation of S^{k} determines an orientation on T_{x_{0}}S^{k} and T_{x_{1}}S^{k}. We have the isomorphism of oriented vector spaces

T_{x_{0}}f_{0} ⊕ T_{x_{1}}f_{1} : T_{x_{0}}S^{k} ⊕ T_{x_{1}}S^{k} ⟶^{≅} T_{d}M.

Define ϵ(d) = 1 if this isomorphism respects the orientations and ϵ(d) = −1 otherwise. The elements g(d) ∈ π and ϵ(d) ∈ {±1} are independent of the choices of u_{0} and u_{1} since S^{k} is simply connected for k ≥ 2. Define

λ(α_{0}, α_{1}) := ∑_{d∈D} ϵ(d) · g(d).

Lift b ∈ M to a base point b̃ ∈ M̃. Let f̃_{i}: S^{k} → M̃ be the unique lift of f_{i} determined by w_{i} and b̃ for i = 0, 1. Let λ_{ℤ}(f̃_{0}, f̃_{1}) be the ℤ-valued intersection number of f̃_{0} and f̃_{1}. This is the same as the algebraic intersection number of the classes in the k-th homology with compact support defined by f̃_{0} and f̃_{1} which obviously depends only on the homotopy classes of f̃_{0} and f̃_{1}. Then

λ(α_{0}, α_{1}) = ∑_{g∈π} λ_{ℤ}(f̃_{0}, l_{g^{−1}} ∘ f̃_{1}) · g,  (4.2)

where l_{g^{−1}} denotes left multiplication with g^{−1}. This shows that λ(α_{0}, α_{1}) depends only on the pointed regular homotopy classes of (f_{0}, w_{0}) and (f_{1}, w_{1}).

In the sequel we use the w = w_{1}(M)-twisted involution on ℤπ which sends ∑_{g∈π} a_{g} · g to ∑_{g∈π} w(g) · a_{g} · g^{−1}. One easily checks

Lemma 4.3 For α, β, β_{1}, β_{2} ∈ I_{k}(M) and u_{1}, u_{2} ∈ ℤπ we have

λ(α, β) = (−1)^{k} · \overline{λ(β, α)};
λ(α, u_{1} · β_{1} + u_{2} · β_{2}) = u_{1} · λ(α, β_{1}) + u_{2} · λ(α, β_{2}).

Remark 4.4 Suppose that the normal bundle of the immersion f: S^{k} → M has a nowhere vanishing section. (In our situation it actually will be trivial.) Suppose that f is regular homotopic to an embedding g. Then the normal bundle of g has a nowhere vanishing section σ. Let g′ be the embedding obtained by moving g a little bit in the direction of this normal vector field σ. Choose a path w_{f} from f(s) to b. Then for appropriate paths w_{g} and w_{g′} we get pointed embeddings (g, w_{g}) and (g′, w_{g′}) such that the pointed regular homotopy classes of (f, w), (g, w_{g}) and (g′, w_{g′}) agree. Since g and g′ have disjoint images, we conclude

λ([f, w], [f, w]) = 0.

Hence the vanishing of λ([f, w], [f, w]) is a necessary condition for finding an embedding in the regular homotopy class of f, provided that the normal bundle of f has a nowhere vanishing section. It is not a sufficient condition. To get a sufficient condition we have to consider self-intersections what we will do next.

4.1.2 Selfintersections of Immersions

Let α ∈ I_{k}(M) be an element. Let (f, w) be a pointed immersion representing α. Recall that we have fixed base points s ∈ S^{k}, b ∈ M and an orientation of T_{b}M. Since we can find arbitrarily close to f an immersion which is in general position, we can assume without loss of generality that f itself is in general position. This means that there is a finite subset D of im(f) such that f^{−1}(y) consists of precisely two points for y ∈ D and of precisely one point for y ∈ im(f) − D and for two points x_{0} and x_{1} in S^{k} with x_{0} ≠ x_{1} and f(x_{0}) = f(x_{1}) we have T_{x_{0}}f(T_{x_{0}}S^{k}) + T_{x_{1}}f(T_{x_{1}}S^{k}) = T_{f_{0}(x_{0})}M. Now fix for any d ∈ D an ordering x_{0}(d), x_{1}(d) of f^{−1}(d). Analogously to the construction above one defines ϵ(x_{0}(d), x_{1}(d)) ∈ {±1} and g(x_{0}(d), x_{1}(d)) ∈ π = π_{1}(M, b). Consider the element ∑_{d∈D} ϵ(x_{0}(d), x_{1}(d)) · g(x_{0}(d), x_{1}(d)) of ℤπ. It does not only depend on f but also on the choice of the ordering of f^{−1}(d) for d ∈ D. One easily checks that the change of ordering of f^{−1}(d) has the following effect for w = w_{1}(M): π → {±1}

g(x_{1}(d), x_{0}(d)) = g(x_{0}(d), x_{1}(d))^{−1};
w(g(x_{1}(d), x_{0}(d))) = w(g(x_{0}(d), x_{1}(d)));
ϵ(x_{1}(d), x_{0}(d)) = (−1)^{k} · w(g(x_{0}(d), x_{1}(d))) · ϵ(x_{0}(d), x_{1}(d));
ϵ(x_{1}(d), x_{0}(d)) · g(x_{1}(d), x_{0}(d)) = (−1)^{k} · ϵ(x_{0}(d), x_{1}(d)) · \overline{g(x_{0}(d), x_{1}(d))}.

Define an abelian group, where we use the w-twisted involution on ℤπ

Q_{(−1)^{k}}(ℤπ, w) := ℤπ/\{u − (−1)^{k} · \overline{u} | u ∈ ℤπ\}.  (4.5)

Define the selfintersection element for α ∈ I_{k}(M)

μ(α) := [∑_{d∈D} ϵ(x_{0}(d), x_{1}(d)) · g(x_{0}(d), x_{1}(d))] ∈ Q_{(−1)^{k}}(ℤπ, w). (4.6)

The passage from ℤπ to Q_{(−1)^{k}}(ℤπ, w) ensures that the definition is independent of the choice of the order on f^{−1}(d) for d ∈ D. It remains to show that it depends only on the pointed regular homotopy class of (f, w). Let h be a pointed regular homotopy from (f, w) to (g, v). We can arrange that h is in general position. In particular each immersion h_{t} is in general position and comes with a set D_{t}. The collection of the D_{t}-s yields a bordism W from the finite set D_{0} to the finite set D_{1}. Since W is a compact one-dimensional manifold, it consists of circles and arcs joining points in D_{0} ∪ D_{1} to points in D_{0} ∪ D_{1}. Suppose that the point e and the point e′ in D_{0} ∪ D_{1} are joint by an arc. Then one easily checks that their contributions to

μ(f, w) − μ(g, w) := [ ∑_{d_{0}∈D_{0}} ϵ(x_{0}(d_{0}), x_{1}(d_{0})) · g(x_{0}(d_{0}), x_{1}(d_{0}))

− ∑_{d_{1}∈D_{1}} ϵ(x_{0}(d_{1}), x_{1}(d_{1})) · g(x_{0}(d_{1}), x_{1}(d_{1}))]

cancel out. This implies μ(f, w) = μ(g, w).

Lemma 4.7 Let μ: I_{k}(M) → Q_{(−1)^{k}}(ℤπ, w) be the map given by the selfintersection element (see (4.6)) and let λ: I_{k}(M) × I_{k}(M) → ℤπ be the intersection pairing (see (4.1)). Then

1. Let (1 + (−1)^{k} · T): Q_{(−1)^{k}}(ℤπ, w) → ℤπ be the homomorphism of abelian groups which sends [u] to u+(−1)^{k} ·\overline{u}. Denote for α ∈ I_{k}(M) by χ(α) ∈ ℤ the Euler number of the normal bundle ν(f) for any representative (f, w) of α with respect to the orientation of ν(f) given by the standard orientation on S^{k} and the orientation on f^{∗}T M given by the fixed orientation on T_{b}M and w. Then

λ(α, α) = (1 + (−1)^{k} · T)(μ(α)) + χ(α) · 1;

2. We get for pr: ℤπ → Q_{(−1)^{k}}(ℤπ, w) the canonical projection and α, β ∈

I_{k}(M)

μ(α + β) − μ(α) − μ(β) = pr(λ(α, β));

3. For α ∈ I_{k}(M) and u ∈ ℤπ we get with respect to the obvious ℤπ-bimodule structure on Q_{(−1)^{k}}(ℤπ, w)

μ(u · α) = uμ(α)\overline{u}.

\underline{\underline{Proof}} : (1) Represent α ∈ I_{k}(M) by a pointed immersion (f, w) which is in general position. Choose a section σ of ν(f) which meets the zero section transversally. Notice that then the Euler number satisfies

χ(f) = ∑_{y∈N(σ)} ϵ(y),

where N(σ) is the (finite) set of zero points of σ and ϵ(y) is a sign which depends on the local orientations. We can arrange that no zero of σ is the preimage of an element in the set of double points D_{f} of f. Now move f a little bit in the direction of this normal field σ. We obtain a new immersion g: S^{k} → M with a path v from b to g(s) such that (f, w) and (g, v) are pointed regularly homotopic.

We want to compute λ(α, α) using the representatives (f, w) and (g, v). Notice that any point in d ∈ D_{f} corresponds to two distinct points x_{0}(d) and x_{1}(d) in the set D = im(f) ∩ im(g) and any element n ∈ N(σ) corresponds to one point x(n) in D. Moreover any point in D occurs as x_{i}(d) or x(n) in a unique way. Now the contribution of d to λ([(f, w)], [(g, v)]) is ϵ(d) · g(d) + (−1)^{k} · ϵ(d) · \overline{g(d)} and the contribution of n ∈ N(σ) is ϵ(n) · 1. Now assertion (1) follows. The elementary proof of assertions (2) and (3) is left to the reader. This finishes the proof of Lemma 4.7.

Theorem 4.8 Let M be a compact connected manifold of dimension n = 2k. Fix base points s ∈ S^{k} and b ∈ M and an orientation of T_{b}M. Let (f, w) be a pointed immersion of S^{k} in M. Suppose that k ≥ 3. Then (f, w) is pointed homotopic to a pointed immersion (g, v) for which g: S^{k} → M is an embedding, if and only μ(f, w) = 0.

\underline{\underline{Proof}} : If f is represented by an embedding, then μ(f, w) = 0 by definition. Suppose that μ(f, w) = 0. We can assume without loss of generality that f is in general position. Since μ(f, w) = 0, we can find d and e in the set of double points D_{f} of f and a numeration x_{0}(d), x_{1}(d) of f^{−1}(d) and x_{0}(e), x_{1}(e) of f^{−1}(e) such that

ϵ(x_{0}(d), x_{1}(d)) = −ϵ(x_{0}(e), x_{1}(e));
g(x_{0}(d), x_{1}(d)) = g(x_{0}(e), x_{1}(e)).

Therefore we can find arcs u_{0} and u_{1} in S^{k} such that u_{0}(0) = x_{0}(d), u_{0}(1) = x_{0}(e), u_{1}(0) = x_{1}(d) and u_{1}(1) = x_{1}(e), the path u_{0} and u_{1} are disjoint from one another, f(u_{0}((0, 1))) and f(u_{1}((0, 1))) do not meet D_{f} and f(u_{0}) and f(u_{1}) are homotopic relative endpoints. We can change u_{0} and u_{1} without destroying the properties above and find a smooth map U: D^{2} → M whose restriction to S^{1} is an embedding (ignoring corners on the boundary) and is given on the upper hemisphere S_{+}^{1} by u_{0} and on the lower hemisphere S_{−}^{1} by u_{1} and which meets im(f) transversally. There is a compact neighborhood K of S^{1} such that U|_{K} is an embedding. Since k ≥ 3 we can find arbitrarily close to U an embedding which agrees with U on K. Hence we can assume without loss of generality that U itself is an embedding. The Whitney trick (see [83, Theorem 6.6 on page 71], [123]) allows to change f within its pointed regular homotopy class to a new pointed immersion (g, v) such that D_{g} = D_{f} − {d, e} and μ(g, v) = 0. By iterating this process we achieve D_{f} = ∅.

Remark 4.9 The condition dim(M) ≥ 5 which arises in high-dimensional manifold theory ensures in the proof of Theorem 4.8 that k ≥ 3 and hence we can arrange U to be an embedding. If k = 2, one can achieve that U is an immersion but not necessarily an embedding. This is the technical reason why surgery in dimension 4 is much more complicated than in dimensions ≥ 5.

4.2 Kernels and Forms

4.2.1 Symmetric Forms and Surgery Kernels

For the rest of this section we fix a normal map of degree one (\overline{f}, f): T M ⊕\underline{ℝ}^{a} → ξ covering f: M → X, where M is a closed connected manifold of dimension n and X is a connected finite Poincaré complex of dimension n. Suppose that f induces an isomorphism on the fundamental groups. Fix a base point b ∈ M together with lifts b̃ ∈ M̃ of b and f̃(b) ∈ X̃ of f(b). We identify π = π_{1}(M, b) = π(X, f(b)) by π_{1}(f, b). The choices of b̃ and f̃(b) determine π-operations on M̃ and on X̃ and a lift f̃: M̃ → X̃ which is π-equivariant.

Definition 4.10 Let K_{k}(M̃) be the kernel of the ℤπ-map H_{k}(f̃): H_{k}(M̃) → H_{k}(X̃). Denote by K^{k}(M̃) the cokernel of the ℤπ-map H^{k}(f̃): H^{k}(X̃) → H^{k}(M̃) which is the ℤπ-map induced by C^{∗}(f̃): C^{∗}(X̃) → C^{∗}(M̃).

Lemma 4.11 1. The cap product with [M] induces isomorphisms

? ∩ [M]: K^{n−k}(M̃) ⟶^{≅} K_{k}(M̃);

2. Suppose that f is k-connected. Then there is the composition of natural ℤπ-isomorphisms

h_{k}: π_{k+1}(f) ⟶^{≅} π_{k+1}(f̃) ⟶^{≅} H_{k+1}(f̃) ⟶^{≅} K_{k}(M̃);

3. Suppose that f is k-connected and n = 2k. Then there is a natural ℤπ-

homomorphism

t_{k}: π_{k}(f) → I_{k}(M).

\underline{\underline{Proof}} : (1) The following diagram commutes and has isomorphisms as vertical arrows

\begin{matrix}H^{n−k}(M̃) & ⟵^{H^{n−k}(f̃)} & H^{n−k}(X̃) \\ ^{?∩[M]}↓≅ & & ≅↓^{?∩[X]} \\ H_{k}(M̃) & ⟶_{H_{k}(f̃)} & H_{k}(X̃) \end{matrix}  (4.12)

Hence the composition K_{k}(M̃) → H_{k}(M̃) ⟶^{(?∩[M])^{−1}} H^{n−k}(M̃) → K^{n−k}(M̃) is bijective.

(2) The commutative square (4.12) above implies that H_{l}(f̃): H_{l}(M̃) → H_{l}(X̃) is split surjective for all l. We conclude from the long exact sequence of C_{∗}(f̃) that the boundary map

∂: H_{k+1}(f̃) := H_{k+1}(cone(C_{∗}(f̃))) → H_{k}(M̃)

induces an isomorphism

∂_{k+1} : H_{k+1}(f̃) ⟶^{≅} K_{k}(M̃).

Since f and hence f̃ is k-connected, the Hurewicz homomorphism

π_{k+1}(f̃) ⟶^{≅} H_{k+1}(f̃)

is bijective [121, Corollary IV.7.10 on on page 181]. The canonical map

π_{k+1}(f̃) → π_{k+1}(f)

is bijective. The composition of the maps above or their inverses yields a natural isomorphism h_{k}: π_{k+1}(f) → K_{k}(M̃).

(3) is analogous to the proof of Theorem 3.59 (2) which was based on Theorem 3.57 provided one takes the base loops into account. Notice that an element in π_{k+1}(f, b) is given by a commutative diagram

S^{k} ⟶^{q} M
↓  ↓
D^{k+1} ⟶_{Q} X

together with a path w from b to f(s) for a fixed base point s ∈ S^{k}.

Suppose that n = 2k. The Kronecker product ⟨ , ⟩: H^{k}(M̃) × H_{k}(M̃) → ℤπ is induced by the evaluation pairing hom_{ℤπ}(C_{p}(M̃), ℤπ) × C_{p}(M̃) → ℤπ which sends (ϕ, x) to ϕ(x). It induces a pairing

⟨ , ⟩: K^{k}(M̃) × K_{k}(M̃) → ℤπ.

Together with the isomorphism

? ∩ [M]: K^{n−k}(M̃) ⟶^{≅} K_{k}(M̃);

of Theorem 4.11 (1) it yields the intersection pairing

s: K_{k}(M̃) × K_{k}(M̃) → ℤπ.  (4.13)

We get from Lemma 4.11 (2) and (3) a ℤπ-homomorphism

α: K_{k}(M̃) → I_{k}(M̃).  (4.14)

We leave it to the reader to check Lemma 4.15 The following diagram commutes, where the upper pairing is defined in (4.13), the lower pairing in (4.1) and the left vertical arrows in (4.14)

K_{k}(M̃) × K_{k}(M̃) ⟶^{s} ℤπ
α×α↓  ↓^{id}
I_{k}(M) × I_{k}(M) ⟶_{λ} ℤπ

The pairing s of (4.13) is the prototype of the following algebraic object which can be defined for any ring R with involution and ϵ ∈ {±1}.

Definition 4.16 An ϵ-symmetric form (P, ϕ) over an associative ring R with unit and involution is a finitely generated projective R-module P together with an

R-map ϕ: P → P^{∗} such that the composition P ⟶^{e(P)} (P^{∗})^{∗} ⟶^{ϕ^{∗}} P^{∗} agrees with ϵ·ϕ. Here and elsewhere e(P) is the canonical isomorphism sending p ∈ P to the element in (P^{∗})^{∗} given by P^{∗} → R, f ↦ \overline{f(p)}. We call (P, ϕ) non-degenerate if ϕ is an isomorphism.

There are obvious notions of isomorphisms and direct sums of ϵ-symmetric forms.

We can rewrite (P, ϕ) as a pairing

λ: P × P → R, (p, q) ↦ ϕ(p)(q).

Then the condition that ϕ is R-linear becomes the condition

λ(p, r_{1} · q_{1} + r_{2} · q_{2},) = r_{1} · λ(p, q_{1}) + r_{2} · λ(p, q_{2});
λ(r_{1} · p_{1} + r_{2} · p_{2}, q) = λ(p_{1}, q) · \overline{r_{1}} + λ(p_{2}, q) · \overline{r_{2}}.

The condition ϕ = ϵ · ϕ^{∗} ∘ e(P) translates to λ(q, p) = ϵ · \overline{λ(p, q)}.

Example 4.17 Let P be a finitely generated projective R-module. The standard hyperbolic ϵ-symmetric form H^{ϵ}(P) is given by the ℤπ-module P ⊕P^{∗} and the R-isomorphism

ϕ: (P ⊕ P^{∗}) \underrightarrow{( \begin{matrix}0 & 1 \\ ϵ & 0 \end{matrix} )} P^{∗} ⊕ P ⟶^{id ⊕e(P)} P^{∗} ⊕ (P^{∗})^{∗} = (P ⊕ P^{∗})^{∗}.

If we write it as a pairing we obtain

(P ⊕ P^{∗}) × (P ⊕ P^{∗}) → R, ((p, μ), (p′, μ′)) ↦ μ(p′) + ϵ · μ′(p).

An example of a non-degenerate (−1)^{k}-symmetric form over ℤπ with the w-twisted involution is K_{k}(M̃) with the pairing s of (4.13), provided that f is k-connected and n = 2k. We have to show that K_{k}(M̃) is finitely generated projective.

Lemma 4.18 Let D_{∗} be a finite projective R-chain complex. Suppose for a fixed integer r that H_{i}(D_{∗}) = 0 for i < r. Suppose that H^{r+1}(hom_{R}(D_{∗}, V)) = 0 for any R-module V. Then im(d_{r+1}) is a direct summand in D_{r} and there are canonical exact sequences

0 → ker(d_{r}) → D_{r} → D_{r−1} → … → D_{0} → 0

and

0 → im(d_{r+1}) → ker(d_{r}) → H_{r}(D_{∗}) → 0.

In particular H_{r}(D_{∗}) is finitely generated projective. If H_{i}(D_{∗}) = 0 for i > r, we obtain an exact sequence

… → D_{r+3} ⟶^{d_{r+3}} D_{r+2} ⟶^{d_{r+2}} D_{r+1} ⟶^{d_{r+1}} im(d_{r+1}) → 0.

\underline{\underline{Proof}} : If we apply the assumption H^{r+1}(hom_{R}(D_{∗}, V)) = 0 in the case V = im(d_{r+1}), we obtain an exact sequence

hom_{ℤπ}(D_{r}, im(d_{r+1})) ⟶^{hom_{ℤπ}(d_{r+1},id)} hom_{ℤπ}(D_{r+1}, im(d_{r+1}))

⟶^{hom_{ℤπ}(d_{r+2},id)} hom_{ℤπ}(D_{r+2}, im(d_{r+1})).

Since d_{r+1} ∈ hom_{ℤπ}(D_{r+1}, im(d_{r+1})) is mapped to zero under hom_{ℤπ}(d_{r+2}, id), we can find an R-homomorphism ρ: D_{r} → im(d_{r+1}) with ρ∘d_{r+1} = d_{r+1}. Hence im(d_{r+1}) is a direct summand in D_{r}. The other claims are obvious.

A R-module V is called stably finitely generated free if for some non-negative integer l the R-module V ⊕ R^{l} is a finitely generated free R-module.

Lemma 4.19 If f: X → Y is k-connected for n = 2k or n = 2k + 1, then K_{k}(M̃) is stably finitely generated free.

\underline{\underline{Proof}} : We only give the proof for n = 2k. The proof for n = 2k + 1 is along the same lines using Poincaré duality for the kernels (see Lemma 4.11 (1)).

Consider the finitely generated free ℤπ-chain complex D_{∗} := cone(C_{∗}(f̃)). Its homology H_{p}(D_{∗}) is by definition H_{p}(f̃). Since f is k-connected and D_{∗} is projective, there is an R-subchain complex E_{∗} ⊂ D_{∗} such that E_{∗} is finite projective, E_{i} = 0 for i ≤ k and the inclusion E_{∗} → D_{∗} is a homology equivalence and hence an R-chain homotopy equivalence. Namely, take E_{i} = D_{i} for i ≥ k + 2, E_{k+1} = ker(d_{k+1}) and E_{i} = 0 for i ≤ k. We get from the commutative square (4.12) a ℤπ-chain homotopy equivalence D^{n+1−∗} → D_{∗}. This implies for any R-module V since hom_{R}(D_{∗}, V) is chain homotopy equivalent to hom_{R}(E_{∗}, V).

H^{n+1−i}(hom_{R}(D_{∗}, V)) = 0  for i ≤ k;
H_{i}(D) = 0  for i ≥ n + 1 − k;

We conclude from Lemma 4.11.

H_{p+1}(D_{∗}) ≅ \{ \begin{matrix}K_{k}(M̃) & , if p = k; \\ 0 & , if p ≠ k; \end{matrix}
H^{k+2}(hom_{R}(D_{∗}, R)) = 0.

Now apply Lemma 4.18 to D_{∗}.

Example 4.20 Consider the normal map (\overline{f}, f): T M ⊕ \underline{ℝ}^{a} → ξ covering the k-connected map of degree one f: M → N of closed n-dimensional manifolds for n = 2k. If we do surgery on the zero element in π_{k+1}(f), then the effect on M is that M is replaced by the connected sum M′ = M♯(S^{k}×S^{k}). The effect on K_{k}(M̃) is that it is replaced by K_{k}(M̃′) = K_{k}(M̃)⊕(ℤπ⊕ℤπ). The intersection pairing on this new kernel is the orthogonal sum of the given intersection pairing on K_{k}(M̃) together with the standard hyperbolic symmetric form H^{(−1)^{k}}(ℤπ). In particular we can arrange by finitely many surgery steps on the trivial element in π_{k+1}(f) that K_{k}(M̃) is a finitely generated free ℤπ-module.

4.2.2 Quadratic Forms and Surgery Kernels

We have already seen that it will not be enough to study intersections of different immersions, we must also deal with selfintersections of one immersion. We have seen in Lemma 4.7 that we can enrich the intersection pairing by the self-intersection pairing. This leads to the following algebraic analog for an associative ring R with unit and involution and ϵ ∈ {±1}. For a finitely generated projective R-module P define an involution of R-modules

T: hom_{R}(P, P^{∗}) → hom(P, P^{∗}),  f ↦ f^{∗} ∘ e(P)  (4.21)

where e(P): P → (P^{∗})^{∗} is the canonical isomorphism.

Definition 4.22 Let P be a finitely generated projective R-module. Define

Q^{ϵ}(P) := ker ((1 − ϵ · T): hom_{R}(P, P^{∗}) → hom_{R}(P, P^{∗})) ;
Q_{ϵ}(P) := coker ((1 − ϵ · T): hom_{R}(P, P^{∗}) → hom_{R}(P, P^{∗})) .

Let

(1 + ϵ · T): Q_{ϵ}(P) → Q^{ϵ}(P)

be the homomorphism which sends the class represented by f: P → P^{∗} to the element f + ϵ · T (f).

An ϵ-quadratic form (P, ψ) is a finitely generated projective R-module P together with an element ψ ∈ Q_{ϵ}(P). It is called non-degenerate if the associated ϵ-symmetric form (P, (1+ϵ·T)(ψ)) is non-degenerate, i.e. (1+ϵ·T)(ψ): P → P^{∗} is bijective.

There is an obvious notion of direct sum of two ϵ-quadratic forms. An isomorphism f: (P, ψ) → (P′, ψ′) of two ϵ-quadratic forms is an R-isomorphism f: P ⟶^{≅} P′ such that the induced map

Q_{ϵ}(P′) → Q_{ϵ}(P), [ϕ: P′ → (P′)^{∗}] ↦ [f^{∗} ∘ ϕ ∘ f: P → P^{∗}]

sends ψ′ to ψ.

We can rewrite this as follows. An ϵ-quadratic form (P, ϕ) is the same as a triple (P, λ, μ) consisting of a pairing

λ: P × P → R

satisfying

λ(p, r_{1} · q_{1} + r_{2} · q_{2},) = r_{1} · λ(p, q_{1}) + r_{2} · λ(p, q_{2});
λ(r_{1} · p_{1} + r_{2} · p_{2}, q) = λ(p_{1}, q) · \overline{r_{1}} + λ(p_{2}, q) · \overline{r_{2}};
λ(q, p) = ϵ · \overline{λ(p, q)}.

and a map

μ: P → Q_{ϵ}(R) = R/\{r − ϵ · \overline{r} | r ∈ R\}

satisfying

μ(rp) = rμ(p)\overline{r};
μ(p + q) − μ(p) − μ(q) = pr(λ(p, q));
λ(p, p) = (1 + ϵ · T)(μ(p)),

where pr: R → Q_{ϵ}(R) is the projection and (1 + ϵ · T): Q_{ϵ}(R) → R the map sending the class of r to r + ϵ · \overline{r}. Namely, put

λ(p, q) = ((1 + ϵ · T)(ψ)) (p)) (q);
μ(p) = ψ(p)(p).

Example 4.23 Let P be a finitely generated projective R-module. The standard hyperbolic ϵ-quadratic form H_{ϵ}(P) is given by the ℤπ-module P ⊕ P^{∗} and the class in Q_{ϵ}(P ⊕ P^{∗}) of the R-homomorphism

ϕ: (P ⊕ P^{∗}) \underrightarrow{( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} )} P^{∗} ⊕ P ⟶^{id ⊕e(P)} P^{∗} ⊕ (P^{∗})^{∗} = (P ⊕ P^{∗})^{∗}.

The ϵ-symmetric form associated to H_{ϵ}(P) is H^{ϵ}(P).

Example 4.24 An example of a non-degenerate (−1)^{k}-quadratic form over ℤπ with the w-twisted involution is given as follows, provided that f is k-connected and n = 2k. Namely, take K_{k}(M̃) with the pairing s of (4.13) and the map

t: K_{k}(M̃) ⟶^{α} I_{k}(M) ⟶^{μ} Q_{(−1)^{k}}(ℤπ, w),  (4.25)

where μ: I_{k}(M) → Q_{(−1)^{k}}(ℤπ, w) is defined in (4.6) and α is defined in (4.14).

Remark 4.26 Suppose that 1/2 ∈ R. Then the homomorphism

(1 + ϵ · T): Q_{ϵ}(P) ⟶^{≅} Q^{ϵ}(P), [ψ] ↦ [ψ + ϵ · T (ψ)]

is bijective. The inverse sends [u] to [u/2]. Hence any ϵ-symmetric form carries a unique ϵ-quadratic structure. Hence there is no difference between the symmetric and the quadratic setting if 2 is invertible in R.

The next result is the key step in translating the geometric question, whether we can change a normal map by a finite sequence of surgery steps into a homotopy equivalence to an algebraic question about quadratic forms. It will lead in a natural way to the definition of the surgery obstruction groups L_{2k}(R).

Theorem 4.27 Consider the normal map (\overline{f}, f): T M ⊕ \underline{ℝ}^{a} → ξ covering the k-connected map of degree one f: M → N of closed connected n-dimensional manifolds for n = 2k. Suppose that k ≥ 3 and that for the non-degenerate (−1)^{k}-quadratic form (K_{k}(M̃), s, t) there are integers u, v ≥ 0 together with an isomorphism of non-degenerate (−1)^{k}-quadratic forms

(K_{k}(M̃), s, t) ⊕ H_{(−1)^{k}}(ℤπ^{u}) ≅ H_{(−1)^{k}}(ℤπ^{v}).

Then we can perform a finite number of surgery steps resulting in a normal map of degree one (\overline{g}, g): T M′ ⊕ \underline{ℝ}^{a+b} → ξ ⊕ \underline{ℝ^{b}} such that g: M′ → X is a homotopy equivalence.

\underline{\underline{Proof}} : If we do a surgery step on the trivial element in π_{k+1}(f), we have explained the effect on (K_{k}(M̃), t) in Example 4.20. The effect on the quadratic form (K_{k}(M̃), s, t) is analogous, one adds a copy of H_{(−1)^{k}}(ℤπ). Hence we can assume without loss of generality that the non-degenerate quadratic form (K_{k}(M̃), s, t) is isomorphic to H_{(−1)^{k}}(ℤπ^{v}). Thus we can choose a ℤπ-basis {b_{1}, b_{2}, …, b_{v}, c_{1}, c_{2}, …, c_{v}} for K_{k}(M̃) such that

\begin{aligned}s(b_{i}, c_{i}) & = & 1 & i ∈ \{1, 2, …, v\}; \\ s(b_{i}, c_{j}) & = & 0 & i, j ∈ \{1, 2, …, v\}, i ≠ j; \\ s(b_{i}, b_{j}) & = & 0 & i, j ∈ \{1, 2, …, v\}; \\ s(c_{i}, c_{j}) & = & 0 & i, j ∈ \{1, 2, …, v\}; \\ t(b_{i}) & = & 0 & i ∈ \{1, 2, …, v\}. \end{aligned}

Notice that f is a homotopy equivalence if and only if the number v is zero. Hence it suffices to explain how we can lower the number v to (v − 1) by a surgery step on an element in π_{k+1}(f). Of course our candidate is the element ω in π_{k+1}(f) which corresponds under the isomorphism h: π_{k+1}(f) → K_{k}(M̃) (see Lemma 4.11 (2)) to the element b_{v}. By construction the composition

π_{k+1}(f) ⟶^{t_{k}} I_{k}(M) ⟶^{μ} Q_{(−1)^{k}}(ℤπ, w)

of the maps defined in (4.6) and in Lemma 4.11 (3) sends ω to zero. Now Theorem 3.59 and Theorem 4.8 ensure that we can perform surgery on ω. Notice that the assumption k ≥ 3 and the quadratic structure on the kernel become relevant exactly at this point. Finally it remains to check whether the effect on K_{k}(M̃) is the desired one, namely, that we get rid of one of the hyperbolic summands H_{ϵ}(ℤπ), or equivalently, v is lowered to v − 1.

We have explained earlier that doing surgery yields not only a new manifold M′ but also a bordism from M to M′. Namely, take W = M × [0, 1] ∪_{S^{k}×D^{n−k}} D^{k+1} × D^{n−k}, where we attach D^{k+1} × D^{n−k} by an embedding S^{k} × D^{n−k} → M × {1}, and M′ := ∂W − M, where we identify M = M × {0}. The manifold W comes with a map F: W → X × [0, 1] whose restriction to M is the given map f: M = M × {0} → X = X × {0} and whose restriction to M′ is a map f′: M′ → X × {1}. The definition of the kernels makes also sense for pair of maps. We obtain an exact braid combining the various long exact sequences of pairs

\xymatrix{0 \ar[rr] \ar[dr] &  & K_{k+1}(W̃, M̃) \ar[rr] \ar[dr] &  & K_{k}(M̃) \ar[rr] \ar[dr] &  & K_{k}(W̃, M̃′) \ar[rr] \ar[dr] &  & 0 \\  & 0 \ar[ur] \ar[dr] &  & K_{k+1}(W̃, ˜∂W) \ar@{-->}[ur] \ar[dr] &  & K_{k}(W̃) \ar@{-->}[ur] \ar[dr] &  & 0 \ar[ur] \\  &  & 0 \ar[ur] \ar[rr] &  & K_{k}(M̃′) \ar@{-->}[ur] \ar[rr] &  & 0 \ar[ur]}

The (k+1)-handle D^{k+1}×D^{n−k} defines an element ϕ^{k+1} in K_{k+1}(W̃, M̃) and the associated dual k-handle (see (1.25)) defines an element ψ^{k} ∈ K_{k}(W̃, M̃′). These elements constitute a ℤπ-basis for K_{k+1}(W̃, M̃) ≅ ℤπ and K_{k}(W̃, M̃′) ≅ ℤπ. The ℤπ-homomorphism K_{k+1}(W̃, M̃) → K_{k}(M̃) maps ϕ to b_{v}. The ℤπhomomorphism K_{k}(M̃) → K_{k}(W̃, M̃′) sends x to s(b_{v}, x) · ψ^{k}. Hence we can find elements b′_{1}, b′_{2} , …, b′_{v} and c′_{1}, c′_{2} , …, c′_{v−1} in K_{k+1}(W̃, ˜∂W) uniquely determined by the property that b′_{i} is mapped to b_{i} and c′_{i} to c_{i} under the ℤπ-homomorphism K_{k+1}(W̃, ˜∂W) → K_{k}(M̃). Moreover, these elements form a ℤπ-basis for K_{k+1}(W̃, ˜∂W) and the element ϕ^{k+1} is mapped to b′_{v} under the ℤπ-homomorphism K_{k+1}(W̃, M̃) → K_{k+1}(W̃, ˜∂W). Define b′′_{i} and c′′_{i} for i = 1, 2, …, (v − 1) to be the image of b′_{i} and c′_{i} under under the ℤπ-homomorphism K_{k+1}(W̃, ˜∂W) → K_{k}(M̃′). Then {b′′_{i} | i = 1, 2 …, (v − 1)} ∐{c′′_{i} | i = 1, 2 …, (v − 1)} is a ℤπ-basis for K_{k}(M̃′). One easily checks for the quadratic structure (s′, t′) on K_{k}(M̃′)

\begin{aligned}s′(b′′_{i}, c′′_{i}) & = & s(b_{i}, c_{i}) & = & 1 & i ∈ \{1, 2, …, (v − 1)\}; \\ s′(b′′_{i}, c′′_{j}) & = & s(b_{i}, c_{j}) & = & 0 & i, j ∈ \{1, 2, …, (v − 1)\}, i ≠ j; \\ s′(b′′_{i}, b′′_{j}) & = & s(b_{i}, b_{j}) & = & 0 & i, j ∈ \{1, 2, …, (v − 1)\}; \\ s′(c′′_{i}, c′′_{j}) & = & s(c_{i}, c_{j}) & = & 0 & i, j ∈ \{1, 2, …, (v − 1)\}; \\ t′(b′′_{i}) & = & t(b_{i}) & = & 0 & i ∈ \{1, 2, …, (v − 1)\}. \end{aligned}

4.3. EVEN DIMENSIONAL L-GROUPS 93

This finishes the proof of Theorem 4.27.

4.3 Even Dimensional L-Groups

Next we define in even dimensions the abelian group, where our surgery obstruction will take values in.

Definition 4.28 Let R be an associative ring with unit and involution. For an even integer n = 2k define the abelian group L_{n}(R) called the n-th quadratic L-group of R to be the abelian group of equivalence classes [(F, ψ)] of nondegenerate (−1)^{k}-quadratic forms (F, ψ) whose underlying R-module F is a finitely generated free R-module with respect to the following equivalence relation. We call (F, ψ) and (F′, ψ′) equivalent if and only if there exists integers u, u′ ≥ 0 and an isomorphism of non-degenerate (−1)^{k}-quadratic forms

(F, ψ) ⊕ H_{ϵ}(R)^{u} ≅ (F′, ψ′) ⊕ H_{ϵ}(R)^{u′}.

Addition is given by the sum of two (−1)^{k}-quadratic forms. The zero element is represented by [H_{(−1)^{k}}(R)^{u}] for any integer u ≥ 0. The inverse of [F, ψ] is given by [F, −ψ].

A morphism u: R → S of rings with involution induces homomorphisms u_{∗}: L_{k}(R) → L_{k}(S) for k = 0, 2 by induction. One easily checks (u∘v)_{∗} = u_{∗}∘v_{∗} and (id_{R})_{∗} = id_{L_{k}(R)} for k = 0, 2.

Before we come to the surgery obstruction, we will present a criterion for an ϵ-quadratic form (P, ψ) to represent zero in L_{1−ϵ}(R) which we will later need at several places. Let (P, ψ) be a ϵ-quadratic form. A subLagrangian L ⊂ P is an R-submodule such that the inclusion i: L → P is split injective, the image of ψ under the map Q_{ϵ}(i): Q_{ϵ}(P) → Q_{ϵ}(L) is zero and L is contained in its annihilator L^{⊥} which is by definition the kernel of

P ⟶^{(1+ϵ·T)(ψ)} P^{∗} ⟶^{i^{∗}} L^{∗}.

A subLagrangian L ⊂ P is called Lagrangian if L = L^{⊥}. Equivalently, a Lagrangian L ⊂ P is an R-submodule L with inclusion i: L → P such that the sequence

0 → L ⟶^{i} P ⟶^{i^{∗}∘(1+ϵ·T)(ψ)} L^{∗} → 0.

is exact.

Lemma 4.29 Let (P, ψ) be an ϵ-quadratic form. Let L ⊂ P be a subLagrangian. Then L is a direct summand in L^{⊥} and ψ induces the structure of a nondegenerate ϵ-quadratic form (L^{⊥}/L, ψ^{⊥}/ψ). Moreover, the inclusion i: L → P extends to an isomorphism of ϵ-quadratic forms

H_{ϵ}(L) ⊕ (L^{⊥}/L, ψ^{⊥}/ψ) ⟶^{≅} (P, ψ).

In particular a non-degenerate ϵ-quadratic form (P, ψ) is isomorphic to H_{ϵ}(Q) if and only if it contains a Lagrangian L ⊂ P which is isomorphic as R-module to Q. The analogous statement holds for ϵ-symmetric forms.

\underline{\underline{Proof}} : Choose an R-homomorphism s: L^{∗} → P such that i^{∗}∘(1+ϵ·T)(ψ)∘s is the identity on L^{∗}. Our first attempt is the obvious split injection i⊕s: L⊕L^{∗} → P. The problem is that it is not necessarily compatible with the ϵ-quadratic structure. To be compatible with the ϵ-quadratic structure it is necessary to be compatible with the ϵ-symmetric structure, i.e. the following diagram must commute

L ⊕ L^{∗}  ⟶^{i⊕s} P
( \begin{matrix}0 & 1 \\ ϵ & 0 \end{matrix} )↓  ↓^{ψ+ϵ·T (ψ)}
L^{∗} ⊕ L ⟵_{(i^{∗}⊕s^{∗})∘Δ_{P^{∗}}} P^{∗}

where Δ_{P^{∗}}: P^{∗} → P^{∗} ⊕ P^{∗} is the diagonal map and we write s^{∗} for the com-

position P^{∗} ⟶^{s^{∗}} (L^{∗})^{∗} ⟶^{e(L)^{−1}} L. The diagram above commutes if and only if s^{∗} ∘ (ψ + ϵ · T (ψ)) ∘ s = 0. Notice that s is not unique, we can replace s by s′ = s + i ∘ v for any R-map v: L^{∗} → L. For this new section s′ the diagram above commutes if and only if s^{∗} ∘(ψ +ϵ·T (ψ))∘s+v^{∗} +ϵ·v = 0. This suggests to take v = −ϵs^{∗} ∘ ψ ∘ s. Now one easily checks that

g := i ⊕ (s − ϵ · i ∘ s^{∗} ∘ ψ ∘ s): L ⊕ L^{∗} → P

is split injective and compatible with the ϵ-quadratic structures and hence induces a morphism g: H_{ϵ}(L) → (P, ψ) of ϵ-quadratic forms.

Let im(g)^{⊥} be the annihilator of im(g). Denote by j: im(g) → P the inclusion. We obtain an isomorphism of ϵ-quadratic forms

g ⊕ j: H_{ϵ}(L) ⊕ (im(g)^{⊥}, j^{∗} ∘ ψ ∘ j) → (P, ψ).

The inclusion L^{⊥} → im(g)^{⊥} induces an isomorphism h: L^{⊥}/L → im(g)^{⊥}. Let ψ^{⊥}/ψ be the ϵ-quadratic structure on L^{⊥}/L for which h becomes an isomorphism of ϵ-quadratic forms. This finishes the proof in the quadratic case. The proof in the symmetric case is analogous.

Finally we state the computation of the even-dimensional L-groups of the integers. Consider an element (P, ϕ) in L_{0}(ℤ). By tensoring over ℤ with ℝ and only taking the symmetric structure into account we obtain a non-degenerate symmetric ℝ-bilinear pairing λ: ℝ ⊗_{ℤ} P × ℝ ⊗_{ℤ} P → ℝ. It turns out that its signature is always divisible by eight.

Theorem 4.30 The signature defines an isomorphism

\frac{1}{8} · sign: L_{0}(ℤ) ⟶^{≅} ℤ, [P, ψ] ↦ \frac{1}{8} · sign(ℝ ⊗_{ℤ} P, λ).

Consider a non-degenerate quadratic form (P, ψ) over the field 𝔽_{2} of two elements. Write (P, ψ) as a triple (P, λ, μ) as explained above. Choose any symplectic basis {b_{1}, b_{2}, …, b_{2m}} for P, where symplectic means that λ(b_{i}, b_{j}) is 1 if i − j = m and 0 otherwise. Define the Arf invariant of (P, ψ) by

Arf(P, ψ) := ∑_{i=1}^{m} μ(b_{i}) · μ(b_{i+m}) ∈ ℤ/2.  (4.31)

The Arf invariant defines an isomorphism

Arf : L_{2}(𝔽_{2}) ⟶^{≅} ℤ/2.

The change of rings homomorphism ℤ → 𝔽_{2} induces an isomorphism

L_{2}(ℤ) ⟶^{≅} L_{2}(𝔽_{2}).

Theorem 4.32 The Arf invariant defines an isomorphism

Arf : L_{2}(ℤ) ⟶^{≅} ℤ/2, [(P, ψ)] ↦ Arf(𝔽_{2} ⊗_{ℤ} (P, ψ)).

For more information about forms over the integers and the Arf invariant we refer for instance to [17], [88]. Implicitly the computation of L_{n}(ℤ) is already in [58]. Notice that Lemma 3.19, Remark 4.26 and Lemma 4.29 imply that the signature defines an isomorphism L_{n}(ℝ) ⟶^{≅} ℤ.

4.4 The Surgery Obstruction in Even Dimensions

Consider a normal map of degree one (\overline{f}, f): T M ⊕\underline{ℝ}^{a} → ξ covering f: M → X, where M is a closed oriented manifold of dimension n and X is a connected finite Poincaré complex of dimension n for even n = 2k. To these data we want to assign an element σ(\overline{f}, f) ∈ L_{n}(ℤπ, w) such that the following holds

Theorem 4.33 (Surgery Obstruction Theorem in Even Dimensions) We get under the conditions above:

1. Suppose k ≥ 3. Then σ(\overline{f}, f) = 0 in L_{n}(ℤπ, w) if and only if we can do a finite number of surgery steps to obtain a normal map (\overline{f′}, f): T M′ ⊕ \underline{ℝ}^{a+b} → ξ ⊕ \underline{ℝ^{b}} which covers a homotopy equivalence f′: M′ → X;

2. The surgery obstruction σ(\overline{f}, f) depends only on the normal bordism class of (\overline{f}, f).

We first explain the definition of the surgery obstruction σ(\overline{f}, f). By finitely many surgery steps we can achieve that f is k-connected (see Theorem 3.61). By a finite number of surgery steps in the middle dimension we can achieve that the surgery kernel K_{k}(M̃) is a finitely generated free ℤπ-module (see Example 4.20). We have already explained in Example 4.24 that K_{k}(M̃) carries the structure (K_{k}(M̃), t, s) of a non-degenerate (−1)^{k}-quadratic form. We want to define the surgery obstruction of (\overline{f}, f)

σ(\overline{f}, f) := [K_{k}(M̃), t, s] ∈ L_{n}(ℤπ, w).  (4.34)

We have to show that this is independent of the surgery steps we have performed to make f k-connected and K_{k}(M̃) finitely generated free. Notice that surgery does not change the normal bordism class. Hence it suffices to show that σ(\overline{f}, f) and σ(\overline{f′}, f′) define the same element in L_{n}(ℤπ, w) if f: M → X and f′: M′ → X are k-connected and normally bordant and their kernels K_{k}(M̃) and K_{k}(M̃′) are finitely generated free ℤπ-modules. Notice that this also will prove Theorem 4.33 (2).

Consider a normal bordism (\overline{F} , F): T W ⊕ \underline{ℝ}^{a} → η covering a map F: W → X × [0, 1] such that ∂W is M^{−} ∐ M′, F (M^{−}) ⊂ X × {0}, F (M′) ⊂ X × {1}, the restriction of \overline{F} to M^{−} is \overline{f} and the restriction of \overline{F} to M′ is \overline{f′}. By surgery on the interior of W we can change W and F leaving ∂W and F |_{∂W} fixed such that F is k-connected. The proof of Theorem 3.61 carries directly over. By a handle subtraction argument (see [119, Theorem 1.4 on page 14, page 50]) we can achieve that K_{k}(W̃, ˜∂W) = 0. This handle subtraction leaves M fixed but may change M′. But the change on the surgery kernel of M′ is adding a standard ϵ-hyperbolic form which does not change the class in the L-group. Moreover, f, f′ and F remain k-connected. So we can assume without loss of generality that K_{i}(W̃, ˜∂W) = 0 for i ≤ k and K_{i}(˜∂W) = 0 for i ≤ k − 1. We know already that K_{k}(˜∂W) is stably finitely generated free by Lemma 4.18. A similar argument shows that K_{k+1}(W̃, ˜∂W) and K_{k}(W̃) are stably finitely generated free.

We obtain an exact sequence of ℤπ-modules

0 → K_{k+1}(W̃, ˜∂W) ⟶^{∂_{k+1}} K_{k}(˜∂W) ⟶^{K_{k}(ĩ)} K_{k}(W̃) → 0.  (4.35)

where ˜∂W and ĩ: ˜∂W → W̃ come from the pullback construction applied to the universal covering W̃ → W and the inclusion i: ∂W → W.

The strategy of the proof is to show that the image of ∂_{k+1} : K_{k+1}(W̃, ˜∂W) → K_{k}(˜∂W) is a Lagrangian for the non-degenerate ϵ-quadratic form K_{k}(˜∂W). Before we do this we explain how the claim follows then. Notice that ∂W = M^{−} ∐ M′. Since F, f and f′ are k-connected and k ≥ 2, we get identifications π = π_{1}(X) = π_{1}(M) = π_{1}(M′) = π_{1}(W) and ˜∂W is the disjoint union of the universal coverings M̃ and M̃′. Hence we get in L_{1−ϵ}(ℤπ, w)

[(K_{k}(˜∂W), t′′, s′′)] = [(K_{k}(M), −t, −s)] + [(K_{k}(M′), s′, t′)]
= −[(K_{k}(M), t, s)] + [(K_{k}(M′), s′, t′)].

If we can construct the Lagrangian above for (K_{k}(˜∂W), t′′, s′′), we conclude [(K_{k}(˜∂W), t′′, s′′)] = 0 in L_{1−ϵ}(R) from Lemma 4.29. This implies the desired equation in L_{1−ϵ}(ℤπ, w)

[(K_{k}(M), −t, −s)] = [(K_{k}(M′), s′, t′)].

It remains to show that im(∂_{k+1}) is a Lagrangian.

This is rather easy for the symmetric structure. We first show for any x, y ∈ K_{k+1}(W̃, ˜∂W) that s(∂_{k+1}(x), ∂_{k+1}(y)) = 0. The following diagram commutes and has isomorphisms as vertical arrows.

0 ⟶ K_{k+1}(W̃, ˜∂W) ⟶^{∂_{k+1}} K_{k}(˜∂W) ⟶^{K_{k}(ĩ)}  K_{k}(W̃)  ⟶ 0
?∩[W,∂W]↑  ^{?∩[∂W]}↑  ^{?∩[W,∂W]}↑
0 ⟶  K^{k}(W̃)  ⟶^{K^{k}(ĩ)} K^{k}(˜∂W) ⟶^{δ^{k}} K^{k+1}(W̃, ˜∂W) ⟶ 0

We have for x, y ∈ K_{k+1}(W̃, ˜∂W)

s(∂_{k+1}(x), ∂_{k+1}(y)) = ⟨(? ∩ [∂W])^{−1} ∘ ∂_{k+1}(x), ∂_{k+1}(y)⟩
= ⟨(K^{k}(ĩ) ∘ (? ∩ [W, ∂W])^{−1}(x), ∂_{k+1}(y)⟩
= ⟨(δ^{k} ∘ K^{k}(ĩ) ∘ (? ∩ [W, ∂W])^{−1}(x), y⟩
= ⟨(0 ∘ (? ∩ [W, ∂W])^{−1}(x), y⟩
= 0.

Now suppose for x ∈ K_{k}(M̃) that s(x, ∂_{k+1}(y)) = 0 for all y ∈ K_{k+1}(W̃, ˜∂W). We must show x ∈ im(∂_{k+1}). This is equivalent to (?∩[W, ∂W])^{−1} ∘K_{k}(ĩ)(x) = 0. Since K_{p}(W̃, ˜∂W) = 0 for p ≤ k and finitely generated projective for p = k+1, the canonical map

K^{k+1}(W̃, ˜∂W) ⟶^{≅} hom_{ℤπ}(K_{k+1}(W̃, ˜∂W), ℤπ), α ↦ ⟨α,?⟩

is bijective by an elementary chain complex argument or by the universal coefficient spectral sequence. Hence the claim follows from the following calculation for y ∈ K_{k+1}(M̃)

⟨(? ∩ [W, ∂W])^{−1} ∘ K_{k}(ĩ)(x), y⟩ = ⟨δ^{k} ∘ (? ∩ [∂W])^{−1}(x), y⟩
= ⟨(? ∩ [∂W])^{−1}(x), ∂_{k+1}(y)⟩
= 0.

Thus we have shown that im(∂_{k+1}) is a Lagrangian for the non-degenerate ϵsymmetric form [K_{k}(˜∂), s)]. It remains to show that it also is a Lagrangian for the non-degenerate ϵ-quadratic form [K_{k}(˜∂), s]. In other words, we must show that t vanishes on im(∂_{k+1}). We sketch the idea of the proof.

Consider x ∈ K_{k+1}(W̃, { \atop ∂W)}. We can find a smooth map (g, ∂g): (S, ∂S) → (W̃, ˜∂W) such that S is obtained from S^{k+1} by removing a finite number of open embedded disjoint discs D^{k+1} and the image of the fundamental class [S, ∂S] under the map on homology induced by (g, ∂g) is x. Moreover we can assume that ∂g: ∂S → ˜∂W is an immersion and g is in general position. We have to show that μ(∂g) is zero, where μ(∂g) the sum of the self intersection numbers of (∂g)|_{C}: C → W̃ for C ∈ π_{0}(∂S). Since g is in general position, the set of double points consists of circles which do not concern us and arcs whose end points are on ˜∂W. Now one shows for each arc that the contributions of its two end points to the self intersection number μ(∂g) cancel out. This proves μ(∂_{k+1}(x)) = 0. This finishes the proof that the surgery obstruction is well-defined and depends only on the normal bordism class. Thus we have proved Theorem 4.33 (2). Assertion (1) of Theorem 4.33 is a direct consequence of Theorem 4.27. This finishes the proof of Theorem 4.33.

Now we can give a complete answer to Problem 3.1 and Problem 3.62 for even-dimensions and in the simply connected case.

Theorem 4.36 1. Let (\overline{f}, f) be a normal map from a closed manifold M to a simply connected finite Poincaré complex X of dimension n = 4k ≥ 5. Then we can change M and f leaving X fixed by finitely many surgery steps to get a normal map (\overline{g}, g) from a closed manifold N to X such that g is a homotopy equivalence if and only if sign(M) = sign(X);

2. Let (\overline{f}, f) be normal map from a closed manifold M to a simply connected finite Poincaré complex X of dimension n = 4k + 2 ≥ 5. Then we can change M and f leaving X fixed by finitely many surgery steps to get a normal map (\overline{g}, g) from a closed manifold N to X such that g is a homotopy equivalence if and only if the Arf invariant taking values in ℤ/2 vanishes;

3. Let X be a simply connected finite Poincaré complex of dimension n = 4k ≥ 5. Then X is homotopy equivalent to a closed manifold if and only if the Spivak normal fibration has a reduction to a vector bundle ξ: E → X, i.e. the set of normal invariants T_{n}(X) is non-empty, such that

⟨L(ξ)^{−1}, [X]⟩ = sign(X);

4. Let X be a simply-connected finite connected Poincaré complex of dimension n = 4k + 2 ≥ 5. Then X is homotopy equivalent to a closed manifold if and only if the Spivak normal fibration has a reduction to a vector bundle ξ: E → X, i.e. the set of normal invariants T_{n}(X) is non-empty such that the Arf invariant of the associated surgery problem, which takes values in ℤ/2, vanishes.

4.5. FORMATIONS AND ODD DIMENSIONAL L-GROUPS 99

\underline{\underline{Proof}} : (1) Because of Theorem 4.33 and Theorem 4.30 we have to show for a 2k-connected normal map of degree one f: M → X from a closed simply connected oriented manifold M of dimension n = 4k to a simply connected Poincaré complex X of dimension n = 4k that for the non-degenerate symmetric bilinear form ℝ ⊗_{ℤ} (K_{2k}(M), λ) induced by the intersection pairing we get

sign (ℝ ⊗_{ℤ} (K_{2k}(M), λ)) = sign(M) − sign(X).

This follows from elementary considerations about signatures and from the commutative diagram with exact rows

0 ⟶ K_{2k}(M) ⟶ H_{2k}(M) ⟶^{H_{2k}(f)} H_{2k}(X) ⟶ 0
?∩[M]↑  ^{?∩[M]}↑  ^{?∩[X]}↑
0 ⟵ K^{2k}(M) ⟵ H^{2k}(M) ⟵_{H^{2k}(f)} H^{2k}(X) ⟵ 0

(2) follows from Theorem 3.48, Theorem 4.33 and Theorem 4.32.

(3) follows from (1) and the Hirzebruch signature formula which implies for a surgery problem (\overline{f}, f): ν(M) → ξ covering f: M → X which is obtained from a reduction ξ of the Spivak normal fibration of X

sign(M) = ⟨L(T M), [M]⟩ = ⟨L(ν(M) ⊕ \underline{ℝ}^{a})^{−1}, [M]⟩ = ⟨L(ξ)^{−1}, [X]⟩.

(4) follows from Theorem 3.48 and (2).

4.5 Formations and Odd Dimensional L-Groups

In this subsection we explain the algebraic objects which describe the surgery obstruction and will be the typical elements in the surgery obstruction group in odd dimensions. Throughout this section R will be an associative ring with unit and involution and ϵ ∈ {±1}.

Definition 4.37 An ϵ-quadratic formation (P, ψ; F, G) consists of a non-degene-rate ϵ-quadratic form (P, ψ) together with two Lagrangians F and G.

An isomorphism f: (P, ψ; F, G) → (P′, ψ′; F,′ , G′) of ϵ-quadratic formations is an isomorphism f: (P, ψ) → (P′, ψ′) of non-degenerate ϵ-quadratic forms such that f(F) = F′ and f(G) = G′ holds.

Definition 4.38 The trivial ϵ-quadratic formation associated to a finitely generated projective R-module P is the formation (H_{ϵ}(P); P, P^{∗}). A formation (P, ψ; F, G) is called trivial if it isomorphic to the trivial ϵ-quadratic formation associated to some finitely generated projective R-module. Two formations are stably isomorphic if they become isomorphic after taking the direct sum of trivial formations.

Remark 4.39 We conclude from Lemma 4.29 that any formation is isomorphic to a formation of the type (H_{ϵ}(P); P, F) for some Lagrangian F ⊂ P ⊕ P^{∗}. Any automorphism f: H_{ϵ}(P) ⟶^{≅} H_{ϵ}(P) of the standard hyperbolic ϵ-quadratic form H_{ϵ}(P) for some finitely generated projective R-module P defines a formation by (H_{ϵ}(P); P, f(P)).

Consider a formation (P, ψ; F, G) such that P, F and G are finitely generated free and suppose that R has the property that R^{n} and R^{m} are R-isomorphic if and only if n = m. Then (P, ψ; F, G) is stably isomorphic to (H_{ϵ}(Q); Q, f(Q)) for some finitely generated free R-module Q by the following argument. Because of Lemma 4.29 we can choose isomorphisms of non-degenerate ϵ-quadratic forms f: H_{ϵ}(F) ⟶^{≅} (P, ψ) and g: H_{ϵ}(G) ⟶^{≅} (P, ψ) such that f(F) = F and g(G) = G. Since F ≅ R^{a} and G ≅ R^{b} by assumption and R^{2a} ≅ F ⊕ F^{∗} ≅ P ≅ G ⊕ G^{∗} ≅ R^{2b}, we conclude a = b. Hence we can choose an R-isomorphism u: F → G. Then we obtain an isomorphism of non-degenerate ϵ-quadratic forms by the composition

v: H_{ϵ}(F) ⟶^{H_{ϵ}(u)} H_{ϵ}(G) ⟶^{g} (P, ψ) ⟶^{f^{−1}} H_{ϵ}(F)

and an isomorphism of ϵ-quadratic formations

f: (H_{ϵ}(F); F, v(F)) ⟶^{≅} (P, ψ; F, G).

Recall that K_{1}(R) is defined in terms of automorphisms of finitely generated free R-modules. Hence it is plausible that the odd-dimensional L-groups will be defined in terms of formations which is essentially the same as in terms of automorphisms of the standard hyperbolic form over a finitely generated free R-module.

Definition 4.40 Let (P, ψ) be a (not necessarily non-degenerate) (−ϵ)-quadratic form. Define its boundary ∂(P, ψ) to be the ϵ-quadratic formation (H_{ϵ}(P); P, Γ_{ψ}), where Γ_{ψ} is the Lagrangian given by the image of the R-homomorphism

P → P ⊕ P^{∗}, x ↦ (x, (1 − ϵ · T)(ψ)(x)) .

One easily checks that Γ_{ψ} appearing in Definition 4.40 is indeed a Lagrangian. Two Lagrangians F, G of a non-degenerate ϵ-quadratic form (P, ψ) are called complementary if F ∩ G = {0} and F + G = P.

Lemma 4.41 Let (P, ψ; F, G) be an ϵ-quadratic formation. Then:

1. (P, ψ; F, G) is trivial if and only F and G are complementary to one another;

2. (P, ψ; F, G) is isomorphic to a boundary if and only if there is a Lagrangian L ⊂ P such that L is a complement of both F and G;

3. There is an ϵ-quadratic formation (P′, ψ′; F′, G′) such that (P, ψ; F, G) ⊕ (P′, ψ′; F′, G′) is a boundary;

4. An (−ϵ)-quadratic form (Q, μ) is non-degenerate if and only if its boundary is trivial.

\underline{\underline{Proof}} : (1) The inclusions of F and G in P induce an R-isomorphism f: F ⊕ G → P. Let

( \begin{matrix}a & b \\ c & d \end{matrix} ) : F ⊕ G → (F ⊕ G)^{∗} = F^{∗} ⊕ G^{∗}

be f^{∗} ∘ ψ ∘ f for some representative ψ: P → P^{∗} of ψ ∈ Q_{ϵ}(P) and let ψ′ ∈ Q_{ϵ}(F ⊕G) be the associated class. Then f is an isomorphism of non-degenerate ϵ-quadratic forms (F ⊕ G, ψ′) → (P, ψ) and F and G are Lagrangians in (F ⊕ G, ψ′). This implies that the isomorphism ψ′ + ϵ · T (ψ′) is given by

( \begin{matrix}a & b \\ c & d \end{matrix} ) + ϵ · T ( \begin{matrix}a & b \\ c & d \end{matrix} ) = ( \begin{matrix}a + ϵ · a^{∗} & b + ϵc^{∗} \\ c + ϵb^{∗} & d + ϵd^{∗} \end{matrix} ) = ( \begin{matrix}0 & e \\ f & 0 \end{matrix} )

Hence (b + ϵ · c^{∗}): G → F^{∗} is an isomorphism. Define an R-isomorphism

u: ( \begin{matrix}1 & 0 \\ 0 & (b + ϵ · c^{∗})^{−1} \end{matrix} ) : F ⊕ F^{∗} ⟶^{≅} P.

One easily checks that u defines an isomorphism of formations

u: (H_{ϵ}(F); F, F^{∗}) ⟶^{≅} (F ⊕ G, ψ′; F, G).

(2) One easily checks that for an (−ϵ)-quadratic form (P, ϕ) the Lagrangian P^{∗} in its boundary ∂(P, ψ) := (H_{ϵ}(P); P, Γ_{ψ}) is complementary to both P and Γ_{ψ}. Conversely, suppose that (P, ψ; F, G) is an ϵ-quadratic formation such that there exists a Lagrangian L ⊂ P which is complementary to both F and G. By the argument appearing in the proof of assertion (1) we find an isomorphism of ϵ-quadratic formations

f : (H_{ϵ}(F); F, F^{∗}) ⟶^{≅} (P, ψ; F, L)

which is the identity on F. The preimage G′ := f^{−1}(G) is a Lagrangian in H_{ϵ}(F) which is complementary to F^{∗}. Write the inclusion of G′ into F ⊕ F^{∗} as (a, b): G′ → F ⊕ F^{∗}. Consider the (−ϵ)-quadratic form (F, ψ′), where ψ′ ∈ Q_{−ϵ}(F) is represented by b∘a^{∗}: F → F^{∗}. One easily checks that its boundary is precisely (H_{ϵ}(F); F, G′) and f induces an isomorphism of ϵ-quadratic formations

∂(F, ψ′) = (H_{ϵ}(F); F, G′) ⟶^{≅} (P, ψ; F, G).

(3) Because of Lemma 4.29 we can find Lagrangians F′ and G′ such that F and F′ are complementary and G and G′ are complementary. Put (P′, ψ′; F′, G′) = (P, −ψ, F′, G′). Then M = {(p, p) | p ∈ P } ⊂ P ⊕ P is a Lagrangian in the direct sum

(P, ψ; F, G) ⊕ (P′, ψ′; F′, G′) = (P ⊕ P, ψ ⊕ (−ψ), F ⊕ F′, G ⊕ G′)

which is complementary to both F ⊕ F′ and G ⊕ G′. Hence the direct sum is isomorphic to a boundary by assertion (2).

(4) The Lagrangian Γ_{ψ} in the boundary ∂(Q, μ) := H_{ϵ}(P); P, Γ_{ψ}) is complementary to P if and only if (1 − ϵ · T)(μ): P → P^{∗} is an isomorphism. This finishes the proof of Lemma 4.41.

Now we can define the odd-dimensional surgery groups.

Definition 4.42 Let R be an associative ring with unit and involution. For an odd integer n = 2k + 1 define the abelian group L_{n}(R) called the n-th quadratic L-group of R to be the abelian group of equivalence classes [(P, ψ; F, G)] of (−1)^{k}-quadratic formations (P, ψ; F, G) such that P, F and G are finitely generated free with respect to the following equivalence relation. We call (P, ψ; F, G) and (P′, ψ′; F′, G′) equivalent if and only if there exist (−(−1)^{k})-quadratic forms (Q, μ) and (Q′, μ′) for finitely generated free R-modules Q and Q′ and finitely generated free R-modules S and S′ together with an isomorphism of (−1)^{k}- quadratic formations

(P, ψ; F, G) ⊕ ∂(Q, μ) ⊕ (H_{ϵ}(S); S, S^{∗})

≅ (P′, ψ′; F′, G′) ⊕ ∂(Q′, μ′) ⊕ (H_{ϵ}(S′); S′, (S′)^{∗}).

Addition is given by the sum of two (−1)^{k}-quadratic formations. The zero element is represented by ∂(Q, μ) ⊕ (H_{(−1)^{k}}(S); S, S^{∗}) for any (−(−1)^{k})-quadratic form (Q, μ) for any finitely generated free R-module Q and any finitely generated free R-module S. The inverse of [(P, ψ; F, G)] is represented by (P, −ψ; F′, G′) for any choice of Lagrangians F′ and G′ in H_{ϵ}(P) such that F and F′ are complementary and G and G′ are complementary.

A morphism u: R → S of rings with involution induces homomorphisms u_{∗}: L_{k}(R) → L_{k}(S) for k = 1, 3 by induction. One easily checks (u∘v)_{∗} = u_{∗}∘v_{∗} and (id_{R})_{∗} = id_{L_{k}(R)} for k = 1, 3.

The odd-dimensional L-groups of the ring of integers vanish.

Theorem 4.43 We have L_{2k+1}(ℤ) = 0.

4.6 The Surgery Obstruction in Odd Dimensions

Consider a normal map of degree one (\overline{f}, f): T M ⊕\underline{ℝ}^{a} → ξ covering f: M → X, where M is a closed oriented manifold of dimension n and X is a connected finite Poincaré complex of dimension n for odd n = 2k + 1. To these data we want to assign an element σ(\overline{f}, f) ∈ L_{n}(ℤπ, w) such that the following holds

Theorem 4.44 (Surgery Obstruction Theorem in Odd Dimensions) We get under the conditions above:

1. Suppose k ≥ 2. Then σ(\overline{f}, f) = 0 in L_{n}(ℤπ, w) if and only if we can do a finite number of surgery steps to obtain a normal map (\overline{f′}, f′): T M′ ⊕ \underline{ℝ}^{a+b} → ξ ⊕ \underline{ℝ^{b}} covering a homotopy equivalence f′: M′ → X;

2. The surgery obstruction σ(\overline{f}, f) depends only on the normal bordism class of (\overline{f}, f).

We can arrange by finitely many surgery steps that f is k-connected (see Theorem 3.61). Consider a normal bordism (\overline{F} , F): T W ⊕ \underline{ℝ^{b}} → η covering a map F: W → X × [0, 1] of degree one from the given normal map (\overline{f}, f): T M ⊕ \underline{ℝ}^{a} → ξ covering f: M → X to a new k-connected normal map (\overline{f′}, f′): T M′ ⊕\underline{ℝ}^{a′} → ξ′ covering f′: M′ → X. We finally want to arrange that f′ is a homotopy equivalence. By applying Theorem 3.61 to the interior of W without changing ∂W we can arrange that W is (k + 1)-connected. The kernels fit into an exact sequence

0 → K_{k+1}(M̃′) ⟶^{K_{k+1}(j̃)} K_{k+1}(W̃) ⟶^{K_{k+1}(ĩ′)} K_{k+1}(W̃, M̃′) ⟶^{∂_{k+1}} K_{k}(M̃′) → 0,

where j: M′ → W and i′: W → (W, M′) are the inclusions. Hence f′ is a homotopy equivalence if and only if K_{k+1}(ĩ′): K_{k+1}(W̃) → K_{k+1}(W̃, M̃′) is bijective. Therefore we must arrange that K_{k+1}(ĩ′): K_{k+1}(W̃) → K_{k+1}(W̃, M̃′) is bijective.

We can associate to the normal bordism (\overline{F} , F): T W ⊕\underline{ℝ^{b}} → η from the given k-connected normal map of degree one (\overline{f}, f) to another k-connected normal map of degree one (\overline{f′}, f′) a (−1)^{k}-quadratic formation (H_{(−1)^{k}}(F); F, G) as follows. The underlying non-degenerate (−1)^{k}-quadratic form is H_{(−1)^{k}}(F) for F := K_{k+1}(W̃, M̃′). The first Lagrangian is F. The second Lagrangian G is given by the image of the map

(K_{k+1}(ĩ′), u ∘ (? ∩ [W, ∂W])^{−1} ∘ K_{k+1}(ĩ): K_{k+1}(W̃)

→ F ⊕ F^{∗} = K_{k+1}(W̃, M̃′) ⊕ K_{k+1}(W̃, M̃′)^{∗},

where (? ∩ [W, ∂W]): K^{k+1}(W̃, M̃′) ⟶^{≅} K_{k+1}(W̃, M̃) is the Poincaré isomorphism, i: W → (W, M) is the inclusion and u is the canonical map

u: K^{k+1}(W̃, M̃′) ⟶^{≅} K_{k+1}(W̃, M̃′)^{∗}, α ↦ ⟨α,?⟩

which is bijective by an elementary chain complex argument or by the universal coefficient spectral sequence.

What is the relation between this formation and the problem whether f′ is a homotopy equivalence? Suppose that f′ is a homotopy equivalence. Then K_{k+1}(ĩ′): K_{k+1}(W̃) → K_{k+1}(W̃, M̃′) is an isomorphism. Define a (−1)^{k+1}- quadratic form (H, ψ) by H = K_{k+1}(W̃) and

ψ: H := K_{k+1}(W̃) ⟶^{K_{k+1}(ĩ)} K_{k+1}(W̃, M̃) ⟶^{(?∩[W,∂W])^{−1}} K^{k+1}(W̃, M̃′)

⟶^{u} K_{k+1}(W̃, M̃′)^{∗} ⟶^{K_{k+1}(ĩ′)^{∗}} K_{k+1}(W̃)^{∗} = H^{∗}.

One easily checks that the isomorphism K_{k+1}(ĩ′): K_{k+1}(W̃) → K_{k+1}(W̃, M̃′) induces an isomorphism of (−1)^{k}-quadratic formations

∂(H, ψ) ⟶^{≅} (H_{(−1)^{k}}(F); F, G).

Hence we see that f′ is a homotopy equivalence only if (H_{(−1)^{k}}(F); F, G) is isomorphic to a boundary.

One decisive step is to prove that the class of this (−1)^{k}-quadratic formation (H_{(−1)^{k}}(F); F, G) in L_{2k+1}(ℤπ, w) is independent of the choice of the (k + 1)- connected nullbordism. We will not give the proof of this fact. This fact enables us to define the surgery obstruction of (\overline{f}, f)

σ(\overline{f}, f) := [(H_{ϵ}(F); F, G)] ∈ L_{2k+1}(ℤπ, w),  (4.45)

where (H_{(−1)^{k}}(F); F, G) is the (−1)^{k}-quadratic formation associated to any normal bordism of degree one (\overline{F} , F) from a normal map (f_{0}, \overline{f_{0}}) to some normal map (f′, \overline{f′}) such that (f_{0}, \overline{f_{0}}) and (f′, \overline{f′}) are k-connected, F is (k+1)-connected and (f_{0}, \overline{f_{0}}) is obtained from the original normal map of degree one (\overline{f}, f) by surgery below the middle dimension. From the discussion above it is clear that the vanishing of σ(\overline{f}, f) is a necessary condition for the existence of a normal map (\overline{f′}, f′) which is normally bordant to f and whose underlying map is a homotopy equivalence. We omit the proof that for k ≥ 2 this condition is also sufficient. This finishes the outline of the proof of Theorem 4.44.

More details can be found in [119, chapter 8]. The reason why we have been very brief in the odd-dimensional case is that the proofs are more complicated but not as illuminating as in the even-dimensional case and that a reader who is interested in details should directly read the approach using Poincaré complexes of Ranicki (see [95], [96], [97]). This approach is more conceptual and treats the even-dimensional and odd-dimensional case simultaneously.

Now we can give a rather complete answer to Problem 3.1 and Problem 3.62 for odd dimensions in the simply connected case.

Theorem 4.46 1. Suppose we have some normal map (\overline{f}, f) from a closed manifold M to a simply connected finite Poincaré complex X of odd dimension n = 2k + 1 ≥ 5. Then we can always change M and f leaving X fixed by finitely many surgery steps to get a normal map (\overline{g}, g) from a closed manifold N to X such that g is a homotopy equivalence;

2. Let X be a simply-connected finite connected Poincaré complex of odd dimension n = 2k + 1 ≥ 5. Then X is homotopy equivalent to a closed manifold if and only if the Spivak normal fibration has a reduction to a vector bundle ξ: E → X, i.e. the set of normal invariants T_{n}(X) is nonempty.

\underline{\underline{Proof}} : (1) follows from Theorem 4.44 and Theorem 4.43.

(2) follows from assertion 1 together with Theorem 3.48.

4.7 Variations of the Surgery Obstruction and the L-Groups

4.7. VARIATIONS OF THE SURGERY OBSTRUCTION AND THE L-GROUPS105

So far we have dealt with Problem 3.1 when a topological space is homotopy equivalent to a closed manifold. This question has motivated and led us to the notions of a finite Poincaré complex, of its Spivak normal fibration, of a normal map of degree one, of L-groups and of the surgery obstruction. This problem is certainly interesting but not our ultimate goal. We have already mentioned in Remark 1.5 our main goal which is to decide whether two closed manifolds are diffeomorphic and we have discussed the so called surgery program which is a strategy to attack it. The surgery program will lead to the surgery sequence in the next chapter (see Theorem 5.12). The surgery program suggests that we have to consider the following variations of the surgery obstruction and the L-groups which will play a role when establishing the surgery sequence.

Consider the third step (3) in the surgery program as explained in Remark 1.5. There are a cobordism (W; M, N) and a map (F, f, id): (W; M, N) → (N × [0, 1]; N × {0}, N × {1}) for f a homotopy equivalence is given and we want to change F into a homotopy equivalence without changing W on the boundary. Thus we have to deal also with surgery problems (\overline{g}, g) where the underlying map g: V → X is a pair of maps (g, ∂g): (V, ∂V) → (X, ∂X) from a compact oriented manifold V with boundary ∂V to a pair of finite Poincaré complexes (X, ∂X) such that ∂f is already a homotopy equivalence. This will be done in Section 4.7.1. If this procedure is successful we obtain a h-cobordism W from M to N. We would like to know whether W is relative M diffeomorphic to M × [0, 1]. This would imply that M and N are diffeomorphic. Because of the s-Cobordism Theorem 1.1 this comes down to the problem to control the Whitehead torsion of the h-cobordism. The Whitehead torsion of the hcobordism is trivial if and only if the Whitehead torsion of τ(F) and of τ(f) in Wh(π_{1}(X)) agree (see Theorem 2.1). We may modify the first step (1) of the surgery program appearing in Remark 1.5 by requiring that f is a simple homotopy equivalence. The existence of a simple homotopy equivalence f: M → N is a necessary condition for M and N to be diffeomorphic. This means that we must modify our surgery obstruction so that its vanishing means that we get a simple homotopy equivalence F: W → N × [0, 1]. This will be outlined in Section 4.7.2.

4.7.1 Surgery Obstructions for Manifolds with Boundary

We want to extend the notion of a normal map from closed manifolds to manifolds with boundary. The underlying map f is a map of pairs (f, ∂f): (M, ∂M) → (X, ∂X), where M is a compact oriented manifold with boundary ∂M and (X, ∂X) is a finite Poincaré pair, the degree of f is one and ∂f: ∂M → ∂X is required to be a homotopy equivalence. The bundle data are unchanged, they consist of a vector bundle ξ over X and a bundle map \overline{f}: T M ⊕ \underline{ℝ^{a}} → ξ. Next we explain what a nullbordism in this setting means.

A manifold triad (W; ∂_{0}W, ∂_{1}W) of dimension n consists of a compact manifold W of dimension n whose boundary decomposes as ∂W = ∂_{0}W ∪∂_{1}W for two compact submanifolds ∂_{0}W and ∂_{1}W with boundary such that ∂_{0}W ∩ ∂_{1}W = ∂(∂_{0}W) = ∂(∂_{1}W). A Poincaré triad (X; ∂_{0}X, ∂_{1}X) of dimension n consists of a finite Poincaré pair (X, ∂X) of dimension n together with (n − 1)-dimensional finite subcomplexes ∂_{0}X and ∂_{1}X such that ∂X = ∂_{0}X ∪ ∂_{1}X, the intersection ∂_{0}X ∩ ∂_{1}X is (n − 2)-dimensional, ∂X is equipped with the structure of a Poincaré complex induced by Poincaré pair structure on (X, ∂X) and both (∂_{0}X, ∂_{0}X ∩ ∂_{1}X) and (∂_{1}X, ∂_{0}X ∩ ∂_{1}X) are equipped with the structure of a Poincaré pair induced by the given structure of a Poincaré complex on ∂X. We use the convention that the fundamental class of [X, ∂X] is sent to [∂X] under the boundary homomorphism H_{n}(X, ∂X) ⟶^{∂_{n}} H_{n−1}(∂X) and [∂X] is sent to ([∂_{0}X, ∂_{0}X ∩ ∂_{1}X], −[∂_{1}X, ∂_{0}X ∩ ∂_{1}X]) under the composite

H_{n−1}(∂X) → H_{n−1}(∂_{0}X ∪ ∂_{1}X, ∂_{0}X ∩ ∂_{1}X)
≅ H_{n−1}(∂_{0}X, ∂_{0}X ∩ ∂_{1}X) ⊕ H_{n−1}(∂_{1}X, ∂_{0}X ∩ ∂_{1}X).

A manifold triad (W; ∂_{0}W, ∂_{1}W) together with an orientation for W is a Poincaré triad. We allow that ∂_{0}X or ∂_{1}X or both are empty.

A normal nullbordism (\overline{F} , F): T W ⊕ \underline{ℝ^{b}} → η for a normal map of degree one (\overline{f}, f) with underlying map (f, ∂f): (M, ∂M) → (X, ∂X) consists of the following data. Put n = dim(M). We have a compact oriented (n + 1)- dimensional manifold triad (W; ∂_{0}W, ∂_{1}W), a finite (n+1)-dimensional Poincaré triad (Y; ∂_{0}Y, ∂Y), a map (F; ∂_{0}F, ∂_{1}F): (W; ∂_{0}W, ∂_{1}W) → (Y; ∂_{0}Y, ∂_{1}Y), an orientation preserving diffeomorphism (u, ∂u): (M, ∂M) → (∂_{0}W, ∂(∂_{0}W)) and an orientation preserving homeomorphism (v, ∂v): (X, ∂X) → (∂_{0}Y, ∂_{0}Y ∩∂_{1}Y). We require that F has degree one, ∂_{1}F is a homotopy equivalence and ∂_{0}F ∘u = v ∘ f. Moreover, we have a bundle map \overline{F}: T W ⊕ \underline{ℝ^{b}} → η covering F and a bundle map \overline{v}: ξ ⊕ \underline{ℝ^{c}} → η covering v such that \overline{F}, T u and \overline{v} fit together.

We call two such normal maps (\overline{f}, f) and (\overline{f′}, f′) normally bordant if the disjoint union of them after changing the orientation for M appearing in (\overline{f}, f) possesses a normal nullbordism.

The definition and the main properties of the surgery obstruction carry over from normal maps for closed manifolds to normal maps for compact manifolds with boundary. The main reason is that we require ∂f: ∂M → ∂X to be a homotopy equivalence so that the surgery kernels “do not feel the boundary”. All arguments such as making a map highly connected by surgery steps and intersection pairings and selfintersection can be carried out in the interior of M without affecting the boundary. Thus we get

Theorem 4.47 (Surgery Obstruction Theorem for Manifolds with Boundary). Let (\overline{f}, f) be a normal map of degree one with underlying map (f, ∂f): (M, ∂M) → (X, ∂X) such that ∂f is a homotopy equivalence. Put n = dim(M). Then

1. We can associate to it its surgery obstruction

σ(\overline{f}, f) ∈ L_{n}(ℤπ, w).  (4.48)

2. The surgery obstruction depends only on the normal bordism class of (\overline{f}, f); 3. Suppose n ≥ 5. Then σ(\overline{f}, f) = 0 in L_{n}(ℤπ, w) if and only if we can do

a finite number of surgery steps on the interior of M leaving the boundary fixed to obtain a normal map (\overline{f′}, f′): T M′ ⊕ \underline{ℝ}^{a′} → ξ′ which covers a homotopy equivalence of pairs (f′, ∂f′): (M′, ∂M′) → (X, ∂X) with ∂M′ = ∂M and ∂f′ = ∂f.

4.7.2 Surgery Obstructions and Whitehead Torsion

Next we want to modify the L-groups and the surgery obstruction so that the surgery obstruction is the obstruction to achieve a simple homotopy equivalence.

We begin with the L-groups. It is clear that this requires to take equivalence classes of bases into account. Suppose that we have specified a subgroup U ⊂ K_{1}(R) such that U is closed under the involution on K_{1}(R) coming from the involution of R and contains the image of the change of ring homomorphism K_{1}(ℤ) → K_{1}(R).

Two bases B and B′ for the same finitely generated free R-module V are called U-equivalent if the change of basis matrix defines an element in K_{1}(R) which belongs to U. Notice that the U-equivalence class of a basis B is unchanged if we permute the order of elements of B. We call an R-module V U-based if V is finitely generated free and we have chosen a U-equivalence class of bases.

Let V be a stably finitely generated free R-module. A stable basis for V is a basis B for V ⊕ R^{u} for some integer u ≥ 0. Denote for any integer v the direct sum of the basis B and the standard basis S^{a} for R^{a} by B ∐ S^{a} which is a basis for V ⊕ R^{u+a}. Let C be a basis for V ⊕ R^{v}. We call the stable basis B and C stably U-equivalent if and only if there is an integer w ≥ u, v such that B ∐ S^{w−u} and C ∐ S^{w−v} are U-equivalent basis. We call an R-module V stably U-based if V is stably finitely generated free and we have specified a stable U-equivalence class of stable basis for V.

Let V and W be stably U-based R-modules. Let f: V ⊕ R^{a} ⟶^{≅} W ⊕ R^{b} be an R-isomorphism. Choose a non-negative integer c together with basis for V ⊕ R^{a+c} and W ⊕ R^{b+c} which represent the given stable U-equivalence classes of basis for V and W. Let A be the matrix of f ⊕ id_{R^{c}}: V ⊕ R^{a+c} ⟶^{≅} W ⊕ R^{b+c} with respect to these bases. It defines an element [A] in K_{1}(R). Define the U-torsion

τ^{U}(f) ∈ K_{1}(R)/U  (4.49)

by the class represented by [A]. One easily checks that τ(f) is independent of the choices of c and the basis and depends only on f and the stable U-basis for V and W. Moreover, one easily checks

τ^{U}(g ∘ f) = τ^{U}(g) + τ^{U}(f);
τ^{U} ( \begin{matrix}f & 0 \\ u & v \end{matrix} ) = τ^{U}(f) + τ^{U}(v);
τ^{U}(id_{V}) = 0

for R-isomorphisms f: V_{0} ⟶^{≅} V_{1}, g: V_{1} ⟶^{≅} V_{2} and v: V_{3} ⟶^{≅} V_{4} and an Rhomomorphism u: V_{0} → V_{4} of stably U-based R-modules V_{i}. Let C_{∗} be a contractible stably U-based finite R-chain complex, i.e. a contractible R-chain complex C_{∗} of stably U-based R-modules which satisfies C_{i} = 0 for |i| > N for some integer N. The definition of Whitehead torsion in (2.7) carries over to the definition of the U-torsion

τ^{U}(C_{∗}) = [A] ∈ K_{1}(R)/U.  (4.50)

Analogously we can associate to an R-chain homotopy equivalence f: C_{∗} → D_{∗} of stably U-based finite R-chain complexes its U-torsion (cf (2.8))

τ^{U}(f_{∗}) := τ(cone_{∗}(f_{∗})) ∈ K_{1}(R)/U.  (4.51)

Theorem 2.9 carries over to U-torsion in the obvious way.

We will consider U-based ϵ-quadratic forms (P, ψ), i.e. ϵ-quadratic forms whose underlying R-module P is a U-based finitely generated free R-module such that U-torsion of the isomorphism (1 + ϵ · T)(ψ): P ⟶^{≅} P^{∗} is zero in K_{1}(R)/U. An isomorphism f: (P, ψ) → (P′, ψ′) of U-based ϵ-quadratic forms is U-simple if the U-torsion of f: P → P′ vanishes in K_{1}(R)/U. Notice that the ϵ-quadratic form H_{ϵ}(R) inherits a basis from the standard basis of R. The sum of two stably U-based ϵ-quadratic forms is again a stably U-based ϵ-quadratic form. It is worthwhile to mention the following U-simple version of Lemma 4.29.

Lemma 4.52 Let (P, ψ) be a U-based ϵ-quadratic form. Let L ⊂ P be a Lagrangian such that L is U-based and the U-torsion of the following 2-dimensional U-based finite R-chain complex

0 → L ⟶^{i} P ⟶^{i^{∗}∘(1+ϵ·T)(ψ)} L^{∗} → 0

vanishes in K_{1}(R)/U. Then the inclusion i: L → P extends to a U-simple isomorphism of ϵ-quadratic forms

H_{ϵ}(L) ⟶^{≅} (P, ψ).

Next we give the simple version of the even-dimensional L-groups.

Definition 4.53 Let R be an associative ring with unit and involution. For ϵ ∈ {±1} define L_{1−ϵ}^{U}(R) to be the abelian group of equivalence classes [(F, ψ)] of U-based non-degenerate ϵ-quadratic forms (F, ψ) with respect to the following equivalence relation. We call (F, ψ) and (F′, ψ′) equivalent if and only if there exists integers u, u′ ≥ 0 and a U-simple isomorphism of non-degenerate ϵ-quadratic forms

(F, ψ) ⊕ H_{ϵ}(R)^{u} ≅ (F′, ψ′) ⊕ H_{ϵ}(R)^{u′}.

Addition is given by the sum of two ϵ-quadratic forms. The zero element is represented by [H_{ϵ}(R)^{u}] for any integer u ≥ 0. The inverse of [F, ψ] is given by [F, −ψ].

For an even integer n define the abelian group L_{n}^{U}(R) called the n-th Udecorated quadratic L-group of R by

L_{n}^{U}(R) := \{ \begin{matrix}L_{0}^{U}(R) & if n ≡ 0 mod 4; \\ L_{2}^{U}(R) & if n ≡ 2 mod 4. \end{matrix}

A U-based ϵ-quadratic formation (P, ψ; F, G) consists of an ϵ-quadratic formation (P, ψ; F, G) such that (P, ψ) is a U-based non-degenerate ϵ-quadratic form, the Lagrangians F and G are U-based R-modules and the U-torsion of the following two contractible U-based finite R-chain complexes

0 → F ⟶^{i} P ⟶^{i^{∗}∘(1+ϵ·T)(ψ)} F^{∗} → 0

and

0 → G ⟶^{j} P ⟶^{j^{∗}∘(1+ϵ·T)(ψ)} G^{∗} → 0

vanish in K_{1}(R)/U, where i: F → P and j: G → P denote the inclusions. An isomorphism f: (P, ψ; F, G) → (P′, ψ′; F′, G′) of U-based ϵ-quadratic formations is U-simple if underlying the U-torsion of the induced R-isomorphisms P ⟶^{≅} P′, F ⟶^{≅} F′ and G ⟶^{≅} G′ vanishes in K_{1}(R)/U. Notice that the trivial ϵ-quadratic formation (H_{ϵ}(R^{u}); R^{u}, (R^{u})^{∗}) inherits a U-basis from the standard basis on R^{u}. Given a U-based (−ϵ)-quadratic form (Q, ψ), its boundary ∂(Q, ψ) is a U-based ϵ-quadratic formation. Obviously the sum of two U-based ϵ-quadratic formations is again a U-based ϵ-quadratic formation. Next we give the simple version of the odd-dimensional L-groups.

Definition 4.54 Let R be an associative ring with unit and involution. For ϵ ∈ {±1} define L_{2−ϵ}^{U}(R) to be the abelian group of equivalence classes [(P, ψ; F, G)] of U-based ϵ-quadratic formations (P, ψ; F, G) with respect to the following equivalence relation. We call two U-based ϵ-quadratic formations (P, ψ; F, G) and (P′, ψ′; F′, G′) equivalent if and only if there exists U-based (−ϵ)-quadratic forms (Q, μ) and (Q′, μ′) and non-negative integers u and u′ together with a U-simple isomorphism of ϵ-quadratic formations

(P, ψ; F, G) ⊕ ∂(Q, μ) ⊕ (H_{ϵ}(R^{u}); R^{u}, (R^{u})^{∗})
≅ (P′, ψ′; F′, G′) ⊕ ∂(Q′, μ′) ⊕ (H_{ϵ}(R^{u′}); R^{u′}, (R^{u′})^{∗}).

Addition is given by the sum of two ϵ-quadratic forms. The zero element is represented by ∂(Q, μ) ⊕ (H_{ϵ}(R^{u}); R^{u}, (R^{u})^{∗}) for any U-based (−ϵ)-quadratic form (Q, μ) and non-negative integer u. The inverse of [(P, ψ; F, G)] is represented by (P, −ψ; F′, G′) for any choice of stably U-based Lagrangians F′ and G′ in H_{ϵ}(P) such that F and F′ are complementary and G and G′ are complementary and the U-torsion of the the obvious isomorphism F ⊕ F′ ⟶^{≅} P and F ⊕ F′ ⟶^{≅} P vanishes in K_{1}(R)/U.

For an odd integer n define the abelian group L_{n}^{U}(R) called the n-th Udecorated quadratic L-group of R

L_{n}^{U}(R) := \{ \begin{matrix}L_{1}^{U}(R) & if n ≡ 1 mod 4; \\ L_{3}^{U}(R) & if n ≡ 3 mod 4. \end{matrix}

Notation 4.55 If R = ℤπ with the w-twisted involution and U ⊂ K_{1}(ℤπ) is the abelian group of elements of the shape (±g) for g ∈ π, then we write

L_{n}^{s}(ℤπ, w) := L_{n}^{U}(ℤπ, w);
L_{n}^{h}(ℤπ, w) := L_{n}(ℤπ, w);

where L_{n}(R) is the L-group introduced in Definitions 4.28 and 4.42 and L_{n}^{U}(R) is the L-group introduced in Definitions 4.53 and 4.54. The L-groups L_{n}^{s}(ℤπ, w) are called simple quadratic L-groups.

Let (\overline{f}, f) be a normal map of degree one with (f, ∂f): (M, ∂M) → (X, ∂X) as underlying map such that (X, ∂X) is a simple finite Poincaré complex and ∂f is a simple homotopy equivalence. Next we want to explain how the definition of the surgery obstruction of (4.48) can be modified to the simple setting. Notice that the difference between the L-groups L_{n}^{h}(ℤπ, w) and the simple L-groups L_{n}^{s}(ℤπ, w) is the additional structure of a U-basis. The definition of the simple surgery obstruction

σ(\overline{f}, f) ∈ L_{n}^{s}(ℤπ, w).  (4.56)

is the same as the one in (4.48) except that we must explain how the various surgery kernels inherit a U-basis.

The elementary proof of the following lemma is left to the reader. Notice that for any stably U-based R-module V and element x ∈ K_{1}(R)/U we can find another stable U-basis C for V such that the U-torsion τ^{U}(id: (V, B) → (V, C) is x. This is not true in the unstable setting. For instance, there exists a ring R with an element x ∈ K_{1}(R)/U for U the image of K_{1}(ℤ) → K_{1}(R) such that x cannot be represented by a unit in R, in other words x is not the U-torsion of any R-automorphism of R.

Lemma 4.57 Let C_{∗} be a contractible finite stably free R-chain complex and r be an integer. Suppose that each chain module C_{i} with i ≠ r comes with a stable U-basis. Then C_{r} inherits a preferred stable U-basis which is uniquely defined by the property that the U-torsion of C_{∗} vanishes in K_{1}(R)/U.

Now Lemma 4.18 has the following version in the simple homotopy setting.

Lemma 4.58 Let D_{∗} be a stably U-based finite R-chain complex. Suppose for a fixed integer k that H_{i}(D_{∗}) = 0 for i ≠ r. Suppose that H^{r+1}(hom_{R}(D_{∗}, V)) = 0 for any R-module V. Then H_{r}(D_{∗}) is stably finitely generated free and inherits a preferred stable U-basis.

\underline{\underline{Proof}} : combine Lemma 4.57 and Lemma 4.18.

Now we can prove the following version of Lemma 4.19

Lemma 4.59 If f: X → Y is k-connected for n = 2k or n = 2k + 1, then K_{k}(M̃) is stably finitely generated free and inherits a preferred stable U-basis.

\underline{\underline{Proof}} : The proof is the same as the one of the proof of Lemma 4.19 except that we apply Lemma 4.58 instead of Lemma 4.18.

In the proof that the surgery obstruction is well-defined, one has to show that the U-equivalence classes of basis on the surgery kernels appearing in the short exact sequence (4.35) are compatible with the short exact sequence in the sense of Lemma 4.57. This follows from the following lemma whose elementary proof is left to the reader.

Lemma 4.60 Let 0 → C_{∗} ⟶^{i_{∗}} D_{∗} ⟶^{q_{∗}} E_{∗} → 0 be a U-based exact sequence of U-based finite R-chain complexes. Here U-based exact means that for each p ≥ 0 the U-torsion of the 2-dimensional U-based finite R-chain complex 0 → C_{p} ⟶^{i_{p}} D_{p} ⟶^{q_{p}} E_{p} → 0 vanishes in K_{1}(R)/U. Let r be a fixed integer. Suppose that H_{i}(C_{∗}) = H_{i}(E_{∗}) = 0 for i ≠ r and H^{r+1}(hom_{R}(C_{∗}, V)) = H^{r+1}(hom_{R}(C_{∗}, V)) = 0 holds for any R-module V. Equip H_{r}(C_{∗}) and H_{r}(E_{∗}) with the U-equivalence class of stable basis defined in Lemma 4.58.

Then H_{i}(D_{∗}) = 0 for i ≠ r and H^{r+1}(hom_{R}(D_{∗}, V)) = 0 holds for any R-module V. We obtain a short exact sequence

0 → H_{r}(C_{∗}) ⟶^{H_{p}(i_{∗})} H_{r}(D_{∗}) ⟶^{H_{r}(q_{∗})} H_{r}(E_{∗}) → 0.

The U-equivalence class of stable basis on H_{r}(D_{∗}) obtained from Lemma 4.58 applied to this exact sequence and the U-equivalence class of stable basis on H_{r}(D_{∗}) obtained from Lemma 4.58 applied to D_{∗} agree.

Next we can give the simple version of the surgery obstruction theorem. Notice that simple normal bordism class means that in the definition of normal nullbordisms the pairs (Y, ∂Y), (∂_{0}Y, ∂_{0}Y ∩ ∂_{1}Y) and (∂_{1}Y, ∂_{0}Y ∩ ∂_{1}Y) are required to be simple finite Poincaré pairs and the map ∂_{1}F: ∂_{1}M → ∂_{1}Y is required to be a simple homotopy equivalence.

Theorem 4.61 (Simple Surgery Obstruction Theorem for Manifolds with Boundary) Let (\overline{f}, f) be a normal map of degree one with underlying map (f, ∂f): (M, ∂M) → (X, ∂X) such that (X, ∂X) is a simple finite Poincaré complex ∂f is a simple homotopy equivalence. Put n = dim(M). Then:

1. The simple surgery obstruction depends only on the simple normal bordism class of (\overline{f}, f);

2. Suppose n ≥ 5. Then σ(\overline{f}, f) = 0 in L_{n}^{s}(ℤπ, w) if and only if we can do a finite number of surgery steps on the interior of M leaving the boundary fixed to obtain a normal map (\overline{f′}, f′): T M′ ⊕ \underline{ℝ}^{a′} → ξ′ which covers a simple homotopy equivalence of pairs (f′, ∂f′): (M′, ∂M′) → (X, ∂X) with ∂M′ = ∂M and ∂f′ = ∂f.

4.8 Miscellaneous

A guide for the calculation of the L-groups for finite groups is presented by Hambleton and Taylor [51], where further references are given.

112 CHAPTER 4. THE ALGEBRAIC SURGERY OBSTRUCTION

The Surgery Exact Sequence

Introduction

In this section we introduce the exact surgery sequence (see Theorem 5.12). It is the realization of the surgery program which we have explained in Remark 1.5. The surgery exact sequence is the main theoretical tool in solving the classification problem of manifolds of dimensions greater than or equal to five.

5.1 The Structure Set

Definition 5.1 Let X be a closed oriented manifold of dimension n. We call two orientation preserving simple homotopy equivalences f_{i}: M_{i} → X from closed oriented manifolds M_{i} of dimension n to X for i = 0, 1 equivalent if there exists an orientation preserving diffeomorphism g: M_{0} → M_{1} such that f_{1} ∘ g is homotopic to f_{0}. The simple structure set S_{n}^{s}(X) of X is the set of equivalence classes of orientation preserving simple homotopy equivalences M → X from closed oriented manifolds of dimension n to X. This set has a preferred base point, namely the class of the identity id: X → X.

The simple structure set S_{n}^{s}(X) is the basic object in the study of manifolds which are diffeomorphic to X. Notice that a simple homotopy equivalence f: M → X is homotopic to a diffeomorphism if and only if it represents the base point in S_{n}^{s}(X). A manifold M is oriented diffeomorphic to N if and only if for some orientation preserving simple homotopy equivalence f: M → N the class of [f] agrees with the preferred base point. Some care is necessary since it may be possible that a given orientation preserving simple homotopy equivalence f: M → N is not homotopic to a diffeomorphism although M and N are diffeomorphic. Hence it does not suffice to compute S_{n}^{s}(N), one also has to understand the operation of the group of homotopy classes of simple orientation preserving selfequivalences of N on S_{n}^{s}(N). This can be rather complicated in general. But it will be no problem in the case N = S^{n}, because any orientation preserving selfhomotopy equivalence S^{n} → S^{n} is homotopic to the identity.

There is also a version of the structure set which does not take Whitehead torsion into account.

Definition 5.2 Let X be a closed oriented manifold of dimension n. We call two orientation preserving homotopy equivalences f_{i}: M_{i} → X from closed oriented manifolds M_{i} of dimension n to X for i = 0, 1 equivalent if there is a manifold triad (W; ∂_{0}W, ∂_{1}W) with ∂_{0}W ∩ ∂_{1}W = ∅ and an orientation preserving homotopy equivalence of triads (F; ∂_{0}F, ∂_{1}F): (W; ∂_{0}W, ∂_{1}W) → (X × [0, 1]; X × {0}, X × {1}) together with orientation preserving diffeomorphisms g_{0} : M_{0} → ∂_{0}W and g_{1} : M_{1}^{−} → ∂_{1}W satisfying ∂_{i}F ∘ g_{i} = f_{i} for i = 0, 1. (Here M_{1}^{−} is obtained from M_{1} by reversing the orientation.) The structure set S_{n}^{h}(X) of X is the set of equivalence classes of orientation preserving homotopy equivalences M → X from a closed oriented manifold M of dimension n to X. This set has a preferred base point, namely the class of the identity id: X → X.

Remark 5.3 If we require in Definition 5.2 the homotopy equivalences F, f_{0} and f_{1} to be simple homotopy equivalences, we get the simple structure set S_{n}^{s}(X) of Definition 5.1, provided that n ≥ 5. We have to show that the two equivalence relations are the same. This follows from the s-Cobordism Theorem 1.1. Namely, W appearing in Definition 5.2 is an h-cobordism and is even an s-cobordism if we require F, f_{0} and f_{1} to be simple homotopy equivalences (see Theorem 2.1). Hence there is a diffeomorphism Φ: ∂_{0}W × [0, 1] → W inducing the obvious identification ∂_{0}W × {0} → ∂_{0}W and some orientation preserving diffeomorphism ϕ_{1} : (∂_{0}W)^{−} = (∂_{0}W × {1})^{−} → ∂_{1}W. Then ϕ: M_{0} → M_{1} given by g_{1}^{−1} ∘ ϕ_{1} ∘ g_{0} is an orientation preserving diffeomorphism such that f_{1} ∘ ϕ is homotopic to f_{0}. The other implication is obvious.

Remark 5.4 As long as we are dealing with smooth manifolds, there is in general no canonical group structure on the structure set.

5.2 Realizability of Surgery Obstructions

In this section we explain that any element in the L-groups can be realized as the surgery obstruction of a normal map (\overline{f}, f): T M ⊕ \underline{ℝ^{a}} → ξ covering a map f: (M, ∂M) → (Y, ∂Y) if we allow M to have non-empty boundary ∂M.

Theorem 5.5 Suppose n ≥ 5. Consider a connected compact oriented manifold X possibly with boundary ∂X. Let π be its fundamental group and let w: π → {±1} be its orientation homomorphism. Consider an element x ∈ L_{n}(ℤπ, w).

Then we can find a normal map of degree one

(\overline{f}, f): T M ⊕ \underline{ℝ^{a}} → T X × [0, 1] ⊕ \underline{ℝ^{a}}

covering a map of triads

f = (f; ∂_{0}f, ∂_{1}f): (M; ∂_{0}M, ∂_{1}M) → (X × [0, 1], X × {0} ∪ ∂X × [0, 1], X × {1})

with the following properties:

1. ∂_{0}f is a diffeomorphism and \overline{f}|_{∂_{0}M} is given by T (∂f_{0}) ⊕ id\underline{ℝ^{a+1}};

2. ∂_{1}f is a homotopy equivalence;

3. The surgery obstruction σ(\overline{f}, f) in L_{n}(ℤπ, w) (see Theorem 4.48) is the given element x.

The analogous statement holds for x ∈ L_{n}^{s}(ℤπ, w) if we require ∂_{1}f to be a simple homotopy equivalence and we consider the simple surgery obstruction (see (4.56)).

\underline{\underline{Proof}} : We give at least the idea of the proof in the case n = 2k, more details can be found in [119, Theorem 5.3 on page 53, Theorem 6.5 on page 66].

Recall that the element x ∈ L_{n}(ℤπ, w) is represented by a non-degenerate (−1)^{k}-quadratic form (P, ψ) with P a finitely generated free ℤπ-module. We fix such a representative and write it as a triple (P, μ, λ) as explained in Subsection 4.2.2. Let {b_{1}, b_{r}, …, b_{r}} be a ℤπ-basis for P.

Choose r disjoint embeddings j_{i}: D^{2k−1} → X − ∂X into the interior of X for i = 1, 2, …, r. Let F^{0} : S^{k−1} × D^{k} → D^{2k−1} be the standard embedding. Define embeddings F_{i}^{0} : S^{k−1} × D^{k} → X by the composition of the standard embedding F^{0} and j_{i} for i = 1, 2, …, r. Let f_{i}^{0} : S^{k−1} → X be the restriction of F_{i} to S^{k−1} × {0} ⊂ S^{k−1} × D^{k}. Fix base points b ∈ X − ∂X and s ∈ S^{k−1} and paths w_{i} from b to f_{i}^{0}(s) for i = 1, 2, …, r. Thus each f_{i}^{0} is a pointed immersion. Next we construct regular homotopies η_{i}: S^{k−1} → X from f_{i}^{0} to a new embedding f_{i}^{1} which are modelled upon (P, μ, λ). These regular homotopies define associated immersions η′_{i}: S^{k−1} × [0, 1] → X × [0, 1], (x, t) ↦ (η_{i}(x), t). Since these immersions η′_{i} are embeddings on the boundary, we can define their intersection and selfintersection number as in (4.1) and (4.6).

We want to achieve that the intersection number of η′_{i} and η′_{j} for i < j is λ(b_{i}, b_{j}) and the selfintersection number μ(η′_{i}) is μ(b_{i}) for i = 1, 2, …, r. Notice that these numbers are additive under stacking regular homotopies together. Hence it suffices to explain how to introduce a single intersection with value ±g between η′_{i} and η′_{j} for i < j or a selfintersection for η′_{i} with value ±g for i ∈ {1, 2, …, r} without introducing other intersections or selfintersections. We explain the construction for introducing an intersection between η′_{i} and η′_{j} for i < j, the construction for a selfintersection is analogous. This construction will only change η_{i}, the other regular homotopies η_{j} for j ≠ i will be unchanged.

Join f_{i}^{0}(s) and f_{j}^{0}(s) by a path v such that v is an embedding, does not meet any of the f_{i}^{0}-s and the composition w_{j} ∗ v ∗ w_{i}^{−} represents the given element g ∈ π. Now there is an obvious regular homotopy from f_{i}^{0} to another embedding along the path v inside a small tubular neighborhood N(v) of v which leaves f_{i}^{0} fixed outside a small neighborhood U(s) of s and moves f_{i}^{0}|_{U(s)} within this small neighborhood N(v) along v very close to f_{i}^{1}(s). Now use a disc D^{k} which meets f_{j}^{0} transversally at the origin to move f_{i}^{0} further, thus introducing the desired intersection with value ±g. Inside the disc the move looks like pushing the upper hemisphere of the boundary down to the lower hemisphere of the boundary, thus having no intersection with the origin at any point of time with one exception. One has to check that there is enough room to realize both possible signs.

Since f_{i}^{1} is regularly homotopic to f_{i}^{0} which is obtained from the trivial embeddings F_{i}^{0}, we can extend f_{i}^{1} to an embedding F_{i}^{1} : S^{k−1} × D^{k} → X − ∂X. Now attach for each i ∈ {1, 2 …, r} a handle (F_{i}^{1}) = D^{k} × D^{k} to X × [0, 1] by F_{i}^{1} × {1}: S^{k−1} × D^{k} → X × {1}. Let M be the resulting manifold. We obtain a manifold triad (M; ∂_{0}M, ∂_{1}M) if we put ∂_{0}M = X ×0∪∂X ×[0, 1] and ∂_{1}M = ∂M − int(∂_{0}X). There is an extension of the identity id: X × [0, 1] → X ×[0, 1] to a map f: M → X ×[0, 1] which induces a map of triads (f; ∂_{0}f, ∂_{1}f) such that ∂_{0}f is a diffeomorphism. This map is covered by a bundle map \overline{f}: T M ⊕ \underline{ℝ^{a}} → T X ⊕ \underline{ℝ}^{a} which is given on ∂_{0}M by the differential of ∂_{0}f. These claims follow from the fact that f_{i}^{1} is regularly homotopic to f_{i}^{0} and the f_{i}^{0} were trivial embeddings so that we can regard this construction as surgery on the identity map X → X. The kernel K_{k}(M̃) = K_{k}(M̃, { \atop X × [0, 1]}) has a preferred basis corresponding to the cores D^{k} × {0} ⊂ D^{k} × D^{k} of the new handles (F_{i}^{1}). These cores can be completed to immersed spheres S_{i} by adjoining the images of the η′_{i} in X × [0, 1] and finally the obvious discs D^{k} in X × {0} whose boundaries are given by f_{i}^{0}(S^{k−1}). Then the intersection of S_{i} with S_{j} is the same as the intersection of η′_{i} with η′_{j} and the selfintersection of S_{i} is the selfintersection of η′_{i} with itself. Hence by construction λ(S_{i}, S_{j}) = λ(b_{i}, b_{j}) for i < j and the selfintersection number μ(S′_{i}) is μ(b_{i}) for i = 1, 2, …, r. This implies λ(S_{i}, S_{j}) = λ(b_{i}, b_{j}) for i, j ∈ {1, 2, …, r} (see Subsection 4.2.2). The isomorphism P → P^{∗} associated to the given non-degenerate (−1)^{k}-quadratic form can be identified with K_{k}(M̃) → K_{k}(M̃, { \atop ∂_{1}M}) if we use for K_{k}(M̃, { \atop ∂_{1}M}) the basis given by the cocores of the handles (F_{i}^{1}). Hence K_{k}({ \atop ∂_{1}M}) = 0 and we conclude that ∂_{1}f is a homotopy equivalence. By definition and construction σ(\overline{f}, f) in L_{n}(ℤπ, w) is the class of the non-degenerated (−1)^{k}-quadratic form (P, μ, λ). This finishes the outline of the proof of Theorem 5.5.

Remark 5.6 It is not true that for any closed oriented manifold N of dimension n with fundamental group π and orientation homomorphism w: π → {±1} and any element x ∈ L_{n}(ℤπ, w) there is a normal map (\overline{f}, f): T M ⊕ \underline{ℝ^{a}} → ξ covering a map of closed oriented manifolds f: M → N of degree one such that σ(\overline{f}, f) = x. Notice that in Theorem 5.5 the target manifold X × [0, 1] is not closed. The same remark holds for L_{n}^{s}(ℤπ, w).

5.3 The Surgery Exact Sequence

Now we can establish one of the main tools in the classification of manifolds, the surgery exact sequence. For this purpose we have to extend the Definition 3.50 of a normal map for closed manifolds to manifolds with boundary. Let (X, ∂X) be a compact oriented manifold of dimension n with boundary ∂X. We consider normal maps of degree one (\overline{f}, f): T M ⊕ \underline{ℝ^{a}} → ξ with underlying map (f, ∂f): (M, ∂M) → (X, ∂X) for which ∂f: ∂M → ∂X is a diffeomorphism. A normal nullbordism for (\overline{f}, f) consists of a bundle map

(\overline{F} , F): T W ⊕ \underline{ℝ^{b}} → η

with underlying map of manifold triads of degree one

(F; ∂_{0}F, ∂_{1}F): (W; ∂_{0}W, ∂_{1}W) → (X × [0, 1]; X × {0}, ∂X × [0, 1] ∪ X × {1})

together with a diffeomorphism

g: (M, ∂M) → (∂_{0}W, ∂_{0}W ∩ ∂_{1}W)

covered by a bundle map \overline{g}: T M ⊕ \underline{ℝ^{a+b+1}} → T W ⊕ \underline{ℝ^{a+b}} and a bundle isomorphism

(\overline{i_{0}}, i_{0}): ξ ⊕ \underline{ℝ^{b+1}} → η

covering the inclusion i_{0} : X → X ×[0, 1], x ↦ (x, 0) such that the composition (\overline{F} , F)∘(\overline{g}, g) agrees with (\overline{i_{0}}, i_{0})∘(\overline{f} ⊕id\underline{ℝ^{b+1}}, f) and ∂_{1}F: ∂_{1}W → ∂X ×[0, 1]∪ X × {1} is a diffeomorphism. Notice that here the target of the bordism is X × [0, 1], whereas we have allowed in Theorem 4.47 (2) a more general notion of normal bordism, where the target manifold also could vary.

Definition 5.7 Let (X, ∂X) be a compact oriented manifold of dimension n with boundary ∂X. Define the set of normal maps to (X, ∂X)

N_{n}(X, ∂X)

to be the set of normal bordism classes of normal maps of degree one (\overline{f}, f): T M⊕ \underline{ℝ^{a}} → ξ with underlying map (f, ∂f): (M, ∂M) → (X, ∂X) for which ∂f: ∂M → ∂X is a diffeomorphism.

Notice that this definition uses tangential bundle data. One could also use normal bundle data (see Lemma 3.51).

Let X be a closed oriented connected manifold of dimension n ≥ 5. Denote by π its fundamental group and by w: π → {±1} its orientation homomorphism. Let N_{n+1}(X ×[0, 1], X ×{0, 1}) and N_{n}(X) be the set of normal maps of degree one as introduced in Definition 3.47. Let S_{n}^{s}(X) be the structure set of Definition 5.1. Denote by L_{n}^{s}(ℤπ) the simple surgery obstruction group (see Notation 4.55). Denote by

σ: N_{n+1}(X × [0, 1], X × {0, 1}) → L_{n+1}^{s}(ℤπ, w);  (5.8)
σ: N_{n}(X) → L_{n}^{s}(ℤπ, w)  (5.9)

the maps which assign to the normal bordism class of a normal map of degree one its simple surgery obstruction (see (4.56)). This is well-defined by Theorem 4.61 (1). Let

η: S_{n}^{s}(X) → N_{n}(X)  (5.10)

be the map which sends the class [f] ∈ S_{n}^{s}(X) represented by a simple homotopy equivalence f: M → X to the normal bordism class of the following normal map of degree one. Choose a homotopy inverse f^{−1} : X → M and a homotopy h: id_{M} ≃ f^{−1} ∘ f. Put ξ = (f^{−1})^{∗}T M. Up to isotopy of bundle maps there is precisely one bundle map (\overline{h}, h): T M × [0, 1] → T M covering h: M × [0, 1] → M whose restriction to T M × {0} is the identity map T M × {0} → T M. The restriction of \overline{h} to X × \{1\} induces a bundle map \overline{f}: T M → ξ covering f: M → X. Put η([f]) := [(\overline{f}, f)]. One easily checks that the normal bordism class of (\overline{f}, f) depends only on [f] ∈ N_{n}^{s}(X) and hence that η is well-defined. Next we define an action of the abelian group L_{n+1}^{s}(ℤπ, w) on the structure set S_{n}^{s}(X)

ρ: L_{n+1}^{s}(ℤπ, w) × S_{n}^{s}(X) → S_{n}^{s}(X).  (5.11)

Fix x ∈ L_{n+1}^{s}(ℤπ, w) and [f] ∈ N_{n}^{s}(X) represented by a simple homotopy equivalence f: M → X. By Theorem 5.5 we can find a normal map (\overline{F} , F) covering a map of triads (F; ∂_{0}F, ∂_{1}F): (W; ∂_{0}W, ∂_{1}W) → (M ×[0, 1], M ×{0}, M ×{1}) such that ∂_{0}F is a diffeomorphism and ∂_{1}F is a simple homotopy equivalence and σ(\overline{F} , F) = x. Then define ρ(x, [f]) by the class [f ∘ ∂_{1}F: ∂_{1}W → X]. We have to show that this is independent of the choice of (\overline{F} , F). Let (\overline{F′}, F′) be a second choice. We can glue W′ and W^{−} together along the diffeomorphism (∂_{0}F)^{−1} ∘ ∂_{0}F′: ∂_{1}W′ → ∂_{1}W and obtain a normal bordism from (\overline{F} |_{∂_{1}W} , ∂_{1}F) to (\overline{F′}|_{∂_{1}W′}, ∂_{1}F′). The simple obstruction of this normal bordism is

σ(\overline{F′}, F′) − σ(\overline{F} , F) = x − x = 0.

Because of Theorem 4.61 (2) we can perform surgery relative boundary on this normal bordism to arrange that the reference map from it to X × [0, 1] is a simple homotopy equivalence. In view of Remark 5.3 this shows that f ∘ ∂_{1}F and f ∘ ∂_{1}F′ define the same element in S_{n}^{s}(X). One easily checks that this defines a group action since the surgery obstruction is additive under stacking normal bordisms together. The next result is the main result of this chapter and follows from the definitions and Theorem 4.61 (2)

Theorem 5.12 (The surgery Exact Sequence) Under the conditions and in the notation above the so called surgery sequence

N_{n+1}(X × [0, 1], X × {0, 1}) ⟶^{σ} L_{n+1}^{s}(ℤπ, w) ⟶^{∂} S_{n}^{s}(X) ⟶^{η} N_{n}(X) ⟶^{σ} L_{n}^{s}(ℤπ, w)

is exact for n ≥ 5 in the following sense. An element z ∈ N_{n}(X) lies in the image of η if and only if σ(z) = 0. Two elements y_{1}, y_{2} ∈ S_{n}^{s}(X) have the same image under η if and only if there exists an element x ∈ L_{n+1}^{s}(ℤπ, w) with ρ(x, y_{1}) = y_{2}. For two elements x_{1}, x_{2} in L_{n+1}^{s}(ℤπ) we have ρ(x_{1}, [id: X → X]) = ρ(x_{2}, [id: X → X]) if and only if there is u ∈ N_{n+1}(X ×[0, 1], X ×{0, 1}) with σ(u) = x_{1} − x_{2}.

There is an analogous surgery exact sequence

N_{n+1}^{h}(X × [0, 1], X × {0, 1}) ⟶^{σ} L_{n+1}^{h}(ℤπ, w) ⟶^{∂} S^{h}(X) ⟶^{η} N_{n}(X) ⟶^{σ} L_{n}^{h}(ℤπ, w)

where S^{h}(X) is the structure set of Definition 5.2 and L_{n}^{h}(ℤπ, w) := L_{n}(ℤπ, w) have been introduced in Definitions 4.28 and 4.42.

Remark 5.13 There is no group structure known for S_{s}^{h}(X) and N_{n}(X) such that S_{s}^{h}(X) ⟶^{η} N_{n}(X) ⟶^{σ} L_{n}^{h}(ℤπ, w) is a sequence of groups (and analogous for the simple version). Notice that the composition (see Theorem 3.52)

[X; G/O] ≅ N_{n}(X) ⟶^{σ} L_{n}^{h}(ℤπ, w)

is a map whose source and target come with canonical group structures but it is not a homomorphism of abelian groups in general. These difficulties are connected to the existence of homotopy spheres and can be overcome in the topological category.

The surgery sequence of Theorem 5.12 can be extended to infinity to the left. In the range far enough to the left it is a sequence of abelian groups.

5.4 Miscellaneous

One can also develop surgery theory in the P L-category or in the topological category [59]. This requires to carry over the notions of vector bundles and tangent bundles to these categories. There are analogs of the sets of normal invariants N_{n}^{P L}(X) and N_{n}^{T OP} (X) and the structure sets S_{n}^{P L,h}(X), S_{n}^{P L,s}(X) S_{n}^{T OP,h}(X) and S_{n}^{T OP,s}(X). There are analogs P L and T OP of the group O. Theorem 3.52 and Theorem 3.48 (see also Remark 3.49) carry over to the P Lcategory and the topological category.

Theorem 5.14 Let X be a connected finite n-dimensional Poincaré complex. Suppose that N_{n}^{P L}(X) is non-empty. Then there is a canonical group structure on the set [X, G/P L] of homotopy classes of maps from X to G/P L and a transitive free operation of this group on N_{n}^{P L}(X). The analogous statement holds for T OP instead of P L.

There are analogs of the surgery exact sequence (see Theorem 5.12) for the P L-category and the topological category.

Theorem 5.15 (The surgery Exact Sequence) There is a surgery sequence

N_{n+1}^{P L}(X × [0, 1], X × {0, 1}) ⟶^{σ} L_{n+1}^{s}(ℤπ, w) ⟶^{∂} S_{n}^{P L,s}(X)

⟶^{η} N_{n}^{P L}(X) ⟶^{σ} L_{n}^{s}(ℤπ, w)

which is exact for n ≥ 5 in the sense of Theorem 5.12. There is an analogous surgery exact sequence

N_{n+1}^{P L}(X × [0, 1], X × {0, 1}) ⟶^{σ} L_{n+1}^{h}(ℤπ, w) ⟶^{∂} S_{n}^{P L,h}(X)

⟶^{η} N_{n}^{P L}(X) ⟶^{σ} L_{n}^{h}(ℤπ, w)

The analogous sequences exists in the topological category.

Notice that the surgery obstruction groups are the same in the smooth category, P L-category and in the topological category. Only the set of normal invariants and the structure sets are different. The set of normal invariants in the smooth category, P L-category or topological category do not depend on the decoration h and s, whereas the structure sets and the surgery obstruction groups depend on the decoration h and s. In particular the structure set depends on both the choice of category and choice of decoration.

As in the smooth setting the surgery sequence above can be extended to infinity to the left.

Some interesting constructions can be carried out in the topological category which do not have smooth counterparts. An algebraic surgery sequence is constructed in [98, §14, §18] and identified with the geometric surgery sequence above in the topological category. Moreover, in the topological situation one can find abelian group structures on S_{s}^{h}(X) and N_{n}(X) such that the surgery sequence becomes a sequence of abelian groups. The main point is to find the right addition on G/T OP. Given a finite Poincaré complex X of dimension ≥ 5, a single obstruction, the so called total surgery obstruction, is constructed in [98, §17]. It vanishes if and only if X is homotopy equivalent to a closed topological manifold. It combines the two stages of the obstruction we have seen before, namely, the problem whether the Spivak normal fibration has a reduction to a T OP -bundle and whether the surgery obstruction of the associated normal map is trivial.

The computation of the homotopy type of the space G/O and G/P L due to Sullivan is explained in detail in [78, Chapter 4]. One obtains homotopy equivalences

G/T OP [\frac{1}{2}] ≃ BO [\frac{1}{2}];
G/T OP_{(2)} ≃ ∏_{j≥1} K(ℤ_{(2)}, 4j) × ∏_{j≥1} K(ℤ/2, 4j − 2),

where K(A; l) denotes the Eilenberg-MacLane space of type (A, l), i.e. a CW - complex such that π_{n}(K(A, l)) is trivial for n ≠ l and is isomorphic to A if n = 2l, _{(2)} stands for localizing at (2), i.e. all primes except 2 are inverted, and [\frac{1}{2}] stands for localization of 2, i.e. 2 is inverted. In particular we get for a space X isomorphisms

[X, G/T OP] [\frac{1}{2}] ≅ K̃O^{0}(X) [\frac{1}{2}];
[X, G/T OP]_{(2)} ≅ ∏_{j≥1} H^{4j}(X; ℤ_{(2)}) × ∏_{j≥1} H^{4j−2}(M; ℤ/2),

where KO^{∗} is K-theory of real vector bundles.

Kirby and Siebenmann [59, Theorem 5.5 in Essay V on page 251] (see also [101]) have proved

Theorem 5.16 The space T OP/P L is an Eilenberg MacLane space of type (ℤ/2, 3).

More information about the the spaces O, P L, G, G/O and G/P L will be given in Section 6.6.

Finally we mention that a different approach to surgery has been developed by Kreck. A survey about his approach is given in [60]. Its advantage is that one does not have to get a complete homotopy classification first. The price to pay is that the L-groups are much more complicated, they are not necessarily abelian groups any more. This approach is in particular successful when the manifolds under consideration are already highly connected. See for instance [61], [62], [112].

122 CHAPTER 5. THE SURGERY EXACT SEQUENCE

Homotopy Spheres

Introduction

Recall that S^{n} is the standard sphere S^{n} = {x ∈ ℝ^{n+1} | ||x|| = 1}. We will equip it with the structure of a smooth manifold for which the canonical inclusion S^{n} → ℝ^{n+1} is an embedding of smooth manifolds. We use the orientation on S^{n} which is compatible with the isomorphism T_{x}S^{n} ⊕ ν(S^{n}, ℝ^{n}) ⟶^{≅} T_{x}ℝ^{n}, where we use on ν_{x}(S^{n}, ℝ^{n}) the orientation coming from the normal vector field pointing to the origin and on T_{x}ℝ^{n} the standard orientation. This agrees with the convention that S^{n} = ∂D^{n+1} inherits its orientation from D^{n+1}. A homotopy n-sphere Σ is a closed oriented n-dimensional smooth manifold which is homotopy equivalent S^{n}. The Poincaré Conjecture says that any homotopy n-sphere Σ is oriented homeomorphic to S^{n} and is known to be true for all dimensions except n = 3. In this chapter we want to solve the problem how many oriented diffeomorphism classes of homotopy n-spheres exist for given n. For n ≠ 3 this is the same as determining how many different oriented smooth structures exist on S^{n}. This is a beautiful and illuminating example. It shows how the general surgery methods which we have developed so far apply to a concrete problem. It illustrates what kind of input from homotopy theory and algebra is needed for the final solution.

The following theorem summarizes what we will prove in this chapter. Here Θ^{n} denotes the abelian group of oriented h-cobordism classes of oriented homotopy n-spheres, bP^{n+1} ⊂ Θ^{n} is the subgroup of those homotopy n-spheres which bound a stably parallelizable compact manifold, J_{n}: π_{n}(SO) → π_{n}^{s} denotes the J-homomorphism and B_{n} is the n-th Bernoulli number. These notions and the proof of the next result will be presented in this chapter (see Theorem 6.39, Corollary 6.43, Theorem 6.44, Theorem 6.46 and Theorem 6.56).

Theorem 6.1 (Classification of Homotopy Spheres) 1. Let k ≥ 2 be an integer. Then bP^{4k} is a finite cyclic group of order

\frac{3 − (−1)^{k}}{2} · 2^{2k−2} · (2^{2k−1} − 1) · numerator(B_{k}/(4k));

2. Let k ≥ 1 be an integer. Then bP^{4k+2} is trivial or isomorphic to ℤ/2. We have

bP^{4k+2} = \{ \begin{matrix}ℤ/2 & 4k + 2 ≠ 2^{l} − 2, k ≥ 1; \\ 0 & 4k + 2 ∈ \{6, 14, 30, 62\}. \end{matrix}

3. If n = 4k + 2 for k ≥ 2 , then there is an exact sequence

0 → Θ^{n} → coker(J_{n}) → ℤ/2.

If n = 4k for k ≥ 2 or n = 4k + 2 with 4k + 2 ≠ 2^{l} − 2, then

Θ^{n} ≅ coker(J_{n});

4. Let n ≥ 5 be odd. Then there is an exact sequence

0 → bP^{n+1} → Θ^{n} → coker(J_{n}) → 0.

If n ≠ 2^{l} − 3, the sequence splits.

The following table taken from [58, pages 504 and 512] gives the orders of the finite groups Θ^{n}, bP^{n+1} and Θ^{n}/bP^{n+1} as far as they are known in low dimensions. The values in dimension 1 and 2 come from obvious adhoc computations. The computation of Θ^{4} requires some additional analysis which we will not present here. Notice that Θ^{4} = 1 does not mean that any homotopy 4-sphere is diffeomorphic to S^{4} (see Lemma 6.2).

\begin{matrix}n & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 \\ Θ^{n} & 1 & 1 & ? & 1 & 1 & 1 & 28 & 2 & 8 & 6 & 992 & 1 & 3 & 2 & 16256 \\ bP^{n+1} & 1 & 1 & ? & 1 & 1 & 1 & 28 & 1 & 2 & 1 & 992 & 1 & 1 & 1 & 8128 \\ Θ^{n}/bP^{n+1} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 2 & 4 & 6 & 1 & 1 & 3 & 2 & 2 \end{matrix}

The basic paper about homotopy spheres is the one of Kervaire and Milnor [58] which contains a systematic study and can be viewed as the beginning of surgery theory. Nearly all the results presented here are taken from this paper. Another survey article about homotopy spheres has been written by Lance [66] and Levine [68].

6.1 The Group of Homotopy Spheres

Define the n-th group of homotopy spheres Θ^{n} as follows. Elements are oriented h-cobordism classes [Σ] of oriented homotopy n-spheres Σ, where Σ and Σ′ are called oriented h-cobordant if there is an oriented h-cobordism (W, ∂_{0}W, ∂_{1}W) together with orientation preserving diffeomorphisms Σ → ∂_{0}W and (Σ′)^{−} → ∂_{1}W. The addition is given by the connected sum. The zero element is represented by S^{n}. The inverse of [Σ] is given by [Σ^{−}], where Σ^{−} is obtained from Σ by reversing the orientation.

Obviously Θ^{n} becomes an abelian semi-group by the connected sum. It remains to check that [Σ^{−}] is an inverse of [Σ]. It is easy to see that for a homotopy n-sphere Σ that there is an h-cobordism W from the connected sum Σ♯Σ^{−} to S^{n}. Let D^{n} ⊂ Σ be an embedded disk. Let Σ − int(D^{n}) be the manifold with boundary obtained by removing this disk. Delete the interior of an embedded disk D^{n+1} ⊂ (Σ − int(D^{n})) × [0, 1]. The result is the desired h-cobordism W.

The fundamental group π of a homotopy n-sphere is trivial for n ≥ 2. This implies that the orientation homomorphism w is always trivial and Wh(π) = {0}. Hence it does not matter whether we work with simple homotopy equivalences or homotopy equivalence since S_{n}^{s}(S^{n}) = S_{n}^{h}(S^{n}) for n ≥ 5 and L_{n}^{s}(ℤπ) = L_{n}^{h}(ℤπ) = L_{n}(ℤ) holds. Therefore we will omit the decoration h or s for the remainder of this chapter.

Lemma 6.2 Let \overline{Θ}^{n} be the set of oriented diffeomorphism classes [Σ] of oriented homotopy n-spheres Σ. The forgetful map

f: \overline{Θ^{n}} → Θ^{n}

is bijective for n ≠ 3, 4.

\underline{\underline{Proof}} : This follows from the s-cobordism Theorem 1.1 for n ≥ 5 and by obvious adhoc computations for n ≤ 2.

Lemma 6.3 There is a natural bijection

α: S_{n}^{s}(S^{n}) ⟶^{≅} \overline{Θ}^{n} [f: M → S^{n}] ↦ [M].

If n ≠ 3, there is an obvious bijection

\{smooth oriented structures on S^{n}\}/oriented diffeomorphic ⟶^{≅} \overline{Θ}^{n}.

\underline{\underline{Proof}} : For any homotopy n-sphere Σ there is up to homotopy precisely one map f: Σ → S^{n} of degree one. For n ≠ 3 the Poincaré Conjecture is true which says that any homotopy n-sphere is homeomorphic to S^{n}.

In dimension 3 there is no difference between the topological and smooth category. Any closed topological 3-manifold M carries a unique smooth structure. Hence any manifold which is homeomorphic to S^{3} is automatically diffeomorphic to S^{3}. The Poincaré Conjecture is at the time of writing open in dimension 3. It is not known whether any closed 3-manifold which is homotopy equivalent to S^{3} is homeomorphic to S^{3}.

Definition 6.4 A manifold M is called stably parallelizable if T M ⊕ \underline{ℝ^{a}} is trivial for some a ≥ 0.

Definition 6.5 Let bP^{n+1} ⊂ Θ^{n} be the subset of elements [Σ] for which Σ is oriented diffeomorphic to the boundary ∂M of a stably parallelizable compact manifold M.

Lemma 6.6 The subset bP^{n+1} ⊂ Θ^{n} is a subgroup of Θ^{n}. It is the preimage under the composition

Θ^{n} ⟶^{(f∘α)^{−1}} S_{n}(S^{n}) ⟶^{η} N_{n}(S^{n})

of the base point [id: T S^{n} → T S^{n}] in N_{n}(S^{n}), where f is the bijection of Lemma 6.2 and α is the bijection of Lemma 6.3.

\underline{\underline{Proof}} : Suppose that Σ bounds W and Σ′ bounds W′ for stably parallelizable manifolds W and W′. Then the boundary connected sum W ♯W′ is stably parallelizable and has Σ♯Σ′ as boundary. This shows that bP^{n+1} ⊂ Θ^{n+1} is a subgroup.

Consider an element [f: Σ → S^{n}] in η^{−1}([id: T S^{n} → T S^{n}]). Then there is a normal bordism from a normal map (\overline{f}, f): T Σ⊕\underline{ℝ^{a}} → ξ covering f: Σ → S^{n} to the normal map id: T S^{n} → T S^{n}. This normal bordism is given by a bundle map (\overline{F} , F): T W ⊕\underline{ℝ^{a+b}} → η covering a map of triads (F; ∂_{0}F, ∂_{1}F): (W; ∂_{0}W, ∂_{1}W) → (S^{n}×[0, 1]; S^{n}×\{0\}, S^{n}×\{1\}), and bundle isomorphisms (\overline{u}, u): T Σ⊕\underline{ℝ^{a+b+1}} → T W ⊕ \underline{ℝ^{b}} covering an orientation preserving diffeomorphism u: Σ → ∂_{0}W, (\overline{u′}, u′): T S^{n} ⊕ \underline{ℝ^{a+b+1}} → T W ⊕ \underline{ℝ^{b}} covering an orientation preserving diffeomorphism u′: (S^{n})^{−} → ∂_{1}W, (\overline{v}, v): ξ ⊕ \underline{ℝ^{b+1}} → η covering the obvious map v: S^{n} → S^{n} × \{0\} and (\overline{v′}, v′): T S^{n} ⊕ \underline{ℝ^{a+b+1}} → η covering the obvious map v′: S^{n} → S^{n} × \{1\} such that (\overline{F} , F) ∘ (\overline{u}, u) = (\overline{v}, v) ∘ (\overline{f} ⊕ id\underline{ℝ^{b+1}}) and (\overline{F} , F) ∘ (\overline{u′}, u′) = (\overline{v′}, v′) holds. Then D^{n+1} ∪_{u′} W is a manifold whose boundary is oriented diffeomorphic to Σ by u and for which the bundle data above yield a stable isomorphism T W ⊕ \underline{ℝ^{a+b}} → \underline{ℝ^{n+1+a+b}}. Hence [Σ] lies in bP^{n+1}.

Conversely, consider [Σ] such that there exists a stably parallelizable manifold W together with an orientation preserving diffeomorphism u: Σ → ∂W. We can assume without loss of generality that W is connected. Choose an orientation preserving homotopy equivalence f: Σ → S^{n}. We can extend ∂_{0}F := f ∘ u^{−1} : ∂W → S^{n} to a smooth map F: W → D^{n+1}. Since f has degree one, the map (F, ∂F): (W, ∂W) → (D^{n+1}, S^{n}) has degree one. Let y ∈ D^{n+1} − S^{n} be a regular value. Then the degree of F is the finite sum ∑_{x∈F^{−1}(y)} ϵ(x), where ϵ(x) = 1, if T_{x}F: T_{x}W → T_{y}D^{n} preserves the induced orientations and ϵ(x) = −1 otherwise. If two points x_{1} and x_{2} in F^{−1}(y) satisfy ϵ(x_{1}) ≠ ϵ(x_{2}), one can change F up to homotopy relative ∂W such that F^{−1}(y) contains two points less than before. Thus one can arrange that F^{−1}(y) consists of precisely one point x and that T_{x}F: T_{x}W → T_{x}D^{n+1} is orientation preserving. Then one can change F up to homotopy in a small neigborhood of x such that there is an embedded disk D_{o}^{n+1} ⊂ W − ∂W such that F induces a diffeomorphism D_{0}^{n+1} → F (D_{0}^{n+1}) and no point outside D_{0}^{n+1} is mapped to F (D_{0}^{n+1}). Define V = W − int(D_{0}^{n+1}). Then F induces a map also denoted by F: V → D^{n+1} − F (D_{0}^{n+1}). If we identify D^{n+1} − F (D_{0}^{n+1}) with S^{n} × [0, 1] by an orientation preserving diffeomorphism, we obtain a map of triads (F; ∂_{0}F, ∂_{1}F): (V; ∂_{0}V, ∂_{1}V) → (S^{n+1} × [0, 1], S^{n} × {0}, S^{n} × {1}) together with diffeomorphisms u: Σ → ∂_{0}V, v: S^{n} → S^{n} × {0}, u′: S^{n} → ∂_{0}V, v′: S^{n} → S^{n} × {1} such that ∂_{1}F is an orientation preserving diffeomorphism and F ∘ u = v ∘ f and F ∘ u′ = v′. Now one covers everything with appropriate bundle data to obtain a normal bordism from (\overline{f}, f): T Σ ⊕ \underline{ℝ^{a}} → ξ to id: T S^{n} → T S^{n}. This shows η ∘ α^{−1} ∘ f^{−1}([Σ]) = [id: T S^{n} → T S^{n}].

6.2 The Surgery Sequence for Homotopy Spheres

In this section we examine the surgery sequence (see Theorem 5.12) in the case of the sphere. In contrast to the general case we will obtain a long exact sequence of abelian groups. We have to introduce the following bordism groups.

Definition 6.7 A stable framing of a closed oriented manifold M of dimension n is a strong bundle isomorphism \overline{u}: T M ⊕ \underline{ℝ^{a}} ⟶^{≅} \underline{ℝ^{n+a}} for some a ≥ 0 which is compatible with the given orientation. (Recall that strong means that \overline{u} covers the identity.) An almost stable framing of a closed oriented manifold M of dimension n is a choice of a point x ∈ M together with a strong bundle isomorphism \overline{u}: T M|_{M−\{x\}} ⊕ \underline{ℝ^{a}} ⟶^{≅} \underline{ℝ^{n+a}} for some a ≥ 0 which is compatible with the given orientation on M − {x}.

Of course any stably framed manifold is in particular an almost stably framed manifold. A homotopy n-sphere Σ is an almost stably framed manifold since for any point x ∈ Σ the complement Σ − {x} is contractible and hence T Σ|_{Σ−{x}} is trivial. We will later show the non-trivial fact that any homotopy n-sphere is stably parallelizable, i.e. admits a stable framing. The standard sphere S^{n} inherits its standard stable framing from its embedding to ℝ^{n+1}.

A stably framed nullbordism for a stably framed manifold (M, \overline{u}) is a compact manifold W with a stable framing \overline{U}: T W ⊕ \underline{ℝ^{a+b}} ⟶^{≅} \underline{ℝ^{n+1+a+b}} and a bundle isomorphism (\overline{v}, v): T M ⊕ \underline{ℝ^{a+1+b}} ⟶^{≅} T W ⊕ \underline{ℝ^{a+b}} coming from the differential of an orientation preserving diffeomorphism v: M → ∂W such that \overline{U} ∘ \overline{v} = \overline{u} ⊕ id\underline{ℝ^{b+1}}. Now define the notion of a stably framed bordism from a stably framed manifold (M, \overline{u}) to another stably framed manifold (M′, \overline{u′}) to be a stably framed nullbordism for the disjoint union of (M^{−}, \overline{u}^{−}) and (M′, \overline{u′}), where M^{−} is obtained from M by reversing the orientation and \overline{u}^{−} is the composition

\overline{u}^{−}: T M ⊕ \underline{ℝ^{a}} ⟶^{\overline{u}} \underline{ℝ^{a+n}} = \underline{ℝ} ⊕ \underline{ℝ^{a+n−1}} ⟶^{− id\underline{ℝ} ⊕ id\underline{ℝ^{a+n−1}}} \underline{ℝ} ⊕ \underline{ℝ^{a+n−1}} = \underline{ℝ^{a+n}}.

Consider two almost stably framed manifolds (M, x, \overline{u}: T M|_{M−\{x\}} ⊕ \underline{ℝ^{a}} ⟶^{≅}

\underline{ℝ^{a+n}}) and (M′, x′, \overline{u′}: T M′|_{M′−\{x′\}} ⊕ \underline{ℝ^{a′}} ⟶^{≅} \underline{ℝ^{a′+n}}). An almost stably framed

bordism from the first to the second consists of the following data. There is a compact oriented (n+1)-dimensional manifold triad (W; ∂_{0}W, ∂_{1}W) with ∂_{0}W ∩ ∂_{1}W = ∅ together with an embedding j: ([0, 1]; {0}, {1}) → (W; ∂_{0}W, ∂_{1}W) such that j is transversal at the boundary. We also need a strong bundle isomorphism \overline{U}: T W_{W −im(j)} ⊕ \underline{ℝ^{b}} ⟶^{≅} \underline{ℝ^{n+1+b}} for some b ≥ a, a′. Furthermore we re-

quire the existence of a bundle isomorphism (\overline{v}, v): T M⊕\underline{ℝ^{b+1}} ⟶^{≅} T W ⊕\underline{ℝ^{b}} coming from the differential of an orientation preserving diffeomorphism v: M^{−} → ∂_{0}W with v(x) = j(0) and of a bundle isomorphism (\overline{v′}, v′): T M ⊕ \underline{ℝ^{b+1}} ⟶^{≅} T W ⊕ \underline{ℝ^{b}} coming from the differential of an orientation preserving diffeomorphism v′: M′ → ∂_{1}W with v′(x′) = j(1) such that \overline{U} ∘ \overline{v} = \overline{u}^{−} ⊕ id\underline{ℝ^{b−a+1}} and \overline{U} ∘ \overline{v′} = \overline{u′} ⊕ id\underline{ℝ^{b+1−a′}} holds.

Definition 6.8 Let Ω_{n}^{fr} be the abelian group of stably framed bordism classes of stably framed closed oriented manifolds of dimension n. This becomes an abelian group by the disjoint union. The zero element is represented by S^{n} with its standard stable framing. The inverse of the class of (M, \overline{u}) is represented by the class of (M^{−}, \overline{u} ⊕ (− id\underline{ℝ})).

Let Ω_{n}^{alm} be the abelian group of almost stably framed bordism classes of almost stably framed closed oriented manifolds of dimension n. This becomes an abelian group by the connected sum at the preferred base points. The zero element is represented by S^{n} with the base point s = (1, 0, …, 0) with its standard stable framing restricted to S^{n} − \{s\}. The inverse of the class of (M, x, \overline{u}) is represented by the class of (M^{−}, x, \overline{u} ⊕ (− id\underline{ℝ})).

Lemma 6.9 There are canonical bijections of pointed sets

β: N_{n}(S^{n}) ⟶^{≅} Ω_{n}^{alm};
γ: N_{n+1}(S^{n} × [0, 1], S^{n} × {0, 1}) ⟶^{≅} N_{n+1}(S^{n+1}).

\underline{\underline{Proof}} : Consider an element r ∈ N_{n}(S^{n}) represented by a normal map (\overline{f}, f): T M ⊕ \underline{ℝ^{a}} → ξ covering a map of degree one f: M → S^{n}. Since f has degree one, one can change (\overline{f}, f) by a homotopy such that f^{−1}(s) consists of one point x ∈ M for a fixed point s ∈ S^{n}. Since S^{n} − {s} is contractible, ξ_{S^{n}−{s}} admits a trivialization which is unique up to isotopy. It induces together with \overline{f} an almost stable framing on (M, x). The class of (M, x) with this stable framing in Ω_{n}^{alm} is defined to be the image of r under β.

The inverse β^{−1} of β is defined as follows. Let r ∈ Ω_{n}^{alm} be represented by the almost framed manifold (M, x, \overline{u}). Let c: M → S^{n} be the collaps map for a small embedded disk D^{n} ⊂ M with origin x. By construction c induces a diffeomorphism c|_{int(D^{n})} : int(D^{n}) → S^{n} −{s} and maps M −int(D^{n}) to {s} for fixed s ∈ S^{n}. The almost stable framing \overline{u} yields a bundle map \overline{c′}: T M|_{M−\{x\}} ⊕ \underline{ℝ^{a}} → \underline{ℝ^{n+a}} covering c|_{M−\{x\}}: M −\{x\} → S^{n}−\{c(x)\}. Since D^{n} is contractible, we obtain a bundle map unique up to isotopy \overline{c′′}: T D^{n} ⊕ \underline{ℝ^{a}} → \underline{ℝ^{n+a}} covering c|_{D^{n}}: D^{n} → S^{n}. The composition of the inverse of the restriction of \overline{c′′} to int(D^{n}) − \{x\} and of the restriction of \overline{c′} to int(D^{n}) − \{x\} yields a strong bundle automorphism of the trivial bundle \underline{ℝ^{n+a}} over S^{n}−\{s, c(x)\}. Let ξ be the bundle obtained by glueing the trivial bundle \underline{ℝ^{a+n}} over S^{n} −\{s\} and the trivial bundle \underline{ℝ^{a+n}} over S^{n} − \{c(x)\} together using this bundle automorphism over S^{n} − \{s, c(x)\}. Then \overline{c′} and \overline{c′′} fit together to a bundle map \overline{c}: T M ⊕ \underline{ℝ^{n+a}} → ξ covering c. Define the image of r under β^{−1} to be the class of (\overline{c}, c).

Consider r ∈ N_{n+1}(S^{n} × [0, 1], S^{n} × {0, 1}) represented by a normal map (\overline{f}, f): T M ⊕ \underline{ℝ^{a}} → ξ covering (f, ∂f): (M, ∂M) → (S^{n} × [0, 1], S^{n} × \{0, 1\}).

Recall that ∂f is a diffeomorphism. Hence one can form the closed manifold N = M ∪_{∂f: ∂M→S^{n}×{0,1}}D^{n+1}×{0, 1}. The map f and the identity on D^{n+1}×{0, 1} induce a map of degree one g: N → S^{n} × [0, 1] ∪_{S^{n}×{0,1}} D^{n+1} × {0, 1} ≅ S^{n+1}. Define the bundle η over S^{n} × [0, 1] ∪_{S^{n}×{0,1}} D^{n+1} × {0, 1} ≅ S^{n+1} by glueing ξ and T (D^{n+1} × \{0, 1\}) ⊕ \underline{ℝ^{a−1}} together over S^{n} × \{0, 1\} by the strong bundle isomorphism

ξ|_{S^{n−1}×\{0,1\}} ⟶^{(\overline{f}|_{S^{n−1}×\{0,1\}})^{−1}} T M|_{∂M} ⊕ \underline{ℝ^{a}} = T (∂M) ⊕ \underline{ℝ^{a+1}}

⟶^{T ∂f} T (S^{n−1} × \{0, 1\}) ⊕ \underline{ℝ^{a+1}} = T (D^{n+1} × \{0, 1\}) ⊕ \underline{ℝ^{a}}.

Then \overline{f} and id: T (D^{n+1} × \{0, 1\}) ⊕ \underline{ℝ^{a}} → T (D^{n+1} × \{0, 1\}) ⊕ \underline{ℝ^{a}} fit together yielding a bundle map \overline{g}: T N ⊕ \underline{ℝ^{a}} → η covering g. Define the image of r under γ by the class of (\overline{g}, g). We leave it to the reader to construct the inverse of γ which is similar to the construction in Lemma 6.6 but now two embedded discs instead of one embedded disc are removed.

Next we want to construct a long exact sequence of abelian groups

… → Ω_{n+1}^{alm} ⟶^{σ} L_{n+1}(ℤ) ⟶^{∂} Θ^{n} ⟶^{η} Ω_{n}^{alm} ⟶^{σ} L_{n}(ℤ) → …

The map

σ: Ω_{n+1}^{alm} → L_{n+1}(ℤ)

is given by the composition

Ω_{n+1}^{alm} ⟶^{β^{−1}} N_{n+1}(S^{n+1}) ⟶^{σ} L_{n+1}(ℤ),

where β is the bijection of Lemma 6.9 and σ: N_{n}(S^{n}) → L_{n}(ℤ) is given by the surgery obstruction and has already appeared in the surgery sequence (see Theorem 5.12). The map

∂: L_{n+1}(ℤ) → Θ^{n}

is the composition of the inverse of the bijection α ∘ f: S_{n}(S^{n}) ⟶^{≅} Θ^{n} coming from Lemma 6.2 and Lemma 6.3 and the map ∂ : L_{n+1}(ℤ) → S_{n}(S^{n}) of the surgery sequence (see Theorem 5.12). The map

η: Θ^{n} → Ω_{n}^{alm}  (6.10)

sends the class of a homotopy sphere Σ to the class of (Σ, x, \overline{u}), where x is any point in Σ and the stable framing of T Σ|_{Σ−{x}} comes from the fact that Σ−{x} is contractible. This map η corresponds to the map η appearing in the surgery exact sequence (see Theorem 5.12) under the identification α∘f: S_{n}(S^{n}) ⟶^{≅} Θ^{n} coming from Lemma 6.2 and Lemma 6.3.

We leave it to the reader to check that all these maps are homomorphisms of abelian groups. The surgery sequence (see Theorem 5.12) implies Theorem 6.11 The long sequence of abelian groups which extends infinitely to the left

… → Ω_{n+1}^{alm} ⟶^{σ} L_{n+1}(ℤ) ⟶^{∂} Θ^{n} ⟶^{η} Ω_{n}^{alm} ⟶^{σ} L_{n}(ℤ) ⟶^{∂} … ⟶^{η} Ω_{5}^{alm} ⟶^{σ} L_{5}(ℤ)

is exact.

Recall that we have shown in Theorem 4.30, Theorem 4.32 and Theorem 4.43 that there are isomorphisms

\frac{1}{8} · sign: L_{0}(ℤ) ⟶^{≅} ℤ

and

Arf : L_{2}(ℤ) ⟶^{≅} ℤ/2

and that L_{2i+1}(ℤ) = 0 for i ∈ ℤ. Consider a normal map (\overline{f}, f): T M ⊕ \underline{ℝ^{a}} → ξ covering a map of degree one (f, ∂f): (M, ∂M) → (X, ∂X) of oriented compact 4i-dimensional manifolds such that X is simply connected and ∂f a homotopy equivalence. Then the isomorphism \frac{1}{8} · sign: L_{0}(ℤ) ⟶^{≅} ℤ sends σ(\overline{f}, f) to \frac{1}{8} · sign(M ∪_{∂f} X^{−}). Hence Theorem 6.11 and Lemma 6.6 imply

Corollary 6.12 There are for i ≥ 2 and j ≥ 3 short exact sequences of abelian groups

0 → Θ^{4i} ⟶^{η} Ω_{4i}^{alm} ⟶^{\frac{sign}{8}} ℤ ⟶^{∂} bP^{4i} → 0

and

0 → Θ^{4i−2} ⟶^{η} Ω_{4i−2}^{alm} ⟶^{Arf} ℤ/2 ⟶^{∂} bP^{4i−2} → 0

and

0 → bP^{2j} → Θ^{2j−1} ⟶^{η} Ω_{2j−1}^{alm} → 0.

Here the map

\frac{sign}{8} : Ω_{4i}^{alm} → ℤ

sends [M] to \frac{1}{8} · sign(M) and Arf : Ω^{alm} → ℤ/2 sends [M] to the Arf invariant of the normal map β^{−1}([M]) ∈ N_{n}(S^{n}) for β the bijection appearing in Lemma 6.9.

6.3 The J-Homomorphism and Stably Framed Bordism

By Theorem 6.11 we have reduced the computation of Θ^{n} to computations about Ω_{n}^{alm} and certain maps to ℤ and ℤ/2 given by the signature divided by 8 and the Arf invariant. This reduction is essentially due to the surgery machinery. The rest of the computation will essentially be homotopy theory. First we try to understand Ω_{∗}^{fr} geometrically. There is an obvious forgetful map

f: Ω_{n}^{fr} → Ω_{n}^{alm}.  (6.13)

6.3. THE J-HOMOMORPHISM AND STABLY FRAMED BORDISM 131

Define the group homomorphism

∂: Ω_{n}^{alm} → π_{n−1}(SO)  (6.14)

as follows. Given r ∈ Ω_{n}^{alm} choose a representative (M, x, \overline{u}: T M|_{M−\{x\}} ⊕ \underline{ℝ^{a}} → \underline{ℝ^{n+a}}). Let D^{n} ⊂ M be an embedded disk with origin x. Since D^{n} is contractible, we obtain a strong bundle isomorphism unique up to isotopy \overline{v}: T M|_{D^{n}} ⊕ \underline{ℝ^{a}} ⟶^{≅} \underline{ℝ^{a+n}}. The composition of the inverse of the restriction of \overline{u} to S^{n−1} = ∂D^{n} and of the restriction of \overline{v} to S^{n−1} is an orientation preserving bundle automorphism of the trivial bundle \underline{ℝ^{a+n}} over S^{n−1}. This is the same as a map S^{n−1} → SO(n + a). Its composition with the canonical map SO(n + a) → SO represents an element in π_{n−1}(SO) which is defined to be the image of r under ∂: Ω_{n}^{alm} → π_{n−1}(SO). Let

\overline{J}: π_{n}(SO) → Ω_{n}^{fr}  (6.15)

be the group homomorphism which assigns to the element r ∈ π_{n}(SO) represented by a map \overline{u}: S^{n} → SO(n + a) the class of S^{n} with the stable framing T S^{n} ⊕ \underline{ℝ^{a}} ⟶^{≅} \underline{ℝ^{a+n}} which is induced by the standard trivialization

T S^{n} ⊕ \underline{ℝ} ≅ T (∂D^{n+1}) ⊕ ν(∂D^{n+1}, D^{n+1}) ≅ T D^{n+1}|_{∂D^{n+1}} ≅ \underline{ℝ^{n+1}}

and the strong bundle automorphism of the trivial bundle \underline{ℝ^{a+n}} over S^{n} given by \overline{u}. One easily checks

Lemma 6.16 The following sequence is a long exact sequence of abelian groups

… ⟶^{∂} π_{n}(SO) ⟶^{\overline{J}} Ω_{n}^{fr} ⟶^{f} Ω_{n}^{alm} ⟶^{∂} π_{n−1}(SO) ⟶^{\overline{J}} Ω_{n−1}^{fr} ⟶^{f} ….

Next we want to interpret the exact sequence appearing in Lemma 6.16 homotopy theoretically. We begin with the homomorphism \overline{J}. Notice that there is a natural bijection

τ′: colim_{k→∞} Ω_{n}(\underline{ℝ^{k}} → \{∗\}) ⟶^{≅} Ω_{n}^{fr}  (6.17)

which is defined as follows. Consider an element

x = [(M, i: M → ℝ^{n+k}, pr: M → \{∗\}, \overline{u}: ν(i) → \underline{ℝ^{k}})] ∈ Ω_{n}(\underline{ℝ^{k}} → \{∗\})

There is a canonical strong isomorphism \overline{u′}: T M ⊕ν(i) ⟶^{≅} \underline{ℝ^{n+k}}. From \overline{u} and \overline{u′} we get an isomorphism \overline{v}: T M ⊕ \underline{ℝ^{k}} → \underline{ℝ^{n+k}} covering the projection M → \{∗\}. Define τ′(x) by the class of (M, \overline{v}). The Thom space Th(\underline{ℝ^{n+k}}) is S^{n+k}. Hence the Pontrjagin-Thom construction (see Theorem 3.26) induces an isomorphism

P: colim_{k→∞} Ω_{n}(\underline{ℝ^{k}} → \{∗\}) ≅ colim_{k→∞} π_{n+k}(S^{k}).  (6.18)

Notice that the stable n-th homotopy group of a space X is defined by

π_{n}^{s}(X) := colim_{k→∞} π_{n+k}(S^{k} ∧ X).  (6.19)

The stable n-stem is defined by

π_{n}^{s} := π_{n}^{s}(S^{0}) := colim_{k→∞} π_{n+k}(S^{k}).  (6.20)

Thus the isomorphism τ′ of (6.17) and the isomorphism P of (6.18) yield an isomorphism

τ: Ω_{n}^{fr} ⟶^{≅} π_{n}^{s}.  (6.21)

Next we explain the so called Hopf construction which defines for spaces X, Y and Z a map

H: [X × Y, Z] → [X ∗ Y, ΣZ]  (6.22)

as follows. Recall that the join X ∗ Y is defined by X × Y × [0, 1]/ ∼, where ∼ is given by (x, y, 0) ∼ (x′, y, 0) and (x, y, 1) ∼ (x, y′, 1), and that the (unreduced) suspension ΣZ is defined by Z × [0, 1]/ ∼, where ∼ is given by (z, 0) ∼ (z′, 0) and (z, 1) ∼ (z′, 1). Given f: X × Y → Z, let H(f): X ∗ Y → ΣZ be the map induced by f × id: X × Y × [0, 1] → Z × [0, 1]. Consider the following composition

[S^{n}, SO(k)] → [S^{n}, aut(S^{k−1})] → [S^{n} × S^{k−1}, S^{k−1}]

⟶^{H} [S^{n} ∗ S^{k−1}, ΣS^{k−1}] = [S^{n+k}, S^{k}].

Notice that π_{1}(SO(k)) acts trivially on π_{n}(SO(k)) and π_{1}(S^{k}) acts trivially on π_{n+k}(S^{k}) for k, n ≥ 1. Hence no base point questions arise in the next definition.

Definition 6.23 The composition above induces for n, k ≥ 1 homomorphisms of abelian groups

J_{n,k}: π_{n}(SO(k)) → π_{n+k}(S^{k}).

Taking the colimit for k → ∞ induces the so called J-homomorphism

J_{n}: π_{n}(SO) → π_{n}^{s}.

One easily checks

Lemma 6.24 The composition of the homomorphism \overline{J}: π_{n}(SO) → Ω_{n}^{fr} of (6.15) with the isomorphism τ: Ω_{n}^{fr} ⟶^{≅} π_{n}^{s} of (6.21) is the J-homomorphism J_{n}: π_{n}(SO) → π_{n}^{s} of Definition 6.23.

The homotopy groups of O are 8-periodic and given by

\begin{matrix}i mod 8 & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ π_{i}(O) & ℤ/2 & ℤ/2 & 0 & ℤ & 0 & 0 & 0 & ℤ \end{matrix}

Notice that π_{i}(SO) = π_{i}(O) for i ≥ 1 and π_{0}(SO) = 1. The first stable stems are given by

\begin{matrix}n & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ π_{n}^{s} & ℤ & ℤ/2 & ℤ/2 & ℤ/24 & 0 & 0 & ℤ/2 & ℤ/240 & ℤ/2 ⊕ ℤ/2 & ℤ/2 ⊕ ℤ/2 ⊕ ℤ/2 \end{matrix}

6.4. COMPUTATION OF BP^{N+1} 133

The Bernoulli numbers B_{n} for n ≥ 1 are defined by

\frac{z}{e^{z} − 1} = 1 − \frac{z}{2} + ∑_{n≥1} \frac{(−1)^{n+1} · B_{n}}{(2n)!} · (z)^{2n}.  (6.25)

The first values are given by

\begin{matrix}n & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ B_{n} & \frac{1}{6} & \frac{1}{30} & \frac{1}{42} & \frac{1}{30} & \frac{5}{66} & \frac{691}{2730} & \frac{7}{6} & \frac{3617}{510} \end{matrix}

The next result is a deep theorem due to Adams [2, Theorem 1.1, Theorem 1.3 and Theorem 1.5].

Theorem 6.26 1. If n ≠ 3 mod 4, then the J-homomorphism J_{n}: π_{n}(SO) → π_{n}^{s} is injective;

2. The order of the image of the J-homomorphism J_{4k−1} : π_{4k−1}(SO) → π_{4k−1}^{s} is denominator(B_{k}/4k), where B_{k} is the k-th Bernoulli number.

6.4 Computation of bP^{n+1}

In this section we want to compute the subgroups bP^{n+1} ⊂ Θ^{n} (see Definition 6.5).

We have introduced the bijection β: N_{n}(S^{n}) ⟶^{≅} Ω_{n}^{alm} in Lemma 6.9 and the map ∂: Ω_{n}^{alm} → π_{n−1}(SO) in (6.14). Let

δ(k): π_{n}(BSO(k)) ⟶^{≅} π_{n−1}(SO(k))  (6.27)

be the boundary map in the long exact homotopy sequence associated to the fibration SO(k) → ESO(k) → BSO(k). It is an isomorphism since ESO(k) is contractible. It can be described as follows. Consider x ∈ π_{n}(BSO(k)). Choose a representative f: S^{n} → BSO(k) for some k. If γ_{k} → BSO(k) is the universal k-dimensional oriented vector bundle, f^{∗}γ_{k} is a k-dimensional oriented vector bundle over S^{n}. Let S_{−}^{n} be the lower and S_{+}^{n} be the upper hemisphere and S^{n−1} = S_{−}^{n}∩S_{+}^{n}. Since the hemispheres are contractible, we obtain up to isotopy unique strong bundle isomorphisms \overline{u}_{−}: f^{∗}γ_{k}|_{S_{−}^{n}} ⟶^{≅} \underline{ℝ^{k}} and \overline{u}_{+} : f^{∗}γ_{k}|_{S_{+}^{n}} ⟶^{≅} \underline{ℝ^{k}}. The composition of the inverse of the restriction of \overline{u}_{−} to S^{n−1} with the restriction of \overline{u}_{+} to S^{n−1} is a bundle automorphism of the trivial bundle \underline{ℝ^{k}} over S^{n−1} which is the same as map S^{n−1} → SO(k). Define its class in π_{n−1}(SO(k)) to be the image of x under δ(k)^{−1} : π_{n}(BSO(k)) → π_{n−1}(SO(k)). Analogously we get an isomorphism

δ: π_{n}(BSO) ⟶^{≅} π_{n−1}(SO).  (6.28)

Define a map

γ: N_{n}(S^{n}) → π_{n}(BSO)  (6.29)

by sending the class of the normal map of degree one (\overline{f}, f): T M ⊕ \underline{ℝ^{a}} → ξ covering a map f: M → S^{n} to the the class represented by the classifying map f_{ξ}: S^{n} → BSO(n + k) of ξ. One easily checks

Lemma 6.30 The following diagram commutes

Ω_{n}^{alm} ⟶^{∂} π_{n−1}(SO)
β^{−1}↓  ↓_{δ^{−1}}
N_{n}(S^{n}) ⟶_{γ} π_{n}(BSO)

The next ingredient is the Hirzebruch signature formula. It says for a closed oriented manifold M of dimension n = 4k that its signature can be computed in terms of the L-class L(M) by

sign(M) = ⟨L(M), [M]⟩.  (6.31)

The L-class is a cohomology class which is obtained from inserting the Pontrjagin classes p_{i}(T M) into a certain polynomial L(x_{1}, x_{2}, …, x_{k}). The Lpolynomial L(x_{1}, x_{2}, …, x_{k}) is the sum of s_{k} ·x_{k} and terms which do not involve x_{k}, where s_{k} is given in terms of the Bernoulli numbers B_{k} by

s_{k} := \frac{2^{2k} · (2^{2k−1} − 1) · B_{k}}{(2k)!} .  (6.32)

Assume that M is almost stably parallelizable. Then for some point x ∈ M the restriction of the tangent bundle T M to M − {x} is stably trivial and hence has trivial Pontrjagin classes. Since the inclusion induces an isomorphism H^{p}(M) ⟶^{≅} H^{p}(M − {x}) for p ≤ n − 2, we get p_{i}(M) = 0 for i ≤ k − 1. Hence (6.31) implies for a closed oriented almost stably parallelizable manifold M of dimension 4k

sign(M) = s_{k} · ⟨p_{k}(T M), [M]⟩.  (6.33)

We omit the proof of the next lemma which is based on certain homotopy theoretical computations (see for instance [68, Theorem 3.8 on page 76]).

Lemma 6.34 Let n = 4k. Then there is an isomorphism

ϕ: π_{n−1}(SO) ⟶^{≅} ℤ.

Define a map

p_{k}: π_{n}(BSO) → ℤ

by sending the element x ∈ π_{n}(BSO) represented by a map f: S^{n} → BSO(m) to ⟨p_{k}(f^{∗}γ_{m}), [S^{n}]⟩ for γ_{m} → BSO(m) the universal bundle. Let δ: π_{n}(BSO) → π_{n−1}(SO) be the isomorphism of (6.27). Put

t_{k} := \frac{3−(−1)^{k}}{2} · (2k − 1)!  (6.35)

Then

t_{k} · ϕ = p_{k} ∘ δ^{−1}.

Lemma 6.36 The following diagram commutes for n = 4k

Ω_{n}^{alm}  ⟶^{\frac{sign}{8}} ℤ
∂↓  ↑^{\frac{s_{k}·t_{k}}{8} ·id}
π_{n−1}(SO) ⟶_{≅}^{ϕ} ℤ

where \frac{sign}{8} is the homomorphism appearing in Corollary 6.12, the homomorphism ∂ has been defined in in (6.14) and the isomorphism ϕ is taken from (6.34).

\underline{\underline{Proof}} : Consider an almost stably parallelizable manifold M of dimension n = 4k. We conclude from Lemma 6.30 and Lemma 6.34

\frac{s_{k} · t_{k}}{8} · ϕ ∘ ∂([M]] = \frac{s_{k}}{8} · p_{k} ∘ δ^{−1} ∘ ∂([M]]
= \frac{s_{k}}{8} · p_{k} ∘ γ ∘ β^{−1}([M]].  (6.37)

By definition the composition

Ω_{n}^{alm} ⟶^{β^{−1}} N_{n}(S^{n}) ⟶^{γ} π_{n}(SO) ⟶^{p_{k}} ℤ

sends the class of M to ⟨p_{k}(ξ), [S^{n}]⟩ for a bundle ξ over S^{n} for which there exists a bundle map (\overline{c}, c): T M ⊕ \underline{ℝ^{a}} → ξ covering a map c: M → S^{n} of degree one. This implies

\frac{s_{k}}{8} · p_{k} ∘ γ ∘ β^{−1}([M]) = \frac{s_{k}}{8} · ⟨p_{k}(ξ), [S^{n}]⟩
= \frac{s_{k}}{8} · ⟨p_{k}(ξ), c_{∗}([M])⟩
= \frac{s_{k}}{8} · ⟨c^{∗}(p_{k}(ξ)), [M]⟩.
= \frac{s_{k}}{8} · ⟨p_{k}(T M), [M]⟩  (6.38)

Now the claim follows from (6.33), (6.37) and (6.38).

Theorem 6.39 Let k ≥ 2 be an integer. Then bP^{4k} is a finite cyclic group of order

\frac{s_{k} · t_{k}}{8} · |im (J_{4k−1} : π_{4k−1}(SO) → π_{4k−1}^{s})|

= \frac{1}{8} · \frac{2^{2k} · (2^{2k−1} − 1) · B_{k}}{(2k)!} · \frac{3 − (−1)^{k}}{2} · (2k − 1)! · denominator(B_{k}/4k)
= \frac{3 − (−1)^{k}}{2} · 2^{2k−2} · (2^{2k−1} − 1) · numerator(B_{k}/(4k)).

\underline{\underline{Proof}} : This follows from Theorem 6.11, Theorem 6.26 (2) and Lemma 6.36.

Next we treat the case n = 4k + 2 for k ≥ 1. Let

Arf : π_{4k+2}^{s} → ℤ/2  (6.40)

be the composition of the inverse of the Pontrjagin-Thom isomorphism τ: Ω_{n}^{fr} ⟶^{≅} π_{n}^{s} of (6.21), the forgetful homomorphism f: Ω_{4k+2}^{fr} → Ω_{4k+2}^{alm} of (6.13) and the map Arf : Ω_{4k+2}^{alm} → ℤ/2 appearing in Corollary 6.12.

Theorem 6.41 Let k ≥ 3. Then bP^{4k+2} is a trivial group if the homomorphism Arf : π_{4k+2}^{s} → ℤ/2 of (6.40) is surjective and is ℤ/2 if the homomorphism Arf : π_{4k+2}^{s} → ℤ/2 of (6.40) is trivial.

\underline{\underline{Proof}} : We conclude from Lemma 6.16, Lemma 6.24 and and Theorem 6.26 (1) that the forgetful map f: Ω_{4k+2}^{fr} → Ω_{4k+2}^{alm} is surjective. Now the claim follows from Corollary 6.12.

The next result is due to Browder [16].

Theorem 6.42 The homomorphism Arf : π_{4k+2}^{s} → ℤ/2 of (6.40) is trivial if 2k + 1 ≠ 2^{l} − 1

The homomorphism Arf : π_{4k+2}^{s} → ℤ/2 of (6.40) is also known to be non-trivial for 4k + 2 ∈ {6, 14, 30, 62} (combine [16], [6] and [79]). Hence Theorem 6.41 and Theorem 6.42 imply

Corollary 6.43 The group bP^{4k+2} is trivial or isomorphic to ℤ/2. We have

bP^{4k+2} = \{ \begin{matrix}ℤ/2 & 4k + 2 ≠ 2^{l} − 2, k ≥ 1; \\ 0 & 4k + 2 ∈ \{6, 14, 30, 62\}. \end{matrix}

We conclude from Corollary 6.12

Theorem 6.44 We have for k ≥ 3

bP^{2k+1} = 0.

Theorem 6.45 For n ≥ 1 any homotopy n-sphere Σ is stably parallelizable.

For an almost parallelizable manifold M the image of its class [M] ∈ Ω_{n}^{alm} under the homomorphism ∂: Ω_{n}^{alm} → π_{n}(SO(n−1)) is exactly the obstruction to extend the almost stable framing to a stable framing. Recall that any homotopy n-sphere is almost stably parallelizable. The map ∂ is trivial for n ≠ 0 mod 4 by Lemma 6.16, Lemma 6.24, Theorem 6.26 (1). If n = 0 mod 4, the claim follows from Lemma 6.16, Lemma 6.24 and and Lemma 6.36 since the signature of a homotopy n-sphere is trivial.

6.5. COMPUTATION OF Θ^{N}/BP^{N+1} 137

6.5 Computation of Θ^{n}/bP^{n+1}

In this section we compute Θ^{n}/bP^{n+1}.

Theorem 6.46 1. If n = 4k + 2, then there is an exact sequence

0 → Θ^{n}/bP^{n+1} → coker (J_{n}: π_{n}(SO) → π_{n}^{s}) → ℤ/2;

2. If n ≠ 2 mod 4 or if n = 4k + 2 with 2k + 1 ≠ 2^{l} − 1, then

Θ^{n}/bP^{n+1} ≅ coker (J_{n}: π_{n}(SO) → π_{n}^{s}) .

\underline{\underline{Proof}} : Lemma 6.16, Lemma 6.24, Theorem 6.26 (1) and Lemma 6.36 imply

ker (∂: Ω_{n}^{alm} → π_{n−1}(SO)) = Ω_{n}^{alm}  n ≠ 0 mod 4;
ker (∂: Ω_{n}^{alm} → π_{n−1}(SO)) = ker (\frac{sign}{8} : Ω_{n}^{alm} → ℤ)  n = 0 mod 4;
ker (∂: Ω_{n}^{alm} → π_{n−1}(SO)) = coker (J_{n}: π_{n}(SO) → π_{n}^{fr}) .

Now the claim follows from Corollary 6.12 and Theorem 6.42.

6.6 The Kervaire-Milnor Braid

We have established in Lemma 6.16 the long exact sequence

… ⟶^{∂} π_{n}(SO) ⟶^{\overline{J}} Ω_{n}^{fr} ⟶^{f} Ω_{n}^{alm} ⟶^{∂} π_{n−1}(SO) ⟶^{\overline{J}} Ω_{n−1}^{fr} ⟶^{f} ….

The long exact sequence

… → L_{n+1}(ℤ) → Θ^{n} → Ω_{n}^{alm} → L_{n}(ℤ) → Θ^{n−1} → …

is taken from Theorem 6.11. Denote by Θ_{fr}^{n} the abelian group of stably framed h-cobordism classes of stably framed homotopy n-spheres. There is a long exact sequence

… → L_{n+1}(ℤ) → Θ_{fr}^{n} → Ω_{n}^{fr} → L_{n}(ℤ) → Θ_{fr}^{n−1} → …

which is defined as follows. The map Θ_{fr}^{n} → Ω_{n}^{fr} assign to the class of a framed homotopy sphere its class in Ω_{n}^{fr}. The map Ω_{n}^{fr} → L_{n}(ℤ) assigns to a class [M] in Ω_{n}^{fr} the surgery obstruction of the normal map (\overline{c}, c): T M ⊕ \underline{ℝ^{a}} → \underline{ℝ^{n+a}} for any map of degree one c: M → S^{n}, where \overline{c} is given by the framing on M. The map L_{n+1}(ℤ) → Θ_{fr}^{n} assigns to x ∈ L_{n+1}(ℤ) the class of the framed homotopy n-sphere (Σ, \overline{v}: T Σ⊕\underline{ℝ^{a}} → \underline{ℝ^{a+n}}), for which there is a normal map of degree one (\overline{U}, U): T W ⊕ \underline{ℝ^{a+b}} → \underline{ℝ^{n+a+b}} covering a map of triads U: (W; ∂_{0}W, ∂_{1}W) → (S^{n} × [0, 1]; S^{n} × \{0\}, S^{n} × \{1\}) and a bundle map (\overline{u_{0}}, u_{0}): T Σ ⊕ \underline{ℝ^{a+b+1}} → T W_{0} ⊕ \underline{ℝ^{a+b}} covering the orientation preserving diffeomorphism u_{0} : Σ → ∂_{0}W such that \overline{U} ∘ \overline{u_{0}} = \overline{v} ⊕ id\underline{ℝ^{b+1}}, U induces a diffeomorphism ∂_{1}W → S^{n} and the surgery obstruction associated to (\overline{U}, U) is the given element x ∈ L_{n+1}(ℤ).

Theorem 6.47 (The Kervaire-Milnor Braid) The long exact sequences above fit together to an exact braid for n ≥ 5

\xymatrix{π_{n}(SO) \ar[rr] \ar[dr] &  & Ω_{n}^{fr} \ar[dr] \ar[rr] &  & L_{n}(ℤ) \ar[rr] \ar[dr] &  & Θ^{n−1} \\  & Θ_{fr}^{n} \ar[ur] \ar[dr] &  & Ω_{n}^{alm} \ar[ur] \ar[dr] &  & Θ_{fr}^{n−1} \ar[ur] \ar@{-->}[dr] \\ L_{n+1}(ℤ) \ar[ur] \ar[rr] &  & Θ^{n} \ar[ur] \ar[rr] &  & π_{n−1}(SO) \ar[ur] \ar[rr] &  & Ω_{n−1}^{fr}}

We have already introduced BO and BG and have defined G/O as the homotopy fiber of this map. There is also a P L-version of BO called BP L . We can define analogously spaces G/P L and P L/O and fibrations G/P L → BP L → BG and P L/O → BP L → BO. Since ΩBO ≃ O, ΩBP L ≃ BP L and ΩBG ≃ G holds, we get fibrations O → G → G/O, P L → G → G/P L and O → P L → P L/O. Notice that for an inclusion of topological groups H ⊂ K there is an obvious fibration H → K → K/H and the fibrations above are in this spirit. But we have to use the classifying spaces since for instance G is not a group and we cannot talk about the homogeneous space G/P L. More information about these spaces and their homotopy theoretic properties can be found for instance in [78].

Theorem 6.48 (Homotopy Theoretic Interpretation of the Kervaire-Milnor Braid) The long exact homotopy sequences of these three fibrations above yield an exact braid

\xymatrix{π_{n}(O) \ar[rr] \ar[dr] &  & π_{n}(G) \ar[dr] \ar[rr] &  & π_{n}(G/P L) \ar[rr] \ar[dr] &  & π_{n−1}(P L/O) \\  & π_{n}(P L) \ar[ur] \ar[dr] &  & π_{n}(G/O) \ar[ur] \ar[dr] &  & π_{n−1}(P L) \ar[ur] \ar[dr] \\ π_{n+1}(G/P L) \ar[ur] \ar[rr] &  & π_{n}(P L/O) \ar[ur] \ar[rr] &  & π_{n−1}(O) \ar[ur] \ar[rr] &  & π_{n−1}(G)}

which is for n ≥ 5 isomorphic to the Kervaire-Milnor braid of Theorem 6.47

\underline{\underline{Proof}} : At least we explain how the two braids are related by isomorphisms. Since O/SO is the discrete group {±1}, the inclusion induces an isomorphism

π_{n}(SO) ⟶^{≅} π_{n}(O)  for n ≥ 1.

The Pontrjagin-Thom isomorphism τ: Ω_{n}^{fr} ⟶^{≅} π_{n}^{s} of (6.21) and the canonical isomorphism π_{n}(G) ≅ π_{n}^{s} yield an isomorphism

Ω_{n}^{fr} ⟶^{≅} π_{n}(G).

The isomorphism β: N_{n}(S^{n}) ⟶^{≅} Ω_{n}^{alm} of Lemma 6.9 and the bijection π_{n}(G/O) ⟶^{≅} N_{n}(S^{n}) (see Theorem 3.52) induce a bijection

Ω_{n}^{alm} ⟶^{≅} π_{n}(G/O).

One can also develop surgery theory in the P L-category instead of the smooth category. We have the surgery exact sequence in the P L-category (see Theorem 5.15.

… → π_{n+1}(G/P L) → L_{n+1}(ℤ) → S_{n}^{P L}(S^{n+1}) → π_{n}(G/P L) → L_{n}(ℤ) → …

Since the Poincare Conjecture is true for n ≥ 5 and any P L-homeomorphism f: S^{n} → S^{n} can be extended to a P L-homeomorphism F: D^{n+1} → D^{n+1} by coning off (what is not possible in the smooth category), any homotopy n-sphere is P L-homeomorphic to S^{n} for n ≥ 5. Hence S_{n}^{P L}(S^{n}) = {∗} for n ≥ 5. This is the main ingredient in the proof for n ≥ 5, the low dimensional cases follow from direct computations, that we obtain isomorphisms of abelian groups

π_{n}(G/P L) ⟶^{≅} L_{n}(ℤ)  n ≥ 1

There is an isomorphism

Θ^{n} ⟶^{≅} π_{n}(P L/O)  n ≥ 0, n ≠ 3

which is defined as follows. Let Σ be a homotopy n-sphere. As explained above, there is an orientation preserving P L-homeomorphism h: Σ → S^{n}. Fix a classifying map f_{S^{n}}: S^{n} → BP L for the P L-tangent bundle of S^{n}. We obtain a classifying map f_{Σ} : Σ → BO of the smooth tangent bundle of Σ together with a homotopy h: Bi ∘ f_{Σ} ≃ f_{S^{n}}, where Bi: BO → BP L is the canonical map. The pair (f_{Σ}, h) yields a map S^{n} → P L/O, since P L/O is the homotopy fiber of Bi: BO → BP L. The bijectivity of this map for n ≥ 5 follows from the five lemma and the comparision of the surgery exact sequence with the long homotopy sequence associated to the fibration P L/O → G/O → G/P L. There is an isomorphism

Θ_{fr}^{n} ⟶^{≅} π_{n}(P L)  n ≥ 5

which is defined as follows. As above we get a pair (f_{Σ}, h). The framing yields also a homotopy g: f_{Σ} ≃ c for c the constant map. Since P L can be viewed as the homotopy fiber of the obvious map P L/O → BO, these data yield a map S^{n} → P L.

6.7 Miscellaneous

Remark 6.49 We have shown in Theorem 6.39 that bP^{4k} for k ≥ 2 is a finite cyclic group. If we go through the construction again, an explicit generator can be constructed as follows. Recall that \frac{sign}{8} : L_{4k}(ℤ) → ℤ is an isomorphism. Let W be any oriented stably parallelizable manifold of dimension 4k whose boundary is a homotopy sphere and whose intersection pairing on H_{2k}(W) has signature 8. Then ∂_{1}W is an exotic sphere representing a generator of bP^{4k}. Such manifolds W can be explicitly constructed by a plumbing construction (see for instance [17, Theorem V.2.9 on page 122]).

Example 6.50 The first example of an exotic sphere, i.e. a closed manifold which is homeomorphic but not diffeomorphic to S^{n}, was constructed by Milnor [81]. See also [87]. The construction and the detection that it is an exotic sphere is summarized below.

There is an isomorphism

ℤ ⊕ ℤ ⟶^{≅} π_{3}(SO(4))
(h, j) ↦ ω(h, j): S^{3} → SO(4) x ↦ (y ↦ x^{h} · y · x^{j})

where we identify ℝ^{4} with the quaternions ℍ by (a, b, c, d) ↦ a + bi + cj + dk and x^{h} · y · x^{j} is to be understood with respect to the multiplication in ℍ. Since π_{3}(SO(4)) ≅ π_{4}(BSO(4)), each pair (h, j) ∈ ℤ⊕ℤ determines an oriented vector bundle E(h, j) with Riemannian metric over S^{4} unique up to orientation and Riemannian metric preserving isomorphism. The Euler number and the first Pontrjagin class of E(j, h) are given by

χ(E(h, j)) = h + j;
⟨p_{1}(E(h, j)), [S^{4}]⟩ = 2(h − j).

The Gysin sequence shows that the sphere bundle SE(h, j) is a homotopy 7sphere if and only if χ(E(h, j)) = h + j = 1.

Let k be any odd integer. Let W (k) be the disk bundle D(E((1 + k)/2, (1 − k)/2)) and Σ(k) be ∂W (k) = S(E((1 + k)/2, (1 − k)/2)). Then Σ(k) is a homotopy 7-sphere. Next we recall Milnor’s argument why Σ(k) cannot be diffeomorphic to S^{7}.

The obvious embedding i: S^{4} → W (k) given by the zero section is a homotopy equivalence and i^{∗}T W (k) is isomorphic to T S^{4} ⊕ E((1 + k)/2, (1 − k)/2). Hence

⟨i^{∗}p_{1}(T W (k)), [S^{4}]⟩ = 2k.

Suppose that there exists a diffeomorphism Σ(k) → S^{7}. Then we can form the closed oriented smooth 8-dimensional manifold M(k) = W (k) ∪_{f} D^{8}. Let j: W (k) → M(k) be the inclusion. Since the inclusion j ∘i: S^{4} → M(k) induces an isomorphism on H_{4}, the signature of M(k) is one. The Hirzebruch signature Theorem says 1 = sign(M) = ⟨L(M), [M]⟩. Since

L(M) = \frac{7}{45} · p_{2}(T M) − \frac{1}{45} · p_{1}(T M)^{2}

we conclude

1 = ⟨\frac{7}{45}·p_{2}(T M)−\frac{1}{45}p_{1}(T M)^{2}, [M]⟩ = \frac{7}{45}⟨p_{2}(T M), [M]⟩−\frac{1}{45}·⟨p_{1}(T M)^{2}, [M]⟩

= \frac{7}{45}⟨p_{2}(T M), [M]⟩ − \frac{1}{45} · ⟨i^{∗}j^{∗}p_{1}(T M), [S^{4}]⟩^{2} = \frac{7}{45}⟨p_{2}(T M), [M]⟩ − \frac{4k^{2}}{45} .

Since ⟨p_{2}(T M), [M]⟩ is an integer, we conclude k^{2} = 1 mod 7. Hence Σ(k) is an exotic homotopy 7-sphere if k^{2} ≠ 1 mod 7.

Remark 6.51 Milnor’s example 6.50 above fits into the general context as follows. Recall that bP^{8} = ℤ/28 and that we have an isomorphism

\frac{sign}{8} : Θ^{7} → ℤ/28

which sends [Σ] to sign(W)/8 for any stably parallelizable manifold W whose boundary is oriented diffeomorphic to Σ. If Σ(k) is the oriented homotopy 8sphere of Example 6.50, then the isomorphism above sends [Σ(k)] to (1 − k^{2}) ∈ ℤ/28.

Example 6.52 Let W^{2n−1}(d) be the subset of ℂ^{n+1} consisting of those points (z_{0}, z_{1}, …, z_{n}) which satisfy the equations z_{0}^{d} + z_{1}^{2} + … + z_{n}^{2} = 0 and ||z_{0}||^{2} + ||z_{1}||^{2} + … + ||z_{n}||^{2} = 1. These turns out to be smooth submanifolds and are called Brieskorn varieties (see [13], [55]). Suppose that d and n are odd, Then W^{2n−1}(d) is a homotopy (2n − 1)-sphere. It is diffeomorphic to the standard sphere S^{2n−1} if d = ±1 mod 8 and it is an exotic sphere representing the generator of bP^{2n} if d = ±3 mod 8 [13, page 11]. In general one can study the intersection K = f^{−1}(0) ∩ {z ∈ ℂ^{n+1} | ||z|| = ϵ} for a polynomial f(z_{0}, z_{1}, , …, z_{n}) with an isolated singularity at the origin and examine when K is a homotopy sphere and when K is an exotic sphere [85, §8, §9].

Remark 6.53 Let Σ be a homotopy n-sphere for n ≥ 5. Let D_{0}^{n} → Σ and D_{1}^{n} → Σ be two disjoint embedded discs. Then W = Σ − (int(D_{0}^{n}) ∐ int(D_{1}^{n})) is a simply-connected h-cobordism. By the h-Cobordism Theorem 1.2 there is a diffeomorphism (F, id, f): ∂D_{0}^{n}×[0, 1], ∂D_{0}^{n}×{0}, ∂D_{0}^{n}×{1}) → (W, ∂D_{0}^{n}, ∂D_{1}^{n}). Hence Σ is oriented diffeomorphic to D^{n} ∪_{f: S^{n−1}→S^{n−1}} (D^{n})^{−} for some orientation preserving diffeomorphism f: S^{n−1} → S^{n−1}. If f is isotopic to the identity, Σ is oriented diffeomorphic to S^{n}. Hence the existence of exotic spheres shows the existence of selfdiffeomorphisms of S^{n−1} which are homotopic but not isotopic to the identity.

The next result is due to Berger and Klingenberg. Its proof and the proof of the following theorem can be found for instance in [27, Theorem 6.1 on page 106, Theorem 7.16 on page 126].

Theorem 6.54 (Sphere Theorem) Let M be a complete simply connected Riemannian manifold whose sectional curvature is pinched by 1 ≥ sec(M) > \frac{1}{4}. Then M is homeomorphic to the standard sphere.

Theorem 6.55 (Differentiable Sphere Theorem) There exists a constant δ with 1 > δ ≥ \frac{1}{4} with the following property: if M is a complete simply connected Riemannian manifold whose sectional curvature is pinched by 1 ≥ sec(M) > δ. then M is diffeomorphic to the standard sphere.

Brumfield and Frank [19] have shown

Theorem 6.56 For n ≠ 2^{k} − 2 or n ≠ 2^{k} − 3 the sequence

0 → bP^{n+1} → Θ^{n} → im ((Θ^{n} → coker(J_{n})) → 0

splits where the map Θ^{n} → im (Θ^{n} → coker(J_{n})) comes from the map Θ^{n} → coker(J_{n}) appearing in Theorem 6.46.

Fake real projective spaces, lens spaces and and fake tori have been investigated by Wall [119, 14D, 14E, 15A]. These are other examples of manifolds which look close to standard models but are not equal to them.

The Farrell-Jones and Baum-Connes Conjectures

In this chapter we explain the Farrell-Jones Isomorphism Conjecture for the algebraic K- and L-theory of group rings and the Baum-Connes Conjecture for the topological K-theory of reduced group C^{∗}-algebras. These conjectures say that certain assembly maps are isomorphisms. The targets of the assembly maps are the groups we are interested in, namely, K_{n}(RG), L_{n}(RG) or K_{n}^{top}(C_{∗}^{r}(G)).

The sources of the assembly maps are certain G-homology theories evaluated at classifying spaces of G with respect to the family of finite subgroups FIN or virtually cyclic subgroups VC. The construction of the sources and of the assembly maps is presented in Section 7.1.

In Section 7.2 we give a unified formulation of these conjectures. We illustrate their importance by explaining that they imply other well-known conjectures like the Novikov Conjecture about the homotopy invariance of higher signatures, the Borel Conjecture about the topological rigidity of closed aspherical topological manifolds and the stable Gromov-Lawson-Rosenberg Conjecture about the existence of Riemannian metrics with positive sectional curvature.

In Section 7.3 we give an abstract characterization of assembly maps as the best approximation of a homotopy invariant functor from the category of G-CW -complexes to the category of spectra from the left by an excisive functor, i.e. a functor which yields a G-homology theory after taking homotopy groups. This is useful for identifying assembly maps.

In Section 7.4 we discuss why the family FIN or VC appear in the Isomorphism Conjectures and under which circumstances one can replace VC by FIN. This is important for explicit calculations which are much easier in the context with FIN as explained in Chapter 8.

7.1 G-Homology Theories

144CHAPTER 7. THE FARRELL-JONES AND BAUM-CONNES CONJECTURES

Let C be a small category. Our main example will be Definition 7.1 The orbit category Or(G) of a group G has as objects homogeneous G-spaces G/H and as morphisms G-maps.

We will always work in the category SPACES of compactly generated spaces (see [111] and [121, I.4]). Denote by SPACES_{+} the category of pointed spaces. We require that the inclusion of the base point is a cofibration.

We define the category SPECTRA of spectra as follows. A spectrum E = {(E(n), σ(n)) | n ∈ ℤ} is a sequence of pointed spaces {E(n) | n ∈ ℤ} together with pointed maps called structure maps σ(n): E(n) ∧ S^{1} → E(n + 1). A map of spectra (sometimes also called function in the literature) f : E → E′ is a sequence of maps of pointed spaces f(n): E(n) → E′(n) which are compatible with the structure maps σ(n), i.e. we have f(n+1)∘σ(n) = σ′(n)∘(f(n) ∧ id_{S^{1}}) for all n ∈ ℤ. This should not be confused with the notion of map of spectra in the stable category (see [3, III.2.]). Recall that the homotopy groups of a spectrum are defined by

π_{p}(E) := colim_{k→∞} π_{p+k}(E(k)),  (7.2)

where the system π_{p+k}(E(k)) is given by the composition

π_{p+k}(E(k)) ⟶^{S} π_{p+k+1}(E(k) ∧ S^{1}) ⟶^{σ(k)_{∗}} π_{p+k+1}(E(k + 1))

of the suspension homomorphism and the homomorphism induced by the structure map. A weak homotopy equivalence of spectra is a map f : E → F of spectra inducing an isomorphism on all homotopy groups. A spectrum E is called Ω-spectrum if for each structure map its adjoint

E(n) → ΩE(n + 1) = map(S^{1}, E(n + 1))

is a weak homotopy equivalence of spaces. We denote by Ω-SPECTRA the corresponding full subcategory of SPECTRA.

Definition 7.3 A covariant C-space is a covariant functor from C to the category of topological spaces. Morphisms are natural transformations. Define analogously covariant pointed space, covariant spectrum, covariant Ω-spectrum and the contravariant notions.

Notice that the notions of coproduct, product, pushout, pullback, colimit, and limit exist in the category of C-spaces. They are constructed by applying these notions in the category SPACES objectwise. We mention that the terms direct limit and inverse limit are sometimes used in the literature instead of colimit and limit.

Example 7.4 Let G be a group. Let G be the category with a single object whose morphism set is G. Composition is given by the group multiplication. A covariant (contravariant) G-space is the same as a left (right) G-space. Maps of G-spaces correspond to G-maps.

7.1. G-HOMOLOGY THEORIES 145

Example 7.5 Let ℤ/p be the cyclic group of order p for a prime number p. A contravariant Or(ℤ/p)-space Y is specified by a ℤ/p-space Y ((ℤ/p)/{1}), a space Y ((ℤ/p)/(ℤ/p)), and a map Y ((ℤ/p)/(ℤ/p)) → Y ((ℤ/p)/{1})^{ℤ/p}.

Definition 7.6 Let X be a contravariant and Y be a covariant C-space. Define their tensor product to be the space

X ⊗_{C} Y = ∐_{c∈ob(C)} X(c) × Y (c)/ ∼

where ∼ is the equivalence relation generated by (xϕ, y) ∼ (x, ϕy) for all morphisms ϕ: c → d in C and points x ∈ X(d) and y ∈ Y (c). Here xϕ stands for X(ϕ)(x) and ϕy for Y (ϕ)(y).

Given C-spaces X and Y, denote by hom_{C}(X, Y) the space of maps of Cspaces from X to Y with the subspace topology coming from the obvious inclusion into ∏_{c∈ob(C)} map(X(c), Y (c)).

The main property of the tensor product is that it is the left adjoint of hom_{C} as explained in

Lemma 7.7 Let X be a contravariant C-space, Y be a covariant C-space and Z be a space. Denote by map(Y, Z) the obvious contravariant C-space whose value at an object c is the mapping space map(Y (c), Z). Then there is a homeomorphism natural in X, Y and Z

T = T (X, Y, Z): map(X ⊗_{C} Y, Z) ⟶^{≅} hom_{C}(X, map(Y, Z)).

\underline{\underline{Proof}} : We only indicate the definition of T. Given a map g: X ⊗_{C} Y → Z, we have to specify for each object c in C a map T (g)(c): X(c) → map(Y (c), Z). This is the same as specifying a map X(c)×Y (c) → Z which is defined to be the composition of g with the obvious map from X(c) × Y (c) to X ⊗_{C} Y.

Definition 7.8 Let Y be a (left) G-space. Define the associated contravariant Or(G)-space map_{G}(?, Y) by

Or(G) → SPACES, G/H ↦ map_{G}(G/H, Y) = Y^{H}.

A C-spectrum E can also be thought of as a sequence {E(n) | n ∈ ℤ} of pointed C-spaces and the structure maps as maps of pointed C-spaces. With this interpretation it is obvious what the tensor product spectrum X ⊗_{C} E of a contravariant pointed C-space and a covariant C-spectrum means. The canonical associativity homeomorphism

(X ⊗_{C} E(n)) ∧ S^{1} ⟶^{≅} X ⊗_{C} (E(n) ∧ S^{1})

is used in order to define the structure maps. It is given on representatives by sending (x ⊗_{C} e) ∧ z to x ⊗_{C} (e ∧ z).

Definition 7.9 A G-CW -complex X is a G-space X together with a filtration

∅ = X_{−1} ⊂ X_{0} ⊂ X_{1} ⊂ … ⊂ X_{n} ⊂ … ⊂ X

such that X = colim_{n→∞} X_{n} and for any n ≥ 0 the n-skeleton X_{n} is obtained from the (n − 1)-skeleton X_{n−1} by attaching equivariant cells, i.e. there exists a pushout of G-spaces of the form

∐_{i∈I_{n}} G/H_{i} × S^{n−1} ⟶ X_{n−1}
↓  ↓
∐_{i∈I_{n}} G/H_{i} × D^{n} ⟶ X_{n}

A G-CW -subcomplex A ⊂ X is a G-CW -complex such that its n-th skeleton A_{n} is the intersection A∩X_{n}. A G-map f: X → Y of G-CW -complexes is called cellular if f(X_{n}) ⊂ Y_{n}. More information about G-CW -complexes can be found for instance in [115, II.1 and II.2] and [69, section 1 and 2].

Definition 7.10 A G-homology theory H_{∗}^{G} consists of a covariant functor from the category of pairs of G-CW -complexes to the category of ℤ-graded abelian groups (X, A) ↦ (H_{p}^{G}(X, A))_{p∈ℤ} with the following properties.

1. Two G-homotopic maps induce the same homomorphisms;

2. There are long natural exact sequences associated to pairs of G-CW - complexes;

3. Consider a cellular G-pushout

X_{0} ⟶^{i_{1}} X_{1}
i_{2}↓  ↓^{j_{1}}
X_{2} ⟶^{j_{2}} X

i.e. a G-pushout such that i_{1} : X_{0} → X_{1} is an inclusion of G-CW - complexes, the map i_{2} : X_{0} → X_{1} is cellular and the G-CW -structure on X is induced by the ones on X_{i} for i = 0, 1, 2 in the obvious way. Then we get for all p ∈ ℤ an isomorphism

H_{p}^{G}(j_{1}, i_{2}): H_{p}^{G}(X_{1}, X_{0}) ⟶^{≅} H_{n}^{G}(X, X_{2}).

It satisfies the disjoint union axiom if for any collection {X_{i} | i ∈ I} of G-CW - complexes the canonical map induced by the various inclusions is bijective for all n ∈ ℤ

⊕_{i∈I}H_{p}^{G}(X_{i}) ⟶^{≅} H_{p}^{G}(∐_{i∈I} X_{i}).

Example 7.11 Let H_{∗} be a ℤ-graded (non-equivariant) homology theory such as singular homology, cobordism or K-homology. Then we obtain two Ghomology theories

X ↦ H_{∗}(G\X);
X ↦ H_{∗}(G\(EG × X)).

The second one is called the associated Borel homology associated to H_{∗}.

Example 7.12 Equivariant cobordism defines a G-homology theory on the category of proper G-CW -complexes. A G-CW -complex is proper if and only if all its isotropy groups are finite. Namely, one assigns to a proper G-CW - complex the equivariant cobordism group Ω_{p}^{G}(X) of cocompact proper smooth G-manifolds M of dimension p with a G-map M → X.

Definition 7.13 Let E be a covariant Or(G)-spectrum. Define for a pair of G-CW -complexes (X, A) its homology with coefficients in E

H_{p}^{G}(X, A; E) := π_{p}((X ∪_{A} cone(A)) ⊗_{Or(G)} E),

where cone(A) is the cone over A.

As usual H_{p}^{G}(X; E) means H_{p}(X, ∅; E). Notice that the cone over the empty set consists of one point so that X ∪_{A} cone(A) is X_{+} for A = ∅. The proof of the next result is presented in [30, Lemma 4.4].

Theorem 7.14 H_{∗}^{G}(−, E) is a G-homology theory for G-CW -complexes which satisfies the disjoint union axiom. We have canonical isomorphisms

H_{p}^{G}(G/H; E) = π_{p}(E(G/H)).

Our main examples of Or(G)-spectra are presented in the next theorem.

Theorem 7.15 There are specific constructions of covariant Or(G)-spectra

K: Or(G) → Ω − SPECTRA;
L: Or(G) → Ω − SPECTRA;
K^{top} : Or(G) → Ω − SPECTRA

satisfying for all p ∈ ℤ

π_{p}(K(G/H)) ≅ K_{p}(RH);
π_{p}(L(G/H)) ≅ L_{p}^{⟨−∞⟩}(RH);
π_{p}(K^{top}(G/H)) ≅ K_{p}^{top}(C_{r}^{∗}(H)).

Here K_{p}(RH) is the algebraic (non-connective) K-theory of the group ring RH with coefficients in the associative ring with unit R, L_{p}^{⟨−∞⟩}(RH) is the algebraic L-theory with respect to the decoration ⟨−∞⟩ of RH with respect to the involution sending ∑_{h∈H} λ_{h} · h to ∑_{h∈H} \overline{λ_{h}} · h^{−1} for fixed involution R → R, r ↦ \overline{r} and K_{p}^{top}(C_{r}^{∗}(H)) is the topological K-theory of the reduced complex C_{r}^{∗}-algebra C_{r}^{∗}(H) of H. An example for the involution on R is the identity for commutative R or R = ℂ with complex conjugation.

All the constructions first send G/H to the groupoid G^{G}(G/H), where for any G-set S the groupoid G^{G}(S) has as objects the elements of S and as morphism from s_{1} to s_{2} the set of elements {g ∈ G | g · s_{1} = s_{2}}. Composition is given by group multiplication. Then certain standard constructions of spectra for modules over the group ring RH are extended to the category of contravariant functors from G^{G}(G/H) to the category of R-modules. In the C^{∗}-case one must use C^{∗}-categories. Notice that one cannot assign directly K(RH), L(RH) or K^{top}(C_{r}^{∗}(H)) to G/H since this conflicts functoriality. Namely, an inner automorphism of H acts trivially on the homotopy groups of the spectrum but it does not act trivially on the spectrum itself and for h ∈ H the automorphism G/H → G/H, gH ↦ ghH is the identity. More details of the construction can be found in [30, Section 2].

7.2 Isomorphisms Conjectures and Assembly Maps

In this section we formulate an abstract isomorphism conjecture and specialize it to the two most important examples, the Farrell-Jones Isomorphism Conjecture and the Baum-Connes Conjecture.

7.2.1 Classifying Spaces of Families

The next definition is due to tom Dieck [114], [115, I.6].

Definition 7.16 Let G be a group. A family of subgroups F is a set of subgroups closed under conjugation and finite intersection. A classifying space of G with respect to F is a G-CW -complex E(G; F) such that E(G, F)^{H} is contractible for H ∈ F and any isotropy group belongs to F.

Theorem 7.17 1. A model for E(G; F) exists. There are functorial constructions;

2. For any G-CW -complex X whose isotropy groups belong to F, there is up to G-homotopy precisely one G-map X → E(G; F). In particular two models for E(G; F) are G-homotopic.

\underline{\underline{Proof}} : see [114], [115, I.6]. A functorial “bar-type” construction is given in [30, section 7].

A group H is called virtually cyclic if it contains a cyclic group of finite index. Denote by VC the family of virtually cyclic subgroups and by FIN the family of finite subgroups.

Remark 7.18 If F consists only of the trivial subgroup, E(G; F) is the same as the total space EG of the universal G-principal bundle EG → BG. If F consists of all groups, then G/G is a model for E(G; F). Sometimes the space E(G; FIN) is denoted by \underline{E}G in the literature. For discrete groups G the G-CW -complex E(G; FIN) is a classifying space for proper G-actions which is studied for instance in [9, section 2].

Sometimes geometry yields small models for E(G; FIN). Let G be a discrete subgroup of a Lie group L with a finite number of components. If K is a maximal compact subgroup of L, then L/K is homeomorphic to ℝ^{n} and L/K with the obvious left G-action can be taken as a model for E(G, FIN) (see [1, Corollary 4.14]). If G is a group of finite virtual cohomological dimension, then there is a finite-dimensional classifying space E(G, FIN) (see [104, Proposition 12]) and hence a finite dimensional model for E(G, FIN). Many examples of such groups are discussed by Serre in [104]. The Rips complex of a word hyperbolic group is a model for E(G, FIN) [80]. More information about E(G; FIN) can be found for instance in [63], [71], [75].

7.2.2 The Formulation of the Isomorphism Conjectures

Now we can formulate the general Isomorphism Conjecture.

Definition 7.19 Let E be a covariant Or(G)-spectrum and let F be a family of subgroups. The assembly map for E with respect to F is the map

asmb: H_{p}^{G}(E(G; F); E) → H_{p}^{G}(G/G; E) = π_{p}(E(G/G)),

which is induced by the canonical projection E(G; F) → G/G and the canonical identification mor_{Or(G)}(?, G/G) ⊗_{Or(G)} E = E(G/G). The Isomorphism Conjecture for E with respect to F says that the assembly map is bijective for all p ∈ ℤ.

Remark 7.20 The point is to find for given E the family F as small as possible. If we take F to be the family of all subgroups, the map above is an isomorphism but this is a trivial and useless fact. The philosophy is to express π_{p}(E(G/G)), which is the group we are interested in, by the groups π_{q}(E(G/H)) for q ≤ p and H ∈ F and all their relations coming from inclusion of subgroups of F, which we hopefully understand. Notice that in principle H_{p}^{G}(E(G; F); E) can be computed from the groups π_{q}(E(G/H)) by an equivariant Atiyah-Hirzebruch spectral sequence. In this sense one may think of the Isomorphism Conjecture as a kind of induction theorem such as Artin’s Theorem about representations of finite groups.

Remark 7.21 The assembly map appearing in Definition 7.19

asmb: H_{p}^{G}(E(G; F); E) → H_{p}^{G}(G/G; E) = π_{p}(E(G/G))

can be identified with the homomorphism induced on homotopy groups by the canonical map

hocolim_{Or(G;F)} E|_{Or(G;F)} → (hocolim_{Or(G)} E) = E(G/G)

as explained in [30, Section 5.2], where Or(G; F) is the full subcategory of Or(G) consisting of objects G/H for H ∈ F.

Definition 7.22 The Farrell-Jones Isomorphism Conjecture for K-theory or L-theory respectively for the group G and the associative ring with unit R is the Isomorphism Conjecture of Definition 7.19 for E = K or E = L respectively and the family F = VC of virtually cyclic subgroups, i.e. the assembly map

H_{p}^{G}(E(G; VC); K) ⟶^{≅} K_{p}(RG)

or

H_{p}^{G}(E(G; VC); L) ⟶^{≅} L_{p}^{⟨−∞⟩}(RG)

respectively is bijective for all p ∈ ℤ.

Definition 7.23 The Baum-Connes Conjecture is the Isomorphism Conjecture of Definition 7.19 for E = K^{top} and the family F = FIN of finite subgroups, i.e. the assembly map

H_{p}^{G}(E(G; FIN); K^{top}) ⟶^{≅} K_{p}^{top}(C_{r}^{∗}(G))

is bijective for all p ∈ ℤ.

Remark 7.24 The Farrell-Jones Isomorphism Conjecture 7.22 for L_{p}^{⟨−∞⟩}(ℤG) for all p ∈ ℤ and the Farrell-Jones Isomorphism Conjecture 7.22 for K-theory for K_{p}(ℤG) for all p ≤ 1 have been proved for groups G, for which there exists a Lie group L with finitely many components and a discrete cocompact subgroup H ⊂ L such that G occurs as subgroup of H [39, Theorem 2.1 and Remark 2.1.3.]. If G is torsionfree and R = ℤ, then the Farrell-Jones Isomorphism Conjecture for K-theory for K_{p}(ℤG) with p ≤ 1 is equivalent to the vanishing of Wh(G), K̃_{0}(ℤG) and K_{i}(ℤG) for i ≤ −1 and has been proved by Farrel and Jones for all groups G which occur as fundamental groups of closed manifold with nonpositive sectional curvature or of complete Riemannian manifolds with pinched negative sectional curvature [42]. The Farrell-Jones Isomorphism Conjecture 7.22 for K-theory for K_{p}(RG) with p ≤ 1 and any associative ring with unit R has been proved for groups G which occur as fundamental groups of closed Riemannian manifolds with negative sectional curvature by Bartels, Farrell, Jones and Reich [7].

The Baum-Connes Conjecture 7.23 has been proved for groups which satisfy the Haagerup property (= a-T-menable group) by Higson and Kasparov [52].

For information about groups having the Haagerup property we refer to [10], where it is shown that this class of groups includes all amenable groups but no infinite group which has Kazhdan’s property T. For information about groups with property T we refer to [31]. Lafforgue proved the Baum-Connes Conjecture 7.23 also for some groups which do have property T (see [64], [65]). The Baum-Connes-Conjecture 7.23 is known for all word-hyperbolic groups [89].

We refer for instance to [11], [37], [38], [39], [40], [41], [42] and [44] for more information about the Farrell-Jones Isomorphism Conjecture 7.22 and for instance to [9], [56], [64] and [116] for more information about the Baum-Connes Conjecture 7.23.

7.2.3 Conclusions from the Isomorphism Conjectures

Next we draw some conclusions from the Farrell-Jones Isomorphism Conjecture 7.22 and the Baum-Connes Conjecture 7.23 in order to illustrate their meaning.

Recall the Hirzebruch signature formula for a closed oriented manifold M

sign(M) = ⟨L(M), [M]⟩

and the obvious fact that sign(M) is an oriented homotopy invariant.

Conjecture 7.25 (Novikov Conjecture for G) Given a closed oriented manifold M and a map f: M → BG and x ∈ H^{∗}(BG), define the higher signature by

sign_{x}(M, f) = ⟨f^{∗}(x) ∪ L(M), [M]⟩.

The Novikov Conjecture for G says that these are oriented homotopy invariants, i.e. for f: M → BG, g: N → BG and an orientation preserving homotopy equivalence u: M → N with g ∘ u ≃ f we have

sign_{x}(M, f) = sign_{x}(N, g).

The next result is discussed for instance in [45].

Theorem 7.26 The Farrell-Jones Isomorphism Conjecture 7.22 for algebraic L-theory for G and R = ℤ as well as the Baum-Connes Conjecture 7.23 for G imply the Novikov Conjecture for G.

Conjecture 7.27 (Borel Conjecture) Let M and N be closed aspherical manifolds. Then any homotopy equivalence f: M → N is homotopic to a homeomorphism. In particular M and N are homeomorphic if and only if they have isomorphic fundamental groups.

Theorem 7.28 If the Farrell-Jones Isomorphism Conjecture 7.22 holds for G and R = ℤ for both K-theory and L-theory, then the Borel Conjecture holds for closed aspherical manifolds M and N of dimension ≥ 5 with π_{1}(M) ≅ π_{1}(N) ≅ G.

\underline{\underline{Proof}} : We give a sketch of the proof. The Borel Conjecture is equivalent to the claim that the structure set of M in the topological category is trivial

S_{n}^{top}(M) = {id: M → M}.

We have the surgery exact sequence

[ΣM, G/T OP] → L_{p+1}^{s}(ℤπ) → S_{n}^{top}(M) → [M, G/O] → L_{p}^{s}(ℤπ).

The K-theory part of the Farrell-Jones Isomorphism Conjecture implies that Wh(π), K̃_{0}(ℤπ) and K_{i}(ℤπ) for i ≤ −1 vanish. Hence the Rothenberg sequence implies that we do not have to take care of the decorations for the L-groups and structure set. The assembly map in the L-theory part in dimension p and p + 1 can be identified with the first map and last map appearing in the part of surgery sequence above using Poincaré duality for the sources.

More information about the Novikov-Conjecture and the Borel Conjecture can be found for instance in [29], [45].

Conjecture 7.29 (Stable Gromov-Lawson-Rosenberg Conjecture for G) Let G be a group. Let M be a closed Spin-manifold with fundamental group G. Let B be some closed simply connected manifold with vanishing Â-genus. Then M ×B^{k} carries a Riemannian metric of positive scalar curvature for some k ≥ 0 if and only if

index_{C_{r}^{∗}(G)}(M̃, D̃) = 0.

Here D is the Dirac operator and D̃ its lift to M̃ and index_{C_{r}^{∗}(G)}(M̃, D̃) is the C_{r}^{∗}(G)-index defined by Mishchenko and Fomenko [90].

Remark 7.30 Notice that by Bott periodicity

index_{C_{r}^{∗}(G)}(M̃, D̃) = index_{C_{r}^{∗}(G)}({ \atop M × B^{k}}, D̃).

Hence the index theory cannot distinguish between M and M × B^{k}. Indeed, Schick [103] has constructed counterexamples to the unstable version of the Gromov-Lawson-Rosenberg Conjecture following ideas of Stolz, where one requires a positive scalar curvature metric on M itself instead of M ×B^{k} for some k ≥ 0.

The next result is due to Stolz.

Theorem 7.31 If the group G satisfies the Baum-Connes Conjecture 7.23, then the stable Gromov-Lawson Conjecture 7.29 holds for G.

Conjecture 7.32 (Kadison Conjecture for G) Let G be torsionfree. Let p ∈ C_{r}^{∗}(G) be an idempotent, i.e. p^{2} = p. Then p = 0, 1.

Remark 7.33 The Baum-Connes Conjecture 7.23 for G implies the Kadison Conjecture for G. This is a consequence of the L^{2}-index theorem of Atiyah. For a discussion of this proof and the relation of the Baum-Connes Conjecture 7.23 to the (modified) Trace Conjecture we refer to [72].

7.3 Characterizing Assembly Maps

Next we give an abstract characterization of assembly maps. It generalizes the non-equivariant characterization of Weiss-Williams [120] to the equivariant setting.

A covariant functor

E: G − CW − COMPLEXES → SPECTRA

is called (weakly) homotopy invariant if it sends G-homotopy equivalences to (weak) homotopy equivalences of spectra. The functor E is (weakly) excisive if it has the following four properties:

1. it is (weakly) homotopy invariant;

2. E(∅) is contractible;

3. it respects homotopy pushouts up to (weak) homotopy equivalence;

4. E sends countable disjoint unions to countable wedges up to (weak) homotopy;

Remark 7.34 E is weakly excisive if and only if π_{q}(E(X)) defines a G-homology theory on the category of G-CW -complexes satisfying the disjoint union axiom for countable disjoint unions.

Definition 7.35 Let E be a covariant Or(G)-spectrum, i.e. a covariant functor

E: Or(G) → SPECTRA .

Define an extension

E_{%} : G − CW − COMPLEXES → SPECTRA, X ↦ map_{G}(?; X)_{+} ⊗_{Or(G)} E.

We conclude from Theorem 7.14.

Lemma 7.36 E_{%} is excisive.

The proof of the next two results can be found in [30, Theorem 6.3].

Lemma 7.37 Let T: E → F be a transformation of (weakly) excisive functors

E, F: G − CW − COMPLEXES → SPECTRA

so that T(G/H) is a (weak) homotopy equivalence of spectra for all H ⊂ G. Then T(X) is a (weak) homotopy equivalence of spectra for all G-CW -complexes X.

Theorem 7.38 For any (weakly) homotopy invariant functor

E: G − CW − COMPLEXES → SPECTRA

there is a (weakly) excisive functor

E^{%} : G − CW − COMPLEXES → SPECTRA

and natural transformations

A_{E} : E^{%} → E;
B_{E} : E^{%} → (E |_{Or(G)})_{%},

which induce (weak) homotopy equivalences of spectra A_{E}(G/H) for all H ⊂ G and (weak) homotopy equivalences of spectra B_{E}(X) for all G-CW -complexes X.

E is (weakly) excisive if and only if A_{E}(X) is a (weak) homotopy equivalence of spectra for all G-CW -complexes X.

Remark 7.39 Theorem 7.38 characterizes the assembly map in the sense that

A_{E} : E^{%} ⟶ E

is the universal approximation from the left by a (weakly) excisive functor of a (weakly) homotopy invariant functor E from G − CW − COMPLEXES to SPECTRA which agrees with E on homogeneous spaces G/H. Namely, let

T: F → E

be a transformation of functors from G − CW − COMPLEXES to SPECTRA such that F is (weakly) excisive and T(G/H) is a (weak) homotopy equivalence for all H ⊂ G. Then for any G-CW -complex X the following diagram commutes

F^{%}(X) ⟶_{≃}^{A_{F}(X)} F(X)
T^{%}(X)↓^{≃}  ↓^{T(X)}
E^{%}(X) ⟶^{A_{E}(X)} E(X)

and A_{F}(X) and T^{%}(X) are (weak) homotopy equivalences. Hence one may say that T(X) factorizes over A_{E}(X).

Remark 7.40 We can apply the construction above to the homotopy invariant functor

E: G − CW − COMPLEXES → SPECTRA which sends X to

K(π(EG ×_{G} X));
L(π(EG ×_{G} X));
K^{top}(C_{r}^{∗}(π(EG ×_{G} X)).

Then the assembly map appearing in the Isomorphism Conjectures above is given by

π_{p}(A_{E}(X)): π_{p}(E^{%}(X)) → π_{p}(E(X))

if one puts X = E(G; VC) or X = E(G, FIN). More generally, if Z is a free G-CW -complex and one replaces EG ×_{G} X by Z ×_{G} X above, one gets the fibered Farrell-Jones Isomorphism Conjecture [39, 1.7 on page 262].

Remark 7.41 The assembly maps appearing in the Farrell-Jones Isomorphism Conjecture 7.22 and in the Baum-Connes Conjecture 7.23 were defined differently, in terms of index theory or as forget control maps. The identification of the original definitions with the one presented here is non-trivial and carried out by Hambleton-Pedersen [50] using the universal characterization of assembly maps of Section 7.3.

7.4 The Choice of the Families

In this subsection we want to illustrate why one uses in the Farrell-Jones Isomorphism Conjecture 7.22 the family VC and in the Baum-Connes Conjecture 7.23 the family FIN and under which circumstances one may replace VC by FIN also in the Farrell-Jones Isomorphism Conjecture 7.22.

Example 7.42 Consider a semi-direct product G ⋊_{ϕ} ℤ. The automorphism ϕ induces a ϕ: G → G-equivariant homotopy equivalence Eϕ: E(G; FIN) → E(G; FIN). The to both sides infinite mapping telescope T_{Eϕ} with the obvious G ⋊_{ϕ} ℤ-action is a model for E(G ⋊_{ϕ} ℤ; FIN). This implies that we get for any covariant Or(G ⋊_{ϕ} ℤ; FIN)-spectrum E a Wang sequence

… → H_{p}^{G}(E(G; FIN); E|_{G}) ⟶^{ϕ_{∗}−id} H_{p}^{G}(E(G; FIN); E|_{G})

→ H_{p}^{G⋊_{ϕ}ℤ}(E(G ⋊_{ϕ} ℤ; FIN); E)

→ H_{p−1}^{G}(E(G; FIN); E|_{G}) ⟶^{ϕ_{∗}−id} H_{p−1}^{G}(E(G; FIN); E|_{G}) → …

In particular we get for ϕ = id an isomorphism

H_{p}^{G×ℤ}(E(G×ℤ; FIN); E) ≅ H_{p}^{G}(E(G; FIN); E|_{G})⊕H_{p−1}^{G}(E(G; FIN); E|_{G}).

Hence the Baum-Connes Conjecture 7.23 can only be true if there is an isomorphism

K_{p}^{top}(C_{r}^{∗}(G × ℤ)) ≅ K_{p}^{top}(G) ⊕ K_{p−1}^{top}(G).

This is indeed the case. More generally, a Wang sequence exists by results of Pimsner-Voiculescu [93, Theorem 3.1 on page 151].

Remark 7.43 The computation of algebraic K-groups involves VC and not only FIN since in the Bass-Heller-Swan decomposition

K_{i}(R[ℤ]) ≅ K_{i−1}(R) ⊕ K_{i}(R) ⊕ Nil_{i−1}(R) ⊕ Nil_{i−1}(R).

Nil-terms occur. These Nil-terms are very hard to compute (see e.g. [36]). At least they vanish if R is regular, for instance if R is ℤ or a field. The difficulties about the Nil-terms are absorbed by taking the family VC instead of FIN. Notice that enlarging the family FIN to VC is a necessary step to take Nil-terms into account but it is a priori not clear that this takes care of this complication in general.

The corresponding result for L-theory is the Shaneson splitting [106] which involves a change of decoration. For instance

L_{p}^{s}(ℤ[G × ℤ]) = L_{p}^{s}(ℤG) ⊕ L_{p−1}^{h}(ℤG).

Therefore one must pass to the L-groups L^{⟨−∞⟩} which do satisfy

L_{p}^{⟨−∞⟩}(ℤ[G × ℤ]) = L_{p}^{⟨−∞⟩}(ℤG) ⊕ L_{p−1}^{⟨−∞⟩}(ℤG).

The version of the Farrell-Jones Isomorphism Conjecture 7.22 for L^{p}, L^{h} and L^{s} is definitely false by a result of Farrell-Jones-Lück [43].

Example 7.44 Let G be the amalgamated product G = G_{1} ∗_{G_{0}} G_{2} over a common subgroup G_{0} of G_{1} and G_{2}. Then one can construct a cellular Gpushout

G ×_{G_{0}} E(G_{0}; FIN) ⟶ G ×_{G_{1}} E(G_{1}; FIN)
↓  ↓
G ×_{G_{2}} E(G_{2}; FIN) ⟶  E(G; FIN)

Hence for any covariant Or(G)-spectrum E there is a Mayer-Vietoris sequence

… → H_{p}^{G_{0}}(E(G_{0}; FIN); E|_{G_{0}})

→ H_{p}^{G_{1}}(E(G_{1}; FIN); E|_{G_{1}}) ⊕ H_{p}^{G_{2}}(E(G_{2}; FIN); E|_{G_{2}})
→ H_{p}^{G}(E(G; FIN); E) → H_{p−1}^{G_{0}}(E(G_{0}; FIN); E|_{G_{0}})

→ H_{p−1}^{G_{1}}(E(G_{1}; FIN); E|_{G_{1}}) ⊕ H_{p−1}^{G_{2}}(E(G_{2}; FIN); E|_{G_{2}}) → …

Hence the Baum-Connes Conjecture 7.23 predicts the existence of a Mayer-Vietoris sequence

… → K_{p}^{top}(C_{r}^{∗}(G_{0})) → K_{p}^{top}(C_{r}^{∗}(G_{1})) ⊕ K_{p}^{top}(C_{r}^{∗}(G_{2})) → K_{p}^{top}(C_{r}^{∗}(G))

→ K_{p−1}^{top}(C_{r}^{∗}(G_{0})) → K_{p−1}^{top}(C_{r}^{∗}(G_{1})) ⊕ K_{p−1}^{top}(C_{r}^{∗}(G_{2})) → …

Such a sequence exists by results of Pimsner [94, Theorem 18 on page 632].

Notice that such a cellular G-pushout does not exist if one replaces FIN by VC.

Remark 7.45 Waldhausen [117] has shown that the corresponding sequence for K-theory does not exists in this from, again certain Nil-terms arise. Again this shows that one needs to deal with VC for algebraic K-theory.

Remark 7.46 A similar complication occurs for the L-groups by the UNilgroups which appear in the formula for free products due to Cappell [20]

L̃_{p}^{⟨−∞⟩}(ℤ[G ∗ H]) ≅ L̃_{p}^{⟨−∞⟩}(ℤG) ⊕ L̃_{p}^{⟨−∞⟩}(ℤH) ⊕ UNil .

The UNil-groups are infinitely generated 2-torsion groups and have not been computed completely. Therefore one has to use the family VC in the Isomorphism Conjecture for L-theory.

The following theorem is useful if one wants to decrease the family F appearing in the isomorphism conjecture.

Theorem 7.47 Let F ⊂ G be families of subgroups of the group G. Let n be an integer and P be a set of prime numbers. Let E be K, L or K^{top}. Suppose for every H ∈ G that the assembly map induces an isomorphism

H_{p}^{H}(E(H; H ∩ F); E) [\frac{1}{P} ] → H_{p}^{H}(H/H; E) [\frac{1}{P} ]

for p ≤ n, where H∩F is the family of subgroups of H given by {H∩K | K ∈ F}. Then the map

H_{p}^{G}(E(G; F); E) [\frac{1}{P} ] → H_{p}^{G}(E(G; G); E) [\frac{1}{P} ]

is an isomorphism for p ≤ n.

\underline{\underline{Proof}} : This follows from [76, Theorem 2.3].

Remark 7.48 In general it is much harder to deal with the family VC instead of the family FIN. Suppose that R = ℤ. In L-theory there is no difference if one inverts 2. Then the decorations do not matter by the Rothenberg sequences and all UNil-terms vanish. Moreover, for any group G we conclude from Theorem 7.47 and [124]

H_{p}^{G}(E(G, FIN); L) [\frac{1}{2}] ≅ H_{p}^{G}(E(G, VC); L) [\frac{1}{2}]

and from [97, page 376] or [98, Proposition 22.34 on page 252]

L_{p}(ℤG) [\frac{1}{2}] ≅ L_{p}(ℚG) [\frac{1}{2}] .

This allows for computations of L_{p}(ℤG)[1/2] to pass to rational coefficients and to use FIN instead of VC. This much easier than to work with ℤ-coefficients and VC.

Remark 7.49 Notice that K_{q}(ℤG) → K_{q}(ℚG) is in general not bijective rationally as the example G = {1} and q = 1 shows. So for computations of K_{q}(ℤG) we have to stick to integral coefficients and to the family VC even if we only want to get rational information.

If F is a field of characteristic zero, then the obvious map

H_{p}^{G}(E(G; FIN); K) ⟶^{≅} H_{p}^{G}(E(G; VC); K)

is bijective, where K is the covariant Or(G)-spectrum with π_{n}(K(G/H)) = K_{n}(F H). This follows from Theorem 7.47 and [117] since any virtually cyclic group maps to ℤ or ℤ/2∗ℤ/2 with finite kernel and hence is a semidirect product of a finite group with ℤ or can be written as an amalgam of two finite groups over a finite subgroup and the relevant Nil-terms vanish because for any finite group (actually poly-cyclic suffices) group H and field F of characteristic zero the ring F H is regular. Moreover, the Farrell-Jones Isomorphism Conjecture 7.22 predicts that K_{i}(F G) = 0 for i ≤ −1 and the canonical map

colim_{G/H∈Or(G;FIN)} K_{0}(F H) → K_{0}(F G)

is bijective, where Or(G; FIN) is the full subcategory of Or(G) consisting of objects G/H with finite H.

It seems to be the case that in general the computation of K_{p}^{top}(C_{r}^{∗}(G)) is easier than the one of L_{p}(ℤG) and the computation of K_{p}(ℤG) is the hardest.

7.5 Miscellaneous

One can construct the following commutative diagram

H_{n}^{G}(E(G; FIN); L(ℤ?)[1/2])  ⟶^{asmb_{1}}  L_{n}(ℤG)[1/2]
i_{1}↓≅  ^{j_{1}}↓≅
H_{n}^{G}(E(G; FIN); L(ℚ?)[1/2])  ⟶^{asmb_{2}} L_{n}(ℚG)[1/2]
i_{2}↓≅  ^{j_{2}}↓
H_{n}^{G}(E(G; FIN); L(ℝ?)[1/2])  ⟶^{asmb_{3}}  L_{n}(ℝG)[1/2]
i_{3}↓≅  ^{j_{3}}↓
H_{n}^{G}(E(G; FIN); L(C_{r}^{∗}(?; ℝ))[1/2]) ⟶^{asmb_{4}} L_{n}(C_{r}^{∗}(?; ℝ))[1/2]
i_{4}↓≅  ^{j_{4}}↓≅
H_{n}^{G}(E(G; FIN); K^{top}(C_{r}^{∗}(?; ℝ))[1/2]) ⟶^{asmb_{5}} K_{n}^{top}(C_{r}^{∗}(?; ℝ))[1/2]

where L(ℤ?)[1/2], L(ℚ?)[1/2], L(ℝ?)[1/2], L(C_{r}^{∗}(?))[1/2] and K^{top}(C_{r}^{∗}(?; ℝ))[1/2] are the variants of the covariant Or(G)-spectra such that the n-th homotopy group of its evaluation at G/H is L_{n}(ℤH)[1/2], L_{n}(ℚH)[1/2], L_{n}(ℝH)[1/2], L_{n}(C_{r}^{∗}(H; ℝ))[1/2], K_{n}^{top}(C_{r}^{∗}(H))[1/2] for C_{r}^{∗}(H; ℝ) the reduced real group C^{∗}- algebra of H. All horizontal maps are assembly maps as appearing in Definition 7.19. The maps i_{k} and j_{k} for k = 1, 2, 3 are given by change of rings map. The isomorphisms i_{4} and j_{4} come from the general isomorphism for any real C^{∗}-algebra A

L_{n}^{p}(A)[1/2] ⟶^{≅} K_{n}^{top}(A)[1/2]

and its spectrum version [99, Theorem 1.11 on page 350]. The maps i_{1}, j_{1}, i_{2} are isomorphisms by [97, page 376] and [98, Proposition 22.34 on page 252]. The map i_{3} is bijective since for a finite group H we have ℝH = C_{r}^{∗}(H).

We see that the injectivity of asmb_{5} which is the assembly map in the Baum-Connes Conjecture 7.23 after inverting 2 implies the injectivity of asmb_{1} which is the assembly map in the Farrell-Jones Isomorphism Conjecture 7.22 for Ltheory and the family FIN after inverting 2. One may conjecture that all right vertical maps are isomorphisms and try to prove this directly. Then the Baum-Connes Conjecture 7.23 after inverting 2 becomes equivalent to the Farrell-Jones Isomorphism Conjecture 7.22 for the family FIN after inverting 2 because of Remark 7.48.

160CHAPTER 7. THE FARRELL-JONES AND BAUM-CONNES CONJECTURES

Computations of K- and L-Groups for Infinite Groups

In this chapter we introduce equivariant Chern characters which compute rationally the G-homology of a proper G-CW -complex in terms of ordinary singular homology of various fixed point sets provided some additional structures (induction structure, Mackey structure on the coefficients) are available on the Ghomology theory. These additional structures exist for the various G-homology theories appearing as the source of the assembly map in the Farrell-Jones Isomorphism Conjecture 7.22 and the Baum-Connes Conjecture 7.23. This is used to give general formulas for the rationalized algebraic K- and L-theory of group rings and the topological K-theory of reduced group C^{∗}-algebras provided the Isomorphism Conjectures hold. We also present some integral computations of K- and L-groups for some special groups. The integral calculations are much harder and there seems to be no general formula as in the rationalized situation.

8.1 Equivariant Chern Characters

As an illustration we recall a construction of a non-equivariant Chern character due to Dold [34]. The actual construction of the equivariant Chern character is technical much more elaborate and will therefore not be presented here but Dold’s construction is an important ingredient.

162CHAPTER 8. COMPUTATIONS OF K- AND L-GROUPS FOR INFINITE GROUPS

Example 8.1 Consider a (non-equivariant) homology theory H_{∗} with values in R-modules for ℚ ⊂ R. Then a (non-equivariant) Chern character for a CW - complex X is given by the following composite

ch_{n}: ⨁_{p+q=n} H_{p}(X; H_{q}(∗)) \underrightarrow{⨁_{p+q=n} α_{p,q}^{−1}} ⨁_{p+q=n} H_{p}(X; R) ⊗_{R} H_{q}(∗)

\underrightarrow{⨁_{p+q=n}(hur ⊗ id)^{−1}} ⨁_{p+q=n} π_{p}^{s}(X_{+}, ∗) ⊗_{ℤ} R ⊗_{R} H_{q}(∗)

\underrightarrow{⨁_{p+q=n} D_{p,q}} H_{n}(X).

Here the canonical map α_{p,q}: H_{p}(X; R) ⊗_{R} H_{q}(∗) → H_{p}(X; H_{q}(∗)) is bijective, since any R-module is flat over ℤ because of the assumption ℚ ⊂ R. The second bijective map comes from the Hurewicz homomorphism π_{p}^{s}(X_{+}, ∗) ⊗_{ℤ} R → H_{p}(X; R) which is bijective because of ℚ ⊂ R. The map D_{p,q} is defined as follows. For an element a ⊗ b ∈ π_{p}^{s}(X_{+}, ∗) ⊗_{ℤ} H_{q}(∗) choose a representative f: S^{p+k} → S^{k} ∧ X_{+} of a. Define D_{p,q}(a ⊗ b) to be the image of b under the composite

H_{q}(∗) ⟶^{σ} H_{p+q+k}(S^{p+k}, ∗) ⟶^{H_{p+q+k}(f)} H_{p+q+k}(S^{k} ∧ X_{+}, ∗) ⟶^{σ^{−1}} H_{p+q}(X),

where σ denotes the suspension isomorphism. This map turns out to be a transformation of homology theories and induces an isomorphism for X = ∗. Hence it is a natural equivalence of homology theories.

We have introduced the notion of a G-homology theory in Definition 7.10. A proper G-homology theory is a G-homology theory which is only defined for proper G-CW -complexes. To construct equivariant Chern characters we need the following additional structure called induction structure which does exist in the examples we are interested in.

Definition 8.2 A (proper) equivariant homology theory H_{∗}^{?} with values in Rmodules consists of a (proper) G-homology theory H_{∗}^{G} with values in R-modules for each group G together with the following so called induction structure: given a group homomorphism α: H → G and a (proper) H-CW -pair (X, A) such that ker(α) acts freely on X, there are for each n ∈ ℤ natural isomorphisms

ind_{α}: H_{n}^{H}(X, A) ⟶^{≅} H_{n}^{G}(ind_{α}(X, A))

satisfying

1. Compatibility with the boundary homomorphisms ∂_{n}^{G} ∘ ind_{α} = ind_{α} ∘∂_{n}^{H};

2. Functoriality Let β: G → K be another group homomorphism such that ker(β ∘ α) acts freely on X. Then we have for n ∈ ℤ

ind_{β∘α} = H_{n}^{K}(f_{1}) ∘ ind_{β} ∘ ind_{α}: H_{n}^{H}(X, A) → H_{n}^{K}(ind_{β∘α}(X, A)), where f_{1} : ind_{β} ind_{α}(X, A) ⟶^{≅} ind_{β∘α}(X, A), (k, g, x) ↦ (kβ(g), x) is the natural K-homeomorphism;

3. Compatibility with conjugation For n ∈ ℤ, g ∈ G and a (proper) G-CW -pair (X, A) the homomorphism ind_{c(g) : G→G}: H_{n}^{G}(X, A) → H_{n}^{G}(ind_{c(g) : G→G}(X, A)) agrees with H_{n}^{G}(f_{2}) for c(g) the conjugation homomorphism which sends g′ to gg′g^{−1} and the G-homeomorphism f_{2} : (X, A) → ind_{c(g) : G→G}(X, A) which sends x to (1, g^{−1}x) in G ×_{c(g)} (X, A).

Example 8.3 Equivariant cobordism yields an equivariant proper homology theory. The G-homology theory Ω_{∗}^{G} has been explained in Example 7.12. The induction structure comes from induction of spaces. Notice that for a group homomorphism α: H → G and a cocompact smooth H-manifold M such that ker(α) acts freely on M, the G-space ind_{α}(M) = G ×_{α} M is a cocompact Gmanifold.

Example 8.4 Let

E: GROUPOIDS^{inj} → SPECTRA

be a functor from the category of groupoids with injective functors as morphisms to spectra, where a functor F: G_{1} → G_{2} of groupoids is called injective if for each object x of G_{1} the group homomorphism aut_{G_{1}}(x) → aut_{G_{2}}(F (x)) induced by F is injective. For any group G and G-set S let G^{G}(S) be the groupoid whose set of objects is S and whose set of morphisms from s_{1} to s_{2} is {g ∈ G | gs_{1} = s_{2}}. Composition in this groupoid is given by the multiplication in G. We obtain a functor

G^{G}: Or(G) → GROUPOIDS^{inj}

Define for each group G a covariant Or(G)-spectrum

E^{G}: Or(G) → SPECTRA, G/H ↦ E(G^{G}(G(H))

Then we obtain an equivariant homology theory H_{∗}^{?} by assigning to a group G the G-homology theory H_{∗}^{G}(−; E^{G}). We leave it to the reader to figure out the induction structure. The key observation is that the obvious functor of groupoids

G^{H}(H/L) → G^{G}(ind_{α} G/L)

is an equivalence of categories if L ∩ ker(α) = {1}. This implies that the sources of the assembly maps appearing in the Farrell-Jones Isomorphism Conjecture 7.22 and the Baum-Connes Conjecture 7.23 are equivariant homology theories and the induction structures on the sources are compatible with the obvious induction homomorphism on the target since the relevant spectra are constructed as mentioned above.

One easily checks (see [70, Lemma 1.2])

Lemma 8.5 Consider finite subgroups H, K ⊂ G and an element g ∈ G with gHg^{−1} ⊂ K. Let R_{g^{−1}} : G/H → G/K be the G-map sending g′H to g′g^{−1}K and let c(g): H → K be the homomorphism sending h to ghg^{−1}. Denote by pr: (ind_{c(g) : H→K} ∗) → ∗ the projection. Then the following diagram commutes

H_{n}^{H}(∗) ⟶^{H_{n}^{K}(pr)∘ind_{c(g)}} H_{n}^{K}(∗)
ind_{H}^{G} ↓≅  ^{ind}_{K}^{G} ↓≅
H_{n}^{G}(G/H) ⟶^{H_{n}^{G}(R_{g^{−1}} )} H_{n}^{G}(G/K)

Let R be an associative commutative ring with unit. Let FGINJ be the category of finite groups with injective group homomorphisms as morphisms. Let M: FGINJ → R − MODULES be a bifunctor, i.e. a pair (M_{∗}, M^{∗}) consisting of a covariant functor M_{∗} and a contravariant functor M^{∗} from FGINJ to R − MODULES which agree on objects. We will often denote for an injective group homomorphism f: H → G the map M_{∗}(f): M(H) → M(G) by ind_{f} and the map M^{∗}(f): M(G) → M(H) by res_{f} and write ind_{H}^{G} = ind_{f} and res_{G}^{H} = res_{f} if f is an inclusion of groups. We call such a bifunctor M a Mackey functor with values in R-modules if

1. For an inner automorphism c(g): G → G we have M_{∗}(c(g)) = id: M(G) → M(G);

2. For an isomorphism of groups f: G ⟶^{≅} H the composites res_{f} ∘ ind_{f} and ind_{f} ∘ res_{f} are the identity;

3. Double coset formula We have for two subgroups H, K ⊂ G

res_{G}^{K} ∘ ind_{H}^{G} = ∑_{KgH∈K\G/H} ind_{c(g) : H∩g^{−1}Kg→K} ∘ res_{H}^{H∩g^{−1}Kg},

where c(g) is conjugation with g, i.e. c(g)(h) = ghg^{−1}.

Example 8.6 Our main examples of Mackey functors will be R_{ℚ}(H), K_{q}(RH), L_{q}(RH) and K_{q}^{top}(C_{∗}^{r}(H)). The Mackey structure comes from induction and restriction of modules.

Define for a finite group H

S_{H} (H_{q}^{H}(∗)) := coker ( ⨁_{\begin{matrix}K⊂H \\ K≠H \end{matrix}} ind_{K}^{H}: ⨁_{\begin{matrix}K⊂H \\ K≠H \end{matrix}} H_{q}^{K}(∗) → H_{q}^{H}(∗)) .  (8.7)

For a subgroup H ⊂ G we denote by N_{G}H := {g ∈ G | gHg^{−1} = H} the normalizer and by C_{G}H = {g ∈ G | gh = hg for all h ∈ H} the centralizer of H in G. Let H · C_{G}H be the subgroup of N_{G}H consisting of elements of the form hc for h ∈ H and c ∈ C_{G}H. Denote by W_{G}H the quotient N_{G}H/H ·C_{G}H. Notice that W_{G}H is finite if H is finite. It acts from the right on C_{G}H\X^{H} and from the left on H_{q}^{H}(∗) = H_{q}^{G}(G/H) and S_{H} (H_{q}^{H}(∗)). Let FGINJ be the category of finite groups with injective group homomorphisms as morphisms. The proof of the following result can be found in [70, Theorem 0.3].

Theorem 8.8 Let R be a commutative ring with ℚ ⊂ R. Let H_{∗}^{?} be a proper equivariant homology theory with values in R-modules. Suppose that the covariant functor

H_{q}^{?}(∗): FGINJ → R − MODULES, H ↦ H_{q}^{H}(∗)

extends to a Mackey functor for all q ∈ ℤ. Then there is for any group G and any proper G-CW -complex X a natural isomorphism

ch_{n}^{G}: ⨁_{p+q=n} ⨁_{(H)∈I} H_{p}(C_{G}H\X^{H}; R) ⊗_{R[W_{G}H]} S_{H} (H_{q}^{H}(∗)) ⟶^{≅} H_{n}^{G}(X),

where I is the set of conjugacy classes (H) of finite subgroups H of G.

Remark 8.9 If H_{∗}^{?} comes with additional structures such as a restriction structure or a multiplicative structure, then one can define the corresponding structure on the source of the equivariant Chern character and the equivariant Chern character respect these. [70, Section 6].

In the next step we want to specialize further and to express the terms S_{H} (H_{q}^{H}(∗)) by easier terms. This can be done in the cases we are interested in by using further structure.

Let ϕ: R → S be a homomorphism of associative commutative rings with unit. Let M be a Mackey functor with values in R-modules and let N and P be Mackey functors with values in S-modules. A pairing M × N → P with respect to ϕ is a family of maps

m(H): M(H) × N(H) → P (H), (x, y) ↦ m(H)(x, y) =: x · y,

where H runs through the finite groups and we require the following properties for all injective group homomorphisms f: H → K of finite groups:

(x_{1} + x_{2}) · y = x_{1} · y + x_{2} · y for x_{1}, x_{2} ∈ M(H), y ∈ N(H);
x · (y_{1} + y_{2}) = x · y_{1} + x · y_{2} for x ∈ M(H), y_{1}, y_{2} ∈ N(H);
(rx) · y = ϕ(r)(x · y)  for r ∈ R, x ∈ M(H), y ∈ N(H);
x · sy = s(x · y)  for s ∈ S, x ∈ M(H), y ∈ N(H);
res_{f}(x · y) = res_{f}(x) · res_{f}(y) for x ∈ M(K), y ∈ N(K);
ind_{f}(x) · y = ind_{f} (x · res_{f}(y)) for x ∈ M(H), y ∈ N(K);
x · ind_{f}(y) = ind_{f} (res_{f}(x) · y) for x ∈ M(K), y ∈ N(H).

A Green functor with values in R-modules is a Mackey functor U together with a pairing U × U → U with respect to id: R → R and elements 1_{H} ∈ U(H) for each finite group H such that for each finite group H the pairing U(H) × U(H) → U(H) induces the structure of an R-algebra on U(H) with unit 1_{H} and for any morphism f: H → K in FGINJ the map U^{∗}(f): U(K) → U(H) is a homomorphism of R-algebras with unit. Let U be a Green functor with values in R-modules and M be a Mackey functor with values in S-modules. A (left) U-module structure on M with respect to the ring homomorphism ϕ: R → S is a pairing U × M → M such that any of the maps U(H) × M(H) → M(H) induces the structure of a (left) module over the R-algebra U(H) on the Rmodule ϕ^{∗}M(H) which is obtained from the S-module M(H) by rx := ϕ(r)x for r ∈ R and x ∈ M(H).

Example 8.10 Our main examples of a Green functor will be R_{ℚ}(?). Our main example of modules over the Green functor R_{ℚ}(H) will be K_{q}(R?), L_{q}(R?) and K_{q}^{top}(C_{∗}^{r}(?)) for a commutative ring R with ℚ ⊂ R. The Mackey structure comes from induction and restriction of modules and the multiplicative structure of a Green functor from the tensor product with the diagonal action (see [70, Section 8]).

Let class_{ℚ}(H) be the ℚ-vector space of functions H → ℚ which are invariant under ℚ-conjugation, i.e. we have f(h_{1}) = f(h_{2}) for two elements h_{1}, h_{2} ∈ H if the cyclic subgroups ⟨h_{1}⟩ and ⟨h_{2}⟩ generated by h_{1} and h_{2} are conjugate in H. Elementwise multiplication defines the structure of a ℚ-algebra on class_{ℚ} with the function which is constant 1 as unit element. Taking the character of a rational representation yields an isomorphism of ℚ-algebras [105, Theorem 29 on page 102]

χ^{H}: ℚ ⊗_{ℤ} R_{ℚ}(H) ⟶^{≅} class_{ℚ}(H).  (8.11)

For a finite cyclic group C define

θ_{C} ∈ ℚ ⊗_{ℤ} Rep_{ℚ}(C) ≅ class_{ℚ}(C)  (8.12)

to be the function which sends h ∈ C to 1 if h is a generator and to 0 otherwise. Obviously θ_{C} is an idempotent.

The proof of the next theorem can be found in [70, Theorem 0.4].

Theorem 8.13 Let R be a commutative ring with ℚ ⊂ R. Let H_{∗}^{?} be a proper equivariant homology theory with values in R-modules. Suppose that the covariant functor FGINJ → R−MODULES sending H to H_{q}^{H}(∗) extends to a Mackey functor for all q ∈ ℤ, which is a module over the Green functor ℚ⊗_{ℤ} R_{ℚ}(?) with respect to the inclusion ℚ → R. Let J be the set of conjugacy classes (C) of finite cyclic subgroups C of G.

Then for any group G and any proper G-CW -complex X there is a natural isomorphism

ch_{n}^{G}: ⨁_{p+q=n} ⨁_{(C)∈J} H_{p}(C_{G}C\X^{C}; R) ⊗_{R[W_{G}C]} (θ_{C} · H_{q}^{C}(∗)) ⟶^{≅} H_{n}^{G}(X).

where θ_{C} · H_{q}^{C}(∗) is the direct summand of H_{q}^{C}(∗) given by the Rep_{ℚ}(C)-module structure on H_{q}^{C}(∗) and the idempotent θ_{C} ∈ Rep_{ℚ}(C).

8.2 Rational Computations

Next we apply the results about equivariant Chern characters to the homology groups H_{p}^{G}(E(G; FIN); K), H_{p}^{G}(E(G; FIN); L), and H_{p}^{G}(E(G; FIN); K^{top}) and discuss what these computations together with the Farrell-Jones Isomorphism Conjecture 7.22 and the Baum-Connes Conjecture 7.23 imply for K_{n}(RG), L_{n}(RG) and K^{top}(C_{∗}^{r}(G)). Notice that we can only handle E(G; FIN) and not E(G; VC) since our equivariant Chern character only works for proper G-CW - complexes.

We begin with a survey about some computations of K- and L-groups for finite groups. Let q(G), r(G) and c(G) be the number of irreducible rational, real and complex representations of a finite group G. Denote by r_{ℂ}(G) the number of irreducible real representations which are of complex type.

Theorem 8.14 For a finite group G, we have the following:

1. There are isomorphisms

Rep_{ℚ}(G) ≅ ℤ^{q(G)};
Rep_{ℝ}(G) ≅ ℤ^{r(G)};
Rep_{ℂ}(G) ≅ ℤ^{c(G)}.

The number q(G) is the number of conjugacy classes of cyclic subgroups in G, the number c(G) is the number of conjugacy classes of elements in G and the number r(G) is the number of ℝ-conjugacy classes of elements in G, where g_{1} and g_{2} in G are ℝ-conjugated if g_{1} and g_{2} or g_{1}^{−1} and g_{2} are conjugated;

2.

K_{q}^{top}(C_{r}^{∗}(G)) ≅ \{ \begin{matrix}Rep_{ℂ}(G) ≅ ℤ^{c(G)} & q = 0; \\ 0 & q = 1; \end{matrix}

3.

L_{q}(ℤG)[1/2] ≅ L_{q}(ℚG)[1/2] ≅ L_{q}(ℝG)[1/2]

≅ \{ \begin{matrix}ℤ[1/2]^{r(G)} & q ≡ 0 (4); \\ ℤ[1/2]^{r_{ℂ}(G)} & q ≡ 2 (4); \\ 0 & q ≡ 1, 3 (4); \end{matrix}

4. K_{q}(ℤG) = 0 for q ≤ −2; 5. K̃_{0}(ℤG) is finite. For a prime number p K_{0}(ℤ[ℤ/p]) is isomorphic to

the ideal class group of the Dedekind domain ℤ[exp(2πi/p)] (which is unknown);

6. The rank of Wh(G) as an abelian group is r(G) − q(G). The torsion of Wh(G) is isomorphic to the kernel of the change of rings homomorphism K_{1}(ℤG) → K_{1}(ℚG). In particular we have

Wh(ℤ/m) ≅ ℤ^{[m/2]+1−δ(m)},

where δ(m) is the number of divisors of m and [m/2] is the largest integer less or equal to m/2;

7. Let D_{2m} be the dihedral group of order 2m. We have

Wh_{2}(π) = 0, for π = {1}, ℤ/2, ℤ/3, ℤ/4;
| Wh_{2}(ℤ/6)| ≤ 2;
Wh_{2}(D_{6}) = ℤ/2;
Wh_{2}((ℤ/2)^{2}) ≥ (ℤ/2)^{2}.

(1) is proved in [105, Theorem 7 on page 19, Corollary 2 on page 96, page 102, page 106].

(2) follows from Morita equivalence applied to ℂ[π] ≅ ∏_{i=1}^{c(π)} M(n_{i}, n_{i}, ℂ) and the computation K_{0}(ℂ) = ℤ and K_{1}^{top}(ℂ) = 0.

(3) follows from [98, Proposition 22.34 on page 253].

(4) is proved in [24].

(5) see [86, page 29-32].

(6) For information about Wh(π) we refer to [91].

(7) is proved in [76, Theorem 3.2].

Theorem 8.15 Let R be an associative ring with unit satisfying ℚ ⊂ R. Let G be a (discrete) group. Let J be the set of conjugacy classes (C) of finite cyclic subgroups C of G. Then the rationalized assembly map in the Farrell-Jones Isomorphism Conjecture 7.22 with respect to FIN for the algebraic Kgroups K_{n}(RG) and the algebraic L-groups L_{n}(RG) and in the Baum Connes Conjecture 7.23 for the topological K-groups K_{n}^{top}(C_{r}^{∗}(G))

asmb: ℚ ⊗_{ℤ} H_{n}^{G}(E(G; FIN); K) → ℚ ⊗_{ℤ} K_{n}(RG);
asmb: ℚ ⊗_{ℤ} H_{n}^{G}(E(G; FIN); L) → ℚ ⊗_{ℤ} L_{n}(RG);
asmb: ℚ ⊗_{ℤ} H_{n}^{G}(E(G; FIN); K^{top}) → ℚ ⊗_{ℤ} K_{n}^{top}(C_{r}^{∗}(G))

can be identified with the homomorphisms

⨁_{p+q=n} ⨁_{(C)∈J} H_{p}(C_{G}C; ℚ) ⊗_{ℚ[W_{G}C]} θ_{C}^{C} · (ℚ ⊗_{ℤ} K_{q}(RC)) → ℚ ⊗_{ℤ} K_{n}(RG);
⨁_{p+q=n} ⨁_{(C)∈J} H_{p}(C_{G}C; ℚ) ⊗_{ℚ[W_{G}C]} θ_{C}^{C} · (ℚ ⊗_{ℤ} L_{q}(RC)) → ℚ ⊗_{ℤ} L_{n}(RG);
⨁_{p+q=n} ⨁_{(C)∈J} H_{p}(C_{G}C; ℚ) ⊗_{ℚ[W_{G}C]} θ_{C}^{C} · (ℚ ⊗_{ℤ} K_{q}^{top}(C_{r}^{∗}(C))) → ℚ ⊗_{ℤ} K_{q}^{top}(C_{r}^{∗}(G))).

In the L-theory case we assume that R comes with an involution R → R, r ↦ \overline{r} and that we use on RG the involution which sends ∑_{g∈G} r_{g} ·g to ∑_{g∈G} \overline{r_{g}} ·g^{−1}.

Remark 8.16 We conclude from Theorem 8.14 (2) and Theorem 8.15 that the Baum-Connes Conjecture 7.23 for a group G implies for q ∈ ℤ

⨁_{\begin{matrix}p∈ℤ \\ p+q=0 mod 2 \end{matrix}} ⨁_{(C)∈J} H_{p}(C_{G}C; ℚ)⊗_{ℚ[W_{G}C]}θ_{C}^{C}·(ℚ ⊗_{ℤ} Rep_{ℂ}(C)) ≅ ℚ⊗_{ℤ}K_{q}^{top}(C_{r}^{∗}(G)).

Remark 8.17 Next we want to compute ℚ ⊗_{ℤ} L_{n}(ℤG). Because of Remark 7.48 and Theorem 8.15 the Farrell-Jones Isomorphism Conjecture 7.22 implies

⨁_{p+q=n} ⨁_{(C)∈J} H_{p}(C_{G}C; ℚ) ⊗_{ℚ[W_{G}C]} θ_{C}^{C} · (ℚ ⊗_{ℤ} L_{q}(ℚC)) ≅ ℚ ⊗_{ℤ} L_{n}(ℤG).

Remark 8.18 In the algebraic K-theory case we face the problem that in the Farrell-Jones Isomorphism Conjecture 7.22 we have to deal with VC instead of FIN (see Remark 7.43 and Remark 7.45) even if we only want to carry out rational computations. If R is a field F of characteristic zero, we can replace VC by FIN as explained in Remark 7.49. Hence the Farrell-Jones Isomorphism Conjecture 7.22 and Theorem 8.15 imply for a field F of characteristic zero

⨁_{p+q=n} ⨁_{(C)∈J} H_{p}(C_{G}C; F) ⊗_{ℚ[W_{G}C]} θ_{C}^{C} · (ℚ ⊗_{ℤ} K_{q}(F C)) ≅ ℚ ⊗_{ℤ} K_{n}(F G).

Further reductions in special cases can be found for instance in [70, Section 8].

Remark 8.19 The rationally computations above show all the same pattern which one may call a separation of variables. Namely, the relevant group G occurs in all cases as group homology terms involving the group homology of the centralizers of finite cyclic subgroups and its contribution is independent of the relevant K- or L-theory and the ring R. The choice of the K- and L-theory and of R appear in a different contribution in terms of the values at RC for all finite cyclic subgroups. This contribution is only affected by the group G in a weak manner, only the possible finite cyclic subgroups C and the finite groups W_{G}C together with their actions on the group homology of C_{G}C and the K-and L-theory of RC matter.

8.3 Some Special Cases

There does not seem to be such general formulas as in Theorem 8.15 for the algebraic K- and L-theory of group rings and of the topological K-theory of reduced group C^{∗}-algebras without rationalizing. We state as illustration several special cases. The methods of these computations are essentially Mayer-Vietoris sequence or spectral sequence arguments applied to the sources of the assembly maps and use the fact that for these special cases the Farrell-Jones Isomorphism Conjecture 7.22 and the Baum-Connes Conjecture 7.23 are known to be true.

The next result is a special case of computations in [76, Theorem 4.4], where more generally compact planar groups are treated. The case of cocompact Fuchsian groups has also been carried out for K-theory by Berkhove, Juan-Pineda and Pearson [12].

Theorem 8.20 Let F be a cocompact Fuchsian group with presentation

F = ⟨a_{1}, b_{1}, …, a_{g}, b_{g}, c_{1}, …, c_{t} | c_{1}^{γ_{1}} = … = c_{t}^{γ_{t}} = c_{1}^{−1} ⋯ c_{t}^{−1}[a_{1}, b_{1}] ⋯ [a_{g}, b_{g}] = 1⟩

for integers g, t ≥ 0 and γ_{i} > 1. Then

1. The inclusions of the maximal subgroups ℤ/γ_{i} = ⟨c_{i}⟩ induce an isomorphism

⨁_{i=1}^{t} Wh_{q}(ℤ/γ_{i}) ⟶^{≅} Wh_{q}(F)

for q ≤ 1;

2. There are isomorphisms

L_{q}(ℤF)[1/2] ≅ \{ \begin{matrix}(1 + ∑_{i=1}^{t} [\frac{γ_{i}}{2} ]) · ℤ[1/2] & q ≡ 0 (4); \\ (2g) · ℤ[1/2] & q ≡ 1 (4); \\ (1 + ∑_{i=1}^{t} [\frac{γ_{i}−1}{2}]) · ℤ[1/2] & q ≡ 2 (4); \\ 0 & q ≡ 3 (4), \end{matrix}

where [r] for r ∈ ℝ denotes the largest integer less than or equal to r;

3. There are isomorphisms

K_{q}^{top}(C_{r}^{∗}(F)) ≅ \{ \begin{matrix}(2 + ∑_{i=1}^{t}(γ_{i} − 1)) · ℤ & q = 0; \\ (2g) · ℤ & q = 1. \end{matrix}

The next result is taken from [76, Theorem 0.2].

Theorem 8.21 Let 1 → ℤ^{n} → G → π → 1 be a group extension for a finite group π such that the conjugation action of π on ℤ^{n} is free, i.e. the only element in π with a fixed point in ℤ^{n} different from zero is the identity element in π. Let {M_{α} | α ∈ A} be a complete system of representatives of conjugacy classes of maximal finite subgroups of G. Then 1. The natural map induced by the inclusions of subgroups

⨁_{α∈A} Wh_{q}(M_{α}) → Wh_{q}(G)

is an isomorphism for q ≤ 1, and K_{q}(ℤG) is trivial for q ≤ −2;

2. There are short exact sequences

0 → ⨁_{α∈A} L̃_{q}(ℤM_{α})[1/2] → L_{q}(ℤG)[1/2]

→ H_{q}(G\E(G, FIN); L(ℤ))[1/2] → 0,

where L(ℤ) is the L-theory spectrum associated to the ring ℤ, H_{∗}(−; L(ℤ)) is the associated homology theory and the first map is induced by the various inclusions M_{α} → G.

If we invert 2|π|, this sequence splits and we obtain isomorphisms

(⨁_{α∈A} L̃_{q}(ℤM_{α}) [\frac{1}{2|π|}]) ⊕ H_{q}(π\T^{n}; L(ℤ)) [\frac{1}{2|π|}] ⟶^{≅} L_{q}(ℤG) [\frac{1}{2|π|}] ,

where the π-action on T^{n} is induced by the conjugation action of π on ℤ^{n};

3. If |π| is odd, then we obtain a short exact sequence which splits after inverting |π|

0 → ⨁_{α} L̃_{q}^{ϵ}(ℤM_{α}) → L_{q}^{ϵ}(ℤG) → H_{q}(G\\underline{E}G; L(ℤ)) → 0;

4. There are short exact sequences

0 → ⨁_{α∈A} K̃_{q}^{top}(C_{r}^{∗}(M_{α})) → K_{q}^{top}(C_{r}^{∗}(G)) → K_{q}^{top}(G\E(G; FIN)) → 0,

where K_{q}^{top}(G\E(G; FIN)) is the topological complex K-homology of the orbit space G\E(G; FIN) and the first map is induced by the various inclusions M_{α} → G.

If we invert |π|, this sequence splits and we obtain isomorphisms

(⨁_{α∈A} K̃_{q}^{top}(C_{r}^{∗}(M_{α})) [\frac{1}{|π|}]) ⊕ K_{q}^{top}(π\T^{n}) [\frac{1}{|π|}]

⟶^{≅}

K_{q}^{top}(C_{r}^{∗}(G)) [\frac{1}{|π|}] .

Next we give a description of the algebraic K- and L-groups of the integral group ring and the topological K-theory of the reduced C^{∗}-algebra of all 2dimensional crystallographic groups. The algebraic K-theory in dimension ≤ 1 has been determined in [92], and the topological K-theory has been computed in [125].

Our notation for the two-dimensional crystallographic groups follows that of [77]. The so called signatures of crystallographic groups which encode their presentations have been listed in [77, page 1204].

For the computations of Wh_{2} we assume that the Farrell-Jones Isomorphism Conjecture 7.22 for algebraic K-theory holds also for q = 2. In some cases we can drop this assumption for some special reasons.

We denote by D_{2n} the dihedral group of order 2n.

Group  Signature  Wh_{q} ≠ 0, q ≤ 2  L_{q}(ℤG)  K_{q}^{top}(C_{r}^{∗}(G))
P 1  (1, +, [ ], { })  L_{0}  = ℤ ⊕ ℤ/2 K_{0} = ℤ^{2}
L_{1}  = ℤ^{2}  K_{1} = ℤ^{2}
L_{2}  = ℤ ⊕ ℤ/2
L_{3}  = (ℤ/2)^{2}
P 2 (0, +, [2, 2, 2, 2], \{ \})  L_{0} [\frac{1}{2}] = ℤ [\frac{1}{2}]^{5} K_{0} = ℤ^{6}
L_{1} [\frac{1}{2}] = 0  K_{1} = 0
L_{2} [\frac{1}{2}] = ℤ [\frac{1}{2}]
L_{3} [\frac{1}{2}] = 0
P 3 (0, +, [3, 3, 3], { })  L_{0}  = ℤ^{4} ⊕ ℤ/2 K_{0} = ℤ^{8}
L_{1}  = 0  K_{1} = 0
L_{2}  = ℤ^{4} ⊕ ℤ/2
L_{3}  = 0
P 4 (0, +, [2, 4, 4], \{ \})  L_{0} [\frac{1}{2}] = ℤ [\frac{1}{2}]^{6} K_{0} = ℤ^{9}
L_{1} [\frac{1}{2}] = 0  K_{1} = 0
L_{2} [\frac{1}{2}] = ℤ [\frac{1}{2}]^{3}
L_{3} [\frac{1}{2}] = 0
P 6 (0, +, [2, 3, 6], \{ \}) K_{−1} = ℤ  L_{0} [\frac{1}{2}] = ℤ [\frac{1}{2}]^{6} K_{0} = ℤ^{10}
Wh_{2} = Wh_{2}(ℤ/6) L_{1} [\frac{1}{2}] = 0  K_{1} = 0
L_{2} [\frac{1}{2}] = ℤ [\frac{1}{2}]^{4}
L_{3} [\frac{1}{2}] = 0
Cm (1, −, [ ], \{( )\}) L_{0} [\frac{1}{2}] = ℤ [\frac{1}{2}]^{2} K_{0} = ℤ^{2}
L_{1} [\frac{1}{2}] = ℤ [\frac{1}{2}]^{2} K_{1} = ℤ^{2}
L_{2} [\frac{1}{2}] = 0
L_{3} [\frac{1}{2}] = 0

Group  Signature  Wh_{q} ≠ 0, q ≤ 2  L_{q}(ℤG)  K_{q}^{top}(C_{r}^{∗}(G))
P m (0, +, [ ], \{( ), ( )\})  L_{0} [\frac{1}{2}] = ℤ [\frac{1}{2}]^{3} K_{0} = ℤ^{3}
L_{1} [\frac{1}{2}] = ℤ [\frac{1}{2}]^{3} K_{1} = ℤ^{3}
L_{2} [\frac{1}{2}] = 0
L_{3} [\frac{1}{2}] = 0
P g (2, −, [ ], { }) L_{0} = ℤ ⊕ ℤ/2 K_{0} = ℤ
L_{1}  = ℤ ⊕ ℤ/2 K_{1} = ℤ ⊕ ℤ/2
L_{2}  = ℤ/2
L_{3}  = (ℤ/2)^{2}
Cmm (0, +, [2], \{(2, 2)\}) Wh_{2} = Wh_{2}(D_{4})^{2}  L_{0} [\frac{1}{2}] = ℤ [\frac{1}{2}]^{6} K_{0} = ℤ^{6}
L_{1} [\frac{1}{2}] = 0  K_{1} = 0
L_{2} [\frac{1}{2}] = 0
L_{3} [\frac{1}{2}] = 0
P mm (0, +, [ ], \{(2, 2, 2, 2)\}) Wh_{2} = Wh_{2}(D_{4})^{4}  L_{0} [\frac{1}{2}] = ℤ [\frac{1}{2}]^{9} K_{0} = ℤ^{9}
L_{1} [\frac{1}{2}] = 0  K_{1} = 0
L_{2} [\frac{1}{2}] = 0
L_{3} [\frac{1}{2}] = 0
P mg (0, +, [2, 2], \{( )\})  L_{0} [\frac{1}{2}] = ℤ [\frac{1}{2}]^{4} K_{0} = ℤ^{4}
L_{1} [\frac{1}{2}] = ℤ [\frac{1}{2}] K_{1} = ℤ
L_{2} [\frac{1}{2}] = 0
L_{3} [\frac{1}{2}] = 0
P gg (1, −, [2, 2], \{ \}) L_{0} [\frac{1}{2}] = ℤ [\frac{1}{2}]^{3} K_{0} = ℤ^{3}
L_{1} [\frac{1}{2}] = 0  K_{1} = ℤ/2
L_{2} [\frac{1}{2}] = 0
L_{3} [\frac{1}{2}] = 0
P 3m1 (0, +, [3], \{(3)\}) Wh_{2} = ℤ/2  L_{0} [\frac{1}{2}] = ℤ [\frac{1}{2}]^{4} K_{0} = ℤ^{5}
L_{1} [\frac{1}{2}] = ℤ [\frac{1}{2}] K_{1} = ℤ
L_{2} [\frac{1}{2}] = ℤ [\frac{1}{2}]
L_{3} [\frac{1}{2}] = 0
P 31m (0, +, [ ], \{(3, 3, 3)\}) Wh_{2} = (ℤ/2)^{3}  L_{0} [\frac{1}{2}] = ℤ [\frac{1}{2}]^{5} K_{0} = ℤ^{5}
L_{1} [\frac{1}{2}] = ℤ [\frac{1}{2}] K_{1} = ℤ
L_{2} [\frac{1}{2}] = 0
L_{3} [\frac{1}{2}] = 0
P 6m (0, +, [ ], \{(2, 3, 6)\}) Wh_{2} = Wh_{2}(D_{4}) ⊕ ℤ/2  L_{0} [\frac{1}{2}] = ℤ [\frac{1}{2}]^{8} K_{0} = ℤ^{8}
⊕ Wh_{2}(D_{12}) L_{1} [\frac{1}{2}] = 0 K_{1} = 0
K_{−1} = ℤ  L_{2} [\frac{1}{2}] = 0
L_{3} [\frac{1}{2}] = 0
P 4m (1, +, [ ], \{(2, 4, 4)\}) Wh_{2} = Wh_{2}(D_{4}) ⊕ Wh_{2}(D_{8})^{2} L_{0} [\frac{1}{2}] = ℤ [\frac{1}{2}]^{9} K_{0} = ℤ^{9}
L_{1} [\frac{1}{2}] = 0  K_{1} = 0
L_{2} [\frac{1}{2}] = 0
L_{3} [\frac{1}{2}] = 0
P 4g (0, +, [4], \{(2)\}) Wh_{2} = Wh_{2}(D_{4}) L_{0} [\frac{1}{2}] = ℤ [\frac{1}{2}]^{5} K_{0} = ℤ^{6}
L_{1} [\frac{1}{2}] = 0  K_{1} = 0
L_{2} [\frac{1}{2}] = ℤ [\frac{1}{2}]^{1}
L_{3} [\frac{1}{2}] = 0

174CHAPTER 8. COMPUTATIONS OF K- AND L-GROUPS FOR INFINITE GROUPS

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Notation

Arf(P, ψ), 95 BG, 65 BP L, 138 BO, 65 B_{n}, 133 bP^{n+1}, 125 C_{G}H, 165 cone(f), 36 cone_{∗}(f_{∗}), 27 cyl(f), 36 cyl_{∗}(f_{∗}), 27 deg(f), 40 DE, 57 DV, 39 e(f), 40 E(G; F), 148 \underline{E}G, 149 e(P), 87 f(V), 40 G(F), 61 G, 65 G(k) := G(S^{k−1}), 61 GL(R), 24 H^{ϵ}(P), 87 H_{ϵ}(P), 90 H_{n}(X; ℤ^{w}), 51 H_{n}^{G}(X, A; E), 147 J_{n}, 132 K_{1}(R), 24 K_{k}(M̃), 85 K^{k}(M̃), 85 K̃_{1}(R), 24 L(V), 39 l(V),40 L_{2k}(R), 93 L_{2k}^{U}(R), 109 L_{n}^{h}(ℤπ, w), 110 L_{n}^{s}(ℤπ, w), 110 L_{2k+1}(R), 102, L_{2k+1}^{U}(R), 109 M^{−}, 1 map_{G}(?, Y), 145 N_{G}H, 165 O, 65 Or(G), 144 Or(G; F), 150 P L, 119 (P, ϕ; G, H), 99 p_{0} ∗ p_{1}, 60 Q^{ϵ}(P), 89 Q_{ϵ}(P), 89 Q_{(−1)^{k}}(ℤπ, w), 83 sign(X), 54 SE, 57 SV, 39 SPACES, 144 SPACES_{+}, 144 SPECTRA, 144 Th(E), 57 T OP, 119 \underline{V}, 58 W_{G}H, 165 Wh(G), 25 Wh(π(Y)), 31 X ⊗_{C} Y, 145 X ∗ Y, 40 X ∧ Y, 57 X ⊗_{C} E, 145 χ(X), 53 μ(α), 83 ν(M), 57 π_{n}^{s}, 132 π_{n}^{s}(X), 131 θ_{C}, 166 Θ^{n}, 124 \overline{Θ}^{n}, 125 Θ_{fr}^{n}, 137 ΣC_{∗}, 27 σ(\overline{f}, f), 96, 104, 106, 110 τ(f), 31 τ^{U}(f), 107 τ(f_{∗}), 28 τ^{U}(f_{∗}), 108 Ω_{n}(ξ), 58 Ω_{n}^{fr}, 128 G^{G}(S), 148 F, 148 FIN, 148 VC, 148 N_{n}(X), 67 N_{n}(X, k), 67 N_{n}^{P L}(X), 119 N_{n}^{T OP} (X), 119 S_{n}^{h}(X), 114 S_{n}^{s}(X), 113 S_{n}^{P L,h}(X), 119 S_{n}^{P L,s}(X), 119 S_{n}^{T OP,h}(X), 119 S_{n}^{T OP,s}(X), 119 T_{n}(X), 65 N_{n}(X, ∂X), 117 T_{n}(X, k), 65 N^{T} (X), 68 ∂(P, ψ), 100

Index

U-torsion

of a contractible finite U-stably based chain complex, 108

acyclic

U-acyclic, 38 Alexander trick, 2 Arf invariant, 95 assembly map, 149 associated contravariant Or(G)-space

of a left G-space, 145 attaching maps, 9

based exact sequence, 28 basis

equivalent basis, 10 stable basis, 107 stably U-equivalent, 107 U-equivalent, 107

Bernoulli numbers, 133 boundary of a non-degenerate ϵ-quadratic

form, 100 Brieskorn variety, 141

cell closed, 9 open, 9

cellular basis, 10 map, 31 pushout, 32

centralizer, 165 chain complex

contractible, 28

finite, 28

finite based free, 28

free, 28

projective, 28 chain contraction, 28 characteristic map, 9 classifying map of a principal G-bundle,

40 classifying space for G, 40 classifying space with respect to a

family of subgroups, 148 cobordism, 1

diffeomorphic relative M_{0}, 1

h-cobordism, 2 over M_{0}, 1 trivial, 2 cocompact, 25 collaps map, 57 Conjecture

Baum-Connes Conjecture, 150 Borel Conjecture, 151 Farrell-Jones Isomorphism Con-

jecture, 150 Isomorphism Conjecture, 149 Kadison Conjecture, 152 Novikov Conjecture, 151 Poincaré Conjecture, 2 Stable Gromov-Lawson-Rosenberg

Conjecture, 152

degree, 40 disk bundle, 57 dual

R-module, 32 chain complex, 33

Eilenberg-MacLane space, 120

elementary collaps, 34 elementary expansion, 34 embedding

trivial, 7 Euler characteristic, 53 exotic sphere, 2

family of finite subgroups, 148 of subgroups, 148 of virtually cyclic subgroups, 148

fibration orientable, 61 strong fiber homotopy equiva-

lent, 61 form

ϵ-quadratic, 89 ϵ-symmetric, 87 non-degenerate ϵ-quadratic, 89 non-degenerate ϵ-symmetric form,

87 standard hyperbolic ϵ-quadratic

form, 90 standard hyperbolic ϵ-symmetric

form, 87 U-based ϵ-quadratic forms, 108

formation ϵ-quadratic formation, 99 stably isomorphic ϵ-quadratic for-

mations, 99 trivial ϵ-quadratic formation, 99

framing almost stable, 127 stable, 127

free

free unitary representation, 39 fundamental class, 51

G-CW -complex, 146

proper, 147 group

virtually cyclic, 148 groups of homotopy spheres, 124

handle, 4 boundary of the core, 4 cocore, 4 core, 4 dual, 18

transverse sphere, 4 handlebody decomposition, 4

dual, 17 homology

(proper) equivariant homology

theory, 162 Borel homology, 147 G-homology theory, 146 proper G-homology theory, 162 with coefficients in a spectrum over the orbit category, 147

homotopy n-sphere, 123 Hopf construction, 132

induction structure, 162 intersection pairing, 54 for immersions, 81 for kernels, 86

involution of rings, 50 w-twisted involution of a group

ring, 50

J-homomorphism, 132

join, 40

fiberwise join of spherical fibrations, 60

k-connected map, 70

L-group

decorated quadratic L-groups in even dimensions, 109 decorated quadratic L-groups in odd dimensions, 109 quadratic L-groups in even di-

mensions, 93 quadratic L-groups in odd di-

mensions, 102 simple quadratic L-groups, 110

Lagrangian, 93 complementary, 100 subLagrangian, 93

Lemma Cancellation Lemma, 6 Diffeomorphism Lemma, 5 Elimination Lemma, 7 Franz’ Independence Lemma, 46 Homology Lemma, 14 Isotopy Lemma, 5 Modification Lemma, 15 Normal Form Lemma, 16

lens space, 39

Mackey functor, 164

module over a Green functor,

166 mapping cone

of a chain map, 28 of a map of spaces, 36

mapping cylinder of a chain map, 28 of a map of spaces, 36

module stably finitely generated free, 88 stably U-based, 107 U-based, 107

normal bordism, 106 normal bundle, 57 normal invariant

normal k-invariant, 64 set of normal k-invariants, 65 set of normal invariants, 65

normal map, 67

with respect to the tangent bun-

dle, 68 normal map of degree one, 67

of manifolds with boundary, 105 with respect to the tangent bun-

dle, 68 normalizer, 165

orbit category, 144 orientation

of an orientable spherical (k −

1)-fibration, 61 orientation homomorphism

of a Poincaré complex, 51 of a spherical fibration, 61

Poincaré complex finite n-dimensional, 51 simple finite n-dimensional, 51

Poincaré pair finite n-dimensional, 54 simple finite n-dimensional, 54 Poincaré ℤπ-chain homotopy equiv-

alence, 51

ring with involution, 32

selfintersection element, 83

set of normal maps to a compact manifold, 117 set of normal maps to a Poincaré complex, 67

signature higher, 151 of a finite oriented 4k-dimensional Poincaré complex, 54

of a symmetric bilinear non-degenerate

pairing, 54 simple homotopy equivalence, 34 smash product, 57 space over a category, 144 spectrum, 59, 144

covariant Ω-spectrum, 144

map of spectra, 144 Ω-spectrum, 144 over a category, 144 weak homotopy equivalence, 144

sphere bundle, 57 spherical fibration, 60 Spivak normal fibration, 62 stable n-th stem, 132 stable homotopy groups, 131 stable homotopy groups of a spectrum, 60, 144

stably parallelizable, 125

structure set, 114 simple, 113 surgery obstruction

for manifolds with boundary, 106 for manifolds with boundary and

simple homotopy equivalences, in even dimensions, 96 in odd dimensions, 104

surgery problem, 77 surgery program, 3 surgery sequence, 118

for the P L-category and topo-

logical category, 120 surgery step, 77 suspension, 28

of a pointed space, 57 of a spherical fibration, 60

tensor product

of a C-spectrum with a C-space,

145 of spaces over a category, 145

Theorem s-Cobordism Theorem, 1 Classification of Homotopy Spheres,

123 Diffeomorphism Classification of

Lens Spaces, 45 Differentiable Sphere Theorem,

142 Existence and Uniqueness of the

Spivak Normal Fibration,

62 h-Cobordism Theorem, 2 Homotopy Classification of Lens

Spaces, 41 Homotopy Theoretic Interpre-

tation of the Kervaire-Milnor Braid, 138

Immersions and Bundle Monomor-

phisms, 74 Kervaire-Milnor Braid, 138 Normal Maps and G/O, 69 Pontrjagin Thom Construction

and Oriented Bordism, 59 Pontrjagin-Thom Construction,

58 Simple Surgery Obstruction The-

orem for Manifolds with Boundary, 111

Sphere Theorem, 142 Surgery Exact Sequence, 118 Surgery Exact Sequence for PL

and TOP, 119 Surgery Obstruction Theorem for Manifolds with Boundary, 106

Surgery Obstruction Theorem in Even Dimensions, 95 Surgery Obstruction Theorem in Odd Dimensions, 102

The Surgery Step, 75 Thom Isomorphism, 61 Whitney’s Approximation The-

orem, 74 Thom class, 61 Thom isomorphism, 61 Thom space

of a spherical fibration, 60 of a vector bundle, 57

triad manifold triad, 105 Poincaré triad, 105

U-torsion

of a stable isomorphism of stably U-based modules, 107

unit disk, 39 unit sphere, 39 universal principal G-bundle, 40

(weakly) excisive, 153

(weakly homotopy invariant, 153 Whitehead group, 25 of a space, 31 Whitehead torsion, 31

of a chain homotopy equivalence, 28