“Stable Prime Decompositions of Four-Manifolds” by Matthias Kreck, Wolfgang Lück and Peter Teichner

We would like to dedicate this paper to Bill Browder on the occasion of his sixtieth birthday.

Abstract: The main result of this paper is a four-dimensional stable version of Kneser’s conjecture on the splitting of three-manifolds as connected sums. Namely, let M be a topological respectively smooth compact connected four-manifold (with orientation or Spin-structure). Suppose that π_{1}(M) splits as ∗_{i=1}^{n}Γ_{i} such that the image of π_{1}(C) in π_{1}(M) is subconjugated to some Γ_{i} for each component C of ∂M. Then M is stably homeomorphic respectively diffeomorphic (preserving the orientation or Spin-structure) to a connected sum ♯_{i=1}^{n}M_{i} with Γ_{i} = π_{1}(M_{i}). Stably means that one allows additional connected sums with some copies of S^{2} × S^{2} on both sides. We also prove a uniqueness statement. As a consequence we obtain the existence and uniqueness of the stable prime decomposition of compact connected four-manifolds (with orientation or Spin-structure). The main technical ingredients are the bordism approach to the stable classification of manifolds due to the first author and the Kurosh Subgroup Theorem.

Key words: Stable splitting of four-manifolds as connected sums, Kneser’s conjecture, stable classification and bordism theory, Kurosh Subgroup Theorem

AMS-classification number: 57M99

Introduction

A compact connected orientable smooth three-manifold M has a so called prime decomposition. Namely, M is oriented diffeomorphic to a connected sum ♯_{i=1}^{n}M_{i} of oriented manifolds M_{i} which are prime, i.e. if M_{i} is diffeomorphic to M′_{i}♯M′′_{i}, then M′_{i} or M′′_{i} is oriented diffeomorphic to S^{3}. The manifolds M_{i} are unique up to order and oriented diffeomorphism.

The corresponding result cannot hold for four-manifolds. For example (S^{2} × S^{2}) ♯ ℂP^{2} is diffeomorphic to ℂP^{2}♯\overline{ℂP^{2}}♯ℂP^{2}. Or, for a simply connected four-dimensional Spinmanifold M with non-trivial signature M♯M^{−} is homeomorphic and often diffeomorphic to a connected sum of (S^{2} × S^{2})’s. The problem here is, that the value of the signature for different pieces is not determined by the large manifold. A natural way to overcome these difficulties is to allow connected sum with an arbitrary simply connected closed four-manifold. Up to connected sum with simply connected closed manifolds, we prove the corresponding result in dimension four.

A stable oriented diffeomorphism from a four-manifold M to N is an orientation preserving diffeomorphism from M ♯ k(S^{2} × S^{2}) to N ♯ \overline{k}(S^{2} × S^{2}) for some non-negative integers k and \overline{k}. If the manifolds are equipped with a Spin-structure, we can in addition require that these structures are preserved. We call a connected compact orientable smooth four-manifold M stably prime if M_{i} stably oriented diffeomorphic to M′_{i}♯M′′_{i} implies that M′_{i} or M′′_{i} is simply connected and closed.

Theorem 0.1 (Stable Prime Decomposition) Let M be a connected compact oriented smooth four-manifold. Then:

1. There are stably prime oriented four-manifolds M_{1}, M_{2}, …, M_{n} and a stable oriented

diffeomorphism

f : M ⟶ ♯_{i=1}^{n}M_{i}.

2. Let f′ : M ⟶ ♯_{i=1}^{n′}M′_{i} be another stable oriented diffeomorphism for stably prime oriented four-manifolds M′_{1}, M′_{2}, …, M′_{n′}. Suppose that none of the M_{i}’s and M′_{i}’s is simply connected and closed. Then n = n′ and M_{i}♯S_{i} and M′_{σ(i)}♯S′_{i} are stably oriented diffeomorphic for i ∈ {1, 2, …, n}, appropriate simply connected closed oriented four-manifolds S_{i} and S′_{i} and a permutation σ.

Closely related to prime decompositions is Kneser’s conjecture. Let M be a compact connected three-manifold with incompressible boundary whose fundamental group admits a splitting α : π_{1}(M) ⟶ Γ_{1} ∗ Γ_{2}. Kneser’s conjecture whose proof can be found in [6, chapter 7] says that there are manifolds M_{1} and M_{2} with Γ_{1} and Γ_{2} as fundamental groups and a homeomorphism M ⟶ M_{1}♯M_{2} inducing α on the fundamental groups. Kneser’s conjecture fails even in the closed case in dimensions ≥ 5 by results of Cappell [2],[3]. Counterexamples of closed orientable four-manifolds which even do not split up to homotopy and examples of closed orientable four-manifolds which split topologically but not smoothly are constructed by the authors of this article in [9]. But again it holds stably. We restrict ourselves to oriented manifolds. For simplicity we state in the introduction only an easy to formulate special case of our more general results whose precise statements are given in section 1. A group π is indecomposable if π is non-trivial and π ≅ Γ_{1} ∗ Γ_{2} implies that Γ_{1} or Γ_{2} is trivial.

Theorem 0.2 (Stable Kneser Decomposition) If M is a closed connected smooth oriented four-manifold with non-trivial fundamental group, then there are oriented smooth fourmanifolds M_{1}, M_{2}, …, M_{n} with indecomposable π_{1}(M_{i}) for i ∈ {1, 2, …, n}, such that M and ♯_{i=1}^{n}M_{i} are stably oriented diffeomorphic.

If we have two splittings ♯_{i=1}^{n}M_{i} and ♯_{i=1}^{n′}M′_{i} of M as above, then n = n′ and M_{i}♯S_{i} and M′_{σ(i)}♯S′_{i} are oriented diffeomorphic for i ∈ {1, 2, …, n}, appropriate simply connected closed smooth manifolds S_{i}, S′_{i} and a permutation σ ∈ Σ_{n}. If M is a Spin-manifold and we equip M_{i} and M′_{i} with the Spin-structures induced from a stable diffeomorphism as in Theorem 0.1, we can take S_{i} and S′_{i} as Spin-manifolds and the diffeomorphisms Spin-structure preserving.

Since the stable diffeomorphism type of a simply connected closed smooth four-manifold is determined by the type of the intersection form (I = odd = non-Spin or II = even = Spin) and the signature, we can take for the manifolds S either r(ℂP^{2}) ♯ p(S^{2} × S^{2}) in the case I or rK ♯ p(S^{2} × S^{2}) in the case II, where ℂP^{2} is the complex projective space of complex dimension two and K is the Kummer surface (note that by Rohlin’s theorem the signature is divisible by 16 = − sign(K)).

Both results have topological versions. All manifolds are topological, “diffeomorphic” must be substituted by “homeomorphic” and in the last paragraph r(ℂP^{2}) ♯ p(S^{2} × S^{2}) respectively rK ♯ p(S^{2} × S^{2}) must be substituted by r(ℂP^{2}) ♯ s(E_{8}) ♯ p(S^{2} × S^{2}) respectively r(E_{8}) ♯ p(S^{2} × S^{2}) where E_{8} is the simply connected closed topological four-manifold with E_{8} as intersection form whose existence is proved by Freedman [5, Theorem 1.7].

We mention that this article is motivated by a paper of Hillman [7] which shows the existence of a stable splitting for a closed connected four-manifold M with fundamental group π_{1}(M) = Γ_{1} ∗ Γ_{2}. (Actually Hillman only proves that after adding ℂP^{2} or \overline{ℂ}P^{2} one gets a splitting but his argument can be modified to give the original statement).

The paper is organized as follows :

0. Introduction
1. Kneser Splittings for Manifolds with Boundary
2. Stable Classification and Bordism Theory
3. Proof of the Existence of a Stable Kneser Splitting
4. Proof of the Uniqueness Result

References

The precise statements of our results, also for compact connected four-manifolds with boundary are given in section 1 and from this we deduce Theorems 0.1 and 0.2 using Kurosh’s Subgroup Theorem. Section 2 summarizes the bordism approach to the stable classification due to the first author in a setup which is adequate for the purposes of this article and contains some preliminary results. One may skip section 2 and turn directly to the proofs of the main theorems in the following sections and get back to section 2 when necessary.

1. Kneser Splittings for Manifolds with Boundary

All manifolds are assumed to be compact. We will formulate and prove our results for smooth manifolds. With the same modifications as explained in the introduction the analogous results hold for topological manifolds. The proofs are also identically the same replacing everywhere the smooth objects by the corresponding topological ones.

We will use the following convention on fundamental groups. Let f : X ⟶ Y be a map of path-connected spaces. If we write π_{1}(X), we mean π_{1}(X, x) after some choice of base point x ∈ X. The homomorphism π_{1}(f) : π_{1}(X) ⟶ π_{1}(Y) is the composition of π_{1}(f, x) : π_{1}(X, x) ⟶ π_{1}(Y, f(x)) and the isomorphism c(w) : π_{1}(Y, f(x)) ⟶ π_{1}(Y, y) given by conjugation with a path w joining f(x) and y. Notice that π_{1}(f) is only well-defined up to inner automorphisms of π_{1}(Y). Given a connected sum ♯_{i=1}^{n}M_{i} we will use the following composition of isomorphisms as identification

∗_{i=1}^{n}π_{1}(M_{i}) ⟶^{j^{−1}} ∗_{i=1}^{n}π_{1}(M_{i} − int(D^{4})) ⟶^{k} π_{1}(♯_{i=1}^{n}M_{i})

where j and k are induced by the inclusions in the obvious way. Notice again that this identification is only well-defined up to inner automorphisms. We call a component C of ∂M π_{1}-null if the inclusion induces the trivial map π_{1}(C) ⟶ π_{1}(M).

Theorem 1.3 (Existence of Stable Splitting) Let M be an oriented connected fourmanifold with non-trivial fundamental group. Let

α : π_{1}(M) ⟶ ∗_{i=1}^{n}Γ_{i}

be a group isomorphism such that each Γ_{i} is non-trivial. Suppose for any component C of ∂M that the image of the composition α ∘ π_{1}(j) : π_{1}(C) ⟶ ∗_{i=1}^{n}Γ_{i} is subconjugated to one of the Γ_{i}’s for j : C ⟶ M the inclusion.

Then there are oriented connected four-manifolds M_{1}, M_{2}, …, M_{n} with identifications π_{1}(M_{i}) ⟶ Γ_{i} and oriented simply connected four-manifolds N_{1}, N_{2}, …, N_{p} and a stable oriented diffeomorphism

f : M ⟶ ♯_{i=1}^{n}M_{i} ♯ ♯_{j=1}^{p}N_{j}

such that the composition

π_{1}(M) ⟶^{π_{1}(f)} π_{1}(♯_{i=1}^{n}M_{i} ♯ ♯_{j=1}^{p}N_{j}) ⟶ ∗_{i=1}^{n}π_{1}(M_{i}) ⟶ ∗_{i=1}^{n}Γ_{i}

agrees with α up to inner automorphisms, no boundary component of the M_{i}’s is π_{1}-null and each N_{i} has a connected non-empty boundary.

Theorem 1.4 (Uniqueness of Stable Splitting) Let M_{1}, M_{2}, …,M_{n} and M′_{1}, M′_{2}, …, M′_{n} be oriented connected four-manifolds with non-trivial fundamental groups Γ_{i} = π_{1}(M_{i}) and Γ′_{i} = π_{1}(M′_{i}) such that no boundary component of them is π_{1}-null. Let N_{1}, N_{2}, …, N_{p} and N′_{1}, N′_{2}, …, N′_{q} be oriented simply connected four-manifolds whose boundaries are connected and non-empty. Let

f : ♯_{i=1}^{n}M_{i} ♯ ♯_{j=1}^{p}N_{j} ⟶ ♯_{i=1}^{n}M′_{i} ♯ ♯_{j=1}^{q}N′_{j}

be a stable diffeomorphism, which is either oriented or Spin-structure preserving, if the underlying manifolds are Spin. Denote the homomorphism induced by f on the fundamental groups by

f_{∗} : ∗_{i=1}^{n}Γ_{i} ⟶ ∗_{i=1}^{n}Γ′_{i}.

Suppose for i ∈ {1, 2, …, n} that pr′_{i} ∘f_{∗} ∘ j_{i} is an isomorphism where pr′_{i} : ∗_{i=1}^{n}Γ′_{i} ⟶ Γ′_{i} is the canonical projection and j_{i} : Γ_{i} ⟶ ∗_{i=1}^{n}Γ_{i} is the canonical inclusion. Then:

For i ∈ {1, 2, …, n} we have f(∂M_{i}) = ∂M′_{i} and there are simply connected oriented closed four-manifolds S_{i} and S′_{i} and oriented diffeomorphisms

f_{i} : M_{i}♯S_{i} ⟶ M′_{i}♯S′_{i}

which extend f|_{∂M_{i}} : ∂M_{i} ⟶ ∂M′_{i} and induce up to inner automorphism pr′_{i} ∘f_{∗} ∘ j_{i} on the fundamental groups. Moreover, we have p = q and there is an appropriate permutation σ such that f(∂N_{j}) = ∂N′_{σ(j)} and there are oriented simply connected closed four-manifolds T_{j} and T′_{j} and oriented diffeomorphisms

g_{j} : N_{j}♯T_{j} ⟶ N′_{σ(j)}♯T′_{j}

which extend f|_{∂N_{j}} : ∂N_{j} ⟶ ∂N′_{j}. If the manifolds are Spin-manifolds we can choose the manifolds S_{i}, S′_{i}, T_{j} and T′_{j} as Spin-manifolds and f_{i} and g_{i} Spin-structure preserving.

We finish this section by deriving Theorems 0.2 and 0.1 from these results and the following version of Kurosh’s Subgroup Theorem (see [4, Theorem 8,chapter 7 on page 175]).

Theorem 1.5 (Kurosh Subgroup Theorem) Let H be a subgroup of the free product G = ∗_{i∈I}G_{i}. There is a suitable chosen set of representatives g ∈ \overline{g} for the double cosets \overline{g} ∈ H\G/G_{i} for each i ∈ I and a free subgroup F ⊂ G satisfying

H = F ∗ ∗_{i∈I} (∗_{\overline{g}∈H\G/G_{i}} gG_{i}g^{−1} ∩ H) .

Let π be a non-trivial finitely generated group. A Kurosh splitting is an isomorphism

α : π ⟶ ∗_{i=0}^{n}Γ_{i}

such that Γ_{0} is free and Γ_{j} is indecomposabel and not infinite cyclic for j = 1, 2, …, n. Recall that Γ_{i} is called indecomposable if Γ_{i} is non-trivial and Γ_{i} ≅ Γ′_{i} ∗ Γ′′_{i} implies that Γ′_{i} or Γ′′_{i} is trivial. The existence and the following uniqueness statement for a second Kurosh splitting α′ : π ⟶ ∗_{i=0}^{n′}Γ′_{i} follow from Kurosh Subgroup Theorem 1.5. If j_{i} : Γ_{i} ⟶ ∗_{i=0}^{n}Γ_{i} and pr′_{i} : ∗_{i=0}^{n}Γ′_{i} ⟶ Γ′_{i} are the inclusion and projection, then n = n′ and there is a permutation σ such that pr′_{σ(i)} ∘α′ ∘ α^{−1} ∘ j_{i} is an isomorphism for i ∈ {0, 1, …, n}. Theorem 0.2 now follows from the above Theorems.

Before we prove Theorem 0.1, we characterize the property ”stably prime” in terms of the fundamental group data.

Lemma 1.6 A connected compact orientable four-manifold M is stably prime if and only if it satisfies the following conditions.

1. There is no isomorphism α : π_{1}(M) ⟶ Γ_{1} ∗ Γ_{2} for non-trivial groups Γ_{1} and Γ_{2} such that for each component C of the boundary the composition of α and the map induced by the inclusion π_{1}(C) ⟶ π_{1}(M) has an image which is subconjugated to Γ_{1} or Γ_{2}.

2. If M has a π_{1}-null boundary component, then M is simply connected and ∂M is nonempty and connected.

\underline{\underline{Proof}} : If one of the conditions above is violated, Theorem 1.3 gives a splitting of M into M_{1}♯M_{2} such that neither M_{1} nor M_{2} is simply connected and closed. Conversely, given such a splitting, one sees immediately, that at least one of the conditions above is not fullfilled.

In particular a connected closed orientable four-manifold is stably prime if and only if π_{1}(M) is trivial or indecomposable. Now, we prove Theorem 0.1.

\underline{\underline{Proof}} : 1.) The existence of f follows from the following inductive process. If M is stably prime, the process stops. If M is not stably prime, choose a stable oriented diffeomorphism M ⟶ M_{1}♯M_{2} such that neither M_{1} nor M_{2} is simply connected and closed. Now apply this process to both M_{1} and M_{2}. It remains to show that this process stops after a finite number of steps. This follows from Lemma 1.6, the Grushko-Neumann Theorem [10, Theorem 1.8 and Corollary 1.9 on page 178] which implies that the rank of a group, i.e. the minimal number of generators, is additive under free products and the simple fact that M has only finitely many π_{1}-null boundary components.

2.) Consider a stable oriented diffeomorphism

f : (♯_{i=1}^{l}L_{i}) ♯ (♯_{i=1}^{n}M_{i}) ♯ (♯_{i=1}^{p}N_{i}) ⟶ (♯_{i=1}^{l′}L′_{i}) ♯ (♯_{i=1}^{n′}M′_{i}) ♯ (♯_{i=1}^{p′}N′_{i})

such that each L_{i}, L′_{i}, M_{i}, M′_{i}, N_{i} and N′_{i} is stably prime, each L_{i} is closed and has infinite cyclic fundamental group, none of the M_{i}’s and M′_{i}’s is simply connected or has both infinite cyclic fundamental group and empty boundary, and each N_{i} and N′_{i} is simply connected and has a non-empty boundary. Notice that any finite connected sum of stably prime connected four-manifolds can be written in this way if none of the summands is simply connected and closed. We conclude from Lemma 1.6 that none of the M_{i}’s and M′_{i}’s has a π_{1}-null boundary component and that the boundary of each N_{i} and N′_{i} is non-empty and connected. We abbreviate in the sequel Γ_{i} = π_{1}(M_{i}) and Γ′_{i} = π_{1}(M′_{i}) and introduce the finitely generated free groups Γ_{0} = ∗_{i=1}^{l}π_{1}(L_{i}) and Γ′_{0} = ∗_{i=1}^{l′}π_{1}(L_{i}). The map induced on the fundamental groups by f is denoted by

f_{∗} : ∗_{i=0}^{n}Γ_{i} ⟶ ∗_{j=0}^{n′}Γ′_{i}.

Fix an index i ∈ {1, 2, …, n}. We apply Kurosh Subgroup Theorem 1.5 to f_{∗}(Γ_{i}) ⊂ ∗_{j=0}^{n}Γ′_{j} and obtain

f_{∗}(Γ_{i}) = F ∗ ∗_{j=0}^{n′} (∗_{\overline{g}∈f_{∗}(Γ_{i})\∗_{j=0}^{n′}Γ′_{j}/Γ′_{j}} gΓ′_{j}g^{−1} ∩ f_{∗}(Γ_{i})) .

Let C be a boundary component of M_{i}. There is an index j ∈ {1, 2, …, n′} such that f(C) ⊂ M′_{j}. If j_{∗} : π_{1}(C) ⟶ Γ_{i} is the map induced by the inclusion, then f_{∗}(j_{∗}(π_{1}(C))) is subconjugated to Γ′_{j}. Hence there is g_{0} ∈ ∗_{i=0}^{n′}Γ′_{i} such that f_{∗}(j_{∗}(π_{1}(C))) ⊂ g_{0}Γ′_{j}g_{0}^{−1} ∩ f_{∗}(Γ_{i}) holds. We conclude from Kurosh Subgroup Theorem 1.5 that f_{∗}(j_{∗}(π_{1}(C))) is subconjugated to gΓ′_{j}g^{−1} ∩ f_{∗}(Γ_{i}) for appropriate \overline{g} ∈ f_{∗}(Γ_{i})\ ∗_{j=0}^{n′} Γ′_{j}/Γ′_{j}. Recall that M_{i} is stably prime, has no π_{1}-null boundary component and it is not true that M_{i} has both infinite cyclic fundamental group and empty boundary. We derive from Lemma 1.6 applied to the isomorphism induced by f_{∗}

π_{1}(M_{i}) = Γ_{i} ⟶ f_{∗}(Γ_{i}) = F ∗ ∗_{j=0}^{n′} (∗_{\overline{g}∈f_{∗}(Γ_{i})\∗_{j=0}^{n′}Γ′_{j}/Γ′_{j}} gΓ′_{j}g^{−1} ∩ f_{∗}(Γ_{i})) .

that there is a unique index σ(i) ∈ \{1, 2, …, n\} and \overline{g} ∈ f_{∗}(Γ_{i})\ ∗_{j=0}^{n′} Γ′_{j}/Γ′_{j} satisfying

f_{∗}(Γ_{i}) = gΓ′_{σ(i)}g^{−1}.

We get a map σ : {1, 2, …, n} ⟶ {1, 2, …, n′}. Completely analogously one defines a map σ′ : {1, 2, …, n′} ⟶ {1, 2, …, n} such that for each j ∈ {1, 2, …, n′} there is g ∈ ∗_{i=0}^{n}Γ_{i} satisfying

f_{∗}^{−1}(Γ′_{j}) = gΓ_{σ′(j)}g^{−1}.

Let pr′_{j} : ∗_{j=0}^{n}Γ′_{i} ⟶ Γ′_{j} be the canonical projection and j_{i} : Γ_{i} ⟶ ∗_{i=0}^{n}Γ_{i} be the canonical inclusion. We conclude for each i ∈ {1, 2, …, n} and j ∈ {0, 2, …, n′} that the composition pr′_{j} ∘f_{∗} ∘ j_{i} is an isomorphism if j = σ(i) and trivial otherwise. Hence σ′∘σ = id and following diagram commutes for appropriate \overline{f_{∗}}

∗_{i=0}^{n}Γ_{i} ⟶^{f_{∗}} ∗_{i=0}^{n}Γ′_{i}
pr^{0}↓  ↓^{pr}′_{0}
Γ_{0} ⟶_{\overline{f_{∗}}} Γ′_{0}

The same argument applied to f_{∗}^{−1} shows that σ ∘ σ′ = id and that \overline{f_{∗}} has an inverse. Hence σ and σ′ are inverse to one another, n = n′ and the composition pr′_{σ(i)} ∘f_{∗} ∘ j_{i} is an isomorphism for i ∈ {0, 1, 2, …, n} if we put σ(0) = 0.

From Theorem 1.4 we conclude that M_{i}♯S_{i} and M_{σ(i)}♯S′_{i} are stably oriented diffeomorphic for each i ∈ {1, 2, …, n}, appropriate simply connected closed oriented four-manifolds S_{i} and S′_{i} and that p = p′ and N_{i}♯T_{i} and N′_{τ(i)}♯T′_{i} are stably oriented diffeomorphic for each i ∈ {1, 2, …, p}, appropriate simply connected closed oriented four-manifolds T_{i} and T′_{i} and permutation τ. Since Γ_{0} and Γ′_{0} are isomorphic, we get l = l′. Each L_{i} and L′_{i} is stably isomorphic to S^{1} × S^{3} after adding simply connected closed oriented four-manifolds for i ∈ {1, 2, …, l} by Theorem 2.1 and Lemma 2.3 since L_{i} and L′_{i} are closed and have infinite cyclic fundamental groups. This finishes the proof of Theorem 0.1.

2. Stable Classification and Bordism Theory

In this section we explain the necessary details of the bordism approach to the stable classification of manifolds due to the first author and prove some preliminary lemmas. Recall that all manifolds are assumed to be compact and we restrict ourselves to smooth manifolds.

We begin with organizing the bookkeeping of the fundamental group data. We consider pairs (π, w_{2}) which consist of a finitely presented group π and an element w_{2} in H^{2}(π; ℤ/2) ∐{∞}. We call two such pairs (π, w_{2}) and (π′, w′_{2}) equivalent if there is an isomorphism f : π ⟶ π′ with the properties that either w_{2} = ∞ and w′_{2} = ∞ or w_{2} ∈ H^{2}(π; ℤ/2), w′_{2} ∈ H^{2}(π′; ℤ/2) and f^{∗}(w′_{2}) = w_{2} holds. A type T is an equivalence class [π, w_{2}] of such pairs.

An oriented manifold determines a type T (M), called the normal 1-type, for which a representative is given as follows. Put π = π_{1}(M). Let g : M ⟶ K(π, 1) be a classifying map of the universal covering and denote by w_{k}(M) ∈ H^{k}(M; ℤ/2) the k-th Stiefel-Whitney class of the normal bundle of M. If w_{2}(M̃) ≠ 0 holds for the universal covering M̃, then put w_{2} = ∞. Otherwise let w_{2} be the unique element satisfying g^{∗}(w_{2}) = w_{2}(M). The unique existence follows from the exact sequence coming from the Serre spectral sequence of the fibration M̃ ⟶ M ⟶ K(π, 1)

0 ⟶ H^{2}(K(π, 1); ℤ/2) ⟶^{g^{∗}} H^{2}(M; ℤ/2) ⟶ H^{2}(M̃; ℤ/2).

Two homotopy equivalent manifolds have the same normal 1-type.

Before we introduce the relevant bordism groups, we recall how to convert a continous map u : X ⟶ K into a fibration u′ : P (u) ⟶ K. Define

P (u) = {(x, w) | w(0) = u(x)} ⊂ X × map(I, K)

and u′(x, w) = w(1). Define the map u′′ : P (u) ⟶ X by sending (x, w) to x and define the homotopy ψ : u ∘ u′′ ≃ u′ by sending ((x, w), t) to w(t). The triple (P (u), u′′, ψ) has the universal property that for any space Z together with maps f′ : Z → K and f′′ : Z ⟶ X and homotopy ϕ : u ∘ f′′ ≃ f′ there is precisely one map g : Z ⟶ P (u) such that

f′′ = u′′ ∘ g, f′ = u′ ∘ g and ϕ = ψ ∘ (g × id).

Namely, define g(z) = (f′′(z), ψ_{z}) for ψ_{z} the path sending t to ψ(z, t). There is a map i : X ⟶ P (u) sending x to (x, c_{u(x)}) where c_{u(x)} is the constant path in K at u(x). It is a homotopy inverse of u′′ and its composition with u′ is u.

A type T determines a fibration B(T ) over BSO or over BSpin, if w_{2} = 0, as follows. Let [π, w_{2}] be a representative of T. If w_{2} = ∞ define it as the trivial fibration

B(T ) = BSO × K(π, ∞) → BSO

over BSO. If w_{2} = 0 we define it as the trivial fibration

B(T ) = BS√⟩\ × K(π, ∞) → BS√⟩\

over BSpin. If w_{2} ≠ 0, ∞ represent w_{2} by a map u : K(π, 1) → K(ℤ/2, 2) with corresponding fibration P (u) over K(ℤ/2, 2). Represent the universal Stiefel Whitney class by a map q : BSO → K(ℤ/2, 2). Then define our fibration by the pullback

B(T ) = ⨿^{∗}(P(⊓)) → BSO.

These fibrations are up to fibre homotopy equivalence uniquely determined by T. In all three cases there are projection maps to K(π, 1) denoted by p_{K(π,1)}.

Suppose that M has normal 1-type T (M) and let g : M → K(π, 1) be a map satisfying g^{∗}w_{2} = w_{2}(ν(M)) if w_{2}(M) = w_{2}(ν(M)) ≠ ∞. Then the normal Gauss map ν : M → BSO or ν : M → BSpin, if M is has a Spin-structure, admits a lift ρ over B(T ) as follows. If w_{2} = ∞ it is given by the normal Gauss map together with g. If w_{2} ≠ ∞ it is given by the normal Gauss map together with g and with a homotopy between the composition of the two maps to K(ℤ/2, 2). We call such a lift ρ a normal structure of M in B(T (M)) compatible with g and the orientation resp. Spin-structure. If a normal structure ρ is a 2-equivalence, it is called a normal 1-smoothing. Notice that a normal structure ρ is a normal 1-smoothing if and only if the underlying map g induces an isomorphism on π_{1}.

Given a fibration B → BSO or B → BS√⟩\ we denote the bordism group of n-

dimensional closed oriented or Spin-manifolds together with a lift of the normal Gauss map over B by

Ω_{n}(B).

If ρ is a normal 1-smoothing of M in B(T (M)), then the pair (M, ρ) determines an element in Ω_{4}(B(T (M))).

Now we can formulate the main result of the bordism approach to the stable classification of connected four-manifolds due to the first author [8].

Theorem 2.1 (Stable Classification of Four-Manifolds by Bordism Theory) Let M_{1} and M_{2} be connected four-manifolds with orientation respectively Spin-structure and ∂f : ∂M_{1} ⟶ ∂M_{2} be a diffeomorphism which preserves the induced orientation respectively Spin-structure. Suppose that the normal 1-type of M_{1} and M_{2} is equal to T and denote by B(T ) any representation of the associated fibration. Let g_{i} : M_{i} → K(π, 1) be classifying maps of the universal covering respecting w_{2} such that g_{2}|_{∂M_{2}} ∘ ∂f = g_{1}|_{∂M_{1}}.

1. There exists a stable oriented (Spin-structure preserving, if M_{i} are Spin) diffeomor-

phism

f : M_{1} ⟶ M_{2}

extending ∂f such that the maps g_{2} ∘ f and g_{1} to K(π, 1) are homotopic if and only if there are normal 1-smoothings ρ_{i} of M_{i} compatible with g_{i} and the orientations resp. Spin-structures, such that ρ_{2}|_{∂M_{2}} ∘ ∂f and ρ_{1}|_{∂M_{1}} agree and

[M_{1}^{−} ∪_{∂f} M_{2}, ρ_{1}^{−} ∪_{∂f} ρ_{2}] = 0 ∈ Ω_{4}(B(T ))

2. Given a manifold with boundary together with a lift of the normal Gauss-map to B(T ), it is bordant relative boundary to a normal 1-smoothing.

The strategy for proving the main Theorems is to analyse how the bordism group decomposes if the fundamental group splits as a free product. For this the following categorial considerations are useful.

Denote K = K(ℤ/2, 2) or K = ∗. Define a category C as follows. An object (X, u) is a map u : X ⟶ K and a morphism (f, ϕ) : (X, u) ⟶ (Y, v) consists of a map f : X ⟶ Y together with a homotopy ϕ : v ∘ f ≃ u. The composition (g, ψ) ∘ (f, ϕ) is defined by (g ∘ f, (ψ ∘ (f × id)) ∗ ϕ) where ∗ denotes the composition of homotopies. If the homotopy ψ is the constant homotopy, we abbreviate (f, ψ) by f. Two morphisms (f_{0}, ψ_{0}) and (f_{1}, ψ_{1}) from (X, u) to (Y, v) are called homotopic if they can be connected by a continuous one parameter family of morphisms (f_{t}, ϕ_{t}). The following elementary facts will frequently be used in the sequel.

Lemma 2.2 Let (f, ϕ) : (X, u) ⟶ (Y, v) be a morphism. If g : X ⟶ Y is a map homotopic to f, then there is a homotopy ψ : v ∘ g ≃ u such that the morphisms (f, ϕ) and (g, ψ) are homotopic. If f : X ⟶ Y is a homotopy equivalence, then there is a morphism (g, ψ) : (Y, v) ⟶ (X, u) such that both compositions of (f, ϕ) and (g, ψ) are homotopic to the identity morphism.

Let q : B → K be a fixed map. Given an object (X, u), we have the pullback

B(⊓) ⟶^{\overline{q}} P (u)
\overline{u}↓  ↓_{u′}
B ⟶_{q} K

For a manifold M and appropriate choices of q and u, the fibration B(⊓) over B corresponds to a normal 1-type as described above. More precisely, if w_{2} = ∞, let B = BSO, K = ∗ and X = K(π_{1}(M), 1). For w_{2} = 0 choose B = BSpin instead of BSO. For w_{2} ≠ 0, ∞ choose B = BSO, K = K(ℤ/2, 2), X = K(π_{1}(M), 1) and q and u maps representing the second Stiefel Whitney classes. Then in all three cases B(T (M)) = B(⊓).

For the special purpose of this paper the formulation of B(⊓) has the advantage that it separates the categorial input, namely the fundamental group data encoded in K(π, 1) from the other data like orientation and Spin-structure.

Define

Ω_{n}(X, u) = Ω_{n}(B(⊓))

A morphism (f, ϕ) : (X, u) ⟶ (Y, v) defines by the universal property of the construction P (−) a fiber map P (f, ϕ) : P (u) ⟶ P (v) where fiber map means P (f, ϕ) ∘ v′ = u′. If we apply Ω_{n} to it, we obtain a homomorphism denoted by

Ω_{n}(f, ϕ) : Ω_{n}(X, u) ⟶ Ω_{n}(Y, v).

Clearly this is a functor on C. Moreover, it is a generalized homology theory in the sense that it has the following properties. It is homotopy invariant, i.e., homotopic morphisms induce the same homomorphism. There is a Mayer-Vietoris sequence in the following sense. Consider the following pushout

X_{0} ⟶^{i_{1}} X_{1}
i_{2}↓  ↓^{j_{1}}
X_{2} ⟶_{j_{2}} X

with i_{1} a cofibration. Put j_{0} = j_{2} ∘ i_{2} = j_{1} ∘ i_{1}. Let (X, u) be an object. We obtain objects (X_{k}, u_{k}) by u_{k} = u ∘ j_{k} and morphisms j_{k} : (X_{k}, u_{k}) ⟶ (X, u) for k = 0, 1, 2. Recall that we omit constant homotopies in our notation for morphisms. Now there is a long exact Mayer-Vietoris sequence

… ⟶^{δ}Ω_{n}(X_{0}, u_{0}) ⟶^{(Ω_{n}(i_{1}),Ω_{n}(i_{2}))} Ω_{n}(X_{1}, u_{1}) ⊕ Ω_{n}(X_{2}, u_{1}) ⟶^{Ω_{n}(j_{1})−Ω_{n}(j_{2})} Ω_{n}(X, u)

⟶^{δ}Ω_{n−1}(X_{0}, u_{0}) ⟶ …

Namely, we obtain a pushout with a cofibration as horizontal upper arrow

\overline{P (u_{0})} ⟶^{\overline{P (i_{1})}} \overline{P (u_{1})}
\overline{P (i_{2})}↓  \sqrt{P (j_{1})}
\overline{P (u_{2})} ⟶_{\overline{P (j_{2})}} \overline{P (u_{0})}

Notice that the bordism group Ω_{n}(B) can be identified with the bordism group of the stable vector bundle over B which is the pullback of the universal bundle over BSO respectively BSpin and thus the existence of the Mayer-Vietoris sequence follows by standard arguments, namely the Pontrjagin-Thom construction and the fact that stable homotopy is a generalized homology theory [1, Kapitel II].

Let (X, u) be an object with path-connected X. Denote by ∗ the space consisting of one point. Consider an object (∗, v) and a morphism (j, μ) : (∗, v) ⟶ (X, u). Define

Ω̃_{n}(X, u) := cok (Ω_{n}(j, μ) : Ω_{n}(∗, v) ⟶ Ω_{n}(X, u))

We want to show that the definition of Ω̃_{n}(X, u) is independent of the choice of v and (j, μ) and that a morphism (f, ϕ) : (X, u) ⟶ (Y, v) induces a homomorphism making the following diagram commute for pr the canonical projection.

Ω_{n}(X, u) ⟶^{pr} Ω̃_{n}(X, u)
Ω_{n}(f,ϕ)↓  ↓^{Ω̃_{n}(f,ϕ)}
Ω_{n}(Y, v) ⟶_{pr} Ω̃_{n}(Y, v)

This follows from the following fact and Lemma 2.2. If (j′, μ′) : (∗, v′) ⟶ (Y, v) is a morphism and (j, μ) and (f, ϕ) are as above, then there is a morphism (id, ψ) : (∗, v) ⟶ (∗, v′) such that (f, ϕ) ∘ (j, μ) and (j′, μ′) ∘ (id, ψ) are homotopic morphisms.

The reduced group Ω̃_{n}(X, u) is relevant for our Uniqueness Theorem 1.4 since there we classify stably up to connected sum with a simply connected oriented resp. Spin-manifold and by Theorem 2.1 this is decided in the reduced bordism group.

Next we make some computations for this generalized homology theory. Recall that sign denotes the signature. The group Ω_{n}(∗, v) is either equal to Ω_{n}^{SO} if B = BSO, or to Ω_{n}^{Spin} if B = BSpin.

Lemma 2.3 1. The following table gives generators and explicite isomorphisms for the various bordism groups:

Ω_{3}^{SO} = 0
Ω_{3}^{Spin} = 0
sign : Ω_{4}^{SO} ⟶^{≅} ℤ  ℂP^{2}
sign : Ω_{4}^{Spin} ⟶^{≅} 16 · ℤ K

2. Ω̃_{4}(K(F, 1), u)) = 0 for F a finitely generated free group and both cases B = BSO or B = BSpin.

\underline{\underline{Proof}} : 1.) is standard. 2.) follows from the Mayer-Vietoris sequence applied to a wedge of S^{1}’s and to the pushout which decribes S^{1} as the identification of the two end points of [0, 1] to one point.

3. Proof of the Existence of a Stable Kneser Splitting

In this section we prove Theorem 1.3. We recall that the normal 1-type of a manifold M determines a fibration B(T (M)). With the notation of the last section, if M has w_{2} = ∞, let B = BSO, K = ∗ and X = K(π_{1}(M), 1). For w_{2} = 0 choose B = BSpin instead of BSO. For w_{2} ≠ 0, ∞ choose B = BSO, K = K(ℤ/2, 2), X = K(π_{1}(M), 1) and q and u maps representing the second Stiefel Whitney classes. Then in all three cases B(T (M)) = B(⊓).

Firstly we show that we can assume without loss of generality that no boundary component C of M is π_{1}-null, i.e. the inclusion induces the trivial map π_{1}(C) ⟶ π_{1}(M). Let C_{1}, C_{2}, …,C_{m} be the π_{1}-null boundary components of M. Since Ω_{3}(∗, v) is trivial by Lemma 2.3, there is a nullbordism N_{i} for each C_{i} with respect to (∗, v). By 0- and 1-dimensional surgery on the interior of N_{i} we can achieve that N_{i} is simply connected. Define

M̂ = M ∪_{C_{1}} N_{1}^{−} ∪_{C_{2}} … ∪_{C_{m}} N_{m}^{−}

By Theorem 2.1 there is a stable oriented diffeomorphism f : M ⟶ M̂ ♯ N_{1} ♯ … ♯ N_{m} which induces on the fundamental groups the isomorphism induced by the inclusion of M in M̂. No boundary component of M̂ is π_{1}-null. Obviously it suffices to prove the claim for M̂.

If C is a component of ∂M, there is by assumption an index i ∈ {1, 2, …, n} such that the image of α ∘ π_{1}(j) for j : C ⟶ M the inclusion is subconjugated to Γ_{i}. Since we also assume that this image is non-trivial, this index is unique. For i ∈ {1, 2, …, n} let ∂_{i}M be the union of those components C of ∂M for which this index is i. Since the inclusion ∂M ⟶ M is a cofibration, we can construct maps g : M ⟶ ∨_{i=1}^{n}K(Γ_{i}, 1) and ∂_{i}g : ∂_{i}M ⟶ K(Γ_{i}, 1) for i ∈ {1, 2, …, n} such that the restriction of g to ∂_{i}M is the composition of g_{i} with the canonical inclusion j_{i} : K(Γ_{i}, 1) ⟶ ∨_{i=1}^{n}K(Γ_{i}, 1) and g induces α on the fundamental groups. Choose pointed maps u_{i} : K(Γ_{i}, 1) ⟶ K such that the composition u = (∨_{i=1}^{n}u_{i}) ∘ g : M ⟶ K corresponds to the Stiefel-Whitney classes of M in the case where K is not a point, but K(ℤ/2, 2). Let ρ be normal 1-smoothings of M in B(⊓) compatible with g and the orientation resp. Spin-structure. Denote the restriction of ρ to ∂_{i}M by ∂_{i}ρ. By construction the homomorphism

⊕_{i=1}^{n}Ω_{3}(j_{i}) : ⊕_{i=1}^{n}Ω_{3}(K(Γ_{i}, 1), u_{i}) ⟶ Ω_{3}(∨_{i=1}^{n}K(Γ_{i}, 1), ∨_{i=1}^{n}u_{i})

sends ([∂_{i}M, ∂_{i}ρ] | i = 1, 2, …, n) to the element [∂M, ρ|_{∂M}] which is zero since (M, ρ) is a nullbordism for its representative. This homomorphism is injective by a Mayer-Vietoris argument and Lemma 2.3. Hence we can find nullbordisms (V_{i}, σ_{i}) for (∂_{i}M, ∂_{i}ρ) with respect to (K(Γ_{i}, 1), u_{i}) for i ∈ {1, 2, …, n}. By the same argument as above the homomorphism

⊕_{i=1}^{n}Ω_{4}(j_{i}) : ⊕_{i=1}^{n}Ω_{4}(K(Γ_{i}, 1), u_{i}) ⟶ Ω_{4}(∨_{i=1}^{n}K(Γ_{i}, 1), ∨_{i=1}^{n}u_{i})

is surjective. Let ([W_{i}, τ_{i}] | i = 1, 2, …, n) be a preimage of − [M^{−} ∪_{∂M} ∐_{i=1}^{n} V_{i}, ρ^{−} ∪ ∐_{i=1}^{n} σ_{i}] and then we get

[M^{−} ∪_{∂M} ∐_{i=1}^{n}(V_{i} ∐ W_{i}), ρ^{−} ∪ ∐_{i=1}^{n}(σ_{i} ∐ τ_{i})] = 0 ∈ Ω_{4}(∨_{i=1}^{n}K(Γ_{i}, 1), ∨_{i=1}^{n}u_{i}).

By Theorem 2.1 (V_{i} ∐ W_{i}), (σ_{i} ∐ τ_{i}) is bordant relative boundary to a normal 1-smoothing (M_{i}, ρ_{i}). We have

[M ∪_{∂M} ♯_{i=1}^{n}M_{i}, ρ ∪ ♯_{i=1}^{n}(j_{i} ∘ ρ_{i})] = 0 ∈ Ω_{4}(∨_{i=1}^{n}K(Γ_{i}, 1), ∨_{i=1}^{n}u_{i}).

By Theorem 2.1 there is a stable oriented diffeomorphism

f : M ⟶ ♯_{i=1}^{n}M_{i}

such that the composition ♯_{i=1}^{n}(j_{i} ∘ g_{i}) ∘ f is homotopic to g : M ⟶ ∨_{i=1}^{n}K(Γ_{i}, 1). This finishes the proof of Theorem 1.3.

4. Proof of the Uniqueness Result

This section is devoted to the proof of Theorem 1.4. We firstly show that we can assume without loss of generality that none of the manifolds N_{j} respectively N′_{j} are present. By counting the π_{1}-null components we conclude p = q. After possibly renumbering the N′_{j}’s, we can assume that f maps ∂N_{j} to ∂N′_{j} for all j ∈ {1, 2, …, q}. Since each N_{j} and N′_{j} is simply connected, the desired oriented diffeomorphism g_{j} : N_{j} ♯ T_{j} ⟶ N′_{j} ♯ T′_{j} exists by Theorem 2.1. Again by Theorem 2.1 there is a stable orientation respectively Spin-structure preserving diffeomorphism

♯_{i=1}^{n}M_{i} ⟶ ♯_{i=1}^{n}M_{i} ♯ ♯_{j=1}^{p}N_{j} ∪_{∂N_{j}} N_{j}^{−}

inducing on the fundamental group the obvious isomorphism and the claim follows.

Suppose for a moment that ♯_{i=1}^{n}M_{i} and ♯_{i=1}^{n}M′_{i} are Spin. We want to show that M_{i} and M′_{i} are diffeomorphic modulo connected sum with appropriate simply connected four-manifolds with Spin-structure. Notice that for all M_{i} and M′_{i} the normal 1-type has w_{2} = 0. Thus we have to show that M_{i} and M′_{i} have same normal 1-type and admit normal 1-smoothings in B_{⟩} which induce the right map on π_{1} as stated in Theorem 1.4 and are compatible with f|_{∂M_{i}} such that M_{i} and M′_{i} are bordant rel. boundary (identified via f|_{∂M_{i}}) in the reduced bordism group corresponding to the normal 1-type, which here is Ω̃_{4}^{Spin}(K(Γ′_{i}, 1)). If ♯_{i=1}^{n}M_{i} and ♯_{i=1}^{n}M′_{i} are just oriented manifolds we are allowed to modify M_{i} and M′_{i} by connected sum with any oriented simply connected four-manifold and after adding copies of ℂP^{2} we can assume that the normal 1-type for all M_{i} and M′_{i} has w_{2} = ∞. Then we have to show that M_{i} and M′_{i} have same normal 1-type and admit normal 1smoothings in B_{⟩} which induce the right map on π_{1} as stated in Theorem 1.4 and are compatible with f|_{∂M_{i}} such that M_{i} and M′_{i} are bordant relative boundary (identified via f|_{∂M_{i}}) in the reduced bordism group corresponding to the normal 1-type, which here is Ω̃_{4}(K(Γ′_{i}, 1)) = Ω̃_{4}^{SO}(K(Γ′_{i}, 1)). In the following proof the argument is identically the same in the Spin-case and in the oriented case and thus we restrict ourselves for simplicity to the oriented case.

We first show f(∂M_{i}) = f(∂M′_{i}). Let C be a component of ∂M_{i} for i ∈ {0, 1, …, n}. Since pr′_{i} ∘f_{∗} ∘ j_{i} is an isomorphism and C is not π_{1}-null in M_{i}, the image of the composition of pr′_{i} and the homomorphism π_{1}(f(C)) ⟶ π_{1}(♯_{i=1}^{n}M′_{i}) = ∗_{i=1}^{n}Γ′_{i} induced by the inclusion is non-trivial. This implies f(C) ⊂ M′_{i}.

Let W′ be obtained from ∐_{i=1}^{n} M′_{i} × [0, 1] by attaching 1-handles to ∐_{i=1}^{n} M′_{i} × {1} such that

∂W′ = ∐_{i=1}^{n}(M′_{i})^{−} ∪_{∐_{i=1}^{n} ∂M′_{i}} ♯_{i=1}^{n}M′_{i}

where we identify M′_{i} with M′_{i} ∪_{∐_{i=1}^{n} ∂M′_{i}×{0}} ∐_{i=1}^{n} ∂M′_{i} × [0, 1]. Define analogously W for the M_{i}’s. Let V = W^{−} ∪_{f} W′ be obtained by glueing W and W′ together along f. Choose a map h′_{i} : M′_{i} ⟶ K(Γ′_{i}, 1) inducing the identity on the fundamental groups and mapping the embedded disk where the 1-handles are attached to the base point. Let h : W′ ⟶ ∨_{i=1}^{n}K(Γ′_{i}, 1) be the map which is on M_{i} × [0, 1] the composition of the projection M_{i} × [0, 1] ⟶ M_{i}, h_{i} and the canonical inclusion of K(Γ′_{i}, 1) into ∨_{i=1}^{n}K(Γ_{i}, 1) and on the one-handles the constant map. Since the inclusion of W into V is 3-connected we can extend this map to a map h : V ⟶ ∨_{i=1}^{n}K(Γ_{i}, 1). Notice for the sequel that the restriction of this map to M′_{i} composed with the projection pr′_{k} : ∨_{i=1}^{n}K(Γ′_{i}, 1) ⟶ K(Γ_{k}, 1) is the constant map for k ≠ i. Restricting a normal structure of V compatible with h to M_{i} ∪_{f|_{∂M_{i}}} M′_{i} yields a normal structure ρ_{i} for M_{i} ∪_{f|_{∂M_{i}}} M′_{i} with respect to ∨_{i=1}^{n}K(Γ′_{i}, 1). In the sequel we abbreviate M_{i}^{−} ∪_{f|_{∂M_{i}}} M′_{i} by M_{i} ∪ M′_{i}. We conclude

Lemma 4.1 We have

∑_{i=1}^{n}[M_{i} ∪ M′_{i}, ρ_{i}] = 0 ∈ Ω̃_{4}(∨_{i=1}^{n}K(Γ′_{i}, 1)).

The projection pr_{j} : ∨_{i=1}^{n}K(Γ′_{i}, 1) ⟶ K(Γ_{j}, 1) induces a homomorphisms
Ω̃_{4}(pr_{j}) : Ω̃_{4}(∨_{i=1}^{n}K(Γ′_{i}, 1)) ⟶ Ω̃_{4}(K(Γ′_{j}, 1)).

Notice that

⊕_{i=1}^{n}Ω̃_{4}(j_{i}) : ⊕_{i=1}^{n}Ω̃_{4}(K(Γ′_{i}, 1)) ⟶ Ω̃_{4}(∨_{i=1}^{n}K(Γ′_{i}, 1))

and

⊕_{i=1}^{n}Ω̃_{4}(pr_{i}) : Ω̃_{4}(∨_{i=1}^{n}K(Γ′_{i}, 1)) ⟶ ⊕_{i=1}^{n}Ω̃_{4}(K(Γ′_{i}, 1))

are isomorphisms, inverse to one another, by a Mayer-Vietoris argument and Lemma 2.3.

Lemma 4.2 For i, k ∈ {1, 2, …, n} with i ≠ k we get

Ω̃_{4}(pr′_{k}) [M_{i} ∪ M′_{i}, ρ_{i}] = 0 ∈ Ω̃_{4}(K(Γ′_{k}, 1)).

Notice that Theorem 1.4 follows from Lemma 4.1 and Lemma 4.2 because they imply together with the pair of inverse isomorphisms above

[M_{i} ∪ M′_{i}, pr′_{i} ∘ρ_{i}] = 0 ∈ Ω̃_{4}(K(Γ′_{i}, 1)).

for i ∈ {1, 2, …, n} and then one can apply Theorem 2.1. So it remains to prove Lemma 4.2.

Let C_{1}, C_{2}, …,C_{m} be the components of ∂M_{i}. Let k_{j} : π_{1}(C_{j}) ⟶ Γ_{i} be the homomorphism induced by the inclusion. Similarly, define k′_{j} : π_{1}(f(C_{j})) ⟶ Γ′_{i}. If g : G ⟶ H is a group homomorphism, denote by H//g the pushout of groups of ∗ ⟵ G ⟶^{g} H. This is the same as the quotient of H by the normal subgroup generated by the image of g.

Lemma 4.3 Suppose that ∂M_{i} is non-empty. Then there is an isomorphism

α : (Γ_{i}//∗_{j=1}^{m} k_{j}) ∗ F ⟶ π_{1}(M_{i}/∂M_{i})

for F a finitely generated free group of rank m − 1 and a map

β : Γ_{i}//∗_{j=1}^{m} k_{j} ⟶ (Γ′_{i}//∗_{j=1}^{m} k′_{j}) ∗ ∗_{1≤l≤n,l≠i} Γ′_{l}

such that the composition of β with the projection

\overline{pr′_{i}} : (Γ′_{i}//∗_{j=1}^{m} k′_{j}) ∗ ∗_{1≤l≤n,l≠i} Γ′_{l} ⟶ Γ′_{i}//∗_{j=1}^{m} k′_{j}

is an isomorphism and the following diagram commutes up to inner automorphisms of Γ′_{k} for j : Γ_{i}//∗_{j=1}^{m} k_{j} ⟶ Γ_{i}//∗_{j=1}^{m} k_{j} ∗ F the canonical inclusion and \overline{g_{i}} induced by pr′_{k} ∘g_{i}

Γ_{i}//∗_{j=1}^{m} k_{j}  ⟶^{α∘j} π_{1}(M_{i}/∂M_{i})
β↓  ↓^{π_{1}(\overline{g_{i}})}
(Γ′_{i}//∗_{j=1}^{m} k′_{j}) ∗ ∗_{1≤l≤n,l≠i}Γ′_{l} ⟶_{pr′_{k}}  Γ′_{k}

Before we prove Lemma 4.3, we explain how Lemma 4.2 and hence Theorem 1.4 follow from it. We only treat the more difficult case where ∂M_{i} is non-empty, the other case is similiar and does not use Lemma 4.3.

The underlying map of the normal structure ρ_{i} is g_{i} ∪ g′_{i} : M_{i} ∪ M′_{i} ⟶ ∨_{i=1}^{n}K(Γ′_{i}, 1). The composition pr′_{k} ∘g′_{i} is the constant map to the base point. Therefore, we obtain a factorization of pr′_{k} ∘(g_{i} ∪ g′_{i}) : M_{i} ∪ M′_{i} ⟶ K(Γ′_{k}, 1) as the composition of the projection q : M_{i} ∪ M′_{i} ⟶ M_{i}/∂M_{i} and the map \overline{g_{i}} : M_{i}/∂M_{i} ⟶ K(Γ′_{k}, 1) induced by pr′_{k} ∘g_{i}. The element Ω̃_{4}(pr′_{k}) [M_{i} ∪ M′_{i}, ρ_{i}] in Ω̃_{4}(K(Γ′_{k}, 1)) lies in the image of the composition Ω̃_{4}(\overline{g_{i}}) ∘ Ω̃_{4}(q). The map \overline{g_{i}} induces a map K(\overline{g_{i}}, 1) : K(π_{1}(M_{i}/∂M_{i}), 1) ⟶ K(Γ′_{k}, 1) which induces a homomorphism

Ω̃_{4}(K(\overline{g_{i}}, 1)) : Ω̃_{4}(K(π_{1}(M_{i}/∂M_{i}), 1)) ⟶ Ω̃_{4}(K(Γ′_{k}, 1))

As \overline{g_{i}} factorizes over K(\overline{g_{i}}, 1), it suffices to show that Ω̃_{4}(K(\overline{g_{i}}, 1)) is trivial. If j : G ⟶ G ∗ F is the inclusion for F a finitely generated free group, then the homomorphism

Ω̃_{4}(K(j, 1)) : Ω̃_{4}(K(G, 1)) ⟶ Ω̃_{4}(K(G ∗ F, 1))

is an isomorphism by a Mayer-Vietoris argument and Lemma 2.3. Hence it suffices to show because of Lemma 4.3 that the homomorphism

Ω̃_{4}(K(pr′_{k} ∘β, 1)) : Ω̃_{4}(K(Γ_{i}//∗_{j=1}^{m} k_{j}, 1)) ⟶ Ω̃_{4}(K(Γ′_{k}, 1))

is trivial. From the Kurosh Subgroup Theorem 1.5 there are subgroups A_{p} and a free subgroup F′ of (Γ′_{i}//∗_{j=1}^{m} k′_{j}) ∗ ∗_{1≤l≤n,l≠i}Γ′_{l} such that each A_{p} is subconjugated to Γ′_{i}//∗_{j=1}^{m} k′_{j} or some of the Γ_{l}’s and the image of β is given by

im(β) = F′ ∗ ∗_{p=1}^{q}A_{p}.

Since the composition of β with \overline{pr′_{i}} : (Γ′_{i}//∗_{j=1}^{m} k′_{j}) ∗ ∗_{1≤l≤n,l≠i} Γ′_{l} ⟶ Γ′_{i}//∗_{j=1}^{m} k′_{j} is injective by Lemma 4.3 each A_{p} is subconjugated to Γ′_{i}//∗_{j=1}^{m} k′_{j}. The inclusion ι : ∗_{p=1}^{q}A_{p} ⟶ im(β) induces by the argument above an isomorphism

Ω̃_{4}(K(ι, 1)) : Ω̃_{4}(∨_{p=1}^{q}K(A_{p}, 1)) ⟶ Ω̃_{4}(im(β))

Since Ω̃_{4}(K(pr′_{k} ∘β, 1)) factorizes through Ω̃_{4}(im(β)) and the composition of ι with the projection pr′_{k} : (Γ′_{i}//∗_{j=1}^{m} k′_{j}) ∗ ∗_{1≤l≤n,l≠i} Γ′_{l} ⟶ Γ′_{k} is trivial, Ω̃_{4}(K(pr ∘β, 1)) is trivial. This finishes the proof that Lemma 4.3 implies Lemma 4.2 and hence Theorem 1.4.

It remains to prove Lemma 4.3. The map α : (Γ_{i}//∗_{j=1}^{m} k_{j}) ∗ F ⟶ π_{1}(M_{i}/∂M_{i}) is given on Γ_{i}//∗_{j=1}^{m} k_{j} by the map which is induced by the projection M_{i} ⟶ M_{i}/∂M_{i}. The free group F has m−1 generators and α sends the i-th generator to the class in π_{1}(M_{i}/∂M_{i}) represented by some path in M_{i} joining C_{1} and C_{j+1}. Notice that M_{i}/∂M_{i} is up to homotopy the same as attaching to each boundary component C_{j} the cone over C_{j} and then attaching m − 1 one-cells such that the j-th one-cell joins the top of the cone of C_{1} and C_{j+1} for 1 ≤ j ≤ m − 1. One easily checks using Seifert-van Kampen Theorem that α is an isomorphism.

For appropriate choices of elements w_{j} in ∗_{i=1}^{n}Γ′_{i} the following diagram commutes

∗_{j=1}^{m}π_{1}(C_{j}) ⟶^{∗_{j=1}^{m}π_{1}(f|_{C_{j}})} ∗_{j=1}^{m}π_{1}(f(C_{j}))
∗_{j=1}^{m}k_{j} ↓  ↓^{∗_{j=1}^{m}c(w_{j})∘k′_{j}}
Γ_{i}  ⟶_{f_{∗}∘j_{i}}  ∗_{i=1}^{n}Γ′_{i}

where c(w_{j}) denotes conjugation with w_{j}. The map β is the homomorphism making the following diagram commute

Γ_{i}  ⟶^{f_{∗}∘j_{i}}  ∗_{i=1}^{n}Γ′_{i}
pr↓  ↓^{pr}
Γ_{i}//∗_{j=1}^{m} k_{j} ⟶_{β} (Γ′_{i}//∗_{j=1}^{m} k′_{j}) ∗ ∗_{1≤l≤n,l≠i} Γ′_{l}

Since ∗_{j=1}^{m}π_{1}(f|_{C_{j}}) is an isomorphism and pr′_{i} ∘f_{∗} ∘ j_{i} is an isomorphism by assumption, the map pr′_{i} ∘f_{∗} ∘ j_{i} induces an isomorphism β_{1} making the following diagram commute

Γ_{i}  ⟶^{pr′_{i} ∘f_{∗}∘j_{i}}  Γ′_{i}
pr↓  ↓^{pr}
Γ_{i}//∗_{j=1}^{m} k_{j} ⟶_{β_{1}} Γ′_{i}//∗_{j=1}^{m} c(pr′_{i}(w_{j})) ∘ k′_{j}

The identity on Γ′_{i} induces an isomorphism

β_{2} : Γ′_{i}//∗_{j=1}^{m} c(pr′_{i}(w_{j})) ∘ k′_{j} ⟶ Γ′_{i}//∗_{j=1}^{m} k′_{j}

The composition of β with the projection \overline{pr′_{i}} : (Γ′_{i}//∗_{j=1}^{m} k′_{j}) ∗ ∗_{1≤l≤n,l≠i} Γ′_{l} ⟶ Γ′_{i}//∗_{j=1}^{m} k′_{j} is the composition of the isomorphisms β_{1} and β_{2} and hence bijective. The diagram commutes up to inner automorphisms of Γ′_{k} by construction. This finishes the proof of Lemma 4.3 and hence of Theorem 1.4.

References

[1] Bröcker, T. and tom Dieck, T.: “Kobordismentheorie”, Lecture Notes in Mathematics 178 (1970)

[2] Cappell, S.E.: “On connected sums of manifolds”, Topology 13, 395 - 400 (1974)

[3] Cappell, S.E.: “A spitting theorem for manifolds”, Inventiones Math. 33, 69 - 170 (1976)

[4] Cohen, M.M.: “Combinatorial group theory: a topological approach”, LMS student texts 14 (1989)

[5] Freedman, M.H.: “The topology of four-dimensional manifolds”, J. of Differential Geometry 17, 357 - 453 (1982)

[6] Hempel, J.: “3-manifolds”, Annals of Mathematics Studies 86, Princeton University Press (1976)

[7] Hillman, J.A.: “Free products and 4-dimensional connected sums”, preprint (1993)

[8] Kreck, M.: “Surgery and duality”, to appear, Vieweg (1994)

[9] Kreck, M., Lück, W. and Teichner, P.: “Counterexamples to the Kneser conjecture in dimension four”, preprint, Mainz (1994)

[10] Lyndon, R.C. and Schupp, P.E.: “ Combinatorial Group Theory”, Ergebnisse der Mathematik und ihrer Grenzgebiete 89, Springer (1977)

Matthias Kreck, Fachbereich Mathematik, Johannes Gutenberg-Universität, 55099 Mainz, Bundesrepublik Deutschland & Mathematisches Forschungsinstitut Oberwolfach, 77709 Oberwolfach-Walke, Bundesrepublik Deutschland

Wolfgang Lück, Fachbereich Mathematik, Johannes Gutenberg-Universität, 55099 Mainz, Bundesrepublik Deutschland

Peter Teichner, University of California, San Diego, Department of Mathematics, 9500 Gilman Drive, LaJolla, CA 92093-0112, U.S.A.

email: kreck/lueck@topologie.mathematik.uni-mainz.de, teichner@euclid.ucsd.edu

FAX: 06131 393867 Version of November 8, 1996

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