On the cut and paste property of higher signatures of a closed oriented manifold

by

Eric Leichtnam and Wolfgang Lück with an appendix

Mapping tori of special diffeomorphisms by Matthias Kreck

Eric Leichtnam^{∗} Institut de Chevaleret Plateau E, 7^{eme} étage (Algèbres d’ opérateurs) 175 Rue de Chevaleret

75013 Paris France

Wolfgang Lück^{†} Mathematisches Institut Universität Münster Einsteinstr. 62 48149 Münster Germany

Matthias Kreck ^{‡} Mathematisches Institut Universität Heidelberg Im Neuenheimer Feld 288 69120 Heidelberg Germany

November 30, 2000

Abstract

We extend the notion of the symmetric signature σ(\overline{M}, r) ∈ L^{n}(R) for a compact n-dimensional manifold M without boundary, a reference map r : M → BG and a homomorphism of rings with involutions β : ℤG → R to the case with boundary ∂M, where (\overline{M}, \overline{∂M}) → (M, ∂M) is the Gcovering associated to r. We need the assumption that C_{∗}(\overline{∂M}) ⊗_{ℤG} R is R-chain homotopy equivalent to a R-chain complex D_{∗} with trivial m-th differential for n = 2m resp. n = 2m + 1. We prove a glueing formula, homotopy invariance and additivity for this new notion.

∗e-mail: leicht@math.jussieu.fr †email: lueck@math.uni-muenster.de www: http://www.math.uni-muenster.de/u/lueck/org/staff/lueck/

‡email: kreck@mathi.uni-heidelberg.de

Let Z be a closed oriented manifold with reference map Z → BG. Let F ⊂ Z be a cutting codimension one submanifold F ⊂ Z and let \overline{F} → F be the associated G-covering. Denote by α_{m}(\overline{F}) the m-th Novikov-Shubin invariant and by b_{m}^{(2)}(\overline{F}) the m-th L^{2}-Betti number. If for the discrete group G the Baum-Connes assembly map is rationally injective, then we use σ(\overline{M}, r) to prove the additivity (or cut and paste property) of the higher signatures of Z, if we have α_{m}(\overline{F}) = ∞^{+} in the case n = 2m and, in the case n = 2m + 1, if we have α_{m}(\overline{F}) = ∞^{+} and b_{m}^{(2)}(\overline{F}) = 0. This additivity result had been proved (by a different method) in [7, Corollary 0.4] when G is Gromov hyperbolic or virtually nilpotent. We give new examples, where these conditions are not satisfied and additivity fails.

We explain at the end of the introduction why our paper is greatly motivated by and partially extends some of the work of Leichtnam-Lott-Piazza [7], Lott [9] and Weinberger [20]. Key words: higher signatures, symmetric signature, additivity 1991 mathematics subject classification: 57R20, 57R67

0 Introduction

Let M be an oriented compact n-dimensional manifold possibly with boundary. Let G be a (discrete) group and r : M → BG be a (continuous) reference map to its classifying space. Fix an (associative) ring R (with unit and) with involution and a homomorphism β : ℤG → R of rings with involution. Let \overline{∂M} → ∂M and \overline{M} → M be the G-coverings associated to the maps r|_{∂M} : ∂M → BG and r : M → BG. Following [8, Section 4.7] and [7, Assumption 1 and Lemma 2.3], we make an assumption about (∂M, r|_{∂M}).

Assumption 0.1 Let m be the integer for which either n = 2m or n = 2m + 1. Let C_{∗}(\overline{∂M}) be the cellular ℤG-chain complex. Then we assume that the Rchain complex C_{∗}(\overline{∂M}) ⊗_{ℤG} R is R-chain homotopy equivalent to a R-chain complex D_{∗} whose m-th differential d_{m} : D_{m} → D_{m−1} vanishes.

We will discuss this assumption later (see Lemma 3.1). We first consider the easier and more satisfactory case n = 2m. Under

Assumption 0.1 we will assign (in Section 2) for n = 2m to (M, r) the element

σ(\overline{M}, r) ∈ L^{2m}(R),  (0.2)

which we will call the symmetric signature, in the symmetric L-group L^{2m}(R) (Here and in the sequel we are considering the projective version and omit in the standard notation L_{p}^{2m}(R) the index p)

This element σ(\overline{M}, r) agrees with the symmetric signature in the sense of [15, Proposition 2.1], [16, page 26], provided that ∂M is empty. If ∂M is nonempty and D_{m} = 0 then σ(\overline{M}, r) was previously considered in [20] and [9, Appendix A].

The main properties of this invariant will be that it occurs in a glueing formula, is a homotopy invariant and is related to higher signatures as explained in Theorem 0.3, Theorem 0.5 and and Corollary 0.7. Given an oriented manifold M, we will denote by M^{−} the same manifold with the reversed orientation.

Theorem 0.3 (a) Glueing formula Let M and N be two oriented compact 2m-dimensional manifolds with boundary and let ϕ : ∂M → ∂N be an orientation preserving diffeomorphism. Let r : M ∪_{ϕ} N^{−} → BG be a reference map. Suppose that (∂M, r|_{∂M}) satisfies Assumption 0.1. Then

σ(\overline{M ∪_{ϕ} N^{−}}, r) = σ(\overline{M}, r|_{M}) − σ(\overline{N}, r|_{N});

(b) Additivity Let M and N be two oriented compact 2m-dimensional manifolds with boundary and let ϕ, ψ : ∂M → ∂N be orientation preserving diffeomorphisms. Let r : M ∪_{ϕ} N^{−} → BG and s : M ∪_{ψ} N^{−} → BG be reference maps such that r|_{M} ≃ s|_{M} and r|_{N} ≃ s|_{N} holds, where ≃ means homotopic. Suppose that (∂M, r|_{∂M}) satisfies Assumption 0.1. Then σ(\overline{M ∪_{ϕ} N^{−}}, r) = σ(\overline{M ∪_{ψ} N^{−}}, s);

(c) Homotopy invariance Let M_{0} and M_{1} be two oriented compact 2m-dimensional manifolds possibly with boundaries together with reference maps r_{i} : M_{i} → BG for i = 0, 1. Let (f, ∂f) : (M_{0}, ∂M_{0}) → (M_{1}, ∂M_{1}) be an orientation preserving homotopy equivalence of pairs with r_{1} ∘ f ≃ r_{0}. Suppose that (∂M_{0}, r_{0}|_{∂M_{0}}) satisfies Assumption 0.1. Then

σ(\overline{M_{0}}, r_{0}) = σ(\overline{M_{1}}, r_{1}).

Next we consider the case n = 2m+1. Then we need besides Assumption 0.1 the following additional input. Assumption 0.1 implies that H_{m}(C_{∗}(\overline{∂M})⊗_{ℤG}R) is a finitely generated projective R-module and that we get from Poincaré duality the structure of a (non-degenerate) (−1)^{m}-symmetric form μ on it. Following [7, Section 3], we will assume that we have specified a stable Lagrangian L ⊂ H_{m}(C_{∗}(\overline{∂M})⊗_{ℤG}R). The existence of a stable Lagrangian follows automatically if 2 is a unit in R (see Lemma 2.4). Under Assumption 0.1 and after the choice of a stable Lagrangian L we can assign for n = 2m + 1 to (M, r, L) an element, which we will call the symmetric signature, in the symmetric L-group L^{2m+1}(R) (see Section 2)

σ(\overline{M}, r, L) ∈ L^{2m+1}(R).  (0.4)

It agrees with the symmetric signature in the sense of [15, Proposition 2.1], [16, page 26], provided that ∂M is empty.

Theorem 0.5 (a) Glueing formula Let M and N be oriented compact (2m + 1)-dimensional manifolds with boundary and let ϕ : ∂M → ∂N be an orientation preserving diffeomorphism. Let r : M ∪_{ϕ} N^{−} → BG be a reference map. Suppose that (∂M, r|_{∂M}) satisfies Assumption 0.1. Suppose that we have fixed two stable Lagrangians K ⊂ H_{m}(C_{∗}(\overline{∂M}) ⊗_{ℤG} R) and L ⊂ H_{m}(C_{∗}(\overline{∂N}) ⊗_{ℤG} R) such that the isomorphism H_{m}(C_{∗}(\overline{∂M}) ⊗_{ℤG} R) ⟶^{≅} H_{m}(C_{∗}(\overline{∂N}) ⊗_{ℤG} R) of (−1)^{m}-symmetric forms induced by ϕ sends K to L stably. Then

σ(\overline{M ∪_{ϕ} N^{−}}, r) = σ(\overline{M}, r|_{M}, K) − σ(\overline{N}, r|_{N}, L);

(b) Additivity Let M and N be oriented compact (2m + 1)-dimensional manifolds with boundary and let ϕ, ψ : ∂M → ∂N be two orientation preserving diffeomorphisms. Let r : M ∪_{ϕ} N^{−} → BG and s : M ∪_{ψ} N^{−} → BG be reference maps together with homotopies h_{M} : r|_{M} ≃ s|_{M} and h_{M} : r|_{N} ≃ s|_{N}. Suppose that (∂M, r|_{∂M}) satisfies Assumption 0.1. Fix a stable Lagrangian K ⊂ H_{m}(C_{∗}(\overline{∂M}) ⊗_{ℤG} R). The restriction of the homotopies h_{M} and h_{N} to ∂M and ∂N induce a homotopy k : r|_{∂M} ∘ ψ^{−1} ∘ ϕ ≃ r|_{∂M}. We get from ψ^{−1} ∘ ϕ and k an automorphism of the (−1)^{m}-symmetric form (H_{m}(C_{∗}(\overline{∂M}) ⊗_{ℤG} R), μ). Let L ⊂ H_{m}(C_{∗}(\overline{∂M}) ⊗_{ℤG} R) be the stable Lagrangian which is the image of K under this automorphism. Thus we get a formation (H_{m}(C_{∗}(\overline{∂M}) ⊗_{ℤG} R), μ, K, L) which defines an element in [H_{m}(C_{∗}(\overline{∂M}) ⊗_{ℤG} R), μ, K, L] ∈ L^{2m+1}(R) by suspension. Then

σ(\overline{M ∪_{ϕ} N^{−}}, r) − σ(\overline{M ∪_{ψ} N^{−}}, s) = [H_{m}(C_{∗}(\overline{∂M}) ⊗_{ℤG} R), μ, K, L];

(c) Homotopy invariance Let M_{0} and M_{1} be oriented compact (2m + 1)-dimensional manifolds possibly with boundaries together with reference maps r_{i} : M_{i} → BG for i = 0, 1. Let (f, ∂f) : (M_{0}, ∂M_{0}) → (M_{1}, ∂M_{1}) be an orientation preserving homotopy equivalence of pairs together with a homotopy h : r_{1} ∘ f ≃ r_{0}. Suppose that (∂M_{0}, r_{0}|_{∂M_{0}}) satisfies Assumption 0.1. Suppose that we have fixed stable Lagrangians L_{0} ⊂ H_{m}(C_{∗}(\overline{∂M_{0}}) ⊗_{ℤG} R) and L_{1} ⊂ H_{m}(C_{∗}(\overline{∂M_{1}}) ⊗_{ℤG} R). Let L′_{0} be the image of L_{0} under the isomorphism of (−1)^{m}-symmetric forms (H_{m}(C_{∗}(\overline{∂M_{0}}) ⊗_{ℤG} R), μ_{0}) ⟶^{≅} (H_{m}(C_{∗}(\overline{∂M_{1}}) ⊗_{ℤG} R), μ_{1}) induced by ∂f and the restriction of the homotopy h to ∂M_{0}. We get a stable formation (H_{m}(C_{∗}(\overline{∂M_{1}})⊗_{ℤG}R), μ_{1}, L′_{0}, L_{1}) and thus by suspension an element

[H_{m}(C_{∗}(\overline{∂M_{1}}) ⊗_{ℤG} R), μ_{1}, L′_{0}, L_{1}] ∈ L^{2m+1}(R). Then

σ(\overline{M_{0}}, r_{0}, L_{0}) − σ(\overline{M_{1}}, r_{1}, L_{1}) = [H_{m}(C_{∗}(\overline{∂M_{1}}) ⊗_{ℤG} R), μ_{1}, L′_{0}, L_{1}].

Of particular interest is the case, where R is the real reduced group C^{∗}- algebra C_{r}^{∗}(G, ℝ) or the complex reduced group C^{∗}-algebra C_{r}^{∗}(G) and β is the canonical map. Then Assumption 0.1 is equivalent to the assertion that the mth Novikov-Shubin invariant of \overline{∂M} is ∞^{+} in the sense of [10, Definition 1.8, 2.1 and 3.1] (see Lemma 3.1), and the symmetric L-groups are 2-periodic. Moreover, the invariant σ(\overline{M}, r) is linked to higher signatures as follows, provided that ∂M is empty.

Recall that the higher signature sign_{u}(M, r) of a closed oriented manifold M with a reference map r : M → BG for a given class u ∈ H^{k}(BG; ℚ) is the rational number ⟨L(M) ∪ r^{∗}u, [M]⟩, where L(M) ∈ ⊕_{i≥0}H^{4i}(M; ℚ) is the L-class of M, [M] ∈ H_{dim(M)}(M; ℚ) is the homological fundamental class of M and ⟨ , ⟩ is the Kronecker pairing. We will consider the following commutative square of ℤ/4-graded rational vector spaces

(Ω_{∗}(BG) ⊗_{Ω_{∗}(∗)} ℚ)_{n} ⟶_{≅}^{\overline{D}} KO_{n}(BG) ⊗_{ℤ} ℚ ⟶^{induc.} K_{n}(BG) ⊗_{ℤ} ℚ
σ↓  ^{A_{ℝ}}↓  ^{A}↓
L^{n}(C_{r}^{∗}(G; ℝ)) ⊗_{ℤ} ℚ ⟶_{≅}^{sign} KO_{n}(C_{r}^{∗}(G; ℝ)) ⊗_{ℤ} ℚ ⟶^{induc.} K_{n}(C_{r}^{∗}(G)) ⊗_{ℤ} ℚ

Some explanations are in order. We denote by ℚ the ℤ-graded vector space which is ℚ in each dimension divisible by four and zero elsewhere. It can be viewed as a graded module over the ℤ-graded ring Ω_{∗}(∗) by the signature. Then the ℤ-graded ℚ-vector space Ω_{∗}(BG)⊗_{Ω_{∗}(∗)} ℚ is four-periodic (by crossing with [ℂℙ^{2}]) and hence can be viewed as a ℤ/4-graded vector space. The map \overline{D} is induced by the ℤ-graded homomorphism

D : Ω_{n}(BG) → KO_{n}(BG),

which sends [r : M → BG] to the K-homology class of the signature operator of the covering \overline{M} → M associated to r. The homological Chern character is an isomorphism of ℤ/4-graded rational vector spaces

ch : KO_{n}(BG) ⊗_{ℤ} ℚ ⟶^{≅} ⊕_{k≥0}H_{4k+n}(BG; ℚ).

By the Atiyah-Hirzebruch index theorem the image ch ∘D([M, id : M → M]) of the K-homology class of the signature operator of M in K_{dim(M)}(M) under the homological Chern character ch is L(M) ∩ [M]. This implies for any class u ∈ H^{k}(BG; ℚ)

sign_{u}(M, r) := ⟨L(M) ∪ r^{∗}u, [M]⟩
= ⟨r^{∗}u, L(M) ∩ [M]⟩
= ⟨u, r_{∗}(L(M) ∩ [M])⟩
= ⟨u, ch ∘D([M, r])⟩.  (0.6)

Hence the composition ch ∘\overline{D} : (Ω_{∗}(BG) ⊗_{Ω_{∗}(∗)} ℚ)_{n} → ⊕_{k≥0}H_{4k+n}(BG; ℚ) sends [r : M → BG] ⊗ 1 to the image under H_{∗}(r) : H_{∗}(M; ℚ) → H_{∗}(BG; ℚ) of the Poincaré dual L(M) ∩ [M] ∈ ⊕_{i≥0}H_{4i−dim(M)}(M; ℚ) of the L-class L(M).

The map \overline{D} is an isomorphism since it is a transformation of homology theories [6, Example 3.4] and induces an isomorphism for the space consisting of one point. The map σ assigns to [M, r] the associated symmetric Poincaré C_{r}^{∗}(G; ℝ)-chain complex C_{∗}(\overline{M}) ⊗_{ℤG} C_{r}^{∗}(G; ℝ). The map A_{ℝ} resp. A are assembly maps given by taking the index with coefficients in C_{r}^{∗}(G; ℝ) resp. C_{r}^{∗}(G). The map sign is in dimension n = 0 mod 4 given by taking the signature of a non-degenerate symmetric bilinear form. Notice that the map sign is bijective by results of Karoubi (see [18, Theorem 1.11]). The maps induc. are given by induction with the inclusion ℝ → ℂ and are injective. Obviously the right square commutes. In order to show that the diagram commutes it suffices to prove this for the outer square. Here the claim follows from the commutative diagram in [5, page 81].

The Novikov Conjecture says that sign_{u}(M, r) is a homotopy invariant, i.e. if r : M → BG and s : N → BG are closed orientable manifolds with reference maps to BG and f : M → N is a homotopy equivalence with s ∘ f ≃ r, then sign_{u}(M, r) = sign_{u}(N, s). Since the homological Chern character is rationally an isomorphism for CW -complexes, one can say by (0.6) that D(M, r) is rationally the same as the collection of all higher signatures. Moreover, the Novikov Conjecture is equivalent to the statement that two elements [M, r] and [N, s] in Ω_{n}(BG) represent the same element in (Ω_{∗}(BG) ⊗_{Ω_{∗}(∗)} ℚ)_{n} resp. KO_{n}(BG) ⊗_{ℤ} ℚ resp. K_{n}(BG) ⊗_{ℤ} ℚ, if they are homotopy equivalent.

Notice that the Baum-Connes Conjecture for C_{r}^{∗}(G) implies that A and hence A_{ℝ} are rationally injective by the following argument (see [1, section 7] for details). The map A can be written as the composition of the map K_{n}(BG) = K_{n}^{G}(EG) → K_{n}(\underline{E}G)^{G}, which is given by the canonical map from EG to the classifying space \underline{E}G of proper G-actions and is always rationally injective, and the Baum-Connes index map K_{∗}^{G}(\underline{E}G) → K_{∗}(C_{r}^{∗}(G)), which is predicted to be bijective by the Baum-Connes Conjecture. Notice that σ is injective if and only if A_{ℝ} is injective and that the injectivity of A implies the injectivity of A_{ℝ} and hence of σ. Since for a closed oriented manifold M with reference map r : M → BG the image of [r : M → BG] under σ (and sign ∘ σ) is a homotopy invariant of r : M → BG, the commutativity of the diagram above and the rational injectivity of A_{ℝ} implies the homotopy invariance of (0.6) and thus the Novikov conjecture. Moreover, if A_{ℝ} is rationally injective, D([M, r]) contains rationally the same information as σ([M, r]). We mention that the Baum-Connes Conjecture and thus the rational injectivity of A is known for a large class of groups, namely for all a-T-menable groups [3]. The rational injectivity of A is also known for all Gromov-hyperbolic groups [21].

From Theorem 0.3 (b) and Theorem 0.5 (b), we obtain the following corollary which extends [7, Corollary 0.4] to more general groups G. Notice that the hypothesis of Corollary 0.7 implies those of Theorem 0.3 (b) and Theorem 0.5 (b) because of Lemma 3.1

Corollary 0.7 Let M and N be two oriented compact n-dimensional manifolds with boundary and let ϕ, ψ : ∂M → ∂N be orientation preserving diffeomorphisms. Let r : M ∪_{ϕ} N^{−} → BG and s : M ∪_{ψ} N^{−} → BG be reference maps such that r|_{M} ≃ s|_{M} and r|_{N} ≃ s|_{N} holds. Denote by \overline{∂M} → ∂M the G-covering associated to r|_{∂M} : ∂M → BG. If n = 2m, we assume for the m-th Novikov Shubin invariant α_{m}(\overline{∂M}) = ∞^{+} in the sense of [10]. If n = 2m + 1, we assume α_{m}(\overline{∂M}) = ∞^{+} and for the m-th L^{2}-Betti number b_{m}^{(2)}(\overline{∂M}) = 0. (We could replace the condition b_{m}^{(2)}(\overline{∂M}) = 0 by the weaker but harder to check assumption that the automorphism in Theorem 0.5 (b) induced by ψ^{−1} ∘ ϕ and the homotopy k preserve (stably) a Lagrangian of H_{m}( C_{∗}(\overline{∂M}) ⊗_{ℤG} C_{r}^{∗}(G) ). Suppose furthermore that the map A_{ℝ} : KO_{n}(BG) → K_{n}(C_{r}^{∗}(G; ℝ) is injective. Then all the higher signatures are additive in the sense that we have for all u ∈ H^{∗}(BG, ℚ)

sign_{u}(M ∪_{ϕ} N^{−}, r) = sign_{u}(M ∪_{ψ} N^{−}, s).  (0.8)

We will construct in Example 1.10 many new examples (especially for odd dimensional manifolds) of pairs of cut and paste manifolds [M ∪_{ϕ} N^{−}, r] and [M ∪_{ψ} N^{−}, s] (even with ∂M connected) such that r|_{M} ≃ s|_{M} and r|_{N} ≃ s|_{N} holds and for which there exist higher signatures which do not satisfy (0.8), ie are not additive. There, the assumptions of Corollary 0.7, are not fully satisfied. The fact that, in general, higher signatures of closed manifolds are not cut and paste invariant over BG in the sense of [4], was known before (see, for instance, [9, Section 4.1]). The relationship to symmetric signatures of manifolds-with-boundary, and to the necessity of Assumption 0.1, was pointed out by Weinberger (see [9, Section 4.1]). The problem was raised in [9, Section 4.1] of determining which higher signatures of closed manifolds are cut and paste invariant; we refer to [9, Section 4.1] for further discussion. It is conceivable that our Lemma 2.8 might help to provide, in the future, an answer to this problem.

Finally we explain why our paper is greatly motivated by and related to the work of Leichtnam-Lott-Piazza [7], Lott [9] and Weinberger [20].

The relevance of a gap condition in the middle degree on the boundary, when considering topological questions concerning manifolds with boundary, comes from Section 4.7 of Lott’s paper [8]. This leads to Assumption 1 in the paper of Leichtnam-Lott-Piazza [7] which, from [7, Lemma 2.3], is virtually identical to our Assumption 0.1 and was the motivation for our Assumption 0.1. Our construction of the invariant σ(\overline{M}, r) by glueing algebraic Poincaré bordisms is motivated by and extends the one of Weinberger [20] (see also [9, Appendix A]) who uses the more restrictive assumption that C_{∗}(\overline{∂M}) ⊗_{ℤG} R is R-chain homotopy equivalent to a R-chain complex D_{∗} with D_{m} = 0. Notice that Weinberger’s assumption implies both our Assumption 0.1 and H_{m}(C_{∗}(\overline{∂M}) ⊗_{ℤG} R) = 0 so that there is only one choice of Lagrangian, namely L = 0. The idea of using a Lagrangian subspace, instead of assuming the vanishing of the relevant middle (co-)homology group, is taken from Section 3 of [7].

In the case when R = C_{r}^{∗}(G) and under Assumption 0.1, an analog of our symmetric signature σ(\overline{M}, r) was previously constructed in [7] as a conic index class σ_{conic} ∈ K_{0}(C_{r}^{∗}(G)). The homotopy invariance of σ_{conic}, i.e. the analog of our Theorem 0.3(c), was demonstrated in [7, Theorem 6.1]. Furthermore, the analog of the right-hand-side of the equation in our Theorem 0.5(c) previously appeared in [7, Proposition 3.7]

If B^{∞} is a smooth subalgebra of C_{r}^{∗}(G), the authors of [7] computed the Chern character ch(σ_{conic}) ∈ \overline{H}_{∗}(B^{∞}) of their conic index explicitly in terms of the L-form of M and a higher eta-form of ∂M. They identified ch(σ_{conic}) with the \overline{H}_{∗}(B^{∞})-valued higher signature of M introduced in [8]. From this the authors of [7] deduced their main result [7, Theorem 0.1], namely the homotopy invariance of the \overline{H}_{∗}(B^{∞})-valued higher signature of a manifold with boundary, as opposed to just the homotopy invariance of the “symmetric signature” σ_{conic}. (In fact, this was the motivation for the use of σ_{conic} in [7], instead of σ(\overline{M}, r).)

As an immediate consequence of its main result, the paper [7] deduced the additivity of ordinary higher signatures of closed manifolds under Assumption 0.1, i.e. our Corollary 0.7, in the case when G is Gromov-hyperbolic or virtually nilpotent, or more generally when C_{r}^{∗}(G) admits a smooth subalgebra B^{∞} with the property that all of the group cohomology of G extends to cyclic cocycles on B^{∞} [7, Corollary 0.4]. It is known that the Baum-Connes assembly map is rationally injective for such groups G.

What is new in our paper is the direct and purely algebraic approach to the cut and paste problem of higher signatures of closed manifolds, through the construction of symmetric signatures for manifolds with boundary under Assumption 0.1. This leads to the main goal of the present paper, namely, to the proof of the additivity of higher signatures under Assumption 0.1 in Corollary 0.7, provided that the Baum-Connes assembly map is rationally injective. In this way, our main result, Corollary 0.7, is an extension of [7, Corollary 0.4].

1 Additivity and mapping tori in the bordism group

Throughout this section X is some topological space. Denote by Ω_{n}(X) the bordism group of closed oriented smooth n-dimensional manifolds M together with a reference map r : M → X. Consider quadruples (F, h, r, H) consisting of a closed oriented (n − 1)-dimensional manifold F together with an orientation preserving self-diffeomorphism h : F → F, a reference map r : F → X and a homotopy H : F × [0, 1] → X such that H(−, 0) = r and H(−, 1) = r ∘ h. The mapping torus T_{h} is obtained from the cylinder F ×[0, 1] by identifying the bottom and the top by h, i.e. (h(x), 0) ∼ (x, 1). This is again a closed smooth manifold and inherits a preferred orientation. The map r and the homotopy H yield a reference map r_{T_{h}} : T_{h} → X in the obvious way. Hence we can associate to such a quadruple an element

[F, h, r, H] := [T_{h}, r_{T_{h}}] ∈ Ω_{n}(X).  (1.1)

Given two quadruples (F, h[0], r[0], H[0]) and (F, h[1], r[1], H[1]) with the same underlying manifold, a homotopy between them is given by a family of such quadruples (F, h[t], r[t], H[t]) for t ∈ [0, 1] such that the family h[t] : F → F is a diffeotopy. One easily checks that for two homotopic quadruples (as above) we have in Ω_{n}(X)

[F, h[0], r[0], H[0]] = [F, h[1], r[1], H[1]].  (1.2)

The required cobordism has as underlying manifold F × [0, 1] × [0, 1]/ ∼, where ∼ is the equivalence relation generated by (x, 0, t) ∼ (h^{−1}[t](x), 1, t).

Given two quadruples of the shape (F, h, r, H) and (F, g, r, G), we can compose them to a quadruple (F, g∘h, r, H∗G), where H∗G is the obvious homotopy r ≃ r ∘ g ∘ h obtained from stacking together H and G × (h × id_{[0,1]}). One easily checks that in Ω_{n}(X)

[F, g ∘ h, r, H ∗ G] = [F, h, r, H] + [F, g, r, G].  (1.3)

The desired cobordism has as underlying manifold F ×[0, 1]×[0, 1]/ ∼, where ∼ is generated by (x, 0, t) ∼ (h^{−1}(x), 1, t) for t ∈ [0, 1/3] and (x, 1, t) ∼ (g^{−1}(x), 0, t) for t ∈ [1/3, 1]. We recognize the mapping torus of g ∘ h as the part of the boundary which is the image under the canonical projection of the union of F ×{0}×[1/3, 2/3], F ×[0, 1]×{1/3}, F ×{1}×[1/3, 2/3] and F ×[0, 1]×{2/3}.

Notice that the class of a quadruple in Ω_{n}(X) does depend on the choice of the homotopy. Namely, consider two quadruples (F, h, r, H) and (F, h, r, G) which differ only in the choice of the homotopy. Let u : F × S^{1} → X be the obvious map induced by r and composition of homotopies H ∗G^{−} : r ≃ r. Then we get from (1.2) and (1.3) in Ω_{n}(X)

[F, h, r, H] − [F, h, r, G] = [u : F × S^{1} → X].  (1.4)

The right side of (1.4) is not zero in general. Take for instance F = ℂℙ^{2} and X = S^{1} and let u_{k} : F × S^{1} → S^{1} be the composition of the projection F × S^{1} → S^{1} with a map S^{1} → S^{1} of degree k ∈ ℤ. Then in this situation the right-handside of 1.4 becomes [u_{k} : ℂℙ^{2} ×S^{1} → S^{1}] for r : ℂℙ^{2} → S^{1} a constant map. This element [u_{k} : ℂℙ^{2} × S^{1} → S^{1}] is mapped under the isomorphism Ω_{5}(S^{1}) ≅ Ω_{4}(∗) ⊕ Ω_{5}(∗) = ℤ ⊕ ℤ/2 to (k, 0).

Let M and N be compact oriented n-dimensional manifolds and let ϕ, ψ : ∂M → ∂N be two orientation preserving diffeomorphisms. By glueing we obtain closed oriented n-dimensional manifolds M ∪_{ϕ} N^{−} and M ∪_{ψ} N^{−}. Let r : M ∪_{ϕ} N^{−} → X and s : M ∪_{ϕ} N^{−} → X be two reference maps such that there exists two homotopies H : r|_{M} ≃ s|_{M} and G : r|_{N} ≃ s|_{N}. By restriction we obtain homotopies H|_{∂M} : r|_{∂M} ≃ s|_{∂M} and G|_{∂N} : r|_{∂N} ≃ s|_{∂N}. Notice that (by construction) r|_{∂N} ∘ ϕ = r|_{∂M} and s|_{∂N} ∘ ψ = s|_{∂M}. Thus H|_{∂M} and G^{−}|_{∂N} ∘ (ψ × id) can be composed to a homotopy K : r|_{∂M} ≃ r|_{∂M} ∘ ϕ^{−1} ∘ ψ. Thus we obtain a quadruple (∂M, ϕ^{−1} ∘ ψ, r|_{∂M}, K) in the sense of (1.1). Lemma 1.5 We get in Ω_{n}(X)

[r : M ∪_{ϕ} N^{−} → X] − [s : M ∪_{ψ} N^{−} → X] = [∂M, ϕ^{−1} ∘ ψ, r|_{∂M}, K].

\underline{\underline{Proof}} : The underlying manifold of the required bordism is obtained by glueing parts of the boundary of M × [0, 1] and of N^{−} × [0, 1] together as described as follows. Identify (x, t) ∈ ∂M × [0, 1] with (ϕ(x), t) in ∂N × [0, 1] if 0 ≤ t ≤ 1/3, and with (ψ(x), t) in ∂N if 2/3 ≤ t ≤ 1.

Corollary 1.6 Suppose in the situation of Lemma 1.5 that X = BG for a discrete group G and that the image H of the composition

π_{1}(∂M) → π_{1}(M) ⟶^{(r|_{M})_{∗}} π_{1}(BG) = G

satisfies H_{i}(BH; ℚ) = 0 for i ≥ 1. Then the higher signatures of r : M ∪_{ϕ}N^{−} → X and s : M ∪_{ψ} N^{−} → X agree.

\underline{\underline{Proof}} : In view of Lemma 1.5 we have to show that the higher signatures of (∂M, ϕ^{−1} ∘ ψ, r|_{∂M}, K) vanish. The homotopy K yields an element g ∈ G such that the composition of c(g) : G → G g′ ↦ gg′g^{−1} with π_{1}(r|_{∂M}) agrees with the composition of π_{1}(r|_{∂M}) with the automorphism π_{1}(ϕ^{−1} ∘ ψ). Obviously c(g) induces an automorphism of H. Denote the associated semi-direct product by H ⋊ ℤ. There is a group homomorphism from H ⋊ ℤ to G which sends h ∈ H to h ∈ G and the generator of ℤ to g ∈ G. Let p : H ⋊ ℤ → ℤ be the canonical projection. Then the reference map from the mapping torus T_{ϕ^{−1}∘ψ} to BG factorizes as a map T_{ϕ^{−1}∘ψ} ⟶^{t} B(H ⋊ ℤ) → BG and the composition

T_{ϕ^{−1}∘ψ} ⟶^{t} B(H ⋊ ℤ) ⟶^{Bp} Bℤ = S^{1} is homotopic to the canonical projection pr : T_{ϕ^{−1}∘ψ} → S^{1}. Notice that Bp induces an isomorphism H^{∗}(Bℤ; ℚ) → H^{∗}(B(H ⋊ ℤ; ℚ)) since H^{i}(BH; ℚ) = 0 for i ≥ 1. Hence it remains to show that all higher signatures of pr : T_{ϕ^{−1}∘ψ} → S^{1} vanish. If 1 ∈ H^{0}(S^{1}; ℚ) ≅ Q and u ∈ H^{1}(S^{1}; ℚ) ≅ ℚ are the obvious generators, it remains to prove that sign_{1}(pr : T_{ϕ^{−1}∘ψ} → S^{1}) and sign_{u}(pr : T_{ϕ^{−1}∘ψ} → S^{1}) are trivial. (Recall that sign_{u} has been defined in the introduction.) These numbers are given by ordinary signatures of T_{ϕ^{−1}∘ψ} and ∂M. Since T_{ϕ^{−1}∘ψ} fibers over S^{1} and ∂M is nullbordant, these numbers are trivial.

Example 1.7 Consider the situation of Lemma 1.6 in the special case, where ∂M looks like ℝℙ^{3}♯ℝℙ^{3} × N for a simply connected oriented closed (n − 3)- dimensional manifold N for n ≥ 6 such that H_{m−1}(N; ℚ) or H_{m−3}(N; ℂ) is non-trivial and the map π_{1}(∂M) → π_{1}(M) is injective. Recall that m is the integer satisfying n = 2m resp. n = 2m + 1. The fundamental group of the connected sum ℝℙ^{3}♯ℝℙ^{3} is the infinite dihedral group D_{∞} = ℤ/2 ∗ ℤ/2 = ℤ ⋊ ℤ/2. Notice that there is a two-fold covering S^{1} × S^{2} → ℝℙ^{3}♯ℝℙ^{3} and the universal covering (ℝℙ^{3}♯ℝℙ^{3})^{∼} of ℝℙ^{3}♯ℝℙ^{3} is ℝ×S^{2}. Hence H_{m−1}((ℝℙ^{3}♯ℝℙ^{3} × N)^{∼}; ℂ) ≅ H_{m−1}(N; ℂ) ⊕ H_{m−3}(N; ℂ) is a non-trivial direct sum of finitely many copies of the trivial D_{∞}-representation ℂ. Since α_{m}((ℝℙ^{3}♯ℝℙ^{3} × N)^{∼}) = α_{m}((S^{1} × S^{2} × N)^{∼}) [10, Remark 3.9] we conclude that α_{m}((ℝℙ^{3}♯ℝℙ^{3} × N)^{∼}) is different from ∞^{+} (see [11, Example 4.3, Theorem 5.4] or [12, Theorem 8.7 (9)]). Hence Assumption 0.1 is not satisfied because of Lemma 3.1 and we cannot conclude the additivity of the higher signatures from Corollary 0.7. But we can conclude the additivity of the higher signatures from Corollary 1.6 since H_{i}(BD_{∞}; ℚ) ≅ H_{i}(Bℤ/2); ℚ) ⊕ H_{i}(Bℤ/2); ℚ) = 0 holds for i ≥ 1.

More generally one can consider glueing tuples (M, r_{M}, N, r_{N}, ϕ, H), which consist of two compact oriented n-dimensional manifolds M and N with boundaries with reference maps r_{M} : M → X and r_{N} : N → X, an orientation preserving diffeomorphism ϕ : ∂M → ∂N and a homotopy H : ∂M × [0, 1] → X between r_{M}|_{∂M} and r_{N}|_{∂N} ∘ ϕ, i.e. r_{M}|_{∂M} ∼ r_{N}|_{∂N} ∘ ϕ. To such a glueing tuple one can associate an element

[M, r_{M}, N, r_{N}, ϕ, H] := [M ∪_{∂M×{0}} (∂M × [0, 1]) ∪_{ϕ} N^{−}, r] ∈ Ω_{n}(X),(1.8)

where r is constructed from r_{M}, H and r_{N} in the obvious way. One gets

Lemma 1.9 Let (M, r_{M}, N, r_{N}, ϕ, H) and (N, r_{N}, P, r_{P} , ψ, G) be two glueing tuples. They can be composed to a glueing tuple (M, r_{M}, P, r_{P} , ψ ∘ ϕ, K), where K is the composition of the homotopies H and G ∘ (ϕ × id). Then we get in Ω_{n}(X)

[M, r_{M}, P, r_{P} , ψ ∘ ϕ, K] = [M, r_{M}, N, r_{N}, ϕ, H] + [N, r_{N}, P, r_{P} , ψ, G].

\underline{\underline{Proof}} : The required bordism has the following underlying manifold. Take the disjoint union of M × [0, 1], N × [0, 1] and P × [0, 1] and identify (x, t) ∈ ∂M × [0, 1/3] with (ϕ(x), t) ∈ ∂N × [0, 1/3] and (y, t) ∈ ∂N × [2/3, 1] with (ψ(y), t) ∈ ∂P × [2/3, 1].

Example 1.10 In odd dimensions additivity of the higher signatures (sometimes also called the cut and paste property) fails as badly as possible in the following sense. Let us consider m ≥ 2, a finitely presented group G and any element ω ∈ Ω_{2m+1}(BG). Then using the last (surjective) map of Theorem 3.2 of ([13]) and also the isomorphism given at the bottom of page 57 of ([13]) (or see the Theorem in the appendix by Matthias Kreck), one can find a quadruple (F, h, r, H) for a 2m-dimensional closed oriented manifold F with reference map r : F → BG such that [F, h, r, H] = ω in Ω_{2m+1}(BG) and [F, r] = 0 in Ω_{2m}(BG) holds. Fix a nullbordism R : W → BG for r : F → BG. In the sequel we identity F = ∂W. Since F admits a collar neighborhood in W, the inclusion F → W is a cofibration and thus we can extend the homotopy H : r ≃ r ∘ h to a homotopy H′ : R ≃ R′ for some map R′ : W → BG such that R′_{|∂M} = r ∘ h. Thus we obtain elements R′ ∪_{h} R : W ∪_{h} W → BG and R∪_{id} R : W ∪_{id} W → BG such that R′ ≃ R. We conclude from Lemma 1.5

[R′ ∪_{h} R : W ∪_{h} W → BG] − [R ∪_{id} R : W ∪_{id} W → BG] = ω. (1.11)

The theorem of Matthias Kreck which he proves in the appendix shows that for m ≥ 2 one can arrange in the situation above that the reference map r : F → BG is 2-connected, provided that BG has finite skeleta. (Since we only want to have 2-connected it suffices that BG has finite 2-skeleton.) Consider the special case m = 2 and G = ℤ. Choose in (1.11) a quadruple (F, h, r, H) such that r : F → Bℤ is 2-connected. Then \overline{F} is the universal covering of F. We conclude from [10, Lemma 3.3] that α_{2}(\overline{F}) = α_{2}(S̃^{1}) = ∞^{+}. Therefore Assumption 0.1 is satisfied for \overline{∂W} = \overline{F} by Lemma 3.1. Notice that there are elements ω ∈ Ω_{5}(Bℤ) whose higher signatures do not all vanish, for instance [ℂP^{2} × S^{1}, r] where r : ℂP^{2} × S^{1} → Bℤ = S^{1} is the projection onto the second factor. Hence, (for such an example), if we set [M_{0}, r_{0}] = [W, R] and [M_{1}, r_{1}] = [W, R′], then formula (1.11) and Theorem 0.5 show that the righthandside of the formula of Theorem 0.5 (b) is not zero. Thus Assumption 0.1 is not enough in the case n = 2m + 1 (in contrast to the case n = 2m as proven in Theorem 0.3 (b)) to ensure the additivity of the higher signatures.

Counterexamples to additivity in odd dimensions yield also counterexamples in even dimensions by crossing with S^{1}. In the situation of (1.11) with ω ∈ Ω_{2m+1}(BG), we get in Ω_{2m+2}(B(G × ℤ)) = Ω_{2m+2}(BG × S^{1})

[R′ × id_{S^{1}} ∪_{h×id_{S^{1}}} R × id_{S^{1}} : W × S^{1} ∪_{h×id_{S^{1}}} W × S^{1} → BG × S^{1}]

−[R × id_{S^{1}} ∪_{id_{∂W ×S^{1}}} R × id_{S^{1}} : W × S^{1} ∪_{id_{∂W ×S^{1}}} W × S^{1} → BG × S^{1}]

= ω × [id_{S^{1}}]

and, since the L−class of ω × S^{1} may be identified to the one of ω, for any u ∈ H^{∗}(BG; ℚ) we have sign_{u×[S^{1}]}(ω × [id_{S^{1}}]) = sign_{u}(ω) where [S^{1}] ∈ H^{1}(S^{1}; ℚ) is the fundamental class. Hence if ω ∈ Ω_{2m+1}(BG) admits at least a higher signature which is not zero, then W ×S^{1} ∪_{h×id} W ×S^{1} admit a higher signature which is not cut and paste invariant.

2 Computations in symmetric L-groups

In this section we carry out some algebraic computations and constructions of classes in symmetric L-groups which correspond on the geometric side to defining higher signatures of manifolds with boundaries (under Assumption 0.1) and to glueing processes along boundaries.

We briefly recall some basic facts about (symmetric) Poincaré chain complexes and the (symmetric) L-groups defined in terms of bordism classes of such chain complexes. For details we refer the reader to [14] and to the Section 1 of [16].

Let R be a ring with involution R → R : r ↦ \overline{r}. Two important examples are the group ring ℤG with the involution given by \overline{g} = g^{−1} and the reduced C^{∗}-algebra C_{r}^{∗}(G) of a group G. Given a left R-module V, let the dual V^{∗} be the (left) R-module hom_{R}(V, R) with the R-multiplication given by (rf)(x) = f(x)\overline{r}. Given a chain complex C_{∗} = (C_{∗}, c_{∗}) of (left) R-modules, define C^{n−∗} to be the R-chain complex whose i-th chain module is (C_{n−i})^{∗} and whose i-th differential is c_{n−i+1}^{∗} : C_{n−i}^{∗} → C_{n−i+1}^{∗}. We call C_{∗} finitely generated projective if C_{i} is finitely generated projective for all i ∈ ℤ and vanishes for i ≤ 0. An n-dimensional (finitely generated projective symmetric) Poincaré R-chain complex (C_{∗}, ϕ) consists of an n-dimensional finitely generated projective R-chain complex C_{∗} together with a R-chain homotopy equivalence ϕ_{∗}^{0} : C^{n−∗} → C_{∗} which the part for s = 0 of a representative {ϕ^{s} | s ≥ 0} of an element in ϕ in the hypercohomology group Q^{n}(C_{∗}) = H^{n}(ℤ/2; hom(C^{∗}, C_{∗})). The element ϕ^{1} is a chain homotopy (ϕ^{0})^{n−∗} ≃ ϕ_{∗}^{0}, where (ϕ^{0})^{n−∗} is obtained from ϕ^{0} in the obvious way using the canonical identification P → (P^{∗})^{∗} for a finitely generated projective R-module P. The elements ϕ^{s+1} are higher homotopies for ϕ_{∗}^{s} ≃ (ϕ^{s})^{n−∗}.

Consider a connected finite CW -complex X with universal covering X̃ and fundamental group π. It is an n-dimensional Poincaré complex if the (up to ℤπ-chain homotopy well-defined) ℤπ-chain map − ∩ [X] : C^{n−∗}(X̃) → C_{∗}(X̃) is a ℤπ-chain homotopy equivalence. Then for any normal covering \overline{X} → X with group of deck transformations G, the fundamental class [X] determines an element in ϕ ∈ Q^{n}(C_{∗}(\overline{X})), for which ϕ_{∗}^{0} is the ℤG-chain map induced by − ∩ [X] and (C_{∗}(\overline{X}), ϕ) is an n-dimensional Poincaré ℤG-chain complex [15, Proposition 2.1 on page 208].

The (symmetric) L-group L^{n}(R) is defined by the algebraic bordism group of n-dimensional finitely generated projective Poincaré R-chain complexes. The algebraic bordism relation mimics the geometric bordism relation. The general philosophy, which we will frequently use without writing down the details, is that any geometric construction for geometric Poincaré pairs, such as glueing along a common boundary with a homotopy equivalence, or taking mapping tori or writing down certain bordisms, can be transferred to the category of algebraic Poincaré chain complexes.

However, there is one important difference between the geometric bordism group Ω_{n}(X) and the L-group L^{n}(R) concerning homotopy invariance. Let G be a group and let M, N be two closed oriented n-dimensional manifolds with reference maps r : M → BG and s : N → BG. Suppose that f : M → N is a homotopy equivalence such that s ∘ f ≃ r. Then this does not imply that the bordism classes [M, r] and [N, s] agree. But the Poincaré ℤG-chain complexes C_{∗}(\overline{M}) and C_{∗}(\overline{N}) are ℤG-chain homotopy equivalent, and this does imply that their classes in L^{n}(ℤG) agree [14, Proposition 3.2 on page 136].

The following lemma explains the role of Assumption 0.1. Its elementary proof is left to the reader. Lemma 2.1 Let C_{∗} be a projective R-chain complex. Then the following assertions are equivalent.

(a) C_{∗} is R-chain homotopy equivalent to a R-chain complex D_{∗} with trivial m-th differential;

(b) im(c_{m}) is a direct summand in C_{m−1} where c_{m} : C_{m−1} → C_{m} is the differential;

(c) There is a finitely generated projective R-subchain complex D_{∗} ⊂ C_{∗} with D_{m} = ker(c_{m}), D_{m−1} ⊕ im(c_{m−1}) = C_{m−1} and D_{i} = C_{i} for i ≠ m, m − 1 such that the m-th differential of D_{∗} is zero and the inclusion D_{∗} → C_{∗} is a R-chain homotopy equivalence;

Fix a non-negative integer n. Let m be the integer for which either n = 2m or n = 2m + 1. Next we give an algebraic construction which allows to assign to a (finitely generated projective symmetric) Poincaré pair (i_{∗} : C_{∗} → \overline{C}_{∗}, (δϕ, ϕ)) of n-dimensional R-chain complexes an element in L^{n}(R), provided that C_{∗} is chain homotopy equivalent to a R-chain complex with trivial m-th differential. In geometry this would correspond to assign to an inclusion i : ∂M → M of a manifold M with boundary ∂M together with a reference map r : M → X an element in Ω_{n}(X), where C_{∗} resp. \overline{C}_{∗} resp. i_{∗} plays the role of C_{∗}(∂M), C_{∗}(M) and C_{∗}(i). The idea would be to glue some preferred nullbordism to the boundary. This can be carried out in the more flexible algebraic setting under rather weak assumptions.

We begin with the case n = 2m. Recall that we assume that C_{∗} is chain homotopy equivalent to an R-chain complex D_{∗} such that d_{m} : D_{m} → D_{m−1} is trivial. Notice that we can arrange that D_{∗} is (2m − 1)-dimensional finitely generated projective by Lemma 2.1. Fix such a chain homotopy equivalence u_{∗} : C_{∗} → D_{∗}. Equip D_{∗} with the Poincaré structure ψ induced by ϕ on C_{∗} and u_{∗}. Define \overline{D_{∗}} as the quotient chain complex of D_{∗} for which \overline{D}_{i} = D_{i} if 0 ≤ i ≤ m − 1 and \overline{D}_{i} = 0 otherwise. Let j_{∗} : D_{∗} → \overline{D}_{∗} be the canonical projection. Notice that it is indeed a chain map since d_{m} vanishes. There is a canonical extension of the Poincaré structure ψ on D_{∗} to a Poincaré structure (δψ, ψ) on the pair j_{∗} : D_{∗} → \overline{D_{∗}}, namely, take δψ to be zero. Now we can glue the Poincaré pairs (i_{∗} : C_{∗} → \overline{C}_{∗}, (δϕ, ϕ)) and (j_{∗} : D_{∗} → \overline{D_{∗}}, (δψ, ψ)) along the R-chain homotopy equivalence u_{∗} : C_{∗} → D_{∗} [14, §3], [16, 1.7]. We obtain a 2m-dimensional Poincaré R-chain complex which presents a class in L^{2m}(R). Since chain homotopy equivalent Poincaré R-chain complexes define the same element in the (symmetric) L-groups, this class is independent of the choice of u_{∗} : C_{∗} → D_{∗}. We denote it by

σ(i_{∗} : C_{∗} → \overline{C}_{∗}, (δϕ, ϕ)) ∈ L^{2m}(R).  (2.2)

Notice that a chain homotopy equivalence u_{∗} : D_{∗} → E_{∗} of (2m−1)-dimensional chain complexes with trivial m-th differential induces a chain equivalence \overline{u}_{∗} : \overline{D}_{∗} → \overline{E}_{∗} such that u_{∗} and \overline{u}_{∗} are compatible with the maps D_{∗} → \overline{D}_{∗} and E_{∗} → \overline{E}_{∗} constructed above. Since chain homotopy equivalent Poincaré Rchain complexes define the same element in the (symmetric) L-groups, the class defined in (2.2) is independent of the choice of u_{∗} : C_{∗} → D_{∗}.

The proof of the next lemma is straightforward in the sense that one has to figure out the argument for the corresponding geometric statements, which is easy, and then to translate it into the algebraic setting (see also [16, Prop. 1.8.2 ii]).

Lemma 2.3 (a) Let (i_{∗} : C_{∗} → \overline{C}_{∗}, (δϕ, ϕ)) and (j_{∗} : D_{∗} → \overline{D}_{∗}, (δψ, ψ)) be 2m-dimensional (finitely generated projective symmetric) Poincaré pairs.

Let u_{∗} : C_{∗} → D_{∗} be a R-chain equivalence such that Q^{2m−1}(u_{∗}) : Q^{2m−1}(C_{∗}) → Q^{2m−1}(D_{∗}) maps ϕ to ψ. Denote by (E_{∗}, ν) the 2mdimensional Poincaŕe chain complex obtained from i_{∗} and j_{∗} by glueing along u_{∗}. Suppose that C_{∗} is R-chain homotopy equivalent to a R-chain complex with trivial m-th differential. Then we get in L^{2m}(R)

σ(E_{∗}, ν) = σ(i_{∗} : C_{∗} → \overline{C}_{∗}, (δϕ, ϕ)) − σ(j_{∗} : D_{∗} → \overline{D}_{∗}, (δψ, ψ));

(b) Let (i_{∗} : C_{∗} → \overline{C}_{∗}, (δϕ, ϕ)) and (j_{∗} : D_{∗} → \overline{D}_{∗}, (δψ, ψ)) be two 2mdimensional (finitely generated projective symmetric) Poincaré pairs. Let (\overline{f}_{∗}, f_{∗}) : i_{∗} → j_{∗} be a chain homotopy equivalence of pairs, i.e. R-chain homotopy equivalences \overline{f}_{∗} : \overline{C}_{∗} → \overline{D}_{∗} and f_{∗} : C_{∗} → D_{∗} with \overline{f}_{∗} ∘ i_{∗} = j_{∗} ∘ f_{∗} such that Q^{n}(\overline{f}_{∗}, f_{∗}) maps (δϕ, ϕ) to (δψ, ψ). Suppose that C_{∗} is R-chain homotopy equivalent to a R-chain complex with trivial m-th differential. Then we get in L^{2m}(R)

σ(i_{∗} : C_{∗} → \overline{C}_{∗}, (δϕ, ϕ)) = σ(j_{∗} : D_{∗} → \overline{D}_{∗}, (δψ, ψ)).

Now the invariant (0.2) is obtained from the invariant (2.2) applied to the Poincaré pair given by the associated chain complexes. Theorem 0.3 (a) and (c) follow from Lemma 2.3 (a) and (b). Theorem 0.3 (b) follows directly from Theorem 0.3 (a) and (c) because the right side of the formula appearing in Theorem 0.3 (a) does not involve the glueing diffeomorphism. Notice that the geometric version of Lemma 2.3 (a) has been considered in Lemma 1.9

Next we deal with the case n = 2m + 1. Recall that we are considering a (2m + 1)-dimensional finitely generated projective Poincaré R-pair (i_{∗} : C_{∗} → \overline{C}_{∗}, (δϕ, ϕ)) and that we assume that C_{∗} is R-chain homotopy equivalent to a R-chain complex D_{∗} with trivial m-th differential. Since C_{∗} ≃ C^{2m−∗} ≃ D^{2m−∗} holds by Poincaré duality, C_{∗} is also R-chain homotopy equivalent to a R-chain complex, namely, D^{2m−∗} whose (m + 1)-th differential is trivial. We conclude from Lemma 2.1 that we can fix a R-chain homotopy equivalence u_{∗} : C_{∗} → D_{∗} to a 2m-dimensional finitely generated projective R-chain complex D_{∗} such that both d_{m+1} and d_{m} vanish. This implies also that H_{m}(C_{∗}) ≅ H_{m}(D_{∗}) ≅ D_{m} is a finitely generated projective R-module and the Poincaré structure on C_{∗} induces the structure of a (−1)^{m}-symmetric (non-degenerate) form μ on H_{m}(C_{∗}). Recall that a (−1)^{m}-symmetric (non-degenerate) form (P, μ) consists of a finitely generated projective R-module P together with an isomorphism μ : P → P^{∗} such that the composition P ⟶^{≅} (P^{∗})^{∗} ⟶^{μ^{∗}} P of μ^{∗} with the canonical isomorphism P → (P^{∗})^{∗} is (−1)^{m} ·μ. The standard (−1)^{m}-symmetric hyperbolic form H(Q) for a finitely generated projective R-module Q is given by

( \begin{matrix}0 & 1 \\ (−1)^{m} & 0 \end{matrix} ) : H(Q) = Q^{∗} ⊕ Q → (Q^{∗} ⊕ Q)^{∗} = Q ⊕ Q^{∗}.

A Lagrangian for a (−1)^{m}-symmetric form (P, μ) is a direct summand L ⊂ P

with inclusion j : L → P such that the sequence 0 → L ⟶^{j} P ⟶^{j^{∗}∘μ} L^{∗} → 0 is exact. Any inclusion j : L → P of a Lagrangian extends to an isomorphism of (−1)^{m}-symmetric forms H(L) → (P, μ). A stable Lagrangian for (P, μ) is a Lagrangian in (P, μ)⊕H(Q) for some finitely generated projective R-module Q. A formation (P, μ, K, L) consists of a (−1)^{m}-symmetric (non-degenerate) form (P, μ) together with two Lagrangians K, L ⊂ P. A stable formation (P, μ, K, L) on (P, μ) is a formation on (P, μ) ⊕ H(Q) for some finitely generated projective R-module Q. For more informations about these notions we refer to [14, §2].

There are natural identifications of L^{0}(R, (−1)^{m}) with the Witt groups of equivalence classes of (−1)^{m}-symmetric forms and of L^{1}(R, (−1)^{m}) with the Witt group of equivalence classes of (−1)^{m}-symmetric formations [14, §5]. There are suspension maps L^{0}(R, (−1)^{m}) → L^{2m}(R) and L^{1}(R, (−1)^{m}) → L^{2m+1}(R). These suspension maps are in contrast to the quadratic L-groups not isomorphism for all rings with involutions, but they are bijective if R contains 1/2 [14, page 152]. The class of (C_{∗}, ϕ) vanishes in L^{2m}(R), an algebraic nullbordism is given by (i_{∗} : C_{∗} → \overline{C}_{∗}, (δϕ, ϕ)). Let u_{∗} : C_{∗} → D_{∗} be a R-chain homotopy equivalence to a 2m-dimensional finitely generated projective R-chain complex with trivial m-th and (m−1)-th differential. Equip D_{∗} with the Poincaré structure ψ induced by the given Poincaré structure ϕ on C_{∗} and u_{∗}. By doing surgery on the projection onto the quotient R-chain complex D_{∗}|_{m−1} whose ith chain module is D_{i} for i ≤ m − 1 and zero otherwise, in the sense of [14, §4], one sees that the class of (C_{∗}, ϕ) in L^{2m}(R) is the image under suspension of the element given by the (−1)^{m}-symmetric form on H_{m}(C_{∗}). If R contains 1/2, the suspension map is bijective. Hence the (−1)^{m}-symmetric (non-degenerate) form on H_{m}(C_{∗}) represents zero in the Witt group of equivalence classes of (−1)^{m}-symmetric forms. This shows

Lemma 2.4 Suppose that (M, r) satisfies Assumption 0.1 and that 1/2 ∈ R. Then there exists a stable Lagrangian L ⊂ H_{m}(C_{∗}(\overline{∂M}) ⊗_{ℤG} R).

Now suppose that we have fixed a stable Lagrangian L ⊂ H_{m}(C_{∗}). By adding the m-fold suspension of H(Q) for some finitely generated projective Rmodule Q to C_{∗}, we can arrange that L ⊂ H_{m}(C_{∗}) is a (unstable) Lagrangian. Equip D_{∗} with the Poincaré structure ψ induced by ϕ and u_{∗}. Let K ⊂ H_{m}(D_{∗}) be the Lagrangian given by L and H_{m}(u_{∗}). Let \overline{D}_{∗} be the quotient R-chain complex of D_{∗} such that \overline{D}_{i} = D_{i} for i ≤ m − 1, \overline{D}_{m} = K^{∗}, \overline{D}_{i} = 0 for i ≥ m + 1, the i-th differential is d_{i} : D_{i} → D_{i−1} for i ≤ m − 1 and all other differentials are zero. Let j_{∗} : D_{∗} → \overline{D}_{∗} be the R-chain map which is the identity in dimensions i ≤ m − 1 and given by the obvious composition D_{m} = H_{m}(D_{∗}) ⟶^{≅} H_{m}(D^{2m−∗}) = H_{m}(D_{∗})^{∗} → K^{∗}. There is a canonical extension of the Poincaré structure ψ on D_{∗} to a structure (δψ, ψ) of a Poincaré pair on j_{∗} : D_{∗} → \overline{D}_{∗}, namely, put δψ to be zero. Now we can glue the pairs i_{∗} : C_{∗} → \overline{C}_{∗} and j_{∗} : D_{∗} → \overline{D}_{∗} along u_{∗} to get a (2m+1)-dimensional Poincaré R-chain complex. Its class in L^{2m+1}(R) does not depend on the choice of Q, D_{∗} and u_{∗} and is denoted by

σ(i_{∗} : C_{∗} → \overline{C}_{∗}, (δϕ, ϕ), L) ∈ L^{2m+1}(R).  (2.5)

Again the proof of the next lemma is straightforward in the sense that one has to figure out the argument for the corresponding geometric statements, which is easy, and then to translate it into the algebraic setting (see also [16, Prop. 1.8.2 ii]).

Lemma 2.6 (a) Let (i_{∗} : C_{∗} → \overline{C}_{∗}, (δϕ, ϕ)) and (j_{∗} : D_{∗} → \overline{D}_{∗}, (δψ, ψ)) be (2m + 1)-dimensional (finitely generated projective symmetric) Poincaré pairs. Let u_{∗} : C_{∗} → D_{∗} be a R-chain equivalences such that Q^{2m}(u_{∗}) : Q^{2m}(C_{∗}) → Q^{2m}(D_{∗}) maps ϕ to ψ. Suppose that C_{∗} is R-chain homotopy equivalent to a R-chain complex with trivial m-th differential. Let K ⊂ H_{m}(C_{∗}) and L ⊂ H_{m}(D_{∗}) be stable Lagrangians such that H_{m}(u_{∗}) : H_{m}(C_{∗}) → H_{m}(D_{∗}) respects them stably. Let (E_{∗}, ν) be the 2m-dimensional Poincaŕe chain complex obtained from i_{∗} and j_{∗} by glueing along u_{∗}. Then we get in L^{2m}(R)

σ(E_{∗}, ν) = σ(i_{∗} : C_{∗} → \overline{C}_{∗}, (δϕ, ϕ), K) − σ(j_{∗} : D_{∗} → \overline{D}_{∗}, (δψ, ψ), L);

(b) Let (i_{∗} : C_{∗} → \overline{C}_{∗}, (δϕ, ϕ)) and (j_{∗} : D_{∗} → \overline{D}_{∗}, (δψ, ψ)) be two (2m + 1)- dimensional (finitely generated projective symmetric) Poincaré pairs. Let (\overline{f}_{∗}, f_{∗}) : i_{∗} → j_{∗} be a chain homotopy equivalence of pairs, i.e. Rchain homotopy equivalences \overline{f}_{∗} : \overline{C}_{∗} → \overline{D}_{∗} and f_{∗} : C_{∗} → D_{∗} with \overline{f}_{∗} ∘ i_{∗} = j_{∗} ∘ f_{∗} such that Q^{2m+1}(\overline{f}_{∗}, f_{∗}) maps (δϕ, ϕ) to (δψ, ψ). Suppose that C_{∗} is R-chain homotopy equivalent to a R-chain complex with trivial m-th differential. Let K ⊂ H_{m}(C_{∗}) and L ⊂ H_{m}(D_{∗}) be stable Lagrangians. Denote by K′ ⊂ H_{m}(D_{∗}) the image of K under H_{m}(f_{∗}). Then we obtain a stable equivalence class of formations (H_{m}(D_{∗}), ν, K′, L). Let [H_{m}(D_{∗}), ν, K′, L] ∈ L^{2m+1}(R) be the image of the element which is represented in the Witt group of equivalence classes of formations under the suspension homomorphism. Then we get in L^{2m+1}(R)

σ(i_{∗} : C_{∗} → \overline{C}_{∗}, (δϕ, ϕ), K)−σ(j_{∗} : D_{∗} → \overline{D}_{∗}, (δψ, ψ), L) = [H_{m}(D_{∗}), ν, K′, L].

Now the invariant (0.4) is obtained from the invariant (2.5) applied to the Poincaré pair given by the associated chain complexes. Theorem 0.5 follows from Lemma 2.6.

The next example shall illustrate that the choice of the homotopies h_{M} and h_{N} in Theorem 0.5 (b) and of the homotopy h in Theorem 0.5 (c) do affect the terms given by the formations. We are grateful to Michel Hilsum who pointed out to us that in an earlier version we did not make this point clear enough.

Example 2.7 Put R = ℤ[ℤ]. Consider (D^{1}, S^{0}) with the following two different reference maps c, e : D^{1} = [−1, 1] → Bℤ = S^{1}, namely c(s) = exp(0) and e(s) = exp(πi(s + 1)). Let h : D^{1} × [0, 1] → Bℤ = S^{1} be the homotopy e ≃ t sending (s, t) to exp(πit(s+1)). Notice that c^{∗}Eℤ|_{S^{0}} and t^{∗}Eℤ|_{S^{0}} agree and that we can choose therefore for both the same Lagrangian L ⊂ H_{0}(\overline{S^{0}}). Obviously Assumption 0.1 is satisfied. We want to show that σ(\overline{D^{1}}, t, L) and σ(\overline{D^{1}}, e, L) are not the same elements in L^{0}(ℤ[ℤ]). Their difference σ(\overline{D^{1}}, t, L)−σ(\overline{D^{1}}, e, L) is given by the class of the formation [H_{m}(C_{∗}(\overline{S^{0}})), μ_{1}, L′_{0}, L_{1}]. From Theorem 0.5 (c) applied to (f, ∂f) = id : (D^{1}, S^{0}) → (D^{1}, S^{0}) and h, we can identify

the form (H_{0}(S^{0}), μ) with ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) : ℤ[ℤ] ⊕ ℤ[ℤ] → ℤ[ℤ] ⊕ ℤ[ℤ] and choose

L = {(x, x) | x ∈ ℤ[ℤ]} ⊂ ℤ[ℤ] ⊕ ℤ[ℤ]. The homotopy h and the identity on S^{0} induce the ℤ-automorphism of c^{∗}Eℤ|_{S^{0}} = t^{∗}Eℤ|_{S^{0}} = S^{0} × ℤ which is the identity on {−1}×ℤ and multiplication with t on {1}×ℤ. The automorphism of

the form (H_{0}(S^{0}), μ) induced by ∂f = id and h|_{S^{0}} is ( \begin{matrix}1 & 0 \\ 0 & t \end{matrix} ) : ℤ[ℤ] ⊕ ℤ[ℤ] →

ℤ[ℤ] ⊕ ℤ[ℤ]. Hence the difference σ(\overline{D^{1}}, t, L) − σ(\overline{D^{1}}, e, L) is represented by the

formation (ℤ[ℤ]⊕ℤ[ℤ], ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) , L′, L) for L′ = \{(x, tx) | x ∈ ℤ[ℤ]\} ⊂ ℤ[ℤ]⊕

ℤ[ℤ]. But the class of this formation under the isomorphism L^{0}(ℤ[ℤ])[1/2] ≅ L^{0}(ℤ)[1/2] ⊕ L^{1}(ℤ)[1/2] ≅ ℤ[1/2] ⊕ 0 = ℤ[1/2] is the generator.

Similarily one can see from Theorem 0.5 (b) that σ(S^{1}, id) and σ(S^{1}, c) are different for the reference maps id = S^{1} → S^{1} = Bℤ and the constant map c : S^{1} → Bℤ by cutting S^{1} open along the embedded S^{0} ⊂ S^{1}. One has to choose homotopies h_{+} : i_{+} ≃ c : S_{+}^{1} → Bℤ and h_{−} : i_{−} ≃ c : S_{+}^{1} → Bℤ for S_{±}^{1} the upper and lower hemispheres and i_{±} : S_{±}^{1} → Bℤ = S^{1} the inclusion. Then the term describing σ(\overline{S_{+}^{1} ∪_{id} S_{−}^{1}}, id) − σ(\overline{S_{+}^{1} ∪_{id} S_{−}^{1}}, c) is given again by a formation which does not represent zero in L^{1}(ℤ[ℤ]). By crossing with ℂℙ^{2n} one gets also examples in dimensions 4n + 1 of this type because crossing with ℂℙ^{2n} induces an isomorphism L^{1}(ℤ[ℤ])[1/2] → L^{4n+1}(ℤ[ℤ])[1/2].

The next lemma is the algebraic version of Lemma 1.5 (see also [16, Prop. 1.8.2 ii]).

Lemma 2.8 Let n be any positive integer. Let (i_{∗} : C_{∗} → \overline{C}_{∗}, (δϕ, ϕ)) and (j_{∗} : D_{∗} → \overline{D}_{∗}, (δψ, ψ)) be two n-dimensional (finitely generated projective symmetric) Poincaré pairs. Let u_{∗}, v_{∗} : C_{∗} → D_{∗} be a R-chain equivalences such that both Q^{n}(u_{∗}) and Q^{n}(v_{∗}) map (δϕ, ϕ) to (δψ, ψ). Let w_{∗} : C_{∗} → C_{∗} be a Rchain map with u_{∗} ∘ w_{∗} ≃ v_{∗}. Let (E_{∗}(u_{∗}), (δν, ν)(u_{∗})) and (E_{∗}(v_{∗}), (δν, ν)(v_{∗})) respectively be the n-dimensional Poincaŕe chain complexes obtained from i_{∗} and j_{∗} by glueing along u_{∗} and v_{∗} respectively. Let (T (w_{∗}), μ) be the algebraic mapping torus of w_{∗}. Its underlying R-chain complex is the mapping cone of cone(id −w_{∗}) (cf. [17, page 264]). Then we get in L^{n}(R)

σ(E_{∗}(u_{∗}), (δν, ν)(u_{∗})) − σ(E_{∗}(v_{∗}), (δν, ν)(v_{∗})) = σ(T (w_{∗}), μ).

In general symmetric signatures and higher signatures are not additive (see Example 1.10). In the situation of Lemma 1.5 the difference of symmetric signatures (and thus of higher signatures) is measured by the symmetric signature of the corresponding mapping torus. If we want to see the difference in L^{n}(C_{r}^{∗}(G)), we only have to consider the algebraic mapping torus as explained in Lemma 2.8. To detect the image of the class of the mapping torus in L^{n}(C_{r}^{∗}(G)) under the isomorphism sign : L^{n}(C_{r}^{∗}(G)) → K_{0}(C_{r}^{∗}(G)) the formula [14, Proposition 4.3] is useful. It reduces the computation of the difference of the element [r : M ∪_{ϕ} N^{−} → X] − [s : M ∪_{ψ} N^{−} → X] under the composition Ω_{n}(BG) ⟶^{D} K_{n}(BG) ⟶^{A} K_{n}(C_{r}^{∗}(G)) to an expression which only involves the chain complex of C_{∗}(\overline{∂M}) and the map induced by the automorphism ϕ^{−1} ∘ ψ in a rather close range around the middle dimension.

Remark 2.9 Let Z be a closed oriented n-dimensional manifold with a reference map r : Z → BG. Suppose that we have for the m-th Novikov-Shubin invariant α_{m}(\overline{Z}) = ∞^{+} in the case n = 2m−1 and, in the case n = 2m, we have α_{m}(\overline{Z}) = ∞^{+} and for the m-th L^{2}-Betti number b_{m}^{(2)}(\overline{Z}) = 0. Then we conclude from the arguments above and Lemma 3.1 that σ : Ω_{n}(BG) → L^{n}(C_{r}^{∗}(G; ℝ)) maps [Z, r] to zero. Namely, we have constructed an explicit algebraic nullbordism above. Hence we conclude that all higher signatures of [Z, r] vanish if the assembly map A_{ℝ} : KO_{n}(BG)⊗_{ℤ}ℚ → KO_{n}(C_{r}^{∗}(G; ℝ)) is injective. This follows from the discussion in the introduction.

3 Novikov-Shubin invariants

Next we reformulate (following [7]) the condition that the middle differential vanishes in terms of spectral invariants.

Let N (G) be the von Neumann algebra associated to G. Let M be a closed Riemannian manifold with normal covering \overline{M} → M with deck transformation group G. Let ν be the flat C_{r}^{∗}(G)-bundle over M whose total space is \overline{M} ×_{G} C_{r}^{∗}(G). Let H^{m}(M; ν) and \overline{H}^{m}(M; ν) resp. be the unreduced and reduced mth cohomology of M, i.e. ker(d^{m})/ im(d^{m−1}) and ker(d^{m})/\overline{im(d^{m−1})} resp. for d the differential in the deRham complex Ω^{∗}(M; ν) of Hilbert C_{r}^{∗}(G)-modules. The next lemma is contained in Lemmas 2.1 and 2.3 of [7]. Lemma 3.1 The following assertions are equivalent for an integer m.

(a) The canonical projection H^{m}(\overline{M}; ν) → \overline{H}^{m}(\overline{M}; ν) is bijective;

(b) The C_{r}^{∗}(G)-chain complex C_{∗}(\overline{M}) ⊗_{ℤG} C_{r}^{∗}(G) is C_{r}^{∗}(G)-chain homotopy equivalent to a finitely generated projective C_{r}^{∗}(G)-chain complex D_{∗} whose m-th differential d_{m} : D_{m} → D_{m−1} is trivial;

(c) The N (G)-chain complex C_{∗}(\overline{M}) ⊗_{ℤG} N (G) is N (G)-chain homotopy equivalent to a finitely generated projective N (G)-chain complex D_{∗} whose m-th differential d_{m} : D_{m} → D_{m−1} is trivial;

(d) The Novikov Shubin invariant α_{m}(\overline{M}) is ∞^{+} (see [10]);

(e) The Laplacian acting on L^{2}(\overline{M}, Ω^{m−1})/ ker(d^{m−1}) has a strictly positive spectrum.

\underline{\underline{Proof}} : (a) ⇔ (b) We can interprete the (a priori purely algebraic) C_{r}^{∗}(G)- cochain complex hom_{ℤG}(C_{∗}(\overline{M}), C_{r}^{∗}(G) as cochain complexes of Hilbert C_{r}^{∗}(G)- chain complexes with adjointable morphisms as differentials by the identification of each cochain module with the direct sum of finitely many copies of C_{∗}^{r}(G) using cellular ℤG-basis. There is a C_{r}^{∗}(G)-chain homotopy equivalence (by bounded chain maps and homotopies) Ω^{∗}(M; ν) → hom_{ℤG}(C_{∗}(\overline{M}), C_{r}^{∗}(G)). Hence the image of the (m − 1)-th differential in Ω^{∗}(M; ν) is closed if and only if the same is true for the one in hom_{ℤG}(C_{∗}(\overline{M}), C_{r}^{∗}(G)). The image of a differential in hom_{ℤG}(C_{∗}(\overline{M}), C_{r}^{∗}(G)) is closed if and only if the image is a direct summand in the purely algebraic sense [19, Corollary 15.3.9]. But this is equivalent to the assertion that hom_{ℤG}(C_{∗}(\overline{M}), C_{r}^{∗}(G)) is C_{r}^{∗}(G)-chain homotopy equivalent to a finitely generated projective C_{r}^{∗}(G)-cochain complex whose (m − 1)-th codifferential is trivial by Lemma 2.1. This is true if and only if C_{∗}(\overline{M}) ⊗_{ℤG} C_{r}^{∗}(G) is C_{r}^{∗}(G)-chain homotopy equivalent to C_{r}^{∗}(G)-chain complex with trivial m-th differential.

(b) ⇒ (c) is obvious.

(c) ⇔ (d) follows directly from the interpretation of Novikov-Shubin invariants in terms of the homology of C_{∗}(\overline{M}) ⊗_{ℤG} N (G) [11].

(d) ⇔ (e) follows from the fact that the dilatational equivalence class of the spectral density function of the simplicial m-th codifferential and the analytic m-th codifferential agree. [2].

(e) ⇔ (a) Assertion (a) can be reformulated to the statement that the spectrum of (d^{m−1})^{∗}d^{m−1} for d^{m−1} (m − 1)-th differential in Ω^{∗}(M; ν) has a gap at zero. But this spectrum is the same as the spectrum of (d^{m−1})^{∗}d^{m−1} for d^{m−1} the (m−1)-th differential in the deRham complex L^{2}Ω^{∗}(\overline{M}) of Hilbert spaces which has a gap at zero if and only if (e) is true.

References

[1] Baum, P., Connes, A. and Higson, N.: “Classifying space for proper actions and K−theory of group C^{∗}−algebras”, Contemp. Math 167, 241– 291, (1994).

[2] Gromov, M. and Shubin, M.A.: “Von Neumann spectra near zero”, Geometric and Functional Analysis 1, 375–404,(1991).

[3] Higson, N. and Kasparov, G.: “Operator K-theory for groups which act properly and isometrically on Hilbert space”, preprint (1997).

[4] Karras, U., Kreck, M., Neumann, W. and Ossa, E.: Cutting and pasting of manifolds; SK-groups, Publish or Perish, Boston (1973)

[5] Kasparov, G.: “Novikov’s conjecture on higher signatures: The operator K-theory approach” , Contemporay Math. 145, 79–99(1993).

[6] Landweber, P.S.: “Homological properties of comodules over MU_{∗}MU and BP_{∗}BP ”, Amer. J. Math. 98, 591 – 610 (1976).

[7] Leichtnam, E., Lott, J. and Piazza, P.: “On the homotopy invariance of higher signatures for manifolds with boundary”, preprint (1999).

[8] Lott. J.: “Higher eta invariants, K-theory 6, 191–223 (1992).

[9] Lott, J.: “Signatures and higher signatures of S^{1}−quotients”, Math. Annalen to appear (1999).

[10] Lott, J. and Lück, W.: “L^{2}-topological invariants of 3-manifolds”, Inventiones Math. 120, 15–60,(1995).

[11] Lück, W.: “Hilbert modules and modules over finite von Neumann algebras and applications to L^{2}-invariants”, Math. Annalen 309, 247–285,(1997).

[12] Lück, W.: “L^{2}-invariants of regular coverings of compact manifolds and CW -complexes”, to appear in handbook of geometry“,editors: Davermann, R.J. and Sher, R.B., Elsevier.

[13] Quinn, F.: “Open book decompositions, and the bordism of automorphisms”, Topology 18, 55–73,(1979).

[14] Ranicki, A.: “The algebraic theory of surgery I: Foundations”, Proc. of LMS 40, 87–192,(1980).

[15] Ranicki, A.: “The algebraic theory of surgery II: Applications to topology”, Proc. of LMS 40, 193–287,(1980).

[16] Ranicki, A.: “Exact sequences in the algebraic theory of surgery”, Princeton University Press (1981).

[17] Ranicki, A.: “High-dimensional knot theory – Algebraic surgery in codimension 2”, Springer Monographs in Mathematics, Springer (1998).

[18] Rosenberg, J.: “Analytic Novikov for topologists”, in: Proceedings of the conference “Novikov conjectures, index theorems and rigidity” volume I, Oberwolfach 1993 , LMS Lecture Notes Series 226, 338–372, Cambridge University Press (1995).

[19] Wegge-Olsen, N.: “K-theory and C^{∗}-algebras – a friendly approach”, Oxford University Press (1993).

[20] Weinberger, S.: “Higher ρ-invariants”, in “Tel Aviv Topology conference, Rothenberg Festschrift”, editors: M. Farber, W. Lück and S. Weinberger, Contemporary Mathematics 231. (1999).

[21] Yu. G.:“The coarse Baum-Connes conjecture for spaces which admit a uniform embedding into Hilbert space”, Invent. Math 139, No 1, 201–240, (2000).

Appendix: Mapping tori of special diffeomorphisms by Matthias Kreck

In this appendix we consider the image of the bordism group of diffeomorphisms on smooth manifolds over a CW -complex X with finite skeleta under the mapping torus construction. By a diffeomorphism over X we mean a quadruple (M, f, g, h), where M is a closed oriented smooth manifold, g an orientation preserving diffeomorphism on M, f : M → X a continuous map and h a homotopy between f and f ∘ g. The role of the homotopy h becomes clear if we consider the mapping torus M_{g} := M × [0, 1]/_{(g(x),0)∼(x,1)}, which is by projection to the second factor a smooth fibre bundle over S^{1}. Then h allows an extension of f on the fibre over 0 to a map h̄([(x, t)]) := h(x, t) and any such extension gives a homotopy h with the properties above.

Following [K1] we denote the bordism group of these quadruples by Δ_{n}(X). Let Ω_{n+1}(X) be the bordism group of oriented smooth manifolds with reference map to X. The mapping torus construction above gives a homomorphism Δ_{n}(X) → Ω_{n+1}(X). It was shown in [K1] for X simply connected and in [Q1] for general X that for n even this map is surjective. Recently Wolfgang Lück and Eric Leichtnam asked whether the same statement holds if we only allow quadruples where the map f is 2-connected and (M, f) represents zero in Ω_{n}(X). We call such a quadruple a special diffeomorphism over X and the subset of Δ_{n}(X) represented by special diffeorphisms by SΔ_{n}(X) (it is not clear to the author whether this subset forms a subgroup). For X simply connected one can conclude from [K1, §9 ], that SΔ_{2n}(X) → Ω_{2n+1}(X) is surjective. In this note we generalize this to arbitrary complexes X.

Theorem: Let X be a CW -complex with finite skeleta. For n ≥ 2 the mapping torus construction gives a surjection SΔ_{2n}(X) → Ω_{2n+1}(X).

Proof: Let (N, g) be an element of Ω_{2n+1}(X). Consider a representative (M, r) of 0 in Ω_{2n}(X). We use the language and results from [K2]. We consider the fibration p_{2} : X × BSO → BSO and denote it by B. The map r × ν : M → B, where ν is the normal Gauss map, is a normal B-structure. By [K2, Corollary 1] we can replace (M, r × ν) up to bordism by a n-equivalence r′ × ν′ : M′ → X × BSO giving a normal (n − 1)-structure on M′. In particular r′ : M′ → X is 2-connected.

Now we form the disjoint union (M′ × I) + N and consider the map q : (M′ × I) + N → X given by r′p_{1} and g, where p_{1} : M′ × I → M′ is the projection. We want to replace this manifold by a manifold W diffeomorphic to M × I which is bordant relative boundary over X to (M′ × I) + N. If this is possible we are finished since then we glue the two boundary components of W and the maps together to obtain a mapping torus and a map to X. This is bordant over X to (N, g) since it is bordant to ((M′ × S^{1}) + N, r′p_{1} + g) (note that (M′ × S^{1}, r′p_{1}) is zero bordant over X).

This idea does not work directly. What we will prove is that there is a bordism W between M′#m(S^{n}×S^{n}) and M′#m(S^{n}×S^{n}) for some m equipped with a map to X which on the two boundary components is the composition of the projection from M#m(S^{n} × S^{n}) to M and r′, such that W is diffeomorphic to (M′#m(S^{n} × S^{n})) × I. We further achieve that the manifold obtained by glueing the boundary components of W together is over X bordant to (N, g). This is by the considerations above enough to prove the theorem, since our map from M#m(S^{n} × S^{n}) to X × BSO is again a n-equivalence.

That this indirect way works follows from [K2, Theorem 2], which says that we can replace (M′ × I) + N by a sequence of surgeries over X × BSO and compatible subtractions of tori by an s-cobordism W between M′#m(S^{n} × S^{n}) and M′#m(S^{n} × S^{n}) (the fact that the number of S^{n} × S^{n}’s one has to add by Theorem 2 to the boundary components of W is equal follows from the equality of the Euler characteristic of the two boundary components). If n > 2 the s-cobordism theorem implies W diffeomorphic to (M′#m(S^{n} × S^{n})) × I. If n = 2 the same is true by the stable s-cobordism theorem of [Q2] after further stabilization of W by forming k times a ”connected sum” between (S^{2} × S^{2}) × I and W along an embeded arc joining the two boundary components of W. To finish the argument one has to note from the definition of compatible subtraction of tori that this process does not affect the bordism class over X for the manifold obtained by glueing the two boundary components together.

To see this we recall the definition of subtraction of tori. Consider two disjoint embeddings of S^{n} × D^{n+1} into W such that the map to X is constant on both S^{n} × 0’s. Join each of these embedded tori by an embedded I × D^{2n} with the two boundary components and subtract the interior of these embedded submanifolds to obtain W′. This is the subtraction of a pair of tori used in [K2, Theorem 2]. The boundary of W consists of two copies of M#(S^{n} × S^{n}). There is an obvious bordism over X between the manifold obtained from W by identifying the two boundary components and the manifold obtained from W′ by identifying the two boundary components.

Remark: In general it is difficult to say much about a special diffeomorphism whose mapping torus is bordant to a given pair (N, g). The main difficulty is the determination of the diffeomorphism. One can obtain some information on M′. For example if X = S^{1} and n = 2 the proof above shows that we can take for M′ the following manifold: S^{1} × S^{3}#ℂℙ^{2}#ℂ̄ℙ^{2} and thus the special diffeomorphism lives on S^{1} × S^{3}#ℂℙ^{2}#ℂ̄ℙ^{2}#m(S^{2} × S^{2}) for some unknown integer m. More generally in dimension 4 for an arbitrary X one can use instead of S^{1} × S^{3} the boundary of any thickening of the 2-skeleton of X in ℝ^{5}.

References

[K1] Kreck, M.: “Bordism of diffeomorphisms and related topics, Springer Lecture Notes 1069 (1984).

[K2] Kreck, M.: “Surgery and duality”, Annals of Math. 149, 707–754 (1999).

[Q1] Quinn,F: “Open book decompositions, and the bordism of automorphisms”, Topology 18, 55–73 (1979).

[Q2] Quinn, F.: “The stable topology of 4-manifolds”, Top. and its Appl. 15, 71–77 (1983).