Reidemeister torsion and the K-theory of von Neumann algebras by Wolfgang Lück and Mel Rothenberg

0. Introduction

The purpose of this paper is to introduce and study a new topological type invariant for cocompact properly discontinous actions of discrete groups of isometries on Riemannian manifolds. This work is inspired in part by the work of Carey and Mathai [4] and our invariant is a generalization of theirs. It is however much more refined and has the advantage of encompasing other powerful invariants such as the Alexander polynomial and the equivariant Reidemeister torsions studied in Lott-Rothenberg [18] and Lück [20], which are themselves useful generalizations of the classical notions.

Let M be a Riemannian manifold and Γ a discrete properly discontinuous group of isometries with Γ\M compact. Properly discontinuous means that for each pair of points (x, y) in M there are neighborhoods U_{x} and U_{y} such that {γ ∈ Γ | γU_{x} ∩ U_{y} ≠ ∅} is finite. An important case to keep in mind is the following. Let p : M ⟶ N be a locally isometric Galois covering over a compact Riemannian manifold N with group of deck transformations π. One may choose Γ to be π. More generally, one can lift an action of a finite group G on N by isometries to an action of a group Γ, such that Γ is an extension of π and G and the Γ-action extends the π-action and covers the G-action.

Let A be a finite von Neumann algebra and V be a finitely generated Hilbert module over A. Consider a unitary representation μ : Γ ⟶ Iso_{A}(V)^{op}. Let K_{1}^{w}(A) be the K-theory of weak automorphisms of finitely generated Hilbert A-modules with the involution given by taking adjoints. We define the Reidemeister von Neumann torsion

ρ(M; V) ∈ K_{1}^{w}(A)^{Z/2}

in 5.7. Some of the main properties of this invariant and the relevant K-theory are listed below :

• The Reidemeister von Neumann torsion ρ(M) is an invariant of the Γ-isometry class of M. If the l^{2}-homology H_{∗}(M; V) vanishes, ρ(M; V) depends only on the simple Γ-homotopy type of M (see theorem 3.11). We remark that in a lot of interesting cases H_{∗}(M; V) indeed vanishes, e.g., the universal covering of a closed hyperbolic manifold M, the universal covering of a compact manifold M admitting a fixed point free S^{1}- action such that the inclusion of one (and hence all orbits) induces an injection on the fundamental groups and the universal covering of a prime Haken 3-manifold provided in all cases that V is l^{2}(π_{1}(M)). The first example follows from Dodziuk [11] and [12], the second statement is proved in theorem 3.20 and the last statement will appear in a forthcoming preprint. Roughly speaking, the image of Whitehead torsion under a change of rings homomorphism is the difference of Reidemeister von Neumann torsion if H_{∗}(M; V) vanishes.

• Let M ∪_{X} N be obtained from M and N by glueing along a common union of components of the boundary. Then ρ(M ∪_{X} N; V) − ρ(M; V) − ρ(N; V) + ρ(X; V) is given by the long Mayer Vietoris homology sequence, more precisely, by its torsion (see theorem 3.14) and is in particular zero if all l^{2}-homology groups vanish.

• There is a product formula ρ(M × N; V ⊗ W) = χ^{A}(M) · [V] ⊗ ρ(N; W) + χ^{B}(N) · ρ(M; V) ⊗ [W], where the integers χ^{A}(M) and χ^{B}(N) are Euler characteristic type invariants (see theorem 3.16).

• The Reidemeister von Neumann torsion is compatible with restriction to subgroups of finite index (see lemma 3.17).

• The Reidemeister von Neumann torsion ρ(M) satisfies Poincaré duality (see theorem 5.13). If Γ acts freely this means ρ(M; V) = (−1)^{1+dim(M)} · ρ(M, ∂M; V).

• Let N(Γ) be the von Neumann algebra of the group Γ. The K-group K_{1}^{w}(N(Γ))^{Z/2} is the group of almost everywhere invertible functions from the r-dimensional torus T^{r} to R if Γ is Z^{r}, and reduces to the ordinary K-group of the complex group ring K_{1}(C[Γ])^{Z/2}, if Γ is finite (see theorem 2.5).

We illustrate these computational tools by proving the following corollary 3.21. Let G be a connected compact Lie group and p : X ⟶ Y be a G-principal bundle of finite CW - complexes such that the image of π_{1}(G) ⟶ π_{1}(X) is infinite. Then X̃ is l^{2}(π_{1}(X))-acyclic, where X̃ denotes the universal covering of X. If G is S^{1}, then ρ(X̃; l^{2}(π_{1}(X))) is given by χ(Y) · [[(w − 1) : l^{2}(π_{1}(X)) ⟶ l^{2}(π_{1}(X))]] in K_{1}^{w}(N(π_{1}(X)))^{Z/2} where N(π_{1}(X)) is the von Neumann algebra of the fundamental group and w ∈ π_{1}(X) is given by an S^{1}-orbit. If G is not S^{1}, then ρ(X; l^{2}(π_{1}(X))) vanishes. Our computations apply to Seifert 3-manifolds (see remark 3.22).

For a finite-dimensional representation V the equivariant Reidemeister von Neumann torsion reduces to the P L-torsion invariants defined in Lott-Rothenberg [18] and Lück [20]. In particular one obtains for free actions the P L-torsion of Ray and Singer [23]. If ∂M is empty, its logarithm is the analytic torsion as shown independently by Cheeger [5] and Müller [22]. This result is extended to the equivariant case and the case with boundary in Lott-Rothenberg [18] and Lück [20]. We will relate Reidemeister von Neumann torsion to the Alexander polynomial of a link (see example 4.7) and to the Lefschetz zeta function of an endomorphism of a finite CW -complex (see example 4.8).

The complex group ring C[Γ] of a group Γ is semisimple if and only if Γ is finite. The semisimplicity is crucial for the definition of classical Reidemeister torsion. If one completes C[Γ] to l^{2}(Γ), one is lead to the theory of Hilbert N(Γ)-modules as established in Atiyah [1], Cheeger and Gromov [6], [7], [8] and Dixmier [10]. The proof that C[Γ] is semisimple for finite Γ is based on the existence of a Hilbert structure. Hence one obtains semisimplicity also for infinite Γ for Hilbert N(Γ)-modules. Now our definition of Reidemeister von Neumann torsion follows the standard pattern. The main technical difficulty comes from the fact that one has to define homology as the quotient of the kernel and the closure of the image of the relevant differentials, so that the vanishing of homology does not imply contractibility. In particular one has to deal in the K-theory instead of isomorphisms with weak isomorphisms, i.e. morphisms with trivial kernel and dense image. Therefore a lot of the material for Hilbert A-modules of section 6 and 7 is essentially standard, but the proofs are different and harder than in the case of modules over the complex numbers.

Here is a short survey of the construction of Reidemeister von Neumann torsion. The relevant K-group K_{1}^{w}(A) has weak automorphisms f : M ⟶ M of finitely generated Hilbert A-modules as generators [f] and the relations are [g ∘ f] = [g] + [f] , [id] = 0 and [f] + [h] = [g], if there is a exact sequence 0 ⟶ (M, f) ⟶ (N, g) ⟶ (P, h) ⟶ 0. Let M be a compact smooth manifold with fundamental group π and μ : π ⟶ Iso_{A}(V)^{op} be a unitary representation. We give the definition of ρ(M; V) in the case that H_{∗}(M; V) is trivial ( for all ∗ ≥ 0). Let C(M; V) be the Hilbert A-chain complex V ⊗_{Z[Γ]} C(M) and Δ_{p} : C_{p}(M; V) ⟶ C_{p}(M; V) be the associated Laplace operator d_{p+1} ∘d_{p+1}^{∗} +d_{p}^{∗} ∘d_{p} where d is the differential of C(M; V). As H_{p}(M; V) vanishes, Δ_{p} is a weak automorphism of finitely generated Hilbert A-modules by the Hodge decomposition theorem. The Reidemeister von Neumann torsion ρ(M; V) ∈ K_{1}^{w}(A)^{Z/2} is given by ∑_{p≥0}(−1)^{p} · p · [Δ_{p}]. We will introduce another definition using weak chain contractions which has some advantages for technical conceptual reasons and is closer to the classical definitions (see definition 7.10). Moreover, it allows in the acyclic case the definition of a refined invariant, the acyclic Reidemeister von Neumann torsion (see 3.24). Both definitions will be identified in lemma 7.12. If H(M; V) is not trivial, one needs a Riemannian metric on M. We remark that H_{∗}(M; l^{2}(Γ)) has a good chance to be trivial (for all ∗ ≥ 0), whereas H_{0}(M; C) never vanishes for non-empty M.

In this article we develop the foundations of Reidemeister von Neumann torsion in the sense that we give its definition, collect its main properties, give computational tools, relate it to known invariants and analyse some examples. The following problems seem to be the natural continuation of these investigations and their relevance and meaning is evident from the discussion above. We will deal with them in forthcoming papers where the results of this article will be applied to these problems.

• Compute the K-groups K_{1}^{w}(N(Γ)) also for non-abelian groups and find detecting homomorphisms into known groups.

• What is the analytic interpretation of the Reidemeister von Neumann torsion in terms of the spectral theory of the Laplace operator?

• Compute Reidemeister von Neumann torsion for certain classes of manifolds (crystallographic manifolds, hyperbolic manifolds, 3-manifolds) and investigate how sharp it is.

If one wants to get a quick survey about the results, one may skip the first two sections and read sections 3 and 4. To get a sufficient impression from section 1 and 2 it suffices to consider von Neumann algebras of a group as explained in example 1.9 and read in section 2 until Theorem 2.5 only assuming A = N(Z^{r}) and X = T^{r}. The material simplifies considerably if one assumes that the action of Γ is free what is true in a lot of interesting examples. Under this assumption one does not need the material about G-CW -complexes and permutation modules in the beginning of section 3 and may start with definition 3.6. In section 5 one can skip the equivariant triangulation theorem as it follows from the non-equivariant one in the case of a free action. Moreover, the definition of Poincaré von Neumann torsion becomes irrelevant in view of theorem 5.13 and the upshot of the discussion about Poincaré duality is that Reidemeister von Neumann torsion satisfies ρ(M; V) = (−1)^{1+dim(M)} · ρ(M, ∂M; V).

The paper is organized as follows :

1. Preliminaries about von Neumann algebras and Hilbert modules

2. Algebraic K-theory of von Neumann algebras

3. Torsion invariants for Γ-CW -complexes

4. Free abelian fundamental groups

5. Torsion invariants for Riemannian Γ-manifolds

6. Hilbert A-chain complexes

7. Torsion invariants for Hilbert A-chain complexes references

1. Preliminaries about von Neumann algebras and Hilbert modules

In this section we collect the basic properties about von Neumann algebras and Hilbert modules.

Let H be a Hilbert space with inner product ⟨ξ, η⟩ and induced norm |ξ | . A Hilbert space is always to be understood as a seperable complex Hilbert space. Let B(H) be the C^{∗}-algebra of bounded linear operators H ⟶ H. Recall that the operator norm is given by ∥f ∥ := sup{ |f(ξ)| | ξ ∈ H, |ξ | = 1} and the involution ∗ : B(H) ⟶ B(H) maps f to its adjoint f^{∗}. The norm topology is the vector space topology induced by the norm ∥f ∥ . The weak resp. strong topology on B(H) is the topology induced by the family of seminorms {p_{ξ,η} | ξ, η ∈ H} resp. {p_{ξ} | ξ ∈ H} defined by p_{ξ,η}(f) := ⟨f(ξ), η⟩ resp. by p_{ξ}(f) := |f(ξ)| . A subalgebra A of B(H) is a subset closed under addition, multiplication with scalars and multiplication and contains the unit of B(H). It is called selfadjoint , if it is closed under the involution.

Definition 1.1 A von Neumann algebra (in H) is a selfadjoint subalgebra A of B(H) which is closed in the weak topology.

The commutant of a subset M of B(H) is M′ := {f ∈ B(H) | fg = gf for all g ∈ M}. Obviously M ⊂ M′′ holds. The following theorem is due to von Neumann (see e.g. Sunder [28] page 12 for a proof).

Theorem 1.2 (Double commutant theorem) Let A be a selfadjoint subalgebra of B(H). Then the following assertions are equivalent :

1.) A = A′′

2.) A is weakly closed.

3.) A is strongly closed.

Given a von Neumann algebra A ⊂ B(H), let A^{+} be the cone of positive elements. Recall that f : H ⟶ H is called positive, if f is selfadjoint and ⟨f(ξ), ξ⟩ ≥ 0 holds for all ξ ∈ H. A map tr : A^{+} ⟶ [0, ∞] is a trace , if for a, b ∈ A^{+} and λ ∈ [0, ∞[ the following holds :

tr(a) + tr(b) = tr(a) + tr(b)  tr(λa) = λtr(a)  tr(aa^{∗}) = tr(a^{∗}a)

A trace tr is finite , if tr(a) < ∞ holds for all a ∈ A^{+}. It is faithful , if tr(a) = 0 implies a = 0 for all a ∈ A^{+}. We call tr normal if tr(a) = sup{tr(a_{i}) | i ∈ I} holds for any monotone increasing net {a_{i} | i ∈ I} in A^{+} with a as supremum. Let M be the ideal in A given by finite sums of products of elements a ∈ A satisfying tr(aa^{∗}) < ∞. Then tr extends uniquely to a C-linear form also denoted tr : M ⟶ C satisfying tr(ab) = tr(ba) for all a, b ∈ M. In particular a finite trace extends uniquely to a C-linear form tr : A ⟶ C.

Definition 1.3 A von Neumann algebra A is called finite , if it possesses a finite, normal and faithful trace.

In the sequel any von Neumann algebra A is assumed to be finite and comes with a finite normal and faithful trace tr : A ⟶ C. Define a pre-Hilbert structure on A by ⟨a, b⟩ = tr(b^{∗}a). Let l^{2}(A) be the Hilbert completion of (A, ⟨ , ⟩). Denote by |a| the induced norm on l^{2}(A). Given a ∈ A, we obtain a linear operator l(a) : A ⟶ A sending b to ab. This operator is bounded and has the operator norm ∥l(a)∥ = |a| . Hence it extends uniquely to a bounded operator \overline{l(a)} : l^{2}(A) ⟶ l^{2}(A) satisfying ∥\overline{l(a)}∥ = |a| . Thus we obtain a left A-module structure on l^{2}(A). Analogously we get r(a) : A ⟶ A sending b to ba and an induced operator \overline{r(a)} : l^{2}(A) ⟶ l^{2}(A). Notice that l(a) and r(b) and hence \overline{l(a)} and \overline{r(b)} commute for a, b ∈ A. In particular we obtain the right regular representation ν_{r} : A^{op} ⟶ B_{A}(l^{2}(A)) from the opposite algebra A^{op} of A into the subalgebra B_{A}(l^{2}(A)) of linear bounded A-operators of B(l^{2}(A)). Recall that the opposite algebra A^{op} is obtained from A by reversing the multiplication, i.e. ab in A^{op} is given by ba in A. The following result is fundamental for the theory of Hilbert modules over a finite von Neumann algebra (see Dixmier [10], page 80 theorem 1, page 99 theorem 2).

Theorem 1.4 Let A be a finite von Neumann algebra. Then the right regular representation

ν = ν_{r} : A^{op} ⟶ B_{A}(l^{2}(A))

is a bijection

Next we introduce the category of Hilbert A-modules over a finite von Neumann algebra A. A Hilbert A-module M is a Hilbert space M together with a continuous left A-module structure such that there exists an isometric linear embedding onto a closed subspace of l^{2}(A) ⊗ H for some Hilbert space H. The embedding is not part of the structure. A morphism between Hilbert A-modules f : M ⟶ N is a bounded linear operator compatible with the A-module structures. We get from the theorem 1.4 above a bijection of C-algebras.

1.5 Ω : hom_{A}(⊕_{i=1}^{n}l^{2}(A), ⊕_{i=1}^{n}l^{2}(A)) ⟶ M(n, n, A)^{op}

Hence we may think of ⊕_{i=1}^{n}l^{2}(A) as the free Hilbert A-module of rank n. A Hilbert A-module M is called finitely generated, if there is an epimorphism of Hilbert A-modules from ⊕_{i=1}^{n}l^{2}(A) onto M for some integer n ≥ 0.

Lemma 1.6 Any finitely generated Hilbert A-module M is projective in the following sense: there is a finitely generated Hilbert A-module N such that there exists an isometric isomorphism of Hilbert A-modules from M ⊕ N to ⊕_{i=1}^{n}l^{2}(A) for appropiate n.

\underline{\underline{Proof}} : By definition there is an epimorphism of Hilbert A-modules f from ⊕_{i=1}^{n}l^{2}(A) to M for appropiate n. It induces a bijective morphism of Hilbert A-modules between ker(f)^{⊥} and M. This is an isomorphism of Hilbert A-modules by the open mapping theorem. The unitary part in its polar decomposition is an isometric isomorphism of Hilbert A-modules from ker(f)^{⊥} to M. Now the claim follows from the orthogonal decomposition of ⊕_{i=1}^{n}l^{2}(A) into ker(f) ⊕ ker(f)^{⊥}.

This lemma 1.6 will enables us to carry over the notion of Reidemeister torsion invariants for finite transformation groups to infinite transformation groups. The construction of Reidemeister torsion for finite groups G is based on the algebraic fact that the group ring C[G] is semi-simple if and only if G is finite. Hence the construction does not go through in the infinite case. However, if one extends the group ring to its Hilbert completion and hence deals with the von Neumann algebra of the group, one gets semi-simplicity again. Recall that the proof of semisimplicity of C[G] for finite G is based on the fact that C[G] is a Hilbert space for finite G.

Let f : M ⟶ N be a morphism of Hilbert A-modules . Its kernel in the categorial sense in just the ordinary kernel ker(f), whereas the cokernel in the categorial sense is given by N/clos(im(f)). We have to divide out the closure of the image and not the image itself, since Hilbert A-modules are required to be complete. Finite coproducts and products are given by finite direct sums. The zero object is given by {0}. However, the category of Hilbert A-modules is not abelian, since it is neither true that any epimorphism is a cokernel nor that any monomorphism is a kernel. This will force us to deal with two different K-theories and to modify the usual definitions of torsion invariants.

A Hilbert A-chain complex C = (C_{∗}, c_{∗}) is a sequence of Hilbert A-modules

… ⟶^{c_{n+1}} C_{n} ⟶^{c_{n}} C_{n−1} ⟶^{c_{n−1}} …

such that c_{n+1} ∘ c_{n} = 0 holds for n ∈ Z. We call C bounded if there is N ∈ Z such that C_{n} is zero for | n |> N. If C is bounded and C_{n} is finitely generated for all n ∈ Z, we say that C is finite. The homology of C is the Z-graded Hilbert A-module H(C) given by H_{n}(C) = ker(c_{n})/clos(im(c_{n+1})). We call C weakly acyclic if H_{n}(C) is zero for all n ∈ Z. Notice that weakly acyclic does not imply acyclic. We call 0 ⟶ M ⟶ N ⟶ P ⟶ 0 weakly exact, if it is weakly acyclic as a Hilbert A-chain complex . A morphism f : M ⟶ N is a weak isomorphism if its kernel is trivial and its image is dense, or, equivalently, if 0 ⟶ M ⟶^{f} N ⟶ 0 ⟶ 0 is weakly exact. This is equivalent to the condition that f and its adjoint f^{∗} are injective because of ker(f^{∗})^{⊥} = clos(im(f)). The following lemma is a direct consequence of the polar decomposition theorem.

Lemma 1.7 Given a weak isomorphism f : M ⟶ N of Hilbert A-modules , there are a positive selfadjoint weak automorphism of Hilbert A-modules h : M ⟶ M and an isometric isomorphism of Hilbert A-modules g : M ⟶ N satisfying f = g ∘ h.

Lemma 1.8 An endomorphism f : M ⟶ M of a finitely generated Hilbert A-module is a weak isomorphism, if and only if f is injective.

\underline{\underline{Proof}} : This follows from dimension theory of von Neumann algebras (see Cheeger-Gromov [8] section 1). Namely, for any morphism g : M ⟶ N of finitely generated Hilbert Amodules we have :

dim(M) + dim(clos(im(g))^{⊥}) = dim(N) + dim(ker(g))

and dim(M) is zero if and only if M is zero. We will give a direct proof of this fact in the abelian case later.

Example 1.9 Our main example will be the finite von Neumann algebra N(Γ) of a countable discrete group Γ. Let C[Γ] be the complex group ring. It becomes a pre-Hilbert space by the inner product

⟨∑_{γ∈Γ} λ_{γ} · γ, ∑_{γ∈Γ} μ_{γ} · γ⟩ = ∑_{γ∈Γ} λ_{γ} · \overline{μ_{γ}}

The Hilbert completion of C[Γ] is denoted by l^{2}(Γ) and consists of square-summable sums ∑_{γ∈Γ} λ_{γ} · g. An element γ ∈ Γ determines an operator \overline{l(γ)} : l^{2}(Γ) ⟶ l^{2}(Γ) whose restriction to the group ring is given by left multiplication with γ. We obtain the left regular representation

ν_{l} : C[Γ] ⟶ B(l^{2}(Γ), l^{2}(Γ))

and the von Neumann algebra N(Γ) is the closure of its image in the weak topology. The trace tr : C[Γ] ⟶ C sending ∑_{γ∈Γ} λ_{γ} · γ to λ_{e} extends to the so called natural trace on N(Γ). Notice that with respect to this trace l^{2}(N(Γ)) can be identified with l^{2}(Γ). One can view N(Γ) as the von Neumann algebra associated to the Hilbert algebra C[Γ] (see Dixmier [10] III.7.6.)

In particular we get using Fourier transforms for Γ = Z^{r} that l^{2}(Z^{r}) is the Hilbert space L^{2}(T^{r}) of square-integrable functions on the r-dimensional torus T^{r} with values in C ∪ {∞} and N(Z^{r}) is the space L^{∞}(T^{r}) of almost everywhere bounded measurable functions on T^{r} with values in C ∪ {∞} and the regular representation is given by the pointwise multiplication of functions. An example of a weak N(Z)-automorphism which is not an automorphism is l^{2}(Z) ⟶ l^{2}(Z) given by multiplication with (z − 1) for z ∈ Z a generator.

We will deal with finite von Neumann algebras only and do not try to give the most general version of our constructions, as in the applications we will use von Neumann algebras of groups and these are always finite.

2. Algebraic K-theory of von Neumann algebras

In this section we define the algebraic K-groups of a finite von Neumann algebra A which will be the value groups for our torsion invariants. If A is an abelian von Neumann algebra, we compute them by a determinant. This computation will be crucial for applications in topology.

Let A be a finite von Neumann algebra. Define K_{0}(A) to be the abelian group generated by isomorphism classes of finitely generated Hilbert A-modules satisfying the relation [M ⊕ N] = [M] + [N]. This can be identified with the Grothendieck group of the abelian semi-group of isomorphism classes of finitely generated Hilbert A-modules with the addition given by ⊕. Let K_{1}(A) resp. K_{1}^{w}(A) be the abelian group generated by conjugation classes of automorphisms, resp. weak automorphisms of finitely generated Hilbert A-modules satisfying the following relations

• [f] + [g] = [h] , if there is an exact sequence of automorphisms

0 ⟶ (M, f) ⟶^{i} (N, g) ⟶^{p} (P, h) ⟶ 0

• [g ∘ f] = [f] + [g] , if f and g are automorphisms resp. weak automorphisms of the same finitely generated Hilbert A-module

• [id : M ⟶ M] = 0

Remark 2.1 The group K_{1}(A) is the abelianization of the general linear group GL(A). If one wants to define also higher K-groups, one can use Waldhausen’s construction (see [30] ) applied to the following category with weak isomorphisms and cofibrations. The underlying category is the category of finitely generated Hilbert A-modules. Cofibrations are split injections of finitely generated Hilbert A-modules and weak isomorphisms are isomorphisms resp. weak isomorphisms of finitely generated Hilbert A-modules. The resulting K-groups in dimension 1 can be identified with the groups defined above.

Remark 2.2 The first relation still holds, if we substitute exact sequence by weak exact sequence. The following relations are sometimes useful. We get [f ∘ g] = [g ∘ f] for weak isomorphisms f : M ⟶ N and g : N ⟶ P and [f] = [g] for weak isomorphisms u : M ⟶ N, f : M ⟶ M and g : N ⟶ N satisfying u ∘ f = g ∘ u.

Taking the adjoint induces involutions ∗ on K_{1}(A) and K_{1}^{w}(A). The forgetful functor induces a homomorphism 2.3 F : K_{1}(A) ⟶ K_{1}^{w}(A)

compatible with the involutions. We will be interested in the fixed point sets K_{1}(A)^{Z/2} and K_{1}^{w}(A)^{Z/2}.

We want to compute these K-groups for abelian von Neumann algebras. Let X be a compact second-countable space together with a positive finite measure ν. Let L^{∞}(X; ν) be the set of almost everywhere bounded measurable functions from X to C ∪ {∞}. In the sequel we abbreviate {x ∈ X | |f(x)| ≤ K} by { |f | ≤ K}. A measurable function is almost everywhere bounded, if there is a real number K such that X − { |f | ≤ K} is contained in a zero set. Define ∥f ∥_{∞} to be the infimum over such numbers K. Denote by L^{2}(X; ν) the (seperable complex) Hilbert space of square integrable functions on X. Given f ∈ L^{∞}(X; ν), we define an operator m_{f} : L^{2}(X; ν) ⟶ L^{2}(X; ν) sending g to f · g. Let A be the subset of B(L^{2}(X; ν)) consisting of all operators m_{f} for f ∈ L^{∞}(X; ν). Then A is an abelian finite von Neumann algebra and m : L^{∞}(X; ν) ⟶ A is an isometric isomorphism of normed algebras. It is compatible with the involutions ∗ on L^{∞}(X; ν) given by complex conjugation and the involution ∗ given on A by taking the adjoint. The abelian von Neumann algebra L^{∞}(X; ν) is finite, namely, a finite faithful and normal trace is given by tr : L^{∞}(X; ν)^{+} ⟶ R^{≥0} sending f to ∫_{X} fdν. In the sequel we will use this trace. Notice that then l^{2}(A) can be identified with L^{2}(X; ν). Any abelian von Neumann algebra A ⊂ B(H) for H a seperable complex Hilbert space is of this type (see Dixmier [10], I.7.3).

Consider the finite abelian von Neumann algebra A = L^{∞}(X; ν). Let Inv(X; ν) be the multiplicative group of almost everywhere invertible measurable functions from X to C ∪ {∞}. Almost everywhere invertible means that the preimage of 0 and the preimage of ∞ are zero-sets. Any element h ∈ Inv(X; ν) can be written as a quotient of almost everywhere bounded invertible functions h = \frac{h_{0}}{h_{1}} . E.g., put h_{1}(x) = 1, if h(x) = 0, h_{1}(x) = h(x)/ |h(x)| , if 0 < | h(x) | ≤ 1 and h_{1}(x) = h(x)^{−1} otherwise, and define h_{0} = h · h_{1}. Let f be a almost everywhere bounded function. If f is almost everywhere invertible, then the associated operators m_{f} and (m_{f})^{∗} = m_{f^{∗}} are injective and hence weak isomorphisms because of clos(im(m_{f})) = ker((m_{f})^{∗})^{⊥}. Hence m_{f} is a weak isomorphism if and only if f is almost everywhere invertible. Therefore we can define homomorphisms :

2.4 i : L^{∞}(X; ν)^{×} ⟶ K_{1}(A) f ↦ [m_{f}] i : Inv(X; ν) ⟶ K_{1}^{w}(A) \frac{h_{0}}{h_{1}} ↦ [m_{h_{0}}] − [m_{h_{1}}]

where L^{∞}(X; ν)^{×} denotes the abelian group of units, i.e. function which are almost everywhere bounded from above and below.

Consider an endomorphism f : M ⟶ M of a finitely generated Hilbert A-module. Because of lemma 1.6 there is a finitely generated Hilbert A-module N, a non-negative integer n and an isomorphism of Hilbert A-modules ϕ : M ⊕ N ⟶ ⊕_{i=1}^{n}l^{2}(A). Define the endomorphism g : ⊕_{i=1}^{n}l^{2}(A) ⟶ ⊕_{i=1}^{n}l^{2}(A) to be ϕ ∘ (f ⊕ id_{N}) ∘ ϕ^{−1}. Let Ω(g) be the (n, n)- matrix over A defined for g in 1.5. Define the determinant det(f) ∈ A to be the ordinary determinant det(Ω(g)) of a quadratic matrix over a commutative ring such as A. We leave it to the reader to check that this is independent of the choices of N, n and ϕ. We will see that this determinant inherits from the ordinary determinant for commutative rings all the expected properties. The main result of this section is:

Theorem 2.5 Let A = L^{∞}(X; ν) be a finite abelian von Neumann algebra. Then the determinant induces isomorphisms :

det : K_{1}(A) ⟶ L^{∞}(X; ν)^{×}
det : K_{1}^{w}(A) ⟶ Inv(X; ν)

These maps are compatible with the involutions given on the K-groups by taking the adjoint and on the targets by complex conjugation. The inverse maps of these isomorphisms are given by the maps of 2.4.

The following Lemma 2.6 and lemma 2.7 imply the main theorem 2.5 of this section. One may say in view of lemma 2.6 that its proof is based on a kind of Euclidean algorithm based on the support of functions in L^{∞}(X; ν). A block matrix G is a matrix whose first row or first column consists of zero entries except the (1, 1)-entry.

Lemma 2.6 Let G be a (n, n)-matrix over A such that the associated homomorphism of Hilbert A-modules is an isomorphism resp. weak isomorphism. Then there are block (n, n)- matrices S_{0}, S_{1}, … S_{n} with the following properties :

1.) S_{i} is an isomorphism for 2 ≤ i ≤ n − 1.

2.) S_{0} and S_{1} are isomorphisms resp. weak isomorphisms.

3.) S_{0} = S_{1} · S_{2} · . . . S_{n} · G.

\underline{\underline{Proof}} : Denote by ∥ G ∥_{∞} the supremum of all ∥ g_{i,j} ∥_{∞} . Recall that ∥ g_{i,j} ∥_{∞} is the infimum over all real numbers K for which { |g_{i,j} | > K} is a zero-set. We have :

∥G · G′ ∥_{∞} ≤ n· ∥G∥_{∞} · ∥G′ ∥_{∞}

Now fix ϵ ≥ 0. Given 2 ≤ i ≤ n, denote by Y the set { |g_{1,1} | ≤ ϵ} ∩ { |g_{i,1} | > ϵ}. Denote by χ_{Y} the characteristic function of Y. Let S_{i} be the matrix having 1 as entries on the diagonal, 2 · χ_{Y} as (1,i)-entry and zero as other entries. Put G′ = S_{i} · G. Since the sum of a number of norm ≥ 2 · ϵ and a number of norm ≤ ϵ has norm ≥ ϵ by the triangle inequality, we have by construction :

{ |g′_{1,1} | ≤ ϵ} ⊂ { |g′_{i,1} | ≤ ϵ}
{ |g′_{1,1} | ≤ ϵ} ⊂ { |g_{1,1} | ≤ ϵ}
{ |g′_{j,1} | ≤ ϵ} = { |g_{j,1} | ≤ ϵ} for j ≠ 1
ffi ∥G′ ∥_{∞} ≤ ∥S ∥_{∞} · ∥G∥_{∞} ·n ≤ max{1, 2· ∥G∥_{∞} } · ∥G∥_{∞} ·n

By iterating this process for i = n, n − 1, … 2 we obtain a matrix G̃ and block matrices S_{i} with the following properties :

S_{i} is an isomorphism

G̃ = S_{2} · S_{3} … S_{n} · G

{ |g̃_{1,1} | ≤ ϵ} ⊂ { |g̃_{i,1} | ≤ ϵ} for 1 ≤ i ≤ n

∥G̃∥_{∞} ≤ max{1, ∥G∥_{∞}}^{2·n} · (2n)^{2·n−1}

Next we finish the proof in the case, where G is a weak isomorphism. Then G̃ is also a weak isomorphism. Choose ϵ to be zero. Notice that G̃ maps (χ_{{ |_{g_{1,1}}| ≤0}}, 0, … 0) to zero. Hence { | g̃_{1,1} | ≤ 0} is a zero set. This shows that g̃_{1,1} is a weak isomorphism. Let S_{1} be the block matrix having g̃_{1,1} as (i, i)-entry and −g̃_{i,1} as (i, 1)-entry for 2 ≤ i ≤ n, 1 as (1, 1)-entry and zero as other entries. Then S_{0} := S_{1} · G̃ is a block matrix and S_{0} and S_{1} are weak isomorphisms by the implication b.) ⇒ a.) in claim 5.) of lemma 2.7.

Finally, we deal with the case where G and hence G̃ are isomorphisms. Then there are elements h_{i} ∈ A such that ∑_{i=1}^{n} h_{i} · g_{i,1} = 1 holds. Choose ϵ > 0 such that ϵ · ∑_{i=1}^{n} ∥h_{i} ∥_{∞} is smaller than 1. Then { |g̃_{1,1} | ≤ ϵ} must be a zero-set. Therefore g_{1,1} is a unit. Let S_{1} be the block matrix having 1 on each diagonal entry, −g_{i,1} · g_{1,1}^{−1} as (i, 1)-entry for 2 ≤ i ≤ n and zero as other entries. Then S_{0} = S_{1} · G is a block matrix and S_{0} and S_{1} are isomorphisms by the implication b.) ⇒ a.) in claim 4.) of lemma 2.7.

Lemma 2.7 1.) Given to endomorphisms f and g of the same finitely generated Hilbert

A-module, we get :

det(f ∘ g) = det(f) · det(g)

2.) Let f : M ⟶ M and g : N ⟶ N be endomorphisms of finitely generated Hilbert Amodules and u : M ⟶ N be a weak isomorphism such that u ∘ f = g ∘ u holds. Then we have :

det(f) = det(g)

3.) If 0 ⟶ (M, f) ⟶^{i} (N, g) ⟶^{p} (P, h) ⟶ 0 is a weakly exact sequence of endomorphisms of finitely generated Hilbert A-modules, we have :

det(f) · det(h) = det(g)

4.) The following assertions are equivalent for an endomorphism f : M ⟶ M of a finitely generated Hilbert A-module M :

a.) f is an isomorphism. b.) det(f) : l^{2}(A) ⟶ l^{2}(A) is an isomorphism. c.) det(f) ∈ A is a unit.

5.) The following assertions are equivalent for an endomorphism f : M ⟶ M of a finitely generated Hilbert A-module M :

a.) f is a weak isomorphism. b.) m_{det(f)} : l^{2}(A) ⟶ l^{2}(A) is a weak isomorphism. c.) det(f) ∈ A = L^{∞}(X; ν) is almost everywhere invertible. d.) f is injective. e.) f has dense image

\underline{\underline{Proof}} : 1.) follows directly from the corresponding property of the determinant for commutative rings.

4.) Obviously det(f_{1} ⊕ f_{2}) = det(f_{1}) · det(f_{2}) holds. Hence we may suppose that fis an endomorphisms of ⊕_{i=1}^{n}l^{2}(A) = ⊕_{i=1}^{n}L^{2}(X; ν). Choose a matrix

G = ( \begin{matrix}g_{1,1} & g_{1,2} & … & g_{1,n} \\ g_{2,1} & g_{2,2} & … & g_{2,n} \\ ⋮ & ⋮ & ... & ⋮ \\ g_{nΥ,1} & g_{n,2} & … & g_{n,n} \end{matrix} )

with entries in A = L^{∞}(X; ν) such that f and G correspond to one another under Ω (see 1.5). By Cramer’s rule there exists a n-n-matrix G′ with entries in A = L^{∞}(X; ν) satisfying G′ · G = G · G′ = det(g) · id. Now assertion 4.) follows.

5.) We have shown c.) ⇔ b.) already before 2.4. We derive b.) ⇒ a.) from Cramer’s rule. Obviously a.) implies d.) and e.).

Next we prove a.) ⇒ c.), by induction over the size of the matrix G associated to f. The induction begin n = 1 is trivial, the induction step from n − 1 to n ≥ 2 done as follows. Let f be a weak isomorphism. Because of lemma 2.6 we can assume without of loss of generality that G is given by a block matrix of the following shape

G = ( \begin{matrix}g_{1,1} & g_{1,2} & … & g_{1,n} \\ 0 \\ ⋮ & & Ĝ \\ 0 \end{matrix} )

where Ĝ is a (n − 1)-(n − 1)-matrix. Then we get:

det(G) = det(Ĝ) · g_{1,1}

As G defines a weak isomorphism, the morphism given by g_{1,1} is injective and the one given by Ĝ has dense image. By induction hypothesis det(Ĝ) and g_{1,1} are almost everywhere invertible and hence the same is true for det(G). This shows a.) ⇒ c.).

Next we show d.) ⇒ c.). If f is injective, f^{∗}f is injective because of the following calculation for x ∈ ker(f^{∗}f)

0 = ⟨f^{∗}f(x), x⟩ = ⟨f(x), f(x)⟩ = |f(x)|^{2} ⇒ f(x) = 0 ⇒ x = 0

As f^{∗}f is selfadjoint, f^{∗}f is a weak isomorphism so that det(f^{∗}f) = |det(f)|^{2} and hence det(f) are almost everywhere invertible by a.) ⇒ c.). This finishes the proof of d.) ⇒ c.). If f has dense image, f^{∗} is injective and hence det(f) = det(f^{∗})^{∗} is almost everywhere invertible. This shows e.) ⇒ c.) and hence 5.) is true.

2.) follows from 1. and 5. Namely, because det(u) is almost everywhere invertible, det(f) · det(u) = det(u) · det(g) implies det(g) = det(f).

3.) Let 0 ⟶ (ker(p), g_{1}) ⟶ N ⟶ (ker(p)^{⊥}, g_{2}) ⟶ 0 be the exact sequence of endomorphisms induced from the given weakly exact sequence. We derive from the corresponding statement for determinants of commutative rings that det(g_{1}) · det(g_{2}) = det(g) holds. We conclude det(f) = det(g_{1}) and det(h) = det(g_{2}) from assertion 2.).

Next we give the proof of theorem 2.5. Obviously det is a well-defined homomorphism and det ∘ i = id if i is the map defined in 2.4. It remains to show surjectivity of i. We prove inductively over the size of a quadratic matrix G representing an element [G] in the K_{1}-group that [G] is given by a sum of elements represented by (1, 1)-matrices. Because of lemma 2.6 we may assume that G is a block matrix. But then [G] is the sum of elements given by matrices of smaller size.

We now drop the assumption that A is abelian. There is a well-defined notion of a tensor product A ⊗ B of von Neumann algebras. Let A and B be finite von Neumann algebras with given finite normal faithful traces. Then A ⊗ B is a finite von Neumann algebra and inherits a finite normal faithful trace. There is a natural isomorphism l^{2}(A) ⊗ l^{2}(B) ⟶ l^{2}(A ⊗ B). If M resp. N is a Hilbert A resp. B- module, then the tensor product of Hilbert spaces M ⊗ N comes with a canonical Hilbert A ⊗ B-structure. If M and N are finitely generated, then also M ⊗ N. This tensor product of Hilbert modules over finite von Neumann algebras is functorial. If A = N(A) and B = N(B) for countable discrete groups A and B, then A ⊗ B can be identified with N(A × B). We obtain a pairings

2.8 ⊗ : K_{0}(A) ⊗ K_{1}(B) ⟶ K_{1}(A ⊗ B) ⊗ : K_{0}(A) ⊗ K_{1}^{w}(B) ⟶ K_{1}^{w}(A ⊗ B)

⊗ : K_{0}(A) ⊗ K_{1}(B)^{Z/2} ⟶ K_{1}(A ⊗ B)^{Z/2}
⊗ : K_{0}(A) ⊗ K_{1}^{w}(B)^{Z/2} ⟶ K_{1}^{w}(A ⊗ B)^{Z/2}

sending [M] ⊗ [f : N ⟶ N] to [id_{M} ⊗ f : M ⊗ N ⟶ M ⊗ N].

Proposition 2.9 Let G be a finite group. Then K_{0}(N(G)) is the complex representation ring Rep_{C}(G) and the pairing 2.8 induces isomorphisms

Rep_{C}(G) ⊗ K_{1}(A) ⟶ K_{1}(C[G] ⊗ A)
Rep_{C}(G) ⊗ K_{1}^{w}(A) ⟶ K_{1}^{w}(C[G] ⊗ A)

\underline{\underline{Proof}} : We construct the inverse isomorphism. Let f : M ⟶ M be a (weak) automorphism of a finitely generated Hilbert C[G] ⊗ A- module. Let I be a complete set of representatives of the isomorphism classes of irreducible unitary G-representations. Let V̄ be the dual unitary G-representation of V. Equip (V̄ ⊗ M)^{G} with the induced Hilbert Astructure. The inverse map sends [f] to the sum ∑_{V ∈I} [V] ⊗ [(idV̄ ⊗ f)^{G}]. This is an inverse as there is a natural isometric C[G] ⊗ A-isomorphism from ∑_{V ∈I} V ⊗ (V̄ ⊗ M)^{G} to M sending v ⊗ w ⊗ m ∈ V ⊗ (V̄ ⊗ M)^{G} to ⟨v, w⟩ · m.

Corollary 2.10 Let G be finite group and r be a non-negative integer. Then there are isomorphisms :

K_{1}(N(G × Z^{r}))  ⟶ Rep_{C}(G) ⊗ L^{∞}(T^{r}, C ∪ {∞})^{×}
K_{1}(N(G × Z^{r}))^{Z/2} ⟶ Rep_{C}(G) ⊗ L^{∞}(T^{r}, R ∪ {∞})^{×}
K_{1}^{w}(N(G × Z^{r}))  ⟶ Rep_{C}(G) ⊗ Inv(T^{r}, C ∪ {∞})
K_{1}^{w}(N(G × Z^{r}))^{Z/2} ⟶ Rep_{C}(G) ⊗ Inv(T^{r}, R ∪ {∞})

3. Torsion invariants for Γ-CW -complexes

In this section we want to introduce Reidemeister von Neumann torsion for finite proper Γ-CW -complexes for a discrete group Γ. We state basic properties like sum and product formulas and relate Reidemeister von Neumann torsion to Whitehead torsion. As an illlustration we compute the Reidemeister von Neumann torsion of spaces carrying appropiate torus actions. Most of the technical proofs are deferred to sections 6 and 7. The definition of Reidemeister von Neumann torsion in this section has the advantage that it is easy to state. For technical purposes, however, we will introduce a different definition in section 7 which will be shown to be equivalent to the one in this section. If one is only interested in the case where Γ acts freely, one may skip the first part and start directly with definition 3.6.

Let Γ be a discrete group. A Γ-CW -complex X is a Γ-space X together with a filtration ∅ = X_{−1} ⊂ X_{0} ⊂ X_{1} ⊂ … ⊂ X_{n} ⊂ … ⊂ X such that X has the weak topology with respect to the filtration {X_{n} | n ≥ −1} and for each n ≥ 0 there exists a Γ-push out

3.1  [figure: ∐_{i∈I_{n}} Γ/Γ_{i} × S^{n−1}; X_{n−1}; ∐_{i∈I_{n}} q_{i}^{n}; ∐_{i∈I_{n}} Γ/Γ_{i} × D^{n}; X_{n}; ∐_{i∈I_{n}} Q_{i}^{n}]

A Γ-CW -complex X is finite , if the set ∐_{n≥0} I_{n} is finite and X is proper, if the map Γ × X ⟶ X × X sending (γ, x) to (x, γ · x) is proper. Since we are working in the category of compactly generated spaces, a map is proper if and only if preimages of compact sets are compact. A Γ-CW -complex X is finite if and only if Γ\X is compact. It is proper if and only if the isotropy subgroup Γ_{x} = {γ ∈ Γ | γx = x} is finite for all x ∈ X (see Lück [19], theorem 1.23). Since Γ is discrete, the Γ-action is proper if and only it is properly discontinuos in the sense of the introduction. Notice that the Γ-push outs appearing in the definition of a Γ-CW - complex are not part of the structure. The universal covering of a compact CW -complex X is a proper free π_{1}(X)-CW -complex. More generally we have the following examples.

Example 3.2 Let G be a finite group and X a compact G-CW -complex. Denote by p : X̃ ⟶ X the universal covering of X and identify π = π_{1}(X, x) with the group of deck transformations. Let Γ be the discrete group 3.3 Γ := {(f̃, g) | f : X̃ → X̃, g ∈ G, p ∘ f̃ = l(g) ∘ p}

where l(g) : X → X denotes multiplication with g. There is an obvious exact sequence 0 ⟶ π ⟶^{i} Γ ⟶^{q} G ⟶ 0 and an operation of Γ on X̃ making the following diagram commute

[figure: π; ×; X̃; X̃; i × id; id; Γ; ×; X̃; X̃; q × p; p; G; ×; X; X]

The G-CW -structure on X induces a finite proper Γ-CW -structure on X̃.

Next we prepare the definition of the cellular Hilbert A-chain complex. Let P be a finitely generated projective C[Γ]-module. Consider left Γ-sets T and S. We call two C[Γ]- isomorphisms α : P ⟶ C[T] and β : P ⟶ C[S] equivalent, if there is a bijective Γ-map f : S ⟶ T and a Γ-invariant map ϵ : S ⟶ {±1} such that β ∘ α^{−1} : C[T] ⟶ C[S] sends ∑_{t∈T} λ_{t} · t to ∑_{s∈S} ϵ(s) · λ_{f(s)} · s. A permutation C[Γ]- module is a C[Γ]-module together with a choice of equivalence classes of C[Γ]-isomorphisms α : P ⟶ C[T] for some Γ-set T. Notice for a Γ-set T that the C[Γ]-module C[T] is finitely generated and projective if and only if Γ\T is finite and the isotropy groups of elements in T under the Γ-action are all finite.

Let X be a finite proper Γ-CW -complex. Then the cellular C[Γ]-chain complex C_{∗}(X̃) inherits from the Γ-CW -structure the structure of a finite permutation C[Γ]-chain complex for all n ≥ 0. Namely, an explicit choice of Γ-push outs 3.1 together with a choice of generators of the homology groups H_{n}(D^{n}, S^{n−1}; Z) determine a C[Γ]-isomorphism C_{n}(X̃) ⟶ C[∐_{i∈I_{n}} Γ/H_{i}]. The associated structure of a permutation C[Γ]-module depends only on the Γ-CW -structure of X (see Lück [19] lemma 13.2).

Let A be a finite von Neumann algebra with a finite normal faithful trace. Let V be a finitely generated Hilbert A-module. A unitary representation of Γ in Iso_{A}(V)^{op} 3.4 μ : Γ ⟶ Iso_{A}(V)^{op}

is a group homomorphism from Γ into the opposite group of the group of isometric A-auto-morphisms of V. The associated algebra homomorphism is denoted in the same way by μ : C[Γ] ⟶ End_{A}(V)^{op}. Notice that Γ acts from the left on C[T] and from the right on V. Hence the tensor product V ⊗_{C[Γ]} C[T] is defined. If we consider Hom_{C[Γ]}(C[T], V) we use on V the induced left module structure given by γ · v := v · g^{−1} for γ ∈ Γ and v ∈ V.

Lemma 3.5 Let P and Q be a finitely generated projective permutation C[Γ]-modules and f : P ⟶ Q a C[Γ]-homomorphism. Then :

1.) There are preferred structures of a finitely generated Hilbert A-module on V ⊗_{C[Γ]} P and Hom_{C[Γ]}(P, V).

2.) The induced maps id_{V} ⊗_{C[Γ]} f and Hom_{C[Γ]}(f, id_{V}) are homomorphisms of Hilbert Amodules.

3.) There is an isometric isomorphism D(P) : V ⊗_{C[Γ]} P ⟶ Hom_{C[Γ]}(P, V). It is natural, i.e., Hom_{C[Γ]}(f, id_{V}) ∘ D(Q) and D(P) ∘ (id_{V} ⊗_{C[Γ]} f)^{∗} agree.

\underline{\underline{Proof}} : Consider a Γ-set T such that Γ\T is finite and the isotropy group Γ_{t} of any t ∈ T under the Γ-action is finite. Now V ⊗_{C[Γ]} C[T] and Hom_{C[Γ]}(C[T], V) inherit Amodule structures from V. As V is a unitary representation, there is a Hilbert structure on V ⊗_{C[Γ]} C[T] given by

⟨m ⊗_{C[Γ]} t, n ⊗_{C[Γ]} s⟩ := \frac{1}{|Γ_{t} |} · ∑_{\{γ∈Γ|γ·t=s\}} ⟨γ · m, n⟩

and on Hom_{C[Γ]}(C[T], V) given by

⟨ϕ, ψ⟩ = ∑_{Γt∈Γ\T} ⟨ϕ(t), ψ(t)⟩

These data determine the preferred structure of a finitely generated Hilbert A-module. One easily checks that the maps induced by f are continuous.

The isometric Hilbert A-isomorphism D(C[T]) : V ⊗_{C[Γ]} C[T] ⟶ Hom_{C[Γ]}(C[T], V) sends v ⊗_{C[Γ]} t to the C[Γ]-map from C[T] to V which assigns | Γ_{t} |^{−1} · ∑_{{γ∈Γ|γ·t=s}} v · γ^{−1} to s ∈ T.

Notation 3.6 Given a pair (X, X_{1}) of proper Γ-CW -complexes, let C_{∗}(X, X_{1}; V) be the Hilbert A-chain complex V ⊗_{C[Γ]} C_{∗}(X, X_{1}). Denote by H_{∗}(X, X_{1}; V) its homology. Define the Hilbert A-cochain complex C^{∗}(X, X_{1}; V) by Hom_{C[Γ]}(C_{∗}(X, X_{1}), V). Denote its cohomology by H^{∗}(X, X_{1}; V).

For the definition of Reidemeister von Neumann torsion we need the following result. Let C be a finite Hilbert A-chain complex. Its Laplace operator Δ_{n} : C_{n} ⟶ C_{n} in dimension n is given by c_{n+1} ∘ c_{n+1}^{∗} + c_{n}^{∗} ∘ c_{n}. Let H_{n}(C) be the kernel of Δ_{n}.

Theorem 3.7 (Hodge decomposition theorem) Let C be a finite Hilbert A- chain complex. Then:

1.) C_{n} = H_{n}(C) ⊕ clos(im(c_{n}^{∗})) ⊕ clos(im(c_{n+1}))

2.) We have H_{n}^{harm}(C) ⊂ ker(c_{n}). The natural projection induces an isometric isomor-

phism :

H_{n}(C) ⟶ H_{n}(C)

\underline{\underline{Proof}} : As Δ_{n} is selfadjoint, we have :

H_{n}(C) ⊕ clos(im(Δ_{n})) = C_{n}

Since ⟨c_{n}^{∗}(x), c_{n+1}(y)⟩ = ⟨x, c_{n}c_{n+1}(y)⟩ holds, the spaces clos(im(c_{n}^{∗})) and clos(im(c_{n+1})) are orthogonal. For x ∈ H_{n}(C) we compute :

0 = ⟨x, Δ_{n}(x)⟩ = ⟨c_{n}(x), c_{n}(x)⟩ + ⟨c_{n+1}^{∗}(x), c_{n+1}^{∗}(x)⟩

As x lies in ker(c_{n}) = clos(im(c_{n}^{∗}))^{⊥} and in ker(c_{n+1}^{∗}) = clos(im(c_{n+1}))^{⊥}, we get

clos(im(c_{n}^{∗}))^{⊥} ∩ clos(im(c_{n+1}))^{⊥} = H_{n}^{harm}(C)

This implies :

clos(im(c_{n}^{∗})) ⊕ clos(im(c_{n+1})) = H_{n}(C)^{⊥} = clos(im(Δ_{n}))

Let C be a finite Hilbert A-chain complex. Let C′ be the orthogonal complement of H(C) in C. We have Δ′ ⊕ 0 = Δ if Δ′ resp. Δ are the Laplace operators of C′ and C. Then the morphism Δ′ is a selfadjoint weak automorphism and defines an element [Δ′_{n}] in K_{1}^{w}(A)^{Z/2}.

Definition 3.8 Define the Reidemister torsion of C by:

ρ(C) = − ∑_{n} (−1)^{n} · n · [Δ′_{n}] ∈ K_{1}^{w}(A)^{Z/2}

Definition 3.9 Let (X, X_{1}) be a pair of finite proper Γ-CW -complexes. Consider a finite von Neumann algebra A with finite normal and faithful trace. Let V be a finitely generated Hilbert A-module and μ : Γ ⟶ Iso_{A}(V)^{op} be a unitary representation of Γ. Define Reidemeister-von Neumann torsion of X

ρ(X, X_{1}; V) ∈ K_{1}^{w}(A)^{Z/2}

by the Reidemeister von Neumann torsion ρ(C_{∗}(X, X_{1}; V)) of C_{∗}(X, X_{1}; V) (see definition 7.10).

As it stands, the definition above makes only sense for connected X, but can easily be generalized to arbitrary X. For a component C of X let Γ_{C} be its isotropy group under the Γ-action on π_{0}(X). Suppose we have assigned to any component a representation μ_{C} : Γ_{C} ⟶ Iso_{A}(V_{C})^{op} such that V_{C} and V_{γC} agree for any γ ∈ Γ and μ_{γC} ∘ c_{γ} = μ_{C} holds for the homomorphism c_{γ} : Γ_{C} ⟶ Γ_{γC} sending δ to γδγ^{−1}. We define ρ(X, X_{1}; V) to be the sum ∑_{ΓC∈Γ\π_{0}(X)} ρ(C, C ∩ X_{1}; V_{C}).

We prefer chain complexes instead of cochain complexes in this definition, although later we have to deal with deRham cohomology and hence with cochain complexes. In many cases of interest the spaces are acyclic, so that no cohomology is involved, and it is more convenient to deal with the cellular chain complex instead of the cochain complex since this is done in related classical cases like Whitehead and Reidemeister torsion. In principal there is no difference because of the following lemma. Recall that the dual cochain complex (C_{∗})^{∗} of a Hilbert A-chain complex C_{∗} has the same underlying chain modules and the codifferentials are the adjoint of the differentials.

Lemma 3.10 There are a natural isometric Hilbert A-isomorphisms

(C_{∗}(X, X_{1}; V))^{∗} ⟶ C^{∗}(X, X_{1}; V)  and  (H_{∗}(X, X_{1}; V))^{∗} ⟶ H^{∗}(X, X_{1}; V)

\underline{\underline{Proof}} : The first identification comes from lemma 3.5. Let C be a finite Hilbert Achain complex.We obtain from the Hodge decomposition theorem 3.7 a natural identification H(C^{∗}) = H(C^{∗}) = H(C) = H(C) and thus the second identification from the first.

If (f, f_{1}) : (X, X_{1}) ⟶ (Y, Y_{1}) is a Γ-homotopy equivalence of pairs of finite proper Γ-CW -complexes, its equivariant Whitehead torsion

τ^{Γ}(f, f_{1}) ∈ Wh(ZΠ/(Γ, Y))

is defined in Lück [19], page 68, page 284 (see also Illman [15] and Dovermann-Rothenberg [13] in the case of a finite group Γ). This is an important invariant, e.g. for the classification of smooth G-manifolds. The following result reflects the general idea that Whitehead torsion is the difference of Reidemeister torsion. The map (f, f_{1}) induces a weak isomorphism on homology

H_{n}(f, f_{1}; V) : H_{n}(X, X_{1}; V) ⟶ H_{n}(Y, Y_{1}; V)

It defines an element [[H_{n}(f, f_{1}; V)]] in K_{1}^{w}(A)^{Z/2} by the class of the weak automorphism H_{n}(f, f_{1}; V)^{∗} ∘ H_{n}(f, f_{1}; V) of the finitely generated Hilbert A-module H_{n}(X, X_{1}; V) (see notation 7.2).

Theorem 3.11 There is a natural homomorphism

Φ = Φ(Y, V) : Wh(ZΠ/(Γ, Y)) ⟶ K_{1}^{w}(A)^{Z/2}

such that

Φ(τ^{Γ}(f, f_{1})) − ∑_{n}(−1)^{n} · [[H_{n}(f, f_{1}; V)]] = ρ(Y, Y_{1}; V) − ρ(X, X_{1}; V)

We do not give the precise definition of Φ here, its construction is obvious, if one is familar with the language of modules over the fundamental category as developed in Lück [19]. In the case, where Γ acts freely, Wh(ZΠ/(Γ, Y)) reduces to the ordinary Whitehead group W h(Γ) and Φ sends an element in W h(Γ) represented by the automorphism g of ⊕_{m}Z[π] to [[id_{V} ⊗_{Z[Γ]} g]]. The following conclusion is of particular interest:

Corollary 3.12 If (X, X_{1}) is V -acyclic, i.e. H(X, X_{1}; V) is trivial, then ρ(X, X_{1}; V) depends only the simple Γ-homotopy type of (X, X_{1}).

The next two results are the basic tools for computations. Consider the cellular Γpush out of finite proper Γ-CW -complexes, where i_{1} is an inclusion of finite proper Γ-CW - complexes.

3.13 [figure: X_{0}; X_{2}; i_{2}; i_{1}; j_{0}; j_{2}; X_{1}; X; j_{1}]

We get an exact sequence of finite Hilbert A-chain complexes

{0} ⟶ C(X_{0}; j_{0}^{∗}V) ⟶^{i_{1∗}⊕i_{2∗}} C(X_{1}; j_{1}^{∗}V) ⊕ C_{∗}(X_{2}; j_{2}^{∗}V) ⟶^{j_{1∗}−j_{2∗}} C_{∗}(X; V) ⟶ {0}

Denote by LHS(X; X_{1}, X_{2}, X_{0}; V) the long weakly exact homology sequence of the sequence above (see 6.1). If k : Y_{1} ⟶ Y is the inclusion of a pair of proper finite G-CW -complexes, let LHS(Y, Y_{1}; V) be the weakly exact long homology sequence of the pair. We derive from theorem 7.16.

Theorem 3.14 (Sum formula ) We have :

ρ(X; V) = ρ(X_{1}; j_{1}^{∗}V) + ρ(X_{2}, j_{2}^{∗}V) − ρ(X_{0}; j_{0}^{∗}V) + ρ(LHS(X; X_{1}, X_{2}, X_{0}; V))

ρ(Y, Y_{1}; V) = ρ(Y; V) − ρ(Y_{1}; k^{∗}V) + ρ(LHS(Y, Y_{1}; V))

Let Γ_{1} and Γ_{2} be discrete groups. Let (X, X_{1}) resp. (Y, Y_{1}) be pairs of a finite proper Γ_{1}- resp. Γ_{2}-CW -complexes. Then the product space (X, X_{1})×(Y, Y_{1}) inherits the structure of a finite proper Γ_{1} ×Γ_{2}-CW -complex. Consider finite von Neumann algebras A and B with finite normal and faithful traces. Let Γ_{1} ⟶ Iso_{A}(V)^{op} and Γ_{2} ⟶ Iso_{B}(W)^{op} be unitary representations. Define

3.15 χ^{A}(X, X_{1}; V) ∈ K_{0}(A)

by χ^{A}(C(X, X_{1}; V)) = ∑_{n}(−1)^{n} · [C_{n}(X, X_{1}; V)] = ∑_{n}(−1)^{n} · [H_{n}(X, X_{1}; V)] (see 7.14). If Γ_{1} acts freely on X, then χ^{A}(X; V) reduces to χ(Γ_{1}\X, Γ_{1}\X_{1}) · [V], where the integer χ(Γ_{1}\X, Γ_{1}\X_{1}) is the ordinary Euler characteristic. Because there is an isometric A ⊗ Bisomorphism from C_{∗}(X, X_{1}; V) ⊗ C_{∗}(Y, Y_{1}; W) to C_{∗}((X; X_{1}) × (Y, Y_{1}); V ⊗ W), we derive from lemma 7.13 and lemma 7.15.

Theorem 3.16 (Product formula) We get under the assumptions above:

1.) ρ((X, X_{1})×(Y, Y_{1}); V ⊗W) = χ^{A}(X, X_{1}; V)⊗ρ(Y, Y_{1}; W)+ρ(X, X_{1}; V)⊗χ^{B}(Y, Y_{1}; W)

2.) H_{∗}(X, X_{1}; V)⊗H_{∗}(Y, Y_{1}; W) and H_{∗}((X; X_{1})×(Y, Y_{1}); V ⊗W) are isometrically A⊗Bisomorphic.

Let i : Γ_{0} ⟶ Γ be the canonical inclusion of a subgroup Γ_{0} of Γ of finite index. Let (X, X_{1}) be a pair of finite Γ-CW -complexes. The restriction (res(X), res(X_{1})) to Γ_{0} inherits the strucure of a pair of finite Γ_{0}-CW -complexes. Let μ : Γ_{0} ⟶ Iso_{A}(V)^{op} be a unitary representation and i_{∗}μ : Γ ⟶ Iso_{A}(V ⊗_{C[Γ_{0}]} C[Γ]) be the induced one.

Lemma 3.17 (Restriction formula) We obtain under the conditions above :

1.) H_{∗}(res(X), res(X_{1}); V) = H_{∗}(X, X_{1}; i_{∗}V) and

ρ(res(X), res(X_{1}); V) = ρ(X, X_{1}; i_{∗}V)

2.) There is a natural restriction homomorphism res : K_{1}^{w}(N(Γ)) ⟶ K_{1}^{w}(N(Γ_{0})) which sends ρ(X, X_{1}; l^{2}(Γ)) to ρ(res(X), res(X_{1}); l^{2}(Γ_{0})).

\underline{\underline{Proof}} : The first assertion follows from the existence of an isometric Hilbert A-chain isomorphism C(res(X), res(X_{1}); V) ⟶ C(X, X_{1}; i_{∗}V). Any finitely generated Hilbert N(Γ)- module V can be viewed by restriction as a finitely generated Hilbert N(Γ_{0})-module denoted by res(V) and similar for morphisms. This induces the restriction homomorphism res : K_{1}^{w}(N(Γ)) ⟶ K_{1}^{w}(N(Γ_{0})) Obviously it sends ρ(X, X_{1}; l^{2}(Γ)) to ρ(X, X_{1}; res(l^{2}(Γ))). One easily checks that the representations of Γ given by res(l^{2}(Γ)) and i_{∗}l^{2}(Γ_{0}) are conjugate by an isometric isomorphism of Hilbert N(Γ_{0})-modules. Hence ρ(X, X_{1}; res(l^{2}(Γ))) and ρ(X, X_{1}; i_{∗}l^{2}(Γ_{0})) agree and the second assertion follows from the first.

For the remainder of the section we use

Notation 3.18 If X is a finite CW -complex and μ : π_{1}(X) ⟶ Iso_{A}(V)^{op} is a unitary representation of its fundamental group, we write ρ(X; V) instead of ρ(X̃; V) and C_{∗}(X; V) for C_{∗}(X̃; V)

We want to illustrate these computational tools by determing the torsion for spaces carrying a torus action. As a preliminary we need the following lemma.

Lemma 3.19 Let T be the n-dimensional torus for n ≥ 1 and μ : π_{1}(T) ⟶ Iso_{A}(V) a unitary representation. Suppose the existence of w ∈ π_{1}(T) such that w ≠ 0 and the endomorphism (μ(w) − 1) : V ⟶ V is a weak isomorphism. Then:

1.) T is weakly V -acyclic, i.e. H_{∗}(T; V) = {0}.

2.) ρ(T; V) = 0 for n ≥ 2.

3.) If n = 1 and w_{1} ∈ π_{1}(T) is a generator we get:

ρ(T; V) = [[(μ(w_{1}) − 1) : V ⟶ V]]

\underline{\underline{Proof}} : Choose a decomposition T = S^{1} × T′ where T′ is the n − 1-dimensional torus such that w = w_{1}^{m} for some m and w_{1} given by S^{1}×{∗}. Since (μ(w) − 1) is a weak isomorphism by assumption and (μ(w_{1}) − 1) ∘ (∑_{j=0}^{m−1} μ(w_{1}^{j})) = (∑_{j=0}^{m−1} μ(w_{1}^{j})) ∘ (μ(w_{1}) − 1) = (μ(w) − 1) is true, (μ(w_{1}) − 1) is a weak isomorphism. Because C(T̃) = C(S̃^{1}) ⊗_{Z} C(T̃′) is the mapping cone of id ⊗ (w_{1} − 1) : Z[Z] ⊗_{Z} C(T̃′) ⟶ Z[Z] ⊗_{Z} C(T̃′). Therefore C(T; V) is the mapping cone of

id_{V} ⊗_{Z[π_{1}(T)]} (id ⊗ (w_{1} − 1)) : V ⊗_{Z[π_{1}(T)]} (Z[Z] ⊗_{Z} C(T̃′)) ⟶ V ⊗_{Z[π_{1}(T)]} (Z[Z] ⊗_{Z} C(T̃′))

In dimension r this chain map is up to conjugation with an isometric isomorphism of Hilbert A-modules given by the weak automorphism (μ(w_{1}) − 1) : ⊕_{b(r)}V ⟶ ⊕_{b(r)}V, where b(r) is the rank of C_{r}(T̃′) over Z[π_{1}(T′)]. We conclude from the long weakly exact homology sequence (see theorem 6.1 and corollary 6.2) that C(T; V) is weakly V -acyclic. We derive from the sum formula for Reidemeister von Neumann torsion in theorem 7.16:

ρ(T; V) = ∑_{r≥0}(−1)^{r} · b(r) · [[(μ(w_{1}) − 1) : V ⟶ V]] = χ(T′) · [[(μ(w_{1}) − 1) : V ⟶ V]]

Now the claim follows since χ(T′) is zero for n ≥ 2 and one for n = 1.

Let X be a finite T -CW -complex for T the n-dimensional torus. (The definition of Γ-CW -complex above carries over to Γ a Lie group). Consider a unitary representation μ : π_{1}(X) ⟶ Iso_{A}(V)^{op}. The space X itself does not carry a canonical CW -complex structure, but has a well-defined simple structure in the sense of Lück [19], page 74 and section 7. Hence ρ(X; V) is well-defined, provided that X is V -acyclic (compare corollary 3.12). Denote by r the rank of the abelian group im(π_{1}(T) ⟶ π_{1}(X)). Let X^{H} be {x ∈ X | H ⊂ T_{x}} and define X^{>H} by {x ∈ X | H ⊂ T_{x}, H ≠ T_{x}}, where T_{x} ⊂ T denotes the isotropy group at x ∈ X. Let I be the set of subgroups H of T of dimension n − 1. Consider H ∈ I and a component C of X^{H}. Given a base point in C, the orbit through X is a circle and thus defines after choosing an orientation w(H, C) ∈ π_{1}(X). Make the assumption that the endomorphism μ(w(H, C) − 1) : V ⟶ V is a weak isomorphism. Hence it defines a class [[(μ(w(H, C) − 1) : V ⟶ V]] in K_{1}^{w}(A)^{Z/2}. Notice that the assumption as well as the class [[(μ(w(H, C) − 1)]] are independent of the choice of base point and choice of orientation of the orbit. Namely, given a weak automorphism f and an isomorphism g with the same source then [[f]] = [[f^{∗}]] = [[g ∘ f ∘ g^{−1}]] and changing the base point corresponds to conjugating with g and changing the orientation corresponds to taking the adjoint of f.

Theorem 3.20 Let T be the n-dimensional torus for n ≥ 1 and μ : π_{1}(T) ⟶ Iso_{A}(V)^{op} be a unitary representation. Let X be a finite T -CW -complex. Assume that there is an element w in im(π_{1}(T) ⟶ π_{1}(X)) such that w ≠ 0 and (μ(w) − 1) : V ⟶ V is a weak isomorphism. (This assumption is fullfilled for A = N(π_{1}(X)) and V the right regular representation l^{2}(π_{1}(X)), provided that w has infinite order). Then :

1.) X is weakly V -acyclic.

2.) If r ≥ 2, then ρ(X; V) vanishes.

3.) ρ(X; V) = ∑_{H∈I} ∑_{C∈π_{0}(X^{H})} χ((C, C ∩ X^{>H})/T) · [[(μ(w(H, C) − 1) : V ⟶ V]]

\underline{\underline{Proof}} : We show for any T -CW -subcomplex k : Y ⟶ X

ρ(Y; k^{∗}V) = ∑_{H∈I} ∑_{C∈π_{0}(X^{H})} χ((k^{−1}(C), k^{−1}(C) ∩ X^{>H})/T) · [[(μ(w(H, C) − 1) : V ⟶ V]]

The induction begin Y = ∅ is trivial. In the induction step we assume that Y is obtained from Z be attaching an equivariant cell T/H × D^{k}. Let j_{1} : T/H × D^{k} ⟶ X, j_{0} : T/H × S^{k−1} ⟶ X and j : Z ⟶ X be the obvious maps induced by k and denote by l : T/H ⟶ T/H × D^{k} the inclusion. We have j_{1}^{∗}V ≃ l^{∗}j_{1}^{∗}V ⊗ C and j_{0}^{∗}V ≃ l^{∗}j_{0}^{∗}V ⊗ C where C denotes the trivial representation. We derive from the product formula 3.16:

H_{∗}(T/H × D^{k}; j_{1}^{∗}V) = H_{k}(T/H; l^{∗}j_{1}^{∗}V) ⊗ H_{k}(D^{k}; C) H_{∗}(T/H × S^{k−1}; j_{1}^{∗}V) = H_{k}(T/H; l^{∗}j_{1}^{∗}V) ⊗ H_{k}(S^{k−1}; C) ρ(T/H × D^{k}; j_{1}^{∗}V) = ρ(T/H; l^{∗}j_{1}^{∗}V) ρ(T/H × S^{k−1}; j_{0}^{∗}V) = (1 + (−1)^{k}) · ρ(T/H, l^{∗}j_{1}^{∗}V)

We obtain from lemma 3.19 that H_{k}(T/H; l^{∗}j_{1}^{∗}V) vanishes and ρ(T/H; l^{∗}j_{1}^{∗}V) is zero, if H does not belong to I and is [[(μ(w(H, C) − 1) : V ⟶ V]] otherwise where C is a component of X^{H} which meets the new cell T/H × D^{k}. Since Z is j^{∗}V acyclic by induction hypothesis, we get from the long weakly exact Mayer Vietoris sequence 6.1 that Y is k^{∗}V -acyclic. One checks directly that the difference

∑_{H∈I} ∑_{C∈π_{0}(X^{H})} χ((k^{−1}(C), k^{−1}(C) ∩ X^{>H})/T) · [[(μ(w(H, C) − 1) : V ⟶ V]] − ∑_{H∈I} ∑_{C∈π_{0}(X^{H})} χ((j^{−1}(C), j^{−1}(C) ∩ X^{>H})/T) · [[(μ(w(H, C) − 1) : V ⟶ V]]

is (−1)^{k} · [[(μ(w(H, C) − 1) : V ⟶ V]], if H is in I and C is a component of X^{H} which meets the new cell G/H × T, and is zero if H is not in I. Now the claim follows from the sum formula 3.14.

Corollary 3.21 Let p : X ⟶ Y be a G-principal bundle over a finite CW -complex Y for G a connected compact Lie group such that the image of π_{1}(G) ⟶ π_{1}(X) is not finite. Then we have :

1.) X is l^{2}(π_{1}(X))-acyclic.

2.) ρ(X; l^{2}(π_{1}(X))) vanishes, if G is not S^{1}.

3.) Let G be S^{1} and w ∈ π_{1}(X) the image of the generator of π_{1}(S^{1}). Then :

ρ(X; l^{2}(π_{1}(X))) = χ(Y) · [[(w − 1) : l^{2}(π_{1}(X)) ⟶ l^{2}(π_{1}(X))]]

\underline{\underline{Proof}} : We restrict the G-action to the action of the maximal torus T and apply theorem 3.20. Notice that SO(3) and SU(2) are the only Lie groups of dimension bigger than one whose maximal torus has dimension one (see Bröcker-tom Dieck [2], page 185) and these Lie groups have finite fundamental groups.

Remark 3.22 The computations above apply in particular to orientable compact 3-manifolds admitting a fixed point free S^{1}-action since for such manifolds π_{1}(S^{1}) ⟶ π_{1}(M) is injective. Any Seifert fibre 3-manifold is covered by such a manifold. Hence any Seifert fibre 3-manifold is L^{2}-acyclic by corollary 3.20 and lemma 3.17.

Let (X, X_{1}) be a pair of finite CW -complexes and μ : π_{1}(X) ⟶ Iso_{A}^{op} be a unitary representation. Under the assumption that H_{∗}(X, X_{1}; V) vanishes, we can improve our invariant, namely, we can drop the symmetrization process.

The Hilbert dimension as defined in Cheeger-Gromov [8], section 1 induces a homomorphism 3.23 dim : K_{0}(A) ⟶ R

Then the Euler characteristic χ(X, X_{1}) ∈ Z is trivial, since χ^{A}(X, X_{1}; V) = χ(X, X_{1}) · [V] ∈ K_{0}(A) vanishes and dim(V) = 0 implies V = {0}. Choose a cellular bases for the finite free vZ[π]-chain complex C(X̃, X̃_{1}). As χ(X, X_{1}) vanishes, the bases yield an Z[π]- isomorphism Φ : C(X̃, X̃_{1})_{ev} ⟶ C(X̃, X̃_{1})_{odd}. It induces an isomorphism of finitely generated Hilbert Amodules Φ_{V} : C(X, X_{1}; V)_{ev} ⟶ C(X, X_{1}; V)_{odd}. As H_{∗}(X, X_{1}; V) is trivial, we can choose a weak chain contraction (γ, u) for C(X, X_{1}; V) (see definition 6.4). Let U_{V} ⊂ K_{1}^{w}(A) be the subgroup of trivial units, element of the shape [±μ(w) : V ⟶ V] for w ∈ π_{1}(X). Define the acyclic Reidemeister von Neumann torsion

3.24 ρ_{a}(X, X_{1}; V) ∈ K_{1}^{w}(A)/U_{V}

by ρ_{a}(X, X_{1}; V) = [Φ_{V} ∘ (uc + γ)_{odd}] − [u_{odd}]. We have to divide out U_{V}, as the cellular bases is not quite unique. One easily checks

3.25 ρ(X, X_{1}; V) = ρ_{a}(X, X_{1}; V) + ∗ρ_{a}(X, X_{1}; V)

If we have a push out of finite CW -complexes as indicated in 3.13 and X_{i} is j_{i}^{∗}V -acyclic for i = 0, 1, 2, then X is V -acyclic and we get :

3.26 ρ_{a}(X; V) = ρ_{a}(X_{1}; j_{1}^{∗}V) + ρ_{a}(X_{2}; j_{2}^{∗}V) − ρ_{a}(X_{0}; j_{0}^{∗}V)

Let (Y, Y_{1}) be a pair of finite CW -complexes and ν : π_{1}(Y) ⟶ Iso_{B}(W) be a unitary representation. Then (X, X_{1}) × (Y, Y_{1}) is V ⊗ W acyclic and we get using the pairing 2.8

3.27 ρ_{a}((X, X_{1}) × (Y, Y_{1}), V ⊗ W) = χ(Y, Y_{1}) · (ρ_{a}(X, X_{1}; V) ⊗ [W])

Theorem 3.11, lemma 3.17, theorem 3.20 and corollary 3.21 have obvious analogues for acyclic Reidemeister von Neumann torsion.

4. Free abelian fundamental groups

In this section we investigate the case of a free abelian fundamental group which is completely understood. Then our torsion invariant is related to the Alexander torsion.

Let (X, X_{1}) be a pair of finite CW -complexes and ϕ : π_{1}(X) ⟶ Z^{r} be an epimorphism into the free abelian group of rank r. Denote by X̂ ⟶ X the corresponding covering. As Z[Z^{r}] is an integral domain, we may consider its quotient field Z[Z^{r}]_{(0)}. We assume (X, X_{1}) is ϕ-acyclic,i.e., C(X̂, X̂_{1})_{(0)} = C(X̂, X̂_{1}) ⊗_{Z[Z^{r}]} Z[Z^{r}]_{(0)} is Z[Z^{r}]_{(0)}-acyclic. We can choose a Z[Z^{r}]_{(0)}-chain contraction γ of C(X̂, X̂_{1})_{(0)},i.e. a map γ of degree 1 satisfying c ∘ γ + γ ∘ c = id where c is the differential. We get an isomorphism of finitely generated free Z[Z^{r}]_{(0)}-modules (c_{(0)} + γ)_{odd} : C(X̂, X̂_{1})_{(0)odd} ⟶ C(X̂, X̂_{1})_{(0)ev}. Fix a Z[Z^{r}]-bases of C(X̂, X̂_{1}) . Let A be the invertible Z[Z^{r}]_{(0)}-matrix of (c_{(0)} + γ)_{odd} with respect to the induced Z[Z^{r}]_{(0)}-bases. Define the Alexander torsion

4.1 a(X, X_{1}; ϕ) ∈ Z[Z^{r}]_{(0)}^{∗}/Z[Z^{r}]^{∗}

by the determinant of A. Since we have divided out the units of Z[Z^{r}], this is welldefined and we get for a homotopy equivalence of pairs of connected finite CW -complexes f : (X′, X′_{1}) ⟶ (X, X_{1}) :

4.2 a(X′, X′_{1}; ϕ ∘ f_{∗}) = a(X, X_{1}; ϕ)

Suppose we have a push out of finite CW -complexes as indicated in 3.13 and X_{i} is ϕ ∘ j_{i∗}-acyclic for i = 0, 1, 2, then X is ϕ-acyclic and we obtain :

4.3 a(X; ϕ) = a(X_{1}, ϕ ∘ j_{1}) + a(X_{2}, ϕ ∘ j_{2}) − a(X_{0}, ϕ ∘ j_{0})

Let (X, X_{1}) and (Y, Y_{1}) be pairs of finite CW -complexes and ϕ : π_{1}(X) ⟶ Z^{r} and ψ : π_{1}(Y) ⟶ Z^{s} be epimorphisms such that (X, X_{1}) is ϕ-acyclic. Then (X, X_{1}) × (Y, Y_{1}) is ϕ × ψ-acyclic. Let i : Z[Z^{r}]_{(0)}^{∗}/Z[Z^{r}]^{∗} ⟶ Z[Z^{r+s}]_{(0)}^{∗}/Z[Z^{r+s}]^{∗} be the inclusion. The product formula for Alexander torsion is given by:

4.4 a((X, X_{1}) × (Y, Y_{1}); ϕ × ψ) = χ(Y, Y_{1}) · i(a(X, X_{1}; ϕ))

Let ν : Z^{r} ⟶ L^{2}(T^{r}) be the right regular representation of Z^{r} where L^{2}(T^{r}) is the Hilbert space of L^{2}-integrable functions from the r-dimensional torus into C ∪ {∞}. Recall that N(Z^{r}) is L^{∞}(T^{r}) and L^{2}(T^{r}) can be identified with l^{2}(N(Z^{r})) = l^{2}(Z^{r}). We get from ϕ a unitary representation \overline{ϕ} : π_{1}(X) ⟶ Iso_{N(Z^{r})}(L^{2}(T^{r})). Suppose that (X, X_{1}) is \overline{ϕ}-acyclic. Then the acyclic Reidemeister von Neumann torsion is defined (see 3.24). By theorem 2.5 it takes values in :

ρ_{a}(X, X_{1}; \overline{ϕ}) ∈ Inv(T^{r})/U′

where U′ is the subgroup of elements of the form

T^{r} ⟶ C (z_{1}, z_{2} … z_{r}) ↦ ±z_{1}^{a_{1}} · z_{2}^{a_{2}} · . . . · z_{r}^{a_{r}}

for integers a_{1}, a_{2}, … a_{r}. There is an obvious injection

4.5 i : Z[Z^{r}]_{(0)}^{∗}/Z[Z^{r}]^{∗} ⟶ Inv(T^{r})/U′

Theorem 4.6 We get under the conditions above :

1.) (X, X_{1}) is ϕ-acyclic if and only if (X, X_{1}) is \overline{ϕ}-acyclic.

2.) Suppose that (X, X_{1}) is ϕ-acyclic. Then :

i(a(X, X_{1}; ϕ)) = ρ_{a}(X, X_{1}; \overline{Φ})

\underline{\underline{Proof}} : Suppose that (X, X_{1}) is ϕ-acyclic. Let γ be a chain contraction of the Z[Z^{r}]_{(0)}- chain complex C(X̂, X̂_{1})_{(0)}. Choose an element u ∈ Z[Z]^{r} with u ≠ 0 and and a Z[Z^{r}]-chain homotopy γ̂ : C(X̂, X̂_{1})_{∗} ⟶ C(X̂, X̂_{1})_{∗+1} satisfying u · id ∘ γ = γ̂_{(0)}. Then we have:

a(X, X_{1}; ϕ) = det ((c_{(0)} + γ)_{odd}) = det (u_{odd} ∘ (c_{(0)} + γ)_{odd}) /det(u_{odd}) = det((uc + γ̂)_{odd})/det(u_{odd})

Tensoring γ̂ and u · id with the regular representation yields a weak chain contraction of C(X, X_{1}; \overline{ϕ}) because of u · id is a weak isomorphism by lemma 2.7. Therefore (X, X_{1}) is \overline{ϕ}-acyclic and i(a(X, X_{1}; ϕ)) is ρ_{a}(X, X_{1}; \overline{ϕ}). It remains to show that \overline{ϕ}-acyclic implies ϕ-acyclic. For n ≥ 0 choose a finitely generated free Z[Z^{r}]-module C_{n} together with a Z[Z^{r}]- map j_{n} : C_{n} ⟶ H_{n}(C(X̂, X̂_{1}; ϕ)) such that (j_{n})_{(0)} is an isomorphism. Lift j_{n} to a Z[Z^{r}]- homomorphism i_{n} : C_{n} ⟶ ker(d_{n}). We have construced a finite free Z[Z^{r}]-chain complex C with trivial differentials and a chain map i : C ⟶ C(X̂) such that H(i)_{(0)} is an isomorphism. Then Cone(i) is Z[Z^{r}]_{(0)}-acyclic. By the argument above and the long weakly exact homology sequence 6.1 the induced N(Z^{r})-chain map from C ⊗_{Z[Z^{r}]} L^{2}(T^{r}) to C(X, \overline{ϕ}) is a weak homology equivalence. But C is Z[Z^{r}]_{(0)}-acyclic, if and only if C ⊗_{Z[Z^{r}]} L^{2}(T^{r}) is weakly acyclic, as C has trivial differentials. Thus \overline{ϕ}-acyclic implies ϕ-acyclic.

Example 4.7 (Alexander polynomial) Let L be an oriented link in an oriented homology 3-sphere M. Denote by M(L) the complement of a tubular neighbourhood of the link. If r is the number of components, the Hopf homomorphism

ϕ : π_{1}(M(L)) ⟶ Z^{r}

maps a loop w to the r-tuple of integers given by the linking numbers of w and the various components of the link in M. Then the Alexander torsion of M(K) defined in 4.1 is just the Alexander polynomial of the link L provided that M(L) is ϕ-acyclic. Notice that the Alexander polynomial is defined to be zero if M(L) is not ϕ-acyclic (see Burde-Zieschang [3] or Turaev [29]). Hence we see by theorem 4.6 that the acyclic Reidemeister von Neumann torsion of the link complement contains the same information as the Alexander polynomial. This identification is very similar to the one in Milnor [21].

The acyclic Reidemeister von Neumann torsion provides the possibility of extending the definition of Alexander polynomial to other coverings of the link complement than abelian coverings. We will investigate this at a different place.

Example 4.8 (Lefschetz zeta function of selfmaps) Given a selfmap f : X ⟶ X of a space of the homotopy type of a finite CW -complex, define the Lefschetz zeta function of f by the formal power series with leading coefficient zero

4.9 L(f) = ∑_{n≥1} \frac{λ(f^{n})}{n}t^{n}

where λ(f^{n}) is the Lefschetz index of the self map f^{n} of X. Denote by T_{f} the mapping torus of f. This space is obtained from X × [0, 1] by the identification (x, 1) ∼ (f(x), 0). There is a natural map T_{f} ⟶ S^{1} inducing a homomorphism ϕ : π_{1}(T_{f}) ⟶ Z. The mapping torus is ϕ-acyclic and the Alexander torsion a(T_{f}; ϕ) can be computed by

a(T_{f}; ϕ) = ∏_{n≥0} det(z − H_{n}(f))^{(−1)^{n}}

Given an endomorphism g : C^{r} ⟶ C^{r} of C-modules, we obtain :

det(id − t · g) = exp(∑_{n≥1} \frac{tr(g^{n})}{n} · t^{n})^{−1}

Hence exp(L(f)) is a rational function \frac{p(t)}{q(t)} in t and \frac{q(z^{−1})}{p(z^{−1})} · z^{χ(X)} represents a(T_{f}, ϕ). This observation is due to Milnor [21]. Theorem 4.6 says that the acyclic Reidemeister von Neumann torsion of the mapping torus and the Lefschetz Zeta function determine one another.

5. Torsion invariants for Riemannian Γ-manifolds

In this section we define Reidemeister von Neumann torsion for Riemannian manifolds with a cocompact properly discontinuous action of a dicrete group Γ by isometries. We investigate Poincaré duality and relate Reidemeister von Neumann torsion to classical Reidemeister torsion and to analytic torsion.

The definition of Reidemeister von Neumann torsion for Riemannian manifolds needs some preparation. We need two ingredients, an equivariant triangulation theorem and a deRham isomorphism. Before we establish these and give the final definition, we investigate an example to illustrate why the Riemannian metric comes in if the manifold is not weakly acyclic. The naive approach to choose a triangulation and to define the Reidemeister von Neumann torsion of a manifold by the Reidemeister von Neumann torsion of the triangulation does only work if the manifold is weakly acyclic. The reason is the appearance of the term depending on the map induced on homology in the formula in theorem 3.11 where Whitehead torsion and Reidemeister von Neumann torsion are compared. If one has two triangulations, they differ by a simple Γ-homotopy equivalence f. Simple means that its equivaraint Whitehead torsion τ^{Γ}(f) is zero. However, the map induced on homology represents in general a non-trivial class in the K-theory unless there is no homology. This will force us to “fix” the homology or cohomology of the given triangulation by comparing it with the space of harmonic l^{2}-forms by the deRham isomorphism.

Example 5.1 Consider the 1-dimensional manifold D^{1} given by the unit intervall. In this example Γ is trivial. Consider for n ≥ 1 the triangulation of D^{1} with \{\{\frac{k}{n}\} | 0 ≤ k ≤ n\} as set of 0-simplices and \{[\frac{k}{n}, \frac{k+1}{n}] | 0 ≤ k < n\} as set of 1-simplices. Let D^{1}(n) be the CW - complex given by this explicit triangulation. We want to compute its Reidemeister von Neumann torsion introduced in 3.9 for A = C and V = C.

The cellular C-chain complex C = C(D^{1}(n)) looks like :

A = ( \begin{matrix}1 & 0 & 0 & … & 0 \\ −1 & 1 & 0 & … & 0 \\ 0 & −1 & 1 & … & 0 \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & 0 & … & −1 \end{matrix} ) : ⊕_{n}C ⟶ ⊕_{n+1}C

Let N be the element (1, 1, …, 1) ∈ C_{0}. Its norm with respect to the standard Hilbert structure is (n + 1)^{1/2} and it is orthogonal to the image of the differential. Hence we obtain an identification of Hilbert spaces by :

C ⟶ H_{0}(C) λ ↦ λ · (n + 1)^{−1/2} · N

Let i_{0} : C ⟶ C_{0} be the C-map sending λ to λ · (n + 1)^{−1/2} · N. We get with the identification above a C-chain map i : H(C) ⟶ C satisfying H(i) = id. The Reidemeister von Neumann torsion of D^{1}(n)

ρ(D^{1}(n)) ∈ (C^{∗})^{Z/2}

is given by ρ(D^{1}(n)) = det(B(n + 1, (n + 1)^{−1/2})) · \overline{det(B(n + 1, (n + 1)^{−1/2}))}, if we denote by B(n + 1, λ) the following (n + 1)-(n + 1)-matrix :

B(n + 1, λ) = ( \begin{matrix}1 & 0 & 0 & … & 0 & λ \\ −1 & 1 & 0 & … & 0 & λ \\ 0 & −1 & 1 & … & 0 & λ \\ ⋮ & ⋮ & ⋮ & ... & ⋮ & ⋮ \\ 0 & 0 & 0 & … & −1 & λ \end{matrix} )

Developing the determinant after the first row gives :

det(B(n + 1, λ)) = 1 · det(B(n, λ)) + (−1)^{n} · λ · (−1)^{n} = det(B(n, λ)) + λ

By induction over n we get det(B(n + 1, λ)) = (n + 1) · λ. This shows :

ρ(D^{1}(n)) = (n + 1)

In particular we see that ρ(D^{1}(n)) does depend on the triangulation.

Next we deal with the equivariant triangulation theorem. Let Γ be a discrete group acting cocompact and properly discontinuosly on a smooth manifold M. Recall that properly discontinuous means that for each pair of points (x, y) in M there are neighborhoods U_{x} and U_{y} such that {γ ∈ Γ | γU_{x} ∩ U_{y} ≠ ∅} is finite, and the action is cocompact if the quotient Γ\M is compact. Our assumptions imply that each point x in int(M) resp. ∂M has a neighborhood U_{x} such that U_{x} is Γ_{x}-invariant and Γ_{x}-diffeomorphic to the unit disk of an orthogonal Γ_{x}-representation resp. half unit disk in the direct sum of an orthogonal Γrepresentation and the trivial representation R and {γ ∈ Γ | γU_{x} ∩ U_{x} ≠ ∅} = Γ_{x} where Γ_{x} is the isotropy subgroup of the point x. Such a neighborhood will be called a nice smooth neighborhood. If we are working in the P L-category, it will be called an nice P L-neighborhood. Because Γ\M is compact, there is a finite set of points {x_{1}, x_{2}, … x_{m}} in M with smooth nice neighborhoods {U_{x_{1}}, U_{x_{2}}, … U_{x_{m}}} such that M = Γ (U_{x_{1}} ∪ U_{x_{2}} ∪ … ∪ U_{x_{m}}). Such a family will be called an Γ-equivariant covering basis for M. A nice simplicial Γ-complex is a locally finite simplicial complex on which Γ acts simplicially, cocompact and properly discontinuously such that the regular neighborhood of each point is a nice P L-neighborhood.

Definition 5.2 A smooth Γ-equivariant triangulation of M is a pair of nice simplicial Γcomplexes (K, ∂K) and a Γ-equivariant homeomorphism (f, ∂f) : (K, ∂K) ⟶ (M, ∂M) of pairs such that h is smooth on each simplex.

Theorem 5.3 There is a smooth Γ-equivariant triangulation of M. Given two such smooth Γ-equivariant triangulations (f_{i}, ∂f_{i}) : (K_{i}, ∂K_{i}) ⟶ (M, ∂M) for i = 1, 2, the composition (h_{2}^{−1} ∘ h_{1}, ∂h_{2}^{−1} ∘ ∂h_{1}) is Γ-isotopic to a Γ-equivariant P L-homeomorphism of pairs. In particular its equivariant Whitehead torsion vanishes.

Illman [16] gives a proof for finite Γ and it generalizes easily to our situation. Notice that the theorem follows from Illman’s provided that Γ contains a normal subgroup π of finite index acting freely. Namely, one applies Illman’s result to the action of Γ/π on π\M and lifts the triangulation to M.

Here is the basic idea. For simplicity assume ∂M = ∅. Choose a Γ-equivariant covering basis for M. Then K_{1} = Γ ×_{Γ_{x_{2}}} U_{x_{2}} has a structure of a nice simplicial Γ-complex. Using the basic affine approximation method of Whitehead move U_{x_{2}} slightly such that the intersection U_{x_{2}} ∩ Γ ×_{Γ_{x_{1}}} K_{1} is a Γ-subcomplex of both. Then K_{2} = K_{1} ∪ Γ ×_{Γ_{x_{1}}} U_{x_{1}} has a structure of a nice simplicial Γ-complex. Continueing in this way yields eventually K = K_{m}. The uniqueness result follows in a similar way directly from Whitehead’s argument.

Next we need a generalized deRham isomorphism. To state this we need some preparation. Let M be a Riemannian manifold of dimension m whose boundary ∂M is the disjoint union ∂_{0}M ∐ ∂_{1}M. Let Γ be a discrete group acting by isometries and cocompact, properly discontinuous on M. Let A be a finite von Neumann algebra and V be a finitely generated Hilbert module over A. Consider a unitary representation μ : Γ ⟶ Iso_{A}(V)^{op}. We will assume that our manifold M is orientable. We can do this without loss of generality since otherwise one can pass to the orientation covering of M. We define the orientation homomorphism

5.4 w = Γ ⟶ {±1}

by sending γ ∈ Γ to +1 resp. ±1 if γ acts orientation preserving resp. reversing.

The fibre over x ∈ M of the Hilbert bundle Alt^{p}(T M, V) consists of alterating pforms on T_{x}M with coefficients in V. The Hilbert structure on the fibres comes from the one on V and the Riemannian metric. Recall that the space Λ^{p}(M; V) of p-forms on M with coefficients in V is the space of smooth sections in Alt^{p}(T M, V). Since Γ acts on M by isometries, Γ acts by isomorphisms of Hilbert bundles on Alt^{p}(T M, V). This induces a action of Γ on Λ^{p}(M; V). Let Λ_{Γ}^{p}(M; V) be the Γ-fixed point set of Λ^{p}(M; V).

Denote by ^{w}V the w-twisted representation given by w(γ) · vγ for γ ∈ Γ and v ∈ V. Let ∗Λ^{p}(M; V) ⟶ Λ^{m−p}(M;^{w} V) be the Hodge star operator, d : Λ^{p}(M; V) ⟶ Λ^{p+1}(M; V) be the differential. We have the product ∧ : Λ^{p}(M; V) ⊗ Λ^{q}(M; V) ⊗ Λ^{p+q}(M;^{w} C) and the volume form dM ∈ Λ^{n}(M;^{w} C). The standard definitions for forms with coefficients in C carry directly over to the case of coefficients in V. Notice that d is compatible with the Γ-action as well as ∗ and dM is Γ-invariant.

We obtain for ω and η in Λ^{p}(T M, V) a function ⟨ω, η⟩_{x} on M from the Hilbert bundles structure on Alt^{p}(T M, V). One easily checks from the definitions ω ∧ (∗η) =< ω, η >_{x} ·dM. Let F be a fundamental domain for the Γ-action on M. Given ω and η in Λ_{Γ}^{p}(M; V), define their inner product

⟨ω, η⟩ = ∫_{F}⟨ω, η⟩_{x}dM = ∫_{F} ω ∧ (∗η)

This is independent of the choice of fundamental domain, since ω and η are Γ-invariant. The adjoint δ^{p} : Λ^{p}(M; V) ⟶ Λ^{p−1}(M; V) of the differential d^{p−1} is (−1)^{mp+p+1} ∗ d^{m−p}∗. Define the Laplace operator Δ^{p} : Λ^{p}(M; V) ⟶ Λ^{p}(M; V) by d^{p−1}δ^{p} + δ^{p+1}d^{p}.

Given a p-form ω ∈ Λ^{p}(M; V), let ω_{tan} be the p-form on ∂M coming from restriction with T i : T ∂M ⟶ T M for the inclusion i. Let ω_{nor} be the (p − 1)-form ∗_{∂M}(∗_{M}ω)_{tan}. We say ω satisfies Dirichlet boundary conditions on ∂_{0}M if ω_{tan} = 0 and (δω)_{tan} = 0 on ∂_{0}M holds, and Neumann boundary conditions on ∂_{1}M, if ω_{nor} = 0 and (dω)_{nor} = 0 on M_{2} holds. Let Λ^{p}(M, ∂_{0}M; V) be the subspace of Λ^{p}(M; V) consisting of p-forms satisfying Dirichlet boundary conditions on ∂_{0}M and Neumann boundary conditions on ∂_{1}M. The space of harmonic p-forms H^{p}(M, ∂_{0}M; V) is the subspace of Λ^{p}(M, ∂_{0}M; V) given by pforms in the kernel of the Laplace operator. Define the space of invariant harmonic pforms H_{Γ}^{p}(M, ∂_{0}M; V) to be the fixed point set (H^{p}(M, ∂_{0}M; V))^{Γ}. It inherits a pre Hilbert structure from Λ_{Γ}^{p}(M; V)

Let (f; f_{0}, f_{1}) : (K; K_{0}, K_{1}) ⟶ (M; ∂_{0}M, ∂_{1}M) be a smooth Γ-equivariant triangulation. Consider a Γ-invariant harmonic p-form ω in H_{Γ}^{p}(M, ∂_{0}M; V). Let σ ⊂ K be a psimplex. The integral ∫_{σ} f^{∗}ω is an element in V. Define the de Rham homomorphism

A^{p}(f, f_{0}; V) : H_{Γ}^{p}(M, ∂_{0}M; V) ⟶ H^{p}(K, K_{0}; f^{∗}V)

by sending ω to the class represented by σ̃ ↦ ∫_{σ} f^{∗}ω.

Theorem 5.5 The deRham homomorphism

A^{p}(f, f_{0}; V) : H_{Γ}^{p}(M, ∂_{0}M; V) ⟶ H^{p}(K, K_{1}; f^{∗}V)

is a well-defined isomorphism of finitely generated Hilbert A-modules. Given another equivariant triangulation (g; g_{0}, g_{1}) : (L; L_{0}, L_{1}) ⟶ (M; ∂_{0}M, ∂_{1}M), we get :

H^{p}((g, g_{0})^{−1} ∘ (f, f_{0}); g^{∗}V) ∘ A^{p}(g, g_{0}; V) = A^{p}(f, f_{0}; V)

If A is the field of complex numbers C and Γ acts freely, the proof of the theorem above can be found in Ray and Singer [23]. Suppose Γ acts freely, M has no boundary, A = N(Γ) and V is the regular right representation l^{2}(Γ). Let L^{2}Λ_{c}^{p}(M) be the Hilbert completion of the space Λ_{c}^{p}(M) of smooth p-forms on M with compact support. Let L^{2}Λ_{Γ}^{p}(M; l^{2}(Γ)) be the Hilbert space completion of Λ_{Γ}^{p}(M; l^{2}(Γ)). There is a canonical isometric isomorphism of Hilbert N(Γ)-modules

5.6 β : L^{2}Λ^{p}(M) ⟶ L^{2}Λ_{Γ}^{p}(M; l^{2}(Γ))

Given ω in Λ^{p}(M), β(ω) assigns to the p-tuple of tangent vectors (v_{1}, v_{2}, …, v_{p}) of M at x the element in l^{2}(Γ) given by ∑_{γ∈Γ} ω_{γ^{−1}x} (T_{x}γ^{−1}(v_{1}), T_{x}γ^{−1}(v_{2}), …, T_{x}γ^{−1}(v_{p})) · γ, where T_{x}γ is the differential of the map given by multiplication with γ. In particular we obtain an identification of the Hilbert space H_{Γ}^{p}(M, ∂_{0}M; V) of invariant harmonic p-forms with coefficients in l^{2}(Γ) with the Hilbert space of smooth harmonic L^{2}-integrable harmonic forms on M. Under the assumptions and identifications the theorem above reduces to the result of Dodzuik [11] applied to Γ\M. The proof of the theorem above is an extension of Dodzuik’s argument combined with the technique or Ray and Singer [23] to handle the boundary.

Now we can define Reidemeister von Neumann torsion for (M, ∂_{0}M) with coefficients in V. Choose a Γ-equivariant tringulation (f; f_{0}, f_{1}) : (K; K_{0}, K_{1}) ⟶ (M, ∂_{0}M, ∂_{1}M).

Definition 5.7 Define the Reidemeister von Neumann torsion

ρ(M, ∂_{0}M; V) ∈ K_{1}^{w}(A)^{Z/2}

by ρ(K, K_{1}; f^{∗}V) − ∑_{p≥0} (−1)^{p} · [[A^{p}(f, f_{0}; V) : H_{Γ}^{p}(M, ∂_{0}M; V) ⟶ H^{p}(K, K_{0}; f^{∗}V)]].

We have to show that this is independent of the choice of equivariant triangulation. If (g; g_{0}, g_{1}) : (L; L_{0}, L_{1}) ⟶ (M, ∂_{0}M, ∂_{1}M) is a second equivariant triangulation, the Whitehead torsion of (g, g_{0})^{−1} ∘ (f, f_{0}) vanishes. We drive from theorem 3.11

ρ(L, L_{0}; g^{∗}V) − ρ(K, K_{0}; f^{∗}V) = −t(H((g, g_{0})^{−1} ∘ (f, f_{0}); g^{∗}V))

Because of lemma 3.10 we have

t(H_{p}((g, g_{0})^{−1} ∘ (f, f_{0}); g^{∗}V)) = t(H^{p}((g, g_{0})^{−1} ∘ (f, f_{0}); g^{∗}V))

We derive from theorem 5.5 and lemma 7.8

t(H^{p}((g, g_{0})^{−1} ∘ (f, f_{0}); g^{∗}V)) = t(A(f, f_{0}; V)) − t(A(g, g_{1}; V))

This implies :

ρ(K, K_{0}; f^{∗}V) − t(A(f, f_{0}; V)) = ρ(L, L_{0}; g^{∗}V) − t(A(g, g_{0}; V))

Now the claim follows as we have

t(A(f, f_{0}; V)) = ∑_{p≥0} (−1)^{p} · [[A^{p}(f, f_{0}; ) : H_{Γ}^{p}(M, ∂_{0}M; V) ⟶ H^{p}(K, K_{0}; f^{∗}V)]]

Example 5.8 Equip D^{1} with a Riemannian metric. Then the constant function on D^{1} with value vol(D^{1})^{−1/2} has norm one in the space of harmonic 0-forms. Hence ρ(D^{1}) is vol(D^{1}) (compare with example 5.1).

Example 5.9 Reidemeister von Neumann-torsion is sensitive to A and V. Consider S^{1} with any Riemannian metric. If we use A = N(Z) and V = l^{2}(Z), the Reidemeister von Neumann torsion is independent of the Riemannian metric and represented in K_{1}^{w}(N(Z)) = Inv(S^{1}) by the function (z − 1)(z^{−1} − 1) (see lemma 3.19 and theorem 2.5). If we choose A = C and V = C the Reidemeister von Neumann torsion is real number in K_{1}^{w}(C)^{Z/2} = R given by the square of the volume of S^{1} and does depend on the Riemannian metric.

Remark 5.10 The results of section 3 for CW -complexes also apply to manifolds. In the sum formula 3.14 the extra term coming from long homology sequences is to be understood with respect to the harmonic forms. The product formula 3.16 and restriction formula 3.17 hold as they stand. In theorem 3.20 and corollary 3.21 one has to substitute finite T -CW - complex by compact smooth manifold with smooth T -action. In the Lie group one has to be more careful with the triangulation theorem. However, we obtain at least a simple structure on M compatible with restriction by Lück [19], lemma 7.4.5, what suffices for theorem 3.20 as the spaces under consideration are V -acyclic.

Finally we examine Poincaré duality following Lück [20] page 26ff. In view of theorem 5.3 we can identify (M, ∂_{0}M) with a triangulation. Let m be dimension of M. Let σ_{1}, σ_{2}, … σ_{r} be the m-simplices of Γ\M. Choose for any 1 ≤ i ≤ r a lift σ̃_{i} in M. Then we obtain a cycle in C_{m}(M, ∂M,^{w} C) by ∑_{i=1}^{r} σ̃_{i} ⊗_{ZΓ} ^{w}C. Its class in H_{m}(M, ∂M;^{w} C) is independent of the choice of the lifts and called the fundamental class [M]. We have a cellular Γ-equivariant approximation of the diagonal map M ⟶ M × M by an equivariant version of the cellular approximation theorem. Hence the standard definitions for cup and cap products go through and we obtain a chain map of Hilbert chain complexes well defined up to homotopy:

5.11 ∩[M] : C^{m−p}(M, ∂_{1}M,^{w} V) ⟶ C_{p}(M, ∂_{0}M, V)

This map turns out to be a chain homotopy equivalence and we refer to it as the Poincaré chain homotopy equivalence. One proof of Poincaré duality for a compact manifold uses handlebody decompositions and a Mayer-Vietoris argument to reduce the claim to a handle body itself. In our case one can use an equivariant handlebody decomposition. Since all isotropy groups are finite, this reduces the claim to an equivariant handle for a finite group and in that case Poincaré duality is well-known. Now we can define:

Definition 5.12 Define the Hilbert Poincaré torsion or briefly Poincaré torsion

ρ_{pd}(M, ∂_{0}M; V) ∈ K_{1}^{w}(A)^{Z/2}

by the torsion of ∩[M] : C^{m−∗}(M, ∂_{1}M;^{w} V) ⟶ C_{∗}(M, ∂_{0}M; V) introduced in 7.4.

If Γ acts freely, then there is another proof of Poincaré duality using dual cells. This does not go through if the action is non-free, because the dual cell of an equivariant cell is not an equivariant cell in general anymore. The Hilbert Poincaré torsion measures this failure. In particular it vanishes if Γ acts freely. Analogously to Lück [20] proposition 5.24 there is a local formula for Hilbert Poincaré torsion in terms of the Euler characteristics of the components of the H-fixed points sets and their normal slices for finite subgroups H of Γ. As in Lück [20] p.26ff one shows:

Theorem 5.13 (Poincaré Duality)

1.) ρ(M, ∂_{0}M; V) + (−1)^{m} · ρ(M, ∂_{1};^{w} V) = ρ_{pd}(M, ∂_{0}M; V)

2.) ρ_{pd}(M, ∂_{0}M; V) = (−1)^{m} · ρ_{pd}(M, ∂_{1}M; V)

3.) χ^{A}(M, ∂_{0}M; V) = (−1)^{m} · χ^{A}(M, ∂_{1}M; V)

4.) If Γ acts freely, ρ_{pd}(M, ∂_{0}M; V) vanishes.

Suppose M is closed and orientable and Γ acts orientation preserving. Then we obtain 2 · ρ(M; V) = ρ_{pd}(M; V), if M has even dimension, and 2 · ρ_{pd}(M; V) = 0, if M has odd dimension. In particular 2 · ρ(M; V) vanishes, if Γ acts freely and orientation preserving and M is an orientable closed Riemannian manifold of even dimension.

Next we want to relate the Reidemeister von Neumann torsion to other well-known invariants Example 5.14 Let X be a finite CW -complex with finite fundamental group π. Consider a finite-dimensional complex representation μ : π ⟶ Iso_{C}(V). This means that we have choosen A to be C. Suppose that X is V -acyclic. The acyclic Reidemeister von Neumann torsion ρ_{a}(X; V) defined in3.24 is essentially a complex number, more precisely, it is an element in C^{∗}/U, where U is given by {det(±μ(g) : V ⟶ V) | g ∈ π}. This is just the classical Reidemeister torsion which was used to classify Lens spaces (see Cohen [9] and Reidemeister [24]).

Example 5.15 Let M be a compact smooth Riemannian manifold with fundamental group π and μ : π ⟶ Iso_{R}(V) be an orthogonal finite-dimensional representation. Then V ⊗_{R} C is a unitary representation of π over the von Numann algebra A = C. The Reidemeister von Neumann torsion ρ(M; V) defined in 5.7 reduces to a positive real number, because K_{1}^{w}(C) is R^{∗}. This is the square of the P L-torsion ρ_{pl}(M; V) introduced by Ray and Singer [23]. Ray and Singer [23] defined also the analytic counterpart ρ_{an}(M; V) based on the zeta function of the Laplace operator. Cheeger [5] and Müller [22] proved independently that ln(ρ_{pl}(M; V)) = ρ_{an}(M; V), provided that M is closed. If M has a boundary, a correction term proportional to the Euler characteristic of the boundary comes in (see Lück [20]).

Example 5.16 The notions of Ray and Singer and the results of Cheeger and Müller were extended to compact Riemannian G-manifolds by Lott and Rothenberg [18] and Lück[20] for a finite group G. Analytic proofs of deRham’s theorem are established that the unit spheres of two orthogonal representations are G diffeomorphic if and only if the representations themselves are isomorphic (cf. deRham [25]). The notion of Reidemeister von Neumann torsion reduces to the notions in Lott and Rothenberg [18] and Lück [20] if Gamma is finite. The meaning of equivariant Reidemeister torsion for finite transformation groups is worked out in Rothenberg [26] and Lück [19].

Remark 5.17 In view of the results above one should try to give an analytic interpretation of the Reidemeister von Neumann torsion. In this context the definition of a real number, the analytic L^{2}- torsion by Lott [17] is very interesting. His invariant should be up to a constant the Fuglede Kadison determinant (see Fuglede and Kadison [14]) applied to the Reidemeister von Neumann torsion. The analytic L^{2}-torsion of Lott is only defined if all the Novikov-Shubin invariants of the manifold are positive whereas the Reidemeister von Neumann torsion is always defined. However, the vanishing of all Novikov-Shubin invariants is needed to apply the Fuglede Kadison determinant to the Reidemeister von Neumann torsion. In a forthcoming preprint the first author and Lott will prove that the Novikov-Shubin invariants are positive for a manifold satisfying the condition of theorem 3.20 and for a compact Haken 3-manifold whose boundary is empty or is a disjoint union of incompressible tori. In particlular under the conditions of theorem 3.20 the Reidemeister von Neumann torsion vanishes so that one would expect that also the analytic L^{2}-torsion vanishes.

We also refer in this context to Carey-Mathai [4], where a real number based on the Kadison-Fuglede determinant is assigned to a smooth closed L^{2}-acyclic manifold whose Novikov-Shubin invariants are positive. Their invariant when defined is just the Fuglede Kadison determinant of the Reidemeister von Neumann torsion. In the original version which appeared in Contemporary Math., vol 105 (1989) there is an error in the definition of their invariant. This has been corrected in the version we refer to.

6. Hilbert A-chain complexes

In this section we generalize basic facts about long homology sequences and chain contractions for chain complexes over a ring to chain complexes over a finite von Neumann algebra A with a finite normal and faithful trace.

The main difficulty lies in the definition of homology as H(C) = ker(c)/clos(im(c)) which does not coincide with the ordinary definition as ker(c)/im(c). Recall that we have to divide out the closure of the image to ensure that the homology consists again of Hilbert A-modules. In the classical situation the constrution of torsion invariants for acyclic chain complexes is based on the existence of a chain contraction γ, i.e., a chain homotopy γ : id ∼ 0. We must look for a weaker notion of chain contraction in the case of Hilbert A-chain complexes which are weakly acyclic. Before we deal with this question, we need some preliminaries. In the sequel module resp. chain complex means Hilbert A-module resp. Hilbert A-chain complex unless explicitly stated differently. Recall that a chain complex is finite if C_{n} is finitely generated for all n ∈ Z and there is a number N such that C_{n} = 0 for |n| ≥ N.

Theorem 6.1 (Cheeger-Gromov [6]) Let 0 ⟶ C ⟶^{i} D ⟶^{p} E ⟶ 0 be an exact sequence of finite chain complexes. Then there is a weakly exact long natural homology sequence :

⟶ H_{n+1} ⟶^{∂} H_{n}(C) ⟶^{H_{n}(i)} H_{n}(D) ⟶^{H_{n}(p)} H_{n}(E) ⟶^{∂} …

\underline{\underline{Proof}} : We firstly define ∂. Since the differential d_{n} of D_{n} maps ker(p_{n}) to ker(p_{n−1}), we may write d_{n} : D_{n} ⟶ D_{n−1} as :

( \begin{matrix}d_{n}^{1} & d_{n}^{2} \\ 0 & d_{n}^{3} \end{matrix} ) : ker(p_{n}) ⊕ ker(p_{n})^{⊥} ⟶ ker(p_{n−1}) ⊕ ker(p_{n−1})^{⊥}

The induced morphisms i_{n} : C_{n} ⟶ ker(p_{n}) and p_{n} |: ker(p_{n})^{⊥} ⟶ E_{n} are isomorphisms by the open mapping theorem. Consider the composition

∂′ = i_{n−1}^{−1} ∘ d_{n}^{2} ∘ (p_{n} |)^{−1} : E_{n} ⟶ C_{n−1}

We claim ∂′_{n}(ker(e_{n})) ⊂ ker(c_{n−1}). Namely, regard x ∈ ker(e_{n}), y ∈ ker(p_{n})^{⊥} and z ∈ C_{n−1} such that p_{n}(y) = z and d_{n}^{2}(y) = i_{n}(z) holds. We have by definition ∂′(x) = z so that we have to check c_{n−1}(z) = 0. Since i_{n−2} is injective, it suffices to show i_{n−2} ∘ c_{n−1}(z) is zero. This element is just d_{n−1} ∘ d_{n}^{2}(y). Now (p_{n−1} |) ∘ d_{n}^{3}(y) is e_{n} ∘ p_{n}(0, y) = e_{n}(x) and hence vanishes. Therefore d_{n}^{3}(y) is zero. This implies d_{n−1} ∘ d_{n}^{2}(y) = d_{n−1} ∘ d_{n}(0, y) = 0. We have shown ∂′_{n}(ker(e_{n})) ⊂ ker(c_{n−1}). Analogously one proves ∂′(im(e_{n+1})) ⊂ im(c_{n}). Hence we can define ∂ to be the map induced by ∂′.

Although the definition of the long homology sequence reduces in the case, where A is C[G] for a finite group G to the classical situation, the verification of weak exactness is much harder and not just a diagram chase. We refer to Cheeger - Gromov [6] theorem 2.1.

For a chain map f : C ⟶ D, denote by Cone(f) its mapping cone given by

c_{n} = ( \begin{matrix}−c_{n−1} & 0 \\ f_{n−1} & d_{n} \end{matrix} ) : C_{n−1} ⊕ D_{n} ⟶ C_{n−2} ⊕ D_{n−1}

Recall that weak homology equivalence is a chain map inducing a weak isomorphism on the homology. A weak chain isomorphism is a chain map f such that f_{n} is a weak isomorphism for each n ∈ Z.

Corollary 6.2

1.) Regard the following commutative diagram of finite chain complexes with weakly exact rows.

[figure: 0; C; i; D; p; E; 0]

[figure: f; g; h; 0; C′; i′; D′; p′; E′; 0]

If two of the vertical arrows are weak homology equivalences, then also the third.

2.) Let f : C ⟶ D and g : D ⟶ E be chain maps of finite chain complexes. If two of the chain maps f, g and g ∘ f are weak homology equivalences, then also the third.

3.) A weak chain isomorphism of finite chain complexes is a weak homology equivalence.

4.) A chain map of finite chain complexs is a weak homology equivalence if and only if its mapping cone Cone(f) is weakly acyclic.

\underline{\underline{Proof}} : Assertion 4.) follows from the weakly exact long homology sequence of

0 ⟶ D ⟶ Cone(f) ⟶ ΣC ⟶ 0

introduced in theorem 6.1, where the suspension ΣC is Cone(C ⟶ {0} .

Next we prove assertion 1.) under the additional assumption that the rows are exact and not only weakly exact. Then we obtain an exact sequence :

0 ⟶ Cone(f) ⟶ Cone(g) ⟶ Cone(h) ⟶ 0

Now the claim follows from assertion 4.) and the weakly exact long homology sequence of theorem 6.1.

The assertion 3.) follows by induction over the dimension n = dim(C). Notice that we can assume without loss of generality that C_{n} is zero for negative n. The induction begin n = 0 is trivial. The induction step follows from 1.) applied to :

0 ⟶ C |_{n}⟶ C ⟶ (n + 1)(C) ⟶ 0

where C |_{n} is C restricted to dimension n and (n + 1)(C) is concentrated in dimension n + 1 and given in this dimension by C_{n+1}.

We obtain assertion 2.) from assertions 1.) and 4.) by constructing a chain map k : Σ^{−1}Cone(g) ⟶ Cone(f) given by

( \begin{matrix}0 & 0 \\ −1 & 0 \end{matrix} ) : D_{n} ⊕ E_{n+1} ⟶ C_{n−1} ⊕ D_{n}

and exact sequences as done in Lück [19] page 245 :

0 ⟶ Cone(f) ⟶ Cone(h) ⟶ Cone(g) ⟶ 0

0 ⟶ Cone(g ∘ f) ⟶ Cone(h) ⟶ Cone(D) ⟶ 0

Finally we prove assertion 1.) in full generality. It suffices to show for a weakly exact sequence 0 ⟶ C ⟶^{i} D ⟶^{p} E ⟶ 0 that C, D and E are weakly acyclic, if two of them are. Consider the induced exact sequence 0 ⟶ ker(p) ⟶ D ⟶ ker(p)^{⊥} ⟶ 0 for which assertion 1.) applies by the argument above. There are weak isomorphisms C ⟶ ker(p) and ker(p)^{⊥} ⟶ E. Now apply assertion 2.) and 3.)

The following example shows that these results are non-trivial and that the finite Hilbert A-module structure is essential.

Example 6.3 Let l^{2} be the Hilbert space {(a_{n} ∈ C)_{n∈N} | ∑ |a_{n} | ^{2} < ∞} with the inner product ⟨(a_{n}), (b_{n})⟩ = ∑ a_{n} · \overline{b_{n}}. Define a linear bounded operator f : l^{2} ⟶ l^{2} by sending (a_{n}) to (1/n · a_{n}). Obviously f is injective and has dense image. But f is not surjective because u := (1/n)_{n} ∈ l^{2} cannot have a preimage. Let pr : l^{2} ⟶ span_{C}(u)^{⊥} be the projection. Then pr ∘ f is injective with dense image. Hence f and pr ∘ f are weak isomorphisms of Hilbert spaces but pr is not. Compare this example with corollary 6.2 assertion 2.). One easily constructs out of it counterexamples to the other claims in corollary 6.2. Since any Hilbert space may be viewed as a Hilbert A-module for the von Neumann algebra C, the example shows that the finiteness conditions are necessary.

Next we introduce an appropiate notion of weak chain contraction.

Definition 6.4 A weak chain contraction for a chain complex C is a pair (γ, u) consisting of a weak chain isomorphism u : C ⟶ C and a chain homotopy γ : u ∼ 0 satisfying γ ∘ u = u ∘ γ.

Lemma 6.5 The following assertions are equivalent for a finite chain complex C :

1.) C is weakly acyclic.

2.) There is a weak chain contraction (γ, u).

3.) There is a weak chain contraction (γ, u) satisfying γ ∘ γ = 0.

4.) There is a weak chain isomorphism u : C ⟶ C and a chain homotopy γ : u ∼ 0.

\underline{\underline{Proof}} : 1.) ⇒ 3.) As C is weakly acyclic, c_{n} : C_{n} ⟶ C_{n−1} induces a weak isomorphism c_{n} |: ker(c_{n})^{⊥} ⟶ ker(c_{n−1}). There is an isomorphism ψ_{n} : ker(c_{n}) ⟶ ker(c_{n+1})^{⊥} because of lemma 1.7. Define γ_{n} : C_{n} ⟶ C_{n+1} by :

( \begin{matrix}0 & 0 \\ ψ_{n} & 0 \end{matrix} ) : ker(c_{n}) ⊕ ker(c_{n})^{⊥} ⟶ ker(c_{n+1}) ⊕ ker(c_{n+1})^{⊥}

Put u_{n} = c_{n+1} ∘ γ_{n} + γ_{n−1} ∘ c_{n}. One easily checks that u is a chain map and γ ∘ γ = 0. As u_{n} is the direct sum of weak isomorphisms, u_{n} itself is a weak isomorphism.

3.) ⇒ 2.) ⇒ 4.) are trivial

4.) ⇒ 1.) As u is nullhomotopic, we have H(u) = 0. Since u is a weak chain isomorphism, u is a weak homology equivalence by lemma 6.2. Hence C is weakly acyclic.

Next we prove some results we will need in the construction of torsion invariants in the following section. The technical condition γ ∘ u = u ∘ γ in the definition 6.4 of a weak chain contraction is needed for the verification of the next lemma. Its proof is a straightforward computation and is left to the reader. In the sequel we write :

6.6  C_{odd} = ⊕_{n∈Z} C_{2n+1}  C_{ev} = ⊕_{n∈Z} C_{2n}

Lemma 6.7 Let (γ, u) and (δ, v) be weak chain contractions for the finite chain complex C. Define Θ : C_{ev} ⟶ C_{ev} by :

Θ := (v ∘ u + δ ∘ γ) = ( \begin{matrix}... & ⋮ & ⋮ & ⋮ & ... \\ … & vu & 0 & 0 & … \\ … & δγ & vu & 0 & … \\ … & 0 & δγ & vu & … \\ ... & ⋮ & ⋮ & ⋮ & ... \end{matrix} )

Then the composition :

Θ′ : C_{odd} ⟶^{(uc+γ)} C_{ev} ⟶^{Θ} C_{ev} ⟶^{(vc+δ)} C_{odd}

is given by the triangle matrix

( \begin{matrix}... & ⋮ & ⋮ & ⋮ & ... \\ … & (v^{2}u^{2})_{2n−1} & 0 & 0 & … \\ … & ∗ & (v^{2}u^{2})_{2n+1} & 0 & … \\ … & ∗ & ∗ & (v^{2}u^{2})_{2n+3} & … \\ ... & ⋮ & ⋮ & ⋮ & ... \end{matrix} )

Lemma 6.8 Let C be a finite chain complex. Consider H(C) as a finite chain complex using the trivial differential.

1.) There is a chain map i : H(C) ⟶ C satisfying H(i) = id.

2.) Let i and j be chain maps H(C) ⟶ C satisfying H(i) = H(j). Then there is a weak chain isomorphism f : C ⟶ C such that f ∘ i and f ∘ j are chain homotopic.

\underline{\underline{Proof}} : 1.) Let s_{n} : H(C)_{n} ⟶ ker(c_{n}) be a section of the projection ker(c_{n}) ⟶ H(C_{n}). Define i_{n} : H(C)_{n} ⟶ C_{n} by the composition of s_{n} and the inclusion. Then i : H(C) ⟶ C is a chain map satisfying H(i) = id.

2.) We may suppose j = 0, otherwise substitute i, j by i − j, 0. Because the map c_{n} |: ker(c_{n})^{⊥} ⟶ clos(im(c_{n})) is a weak isomorphism, we can choose an isomorphism ψ_{n−1} : clos(im(c_{n})) ⟶ ker(c_{n})^{⊥} by lemma 1.7. Define f_{n} : C_{n} ⟶ C_{n} to be the orthogonal sum of

c_{n+1} | ∘ψ_{n} : clos(im(c_{n+1})) ⟶ clos(im(c_{n+1}))
ψ_{n−1} ∘ c_{n} | : ker(c_{n})^{⊥} ⟶ ker(c_{n})^{⊥}
id : ker(c_{n}) ∩ clos(im(c_{n+1}))^{⊥} ⟶ ker(c_{n}) ∩ clos(im(c_{n+1}))^{⊥}

Let γ_{n} : H(C)_{n} ⟶ C_{n+1} be given by ψ_{n} ∘ i_{n}. One easily checks that f is a weak chain isomorphism with H(f) = id and γ a chain homotopy f ∼ 0 ∘ i.

Lemma 6.9 Let f : C ⟶ D be a weak chain isomorphism of finite chain complexes. Then there is weak chain isomorphism g : D ⟶ C.

\underline{\underline{Proof}} : As f and the restriction f |_{n}: C |_{n}⟶ D |_{n} to dimension n are weak chain isomorphism, corollary 6.2 implies that f_{n} induces weak isomorphisms

ker(c_{n}) ⟶ ker(d_{n})
ker(c_{n})^{⊥} ⟶ ker(d_{n})^{⊥}
ker(c_{n}) ∩ clos(im(c_{n+1}))^{⊥} ⟶ ker(d_{n}) ∩ clos(im(d_{n+1}))^{⊥}

The differential d_{n} induces a weak isomorphism d_{n} | : ker(d_{n})^{⊥} ⟶ clos(im(d_{n})). By lemma 1.7 we can choose isomorphisms :

α_{n} : ker(d_{n})^{⊥} ⟶ ker(c_{n})^{⊥}
β_{n} : clos(im(d_{n})) ⟶ ker(d_{n})^{⊥}

Define weak isomorphisms :

u_{n} : ker(d_{n})^{⊥} ⟶ ker(c_{n})^{⊥}
v′_{n−1} : clos(im(d_{n})) ⟶ clos(im(c_{n}))

by u_{n} = α_{n} ∘ β_{n} ∘ d_{n} | and v′_{n−1} = c_{n} ∘ α_{n} ∘ β_{n−1}. Then c_{n} ∘ u_{n} = v′_{n−1} ∘ d_{n} is valid. Extend v′_{n−1} to a weak isomorphism v_{n−1} : ker(d_{n−1}) ⟶ ker(c_{n−1}). Then c_{n} ∘ u_{n} = v_{n−1} ∘ d_{n} remains true. Now we obtain a weak chain isomorphism g : D ⟶ C by :

g_{n} = u_{n} ⊕ v_{n} : ker(d_{n})^{⊥} ⊕ ker(d_{n}) ⟶ ker(c_{n})^{⊥} ⊕ ker(c_{n})

Lemma 6.10 Let 0 ⟶ C ⟶^{i} D ⟶^{p} E ⟶ 0 be a weakly exact sequence of finite chain complexes such that E is weakly acyclic. Then there is a chain map s : E ⟶ D such that p ∘ s : E ⟶ E is a weak chain isomorphism.

\underline{\underline{Proof}} : Suppose that 0 ⟶ C ⟶^{i} D ⟶^{p} E ⟶ 0 is exact. Choose for n ∈ Z a homomorphism σ_{n} : E_{n} ⟶ D_{n} satisfying p_{n} ∘ s_{n} = id. Choose a weak chain contraction (ϵ, w) for E. Define s_{n} : E_{n} ⟶ D_{n} by d_{n+1} ∘ σ_{n+1} ∘ ϵ_{n} + σ_{n} ∘ ϵ_{n−1} ∘ e_{n}. Then s : E ⟶ D is a chain map and p ∘ s = w.

In the general case, where 0 ⟶ C ⟶^{i} D ⟶^{p} E ⟶ 0 is only weakly exact, one applies the argument above to 0 ⟶ ker(p) ⟶ D ⟶ ker(p)^{⊥} ⟶ 0 and uses lemma 6.9 to get a weak chain isomorphism E ⟶ ker(p)^{⊥} from the canonical weak chain isomorphism ker(p)^{⊥} ⟶ E.

7. Torsion invariants for Hilbert A-chain complexes

Recall that A is a finite von Neumann algebra together with a finite faithful normal trace and all modules and chain complexes are understood to be Hilbert A-modules and Hilbert A-chain complexes. We introduce Reidemeister von Neumann torsion for chain complexes and torsion for weak homology equivalences. We verify the basic properties like sum formula, product formula, composition formula and compare these invariants.

Let C be a weakly acyclic chain complex. Because of lemma 6.5 we can choose a weak chain contraction (γ, u) of C. Recall that (γ, u) consists of a weak chain isomorphism u : C ⟶ C and a chain homotopy γ : u ∼ 0 satisfying u ∘ γ = γ ∘ u. Let (δ, v) be a second weak chain contraction for C. We have introduced in section 2 the K-groups K_{1}(A) resp. K_{1}^{w}(A) of automorphisms resp. weak automorphisms of finitely generated Hilbert A-modules and the involution on them given by taking the adjoint.

If c denotes the differential of C, we derive (gc + δ)_{ev} ∘ Θ ∘ (fc + γ)_{odd} = Θ′ from lemma 6.7 in the notation used there, where Θ and Θ′ are triangular matrices. We also have (uc + γ)_{odd} ∘ u_{odd} = u_{ev} ∘ (uc + γ)_{odd}. Hence corollary 6.2 and the relations in the K_{1}-groups imply that the maps

(uc + γ)_{odd} : C_{odd} ⟶ C_{ev} (vc + δ)_{ev} : C_{ev} ⟶ C_{odd}

are weak isomorphism and satisfy :

7.1 [[(uc + γ)_{odd})]] − [[u_{odd}]] = −[[(vc + δ)_{ev}]] + [[v_{odd}]] [u_{odd}] = [u_{ev}]

where we use the following notation:

Notation 7.2 Given a weak automorphism g : M ⟶ M of a finitely generated Hilbert Amodule, we denote its class in K_{1}^{w}(A) by [g]. If g : M ⟶ N is a weak isomorphism of finitely generated Hilbert A-modules, let [[g]] ∈ K_{1}^{w}(A)^{Z/2} be [g^{∗} ∘ g]

Hence the following definition is independent of the choice of (u, γ).

Definition 7.3 Let C be a finite weakly acyclic chain complex. Define the Reidemeister von Neumann torsion of C

ρ(C) ∈ K_{1}^{w}(A)^{Z/2}

by ρ(C) = [[(uc + γ)_{odd}]] − [[u_{odd}]].

We will later generalize this definition to finite chain complexes, dropping the assumption about weak acyclicity and identify it with the definition 3.8 we have used earlier. Recall from corollary 6.2 that the mapping cone Cone(f) of a weak homology equivalence f is weakly acyclic.

Definition 7.4 Define the torsion of a weak homology equivalence f : C ⟶ D of finite chain complexes

t(f) ∈ K_{1}^{w}(A)^{Z/2}

by t(f) = ρ(Cone(f)).

Notation 7.5 Let 0 ⟶ M ⟶^{i} N ⟶^{p} P ⟶ 0 be a weakly exact sequence of finitely generated modules. Define

ρ(M, N, P) ∈ K_{1}^{w}(A)^{Z/2}

by the Reidemeister von Neumann torsion of the corresponding weakly acyclic finite chain complex having P in dimension 0. Explicitely we get for any map s : P ⟶ N such that p ∘ s is a weak isomorphism :

ρ(M, N, P) = −[[i ⊕ s : M ⊕ P ⟶ N]] + [[p ∘ s : P ⟶ P]]

Let 0 ⟶ C ⟶^{i} D ⟶^{p} E ⟶ 0 be a weakly exact sequence of finite chain complexes. Define :

ρ(C, D, E) = ∑_{n} (−1)^{n} · ρ(C_{n}, D_{n}, E_{n})

Lemma 7.6 Let f : C ⟶ D be a weak chain isomorphism of finite weakly acyclic chain complexes. Then

ρ(D) − ρ(C) = ∑_{n}(−1)^{n} · [[f_{n}]]

\underline{\underline{Proof}} : We use induction over the dimension n of C. The induction begin n = 0 is trivial, since in this case C and D are the trivial chain complexes. The case n = 1 is trivial as well, since f_{0} ∘ c_{1} = d_{1} ∘ f_{1} implies:

[[f_{0}]] + ρ(C) = [[f_{0}]] + [[c_{1}]] = [[d_{1}]] + [[f_{1}]] = ρ(D) + [[f_{1}]]

The induction step from n − 1 ≥ 1 to n is done as follows.

Let C′ be the chain subcomplex of C concentrated in dimension n and n − 1 satisfying C′_{n} = C_{n} and C′_{n−1} = clos(im(c_{n})). Let C′′ be the quotient complex so that we have a canonical exact sequence 0 ⟶ C′ ⟶^{i_{C}} C ⟶^{q_{C}} C′′ ⟶ 0. Construct an exact sequence 0 ⟶ D′ ⟶^{i_{D}} D ⟶^{q_{D}} D′′ ⟶ 0 for D similiarly. Then f restricts to a weak chain isomorphism f′ : C′ ⟶ D′ and f and f′ induce a weak chain isomorphism f′′ : C′′ ⟶ D′′. Now one easily checks for i ≥ 0:

[[f′_{i}]] − [[f_{i}]] + [[f′′_{i}]] = 0

Let γ be a weak chain contraction for C. One easily constructs weak chain contractions γ′ and γ′′ for C′ and C′′ compatible with γ and the chain maps i_{C} and q_{C}. This implies:

ρ(C′) − ρ(C) + ρ(C′′)

Since a similiar equation holds for D, D′ and D′′ and the induction hypothesis applies to f′ and f′′, the lemma follows.

Lemma 7.7 (Sum formula) Consider the commutative diagram of finite chain complexes with weakly exact rows and weak homology equivalences as vertical arrows.

[figure: 0; C; i; D; p; E; 0]

[figure: f; g; h; 0; C′; i′; D′; q′; E′; 0]

Then we get :

t(f) − t(g) + t(h) = ρ(C′, D′, E′) − ρ(C, D, E)

\underline{\underline{Proof}} : The diagram above induces a weakly exact sequence of weakly acyclic finite chain complexes by corollary 6.2.

0 ⟶ Cone(f) ⟶^{j} Cone(g) ⟶^{q} Cone(h) ⟶ 0

By lemma 6.10 there is a chain map s : Cone(h) ⟶ Cone(g) such that we obtain a weak chain equivalence i ⊕ s : Cone(f) ⊕ Cone(h) ⟶ Cone(g). We get from lemma 7.6:

ρ(Cone(f) ⊕ Cone(h)) − ρ(Cone(g)) = [[(i ⊕ s)_{odd}]] − [[(i ⊕ s)_{ev}]]

We derive from the definitions

ρ(Cone(f) ⊕ ρ(Cone(h)) = t(f) + t(h)
ρ(Cone(g)) = t(g)
ρ(C′, D′, E′) − ρ(C, D, E) = [[(i ⊕ s)_{odd}]] − [[(i ⊕ s)_{ev}]]

This finishes the proof of lemma 7.7

Lemma 7.8 (Composition formula) If f : C ⟶ D and g : D ⟶ E are weak homology equivalences of finite chain complexes, then :

t(g ∘ f) = t(g) + t(f)

\underline{\underline{Proof}} : We first prove by induction over the dimension of D that ρ(Cone(D)) is zero. The induction begin dim(D) ≤ 0 is trivial. In the induction step let D |_{n} be the restriction of D to dimension n and D′ be the quotient of D and D |_{n}. There is an obvious exact sequence

0 ⟶ Cone(D |_{n}) ⟶ Cone(D) ⟶ Cone(D′) ⟶ 0

Now the claim follows from the induction hypothesis applied to Cone(D |_{n}) and Cone(D′) and lemma 7.7.

Consider the chain map h : Σ^{−1}Cone(g) ⟶ Cone(f) given by

( \begin{matrix}0 & 0 \\ −1 & 0 \end{matrix} ) : D_{n} ⊕ E_{n+1} ⟶ C_{n−1} ⊕ D_{n}

There are exact sequences (see Lück [19] page 245 :

0 ⟶ Cone(f) ⟶ Cone(h) ⟶ Cone(g) ⟶ 0
0 ⟶ Cone(g ∘ f) ⟶ Cone(h) ⟶ Cone(D) ⟶ 0

Since ρ(Cone(D)) vanishes, the claim follows from lemma 7.7 applied to these two exact sequences.

Lemma 7.9 (weak homotopy invariance) Let f, g : C ⟶ D be weak homology equivalences of finite chain complexes. Suppose the existence of weak homology equivalences of finite chain complexes u : C′ ⟶ C and v : D ⟶ D′ such that v ∘ f ∘ u and v ∘ g ∘ u are chain homotopic. Then :

t(f) = t(g)

\underline{\underline{Proof}} : Because of lemma 7.8 we have

t(f) − t(g) = t(v ∘ f ∘ u) − t(v ∘ g ∘ u)

Hence we may suppose that there is a chain homotopy h : f ∼ g. Consider the chain isomorphism I : Cone(f) ⟶ Cone(g) given by :

( \begin{matrix}id & o \\ h_{n−1} & id \end{matrix} ) : C_{n−1} ⊕ D_{n} ⟶ C_{n−1} ⊕ D_{n}

We derive from the composition formula 7.8 and sum formula 7.7 :

t(g) − t(f) = t(0 ⟶ Cone(g)) − t(0 ⟶ Cone(f)) = t(I) = ∑_{n}(−1)^{n} · [[I_{n}]] = 0

Next we define Reidemeister von Neumann torsion for finite not necessarily weakly acyclic chain complexes. We get from lemma 6.8 the existence of a chain map i : H(C) ⟶ C satisfying H(i) = id. Recall that we view H(C) as a chain complex using the trivial differential. The following definition makes sense because of lemma 6.8 and lemma 7.9.

Definition 7.10 Let C be a finite chain complex. Define its Reidemeister von Neumann torsion

ρ(C) ∈ K_{1}^{w}(A)^{Z/2}

by ρ(C) := t(i) for any chain map i : H(C) ⟶ C satisfying H(i) = id.

Lemma 7.11 (Comparision formula) Let f : C ⟶ D be a weak homology equivalence of finite chain complexes. Then :

t(f) − t(H(f)) = ρ(D) − ρ(C)

\underline{\underline{Proof}} : Choose chain maps i_{C} : H(C) ⟶ C and i_{D} : H(D) ⟶ D satisfying H(i_{C}) = id and H(i_{D}) = id. There is a weak chain isomorphism g : H(D) ⟶ H(C) because of lemma 1.7. Then H(f ∘ i_{C} ∘ g ∘ f) is the same as H(i_{D} ∘ f ∘ g ∘ f). Hence by lemma 6.8 there is a weak chain isomorphism h : D ⟶ D such that h ∘ i_{D} ∘ H(f) ∘ g ∘ H(f) is chain homotopic to h ∘ f ∘ i_{C} ∘ g ∘ H(f). We get from the composition formula 7.8 and weak homotopy invariance 7.9 :

t(f) + ρ(C) = t(f) + t(i_{C}) = t(f ∘ i_{C}) = t(i_{D} ∘ H(f)) = t(H(f)) + t(i_{D}) = t(H(f)) + ρ(D)

Next we show that the definitions of Reidemeister von Neumann torsion 7.10 and 3.8 agree. The first one is more appropiate for technical purposes, whereas the second one is easier to state. Recall that H_{p}(C) is the kernel of the Laplace operator Δ_{p}.

Lemma 7.12 Let C be a finite chain complex. Let C′ be the orthogonal complement of H(C) in C. We have Δ′ ⊕ 0 = Δ if Δ′ resp. Δ are the Laplace operators of C′ and C. Then the morphism Δ′ is a selfadjoint weak automorphism and we get :

ρ(C) = − ∑_{n} (−1)^{n} · n · [Δ′_{n}] ∈ K_{1}^{w}(A)^{Z/2}

\underline{\underline{Proof}} : Since ρ(C) = ρ(C′ ⊕ H(C)) = ρ(C′) + ρ(H(C)) holds and ρ(H(C)) vanishes, we may suppose C = C′, or, equivalently, that C is weakly acyclic. By the Hodge decomposition theorem 3.7 Δ_{n} : C_{n} ⟶ C_{n} is a selfadjoint weak automorphism. Let f_{n} : C_{n} ⟶ C_{n} be the n-fold composition (Δ_{n})^{n} = Δ_{n} ∘ … ∘ Δ_{n}. Then the following square commutes :

[figure: C_{odd}; C_{ev}; (Δc + c^{∗})_{odd}; f_{odd}; f_{ev}; C_{odd}; C_{ev}; (c + Δc^{∗})_{odd}]

This shows :

[[(Δc + c^{∗})_{odd}]]
= [(Δc + c^{∗})_{odd}^{∗} ∘ (Δc + c^{∗})_{odd}]
= [(Δc + c^{∗})_{odd}^{∗} ∘ f_{ev} ∘ (Δc + c^{∗})_{odd}] − [f_{ev}]
= [(Δc + c^{∗})_{odd}^{∗} ∘ (c + Δc^{∗}) ∘ f_{odd}] − [f_{ev}]
= [(c + Δc^{∗})_{ev} ∘ (c + Δc^{∗})_{odd}] + [f_{odd}] − [f_{ev}]

As (c^{∗}, Δ) is a weak chain contraction of C, we obtain :

ρ(C) = [[(Δc + c^{∗})_{odd}]] − [[Δ_{odd}]]
= [f_{odd}] − [f_{ev}] + [(c + Δc^{∗})_{ev} ∘ (c + Δc^{∗})_{odd}] − 2 · [Δ_{odd}]
= − ∑_{n} (−1)^{n} · n · [Δ_{n}] + [(c + Δc^{∗})_{ev} ∘ (c + Δc^{∗})_{odd}] − 2 · [Δ_{odd}]

Hence it remains to show :

[(c + Δc^{∗})_{ev} ∘ (c + Δc^{∗})_{odd}] = 2 · [Δ_{odd}]

Consider the dual chain complex C^{∗} given by :

… ⟶ C_{−1}^{∗} ⟶^{c_{0}^{∗}} C_{0}^{∗} ⟶^{c_{1}^{∗}} C_{1}^{∗} …

It has the weak chain contraction (c, Δ). Now the claim follows from lemma 6.7 and the fact that [Δ_{odd}] = [Δ_{ev}] holds (see 7.1).

Let A resp. B a be finite von Neumann algebra and C resp. D a finite Hilbert A- resp. B-chain complex.

Lemma 7.13 There is a natural isometric isomorphism of graded A ⊗ B modules

H_{∗}(C) ⊗ H_{∗}(D) ⟶ H_{∗}(C ⊗ D)

\underline{\underline{Proof}} : Because of the Hodge decomposition theorem 3.7 it suffices to prove that H_{∗}(C) ⊗ H_{∗}(D) is H_{∗}(C ⊗ D). This follows from the fact that the Laplace operator on C ⊗ D is (Δ_{C} ⊗ id_{D}) ⊕ (id_{C} ⊗ Δ_{D})

Define the Euler characteristic

7.14 χ^{A}(C) := ∑_{n≥0}(−1)^{n} · [C_{n}] = ∑_{n≥0}(−1)^{n} · [H_{n}(C)] ∈ K_{0}(A)

Lemma 7.15 (Product formula) Let f : C′ ⟶ C resp. g : D′ ⟶ D be weak chain homology equivalence of finite A- resp. B-chain complexes. Then we get using the pairing 2.8

1.) t(f ⊗ g) = χ^{A}(C) ⊗ t(g) + t(f) ⊗ χ^{B}(D)

2.) ρ(C ⊗ D) = χ^{A}(C) ⊗ ρ(D) + ρ(C) ⊗ χ^{B}(D)

\underline{\underline{Proof}} : Since f ⊗ g = (f ⊗ id) ∘ (id ⊗ g) holds, the composition formula 7.8 reduces the claim 1.) to the case g = id. Now the claim 1.) follows from the sum formula 7.7 and claim 2.) is a consequence of claim 1.) and the Künneth formula 7.13.

The next result is the main theorem of this section. The remainder of the section is devoted to its proof.

Theorem 7.16 (Sum formula) Let 0 ⟶ C ⟶^{i} D ⟶^{p} E ⟶ 0 be an exact sequence of finite chain complexes. Let LHS(C, D, E) be the weakly acyclic finite chain complex given by the long homology sequence 6.1. The we have :

ρ(C) − ρ(D) + ρ(E) = ρ(C, D, E) − ρ(LHS(C, D, E))

We have defined ρ(C, D, E) in 7.5. Notice that in theorem 7.16 we demand the exactness and not only the weak exactness of 0 ⟶ C ⟶^{i} D ⟶^{p} E ⟶ 0. This is needed to ensure that the long weakly exact homology sequence is defined (see theorem 6.1). The proof of theorem 7.16 is broken into a sequence of lemmas. We start with analysing a special case where the sequence of chain complexes is only weakly exact but nevertheless the long homology sequence exists and is weakly exact.

Lemma 7.17 Let 0 ⟶ C ⟶^{i} D ⟶^{p} E ⟶ 0 be a weakly exact sequence of finite chain complexes. Then :

1.) The sequence H(C) ⟶^{H(i)} H(D) ⟶^{H(p)} H(E) is weakly exact at H(D).

2.) Suppose that H(i) : H(C) ⟶ H(D) is injective. Define ∂_{n} : H_{n}(E) ⟶ H_{n−1}(C) to be zero. Then we obtain a weakly exact long homology sequence

⟶ H_{n+1} ⟶^{∂} H_{n}(C) ⟶^{H_{n}(i)} H_{n}(D) ⟶^{H_{n}(p)} H_{n}(E) ⟶^{∂} …

\underline{\underline{Proof}} : The sequence of finite chain complexes 0 ⟶ ker(p) ⟶^{j} D ⟶^{q} ker(p)^{⊥} ⟶ 0 is exact. The chain maps i and p induce weak chain isomorphisms ī: C ⟶ ker(p) and p̄: ker(p)^{⊥} ⟶ D. Because of the weakly exact homology sequence 6.1 and corollary 6.2 we get a commutative diagram with weakly exact lower row and weak isomorphism as vertical maps.

[figure: H(C); H(i); H(D); H(p); H(E); H(C); H(i); H(D); H(ī); id; H(p̄); H(ī); id; H(ker(p)); H(j); H(D); H(q); H(ker(p)^{⊥}; ∂; H(ker(p)); H(j); H(D)]

One easily checks weak exactness of the upper row at H(D). Suppose that H(i) is injective. Then

H(j) ∘ H(ī) = H(i) : H(C) ⟶ clos(im(H(i)))

is a weak isomorphism. Hence H(j) : H(ker(p)) ⟶ clos(im(H(i)) is a weak isomorphism by corollary 6.2 and in particular H(j) : H(ker(p)) ⟶ H(D) is injective. By the weak exactness of the lower sequence H(q) has dense image in H(ker(p)^{⊥}). Hence also H(p) has dense image in H(E). This finishes the proof of lemma 7.17.

Lemma 7.18 Let 0 ⟶ C ⟶^{i} D ⟶^{p} E ⟶ 0 be a weakly exact sequence of chain complexes. Equip ker(H(i))^{⊥}, H(D) and clos(im(H(p))) with the trivial differential. Then there is a commutative diagram of chain complexes

[figure: 0; ker(H(i))^{⊥}; H(i); H(D); H(p); clos(im(H(p))); 0]

[figure: i_{C}; i_{D}; i_{E}; 0; C; i; D; p; E; 0]

such that the rows are weakly exact and the maps i_{C}, i_{D} and i_{E} induce weak isomorphisms

H(i_{C}) : ker(H(i))^{⊥} ⟶ ker(H(i))^{⊥} H(i_{E}) : clos(im(H(p))) ⟶ clos(im(H(p))) H(i_{D}) : H(D) ⟶ H(D)

\underline{\underline{Proof}} : Choose chain maps

r : H(D) ⟶ ker(H(i))^{⊥}
s : clos(im(H(p)))^{⊥} ⟶ H(D)

such that r ∘ H(i) and H(p) ∘ s are weak chain isomorphisms. By lemma 6.8 there are chain maps

î_{C} : H(C) ⟶ C
î_{E} : H(E) ⟶ E

which induce the identity on homology. Define :

i_{C} = î_{C} ∘ r ∘ H(i)
i_{D} = i ∘ î_{C} ∘ r + î_{D} ∘ s ∘ H(p)
i_{E} = p ∘ î_{D} ∘ s

We compute :

i_{D} ∘ H(i) = i ∘ î_{C} ∘ r ∘ H(i) + î_{D} ∘ s ∘ H(p) ∘ H(i) = i ∘ i_{C}
p ∘ i_{D} = p ∘ i ∘ î_{C} ∘ r + p ∘ î_{D} ∘ s ∘ H(p) = i_{E} ∘ H(p)

Since H(i_{C}) = r ∘ H(i) and H(i_{E}) = H(p) ∘ s is valid, i_{C} and i_{D} induce weak isomorphisms ker(H(i))^{⊥} ⟶ ker(H(i))^{⊥} and clos(im(H(p))) ⟶ clos(im(H(p))) by corollary 6.2. The sequence

0 ⟶ ker(H(i))^{⊥} ⟶^{H(i)} H(D) ⟶ clos(im(H(p))) ⟶ 0

is weakly exact by lemma 7.17. By corollary 6.2 also H(i_{D}) : H(D) ⟶ H(D) is a weak isomorphism. This finishes the proof of lemma 7.18.

Next we verify the sum formula in a special case where the sequence of chain complexes is only required to be weakly exact.

Lemma 7.19 Let 0 ⟶ C ⟶^{i} D ⟶^{p} E ⟶ 0 be a weakly exact sequence of finite chain complexes such that H(i) is injective. Then we get a weakly exact long homology sequence LHS(C, D, E) if we put the boundary operator ∂ to be zero, and get :

ρ(C) − ρ(D) + ρ(E) = ρ(C, D, E) − ρ(LHS(C, D, E))

\underline{\underline{Proof}} : Because of lemma 7.17 and lemma 7.18 there is a commutative diagram of chain complexes with weakly exact rows and weak homology equivalences as vertical maps

[figure: 0; H(C); H(i); H(D); H(p); H(E); 0]
[figure: i_{C}; i_{D}; i_{E}; 0; C; i; D; p; E; 0]

We get from the sum formula for torsion 7.7 :

t(i_{C}) − t(i_{D}) + t(i_{E}) = ρ(C, D, E) − ρ(H(C), H(D), H(E)) The comparision formula 7.11 shows:

ρ(C) − ρ(D) + ρ(E) =

t(i_{C}) − t(i_{D}) + t(i_{E}) − (t(H(i_{C})) − t(H(i_{D})) + t(H(i_{E})))
+ ρ(H(C)) − ρ(H(D)) + ρ(H(E))

Obviously ρ(H(C)), ρ(H(D)) and ρ(H(E)) are zero. One easily checks :

ρ(H(C), H(D), H(E)) = ρ(LHS(C, D, E)) Hence it remains to verify :

t(H(i_{C})) − t(H(i_{D})) + t(H(i_{E})) = 0 But this follows from the sum formula of torsion applied to

[figure: 0; H(C); H(i); H(D); H(p); H(E); 0]
[figure: H(i_{C}); H(i_{D}); H(i_{E}); 0; H(C); H(i); H(D); H(p); H(E); 0]

This finishes the proof of lemma 7.19 Lemma 7.20 Theorem 7.16 is true if D is weakly acyclic.

\underline{\underline{Proof}} : Consider the canonical exact sequences of finite chain complexes, where Cyl is the mapping cylinder.

0 ⟶ E ⟶^{j} Cyl(p) ⟶ Cone(D) ⟶ 0

0 ⟶ ΣC ⟶^{k} Cone(p) ⟶ Cone(E) ⟶ 0

0 ⟶ D ⟶ Cyl(p) ⟶^{pr} Cone(p) ⟶ 0

As Cone(D), Cone(E) and D are weakly acyclic, we get from theorem 6.1 and lemma 7.19

ρ(E) − ρ(Cyl(p)) + ρ(Cone(D)) = t(H(j))
− ρ(C) − ρ(Cone(p)) + ρ(Cone(E)) = t(H(k)) − ρ(C, D, E)
ρ(D) − ρ(Cyl(p)) + ρ(Cone(p)) = −t(H(pr))

One checks directly ρ(Cone(D)) = ρ(Cone(E)) = 0. We obtain

ρ(C) − ρ(D) + ρ(E) = ρ(C, D, E) + t(H(pr)) + t(H(j)) − t(H(k))

Since t(∂ : H(E) ⟶ ΣH(C)) is −ρ(H(C, D, E)), it suffices to show

t(∂) + t(H(k)) = t(H(pr)) + t(H(j))

We conclude from the definitions

H(k) ∘ ∂ = H(pr) ∘ H(j)

Now an application of the composition formula 7.8 finishes the proof of lemma 7.20.

Now we are ready to prove theorem 7.16. Consider the commutative diagram appearing in lemma 7.18. Let D′ be clos(im(i_{D}))^{⊥} and p_{D} : D ⟶ D′ be the projection. Let C′ ⊂ D′ be clos(im(p_{D} ∘ i)) and i′ : C′ ⟶ D′ be the inclusion. Put E′ = (C′)^{⊥}. Let p′ : D′ ⟶ E′ be the projection. Then we obtain an exact sequence of finite chain complexes 0 ⟶ C′ ⟶ D′ ⟶ E′ ⟶ 0. As the sequence 0 ⟶ C ⟶^{i} D ⟶^{p} E ⟶ 0 is exact, p′ ∘ p_{D} : D ⟶ E′ factorizes over p : D ⟶ E into a map p_{E} : E ⟶ E′ and p_{D} induces p_{C}; C ⟶ C′. Thus we have constructed a commutative diagram of finite chain complexes with weakly exact middle column, exact lower two rows and weakly exact top row.

0  0  0

[figure: 0; ker(H(i))^{⊥}; H(i); H(D); H(p); clos(im(H(p))); 0]

[figure: i_{C}; i_{D}; i_{E}; 0; C; i; D; p; E; 0]

[figure: p_{C}; p_{D}; p_{E}; 0; C′; i′; D′; p′; E′; 0]

[figure: 0; 0; 0]

We claim that also the outer columns are weakly exact. We substitute in the diagram above for the upper row

{0} ⟶ ker(H(p)) ⟶ H(D) ⟶ ker(H(p))^{⊥} ⟶ {0}

in the obvious way. It suffices to consider this new diagram whose rows are exact. We may interprete it as a short exact sequence of two-dimensional chain complexes given by the columns. Since the middle row is weakly exact, image p_{C} is dense in C′, and i_{E} is injective, the claim follows from the weakly exact long homology sequence of theorem 6.1.

We can apply lemma 7.19 to the columns and obtain :

ρ(C′) − ρ(C) = ρ(ker(H(i))^{⊥}, C, C′) − ρ(LHS(ker(H(i))^{⊥}, C, C′))

ρ(D′) − ρ(D) = ρ(H(D), D, D′) − ρ(LHS(H(D), D, D′))

ρ(E′) − ρ(E) = ρ(clos(im(H(p))), E, E′) − ρ(LHS(clos(im(p))), E, E′) Moreover, D′ is weakly acyclic. Therefore we can apply lemma 7.20 to the bottom exact row and obtain :

ρ(C′) − ρ(D′) + ρ(E′) = ρ(C′, D′, E′) − ρ(LHS(C′, D′, E′)) We derive from lemma 7.19 applied to the short weakly exact sequence of weakly acyclic chain complexes given by the long homology sequences of the rows

ρ(LHS(ker(H(i))^{⊥}, H(D), clos(im(H(p))))) − ρ(LHS(C, D, E)) + ρ(LHS(C′, D′, E′)) =

ρ(ker(H(i))^{⊥}; C, C′) − ρ(H(D), D, D′) + ρ(clos(im(H(p))), E, E′) Interpreting the diagram above as a short weakly exact sequence of two-dimensional weakly acyclic chain complexes given by the rows, we derive from lemma 7.19

ρ(ker(H(i))^{⊥}, C, C′) − ρ(H(D), D, D′) + ρ(clos(im(H(p))), E, E′) =

ρ(ker(H(i))^{⊥}, H(D), clos(im(H(p)))) − ρ(C, D, E) + ρ(C′, D′, E′)

Obviously we have :

ρ(ker(H(i))^{⊥}, H(D), clos(im(H(p)))) = ρ(LHS(ker(H(i))^{⊥}, H(D), clos(im(H(p))))) Now theorem 7.16 follows from the equations above.

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Current adresses :

Wolfgang Lück^{∗} Mel Rothenberg^{∗∗} Fachbereich Mathematik Department of mathematics Johannes Gutenberg-Universität The University of Chicago Staudingerweg 9 5734 South University Avenue 6500 Mainz 1 Chicago Illinois 60637 Germany U. S. A.

* Partially supported by NSF Grant DMS - 9103327

* Partially supported by NSF Grant DMS - 8903168