Relative torsion

D. Burghelea (Ohio State University) L. Friedlander (University of Arizona) T. Kappeler (University of Zürich)

June 17, 1999

Abstract: This paper achieves, among other things, the following:

• It frees the main result of [BFKM] from the hypothesis of determinant class and extends this result from unitary to arbitrary representations.

• It extends (and at the same times provides a new proof of) the main result of Bismut and Zhang [BZ] from finite dimensional representations of Γ to representations on an A−Hilbert module of finite type ( A a finite von Neumann algebra). The result of [BZ] corresponds to A = IC.

• It provides interesting real valued functions on the space of representations of the fundamental group Γ of a closed manifold M. These functions might be a useful source of topological and geometric invariants of M.

These objectives are achieved with the help of the relative torsion R, first introduced by Carey, Mathai and Mishchenko [CMM] in special cases. The main result of this paper calculates explicitly this relative torsion (cf Theorem 0.1).

Table of contents

0. Introduction 0.1 Summary of the results 0.2 Definitions and outline of the paper

1. Operators and complexes 1.1 Operators 1.2 Complexes

2. Relative torsion and its Witten deformation 2.1 Relative torsion 2.2 Witten deformation of the relative torsion

3. Proof of Proposition 0.1 in the case g = g′ 3.1 Outline of the proof 3.2 Asymptotic expansion of log R_{sm}(t) 3.3 Asymptotic expansion of log T_{la}(t)

4. Anomalies of the relative torsion 4.1 Metric and Hermitian anomalies 4.2 Proof of Proposition 0.1 4.3 Anomaly with respect to triangulation

5. Proof of Theorem 0.1 5.1 Proof of Theorem 0.1 5.2 An invariant for odd dimensional manifolds.

Appendix A Lemma of Carey-Mathai-Mishchenko Appendix B Determinant class property

0. Introduction

0.1 Summary of the results

Let (M, x_{0}) be a closed, connected manifold with base point x_{0} and ρ : Γ → Gl_{A}(W) a representation of Γ := π_{1}(M; x_{0}) on an A-Hilbert module W of finite type with A a finite von Neumann algebra. Denote by E → M the vector bundle, associated to M and ρ, equipped with the canonical flat connection ∇.

Let μ be a Hermitian structure of E and τ = (h, g′) a (generalized) triangulation of M (cf subsection 0.2). The relative torsion is a numerical invariant associated to F = (M, ρ, μ, g, τ). Unlike the analytic torsion T_{an} associated to (M, ρ, μ, g), or the Reidemeister torsion T_{Reid} associated to F (cf subsection 0.2), both of which are defined only when (M, ρ) is of determinant class (cf Appendix B) , R is always defined. If (M, ρ) is of determinant class the relative torsion turns out to be the quotient of the analytic and Reidemeister torsion (cf (2.15)). However, as shown in Appendix B, there are many pairs (M, ρ) which are not of determinant class.

Proposition 0.1 Assume μ_{0} is a Hermitian structure on the bundle induced by ρ which is parallel in the neighborhood of Cr(h).

(i) log R = ∫_{M\Cr(h)} α_{F} where α_{F} is a density on M \Cr(h), which vanishes on a neighborhood of Cr(h) and is a local quantity (cf subsection 0.2), depending on ρ, μ, g, and X := −grad_{g′}h. (ii) If μ is parallel then R = 1.

Following the work of Bismut Zhang [BZ], we consider the closed one-form θ(ρ, μ) ∈ Ω^{1}(M), the form Ψ(T M, g) ∈ Ω^{n−1}(T M \ M; O_{T M}) (O_{T M} the orientation bundle of T M), and for two Hermitian structures μ_{1} and μ_{2} on the bundle E → M induced from ρ, the smooth function V (ρ, μ_{1}, μ_{2}) ∈ Ω^{0}(M) (cf section 4). When μ is parallel with respect to ∇ on an open set U, U ⊂ M, then θ(ρ, μ) = 0 on U. Denote by e(M, g) the Euler form associated with the Riemannian metric g. The triangulation τ = (h, g′) provides the vector field X = −grad_{g′}h which will be regarded as a smooth map X : M \ Cr(h) → T M \ M. The above quantities are used in formulae expressing the change of the relative torsion when one varies the Riemannian metric g, the Hermitian structure μ, or the generalized triangulation τ (cf Propositions 4.1-4.3). These formulae are used to derive the following

Theorem 0.1 Assume μ_{0} is a Hermitian structure on the bundle induced by ρ which is parallel in the neighborhood of Cr(h) and μ is an arbitrary Hermitian structure. Then R = R(M, ρ, μ, g, τ) is given by

log R = (−1)^{n+1} ∫_{M\Cr(h)} θ(ρ, μ_{0}) ∧ X^{∗}(Ψ(T M, g)) + ∫_{M} V (ρ, μ, μ_{0})e(M, g)−

− ∑_{x∈Cr(h)}(−1)^{ind(x)}V (ρ, μ, μ_{0})(x).

In the case M is of odd dimension, the above formula for log R simplifies,

log R = ∫_{M\Cr(h)} θ(ρ, μ_{0}) ∧ X^{∗}(Ψ(T M, g)) − ∑_{x∈Cr(h)}(−1)^{ind(x)}V (ρ, μ, μ_{0})(x).

Since μ_{0} is parallel in the neighborhood of the critical points the first integral in the formula evaluating R is convergent, and this is the reasons μ_{0} is necessary. The formulas in Theorem 0.1 have been discovered by Bismut-Zhang in the case A = IC (we choose a slightly different definition of θ(ρ, μ) which is half of Bismut-Zhang’s). Theorem 0.1, applied in this case, implies the main result in [BZ], as in this case R = T_{an}/T_{Reid}. The proof of Theorem 0.1 provides a conceptually new proof of the formula of [BZ]. In particular Theorem 0.1 also shows that the formula of Bismut-Zhang for the quotient of analytic and Reidemeister torsion remains true for the quotient of L_{2}− analytic and Reidemeister torsions, cf subsection 5.1, Remark to Theorem 0.1. This answers a question, raised by several experts in the field. In [BFKM], we have shown that for a unitary representations ρ of Γ with (M, ρ) of determinant class, T_{an} = T_{Reid}. Theorem 0.1 extends this result to an arbitrary representation and reformulates this result in a way that it holds without the assumption of (M, ρ) being of determinant class. Denote by Rep(Γ; W) the space of representations of Γ on W. The group Γ is a finitely presented, let us say with generators γ_{1}, ..., γ_{N_{1}} and relations R_{j}(γ_{1}, ..., γ_{N_{1}}) = e_{Γ} (1 ≤ j ≤ N_{2}). Hence Rep(Γ; W) is a level set of the holomorphic map induced by the relations, Gl_{A}(W)^{N_{1}} → Hom_{A}(W)^{N_{2}} and thus a complex analytic space (of infinite dimension if dimICW is infinite). Using R(M, ρ, μ, g, τ) we will construct in section 5.2 a smooth invariant for a pair of a closed odd dimensional manifold with fundamental group Γ and an Euler structure (cf section 5.2 for the definition of Euler structure). This invariant will be a real analytic function on Rep(Γ; W). It will be shown in a forthcoming paper that this function and other similarly defined functions associated with even dimensional manifolds, are locally the real part of holomorphic functions on Rep(Γ; W). As an example (cf section 5.2), we will calculate this function in the case M = N × S^{1} with N a simply connected manifold of even dimension and E the canonical Euler structure on N × S^{1}. In this case, Rep(Γ; W) ≃ Gl_{A}(W) and the function is equal to (cf Proposition 5.1) −\frac{1}{2}χ(N) log det(A^{∗}A)^{1/2} where A ∈ Gl_{A}(W), det is the determinant in the sense of Fuglede- Kadison (cf.[BFKM]), and χ(N) is the Euler-Poincaré characteristic of N. This calculation shows that the invariant is nontrivial. In [BFK 5] we extend the results of this paper to the relative torsion for a homotopy triangulation.

0.2 Definitions and outline of the paper

Let (M, x_{0}) be a closed manifold with base point x_{0}, ρ : Γ → Gl_{A}(W) a representation of Γ := π_{1}(M; x_{0}) on an A-Hilbert module W, E → M the vector bundle associated to ρ, equipped with the canonical flat connection ∇, and μ a Hermitian structure of E.

deRham complex: Denote by (Ω(M; E), d) the deRham complex of smooth forms with values in E. Here d_{k} : Ω^{k} → Ω^{k+1} is the exterior differential determined by ∇. A Riemannian metric g on M together with the Hermitian structure μ on E allow to introduce an inner product on Ω^{k} and therefore determine an adjoint d_{k}^{∗} of d_{k}. Denote by Δ_{k} = d_{k}^{∗}d_{k} + d_{k−1}d_{k−1}^{∗} the Laplacian on k-forms. These Laplacians can be used to introduce s− Sobolev norms on Ω. We denote by H_{s}(Λ^{k}(M; E)) the completion of Ω^{k} with respect to the s-Sobolev norm. This leads to a family of complexes H_{s}(Λ(M; E)) of A-Hilbert modules.

Analytic torsion T_{an} : The Laplacians Δ_{k} admit a regularized determinant det Δ_{k} in the von Neumann sense. If (M, ρ) is of determinant class (cf Appendix B), these determinants do not vanish so that the analytic torsion can be defined by

log T_{an}(M, ρ, g, μ) := 1/2 ∑_{q}(−1)^{q+1}q log det Δ_{q}.

(The concept of determinant class was introduced in [BFKM] Definition 4.1 and will be reviewed in Appendix B.)

Combinatorial complex: A generalized triangulation of M is a pair τ = (h, g′) with the following properties: (T1) h : M → IR is a smooth Morse function; ^{1} (T2) g′ is a Riemannian metric so that −grad_{g′}h satisfies the Morse - Smale condition (for any two distinct critical points x and y of h, the stable manifold W_{x}^{+} and the unstable manifold W_{y}^{−}, with respect to −grad_{g′}h, intersect transversely); (T3) in a neighborhood of any critical point of h one can introduce local coordinates such that, with q denoting the index of this critical point,

h(x) = h(0) − (x_{1}^{2} + … + x_{q}^{2})/2 + (x_{q+1}^{2} + … + x_{d}^{2})/2

and the metric g′ is Euclidean in these coordinates.

A generalized triangulation τ and a Let p : M̃ → M be the universal covering of M and denote by g̃ and τ̃ = (h̃, g̃′) the lifts of g and τ on M̃. Denote by Cr_{q}(h̃) ⊂ M̃, resp. Cr_{q}(h) ⊂ M, the set of critical points of index q of h̃, resp. h, and let Cr(h̃) = ∪_{q}Cr_{q}(h̃). Notice that the group Γ acts freely on Cr_{q}(h̃) and the quotient set can be identified with Cr_{q}(h).

An orientation

O_{h} := {Ox̃|x̃ ∈ Cr(h̃)}

is provided by a collection of orientations Ox̃ for the unstable manifolds W − ; x̃ that are Γ-invariant.

1In [BFKM], the Morse function h was supposed to be selfindexing (h(x) = index(x) for any critical point x of h). For our proof of Theorem 0.1 (step 2), it is convenient to drop this condition (cf [BFK2]).

To the system (M, ρ, μ, τ, O_{h},), we associate a cochain complex of finite type over the von Neumann algebra A, C ≡ C(M, ρ, μ, τ, O_{h,}) = {C^{q}, δ_{q}}, where C^{q} = ⊕_{x∈Cr_{q}(h)}E_{x}, which can be identified with the module of Γ-equivariant maps f : Cr_{q}(h̃) → W, and the maps δ_{q} : C^{q} → C^{q+1} are given by

δ_{q}(f)(x̃) := ∑_{ỹ∈Cr_{q}(h̃)} ν_{q+1}(x̃, ỹ)f(ỹ)

where ν_{q} : Cr_{q}(h̃)×Cr_{q−1}(h̃) → ZZ is defined by ν_{q}(x̃, ỹ) := intersection number

(W − ; x̃ ∩V, W + ; ỹ ∩V) with V := h̃^{−1}(q − \frac{1}{2}). The cochain complex C(M, ρ, μ, τ, O_{h})

depends on μ only via μ_{x}, x ∈ Cr(h). Let Δ_{q}^{comb} := δ_{q}^{∗}δ_{q} + δ_{q−1}δ_{q−1}^{∗} denote the combinatorial Laplacian.

Reidemeister torsion T_{Reid} : Δ_{q}^{comb} admits a regularized determinant det Δ_{q}^{comb} in the von Neumann sense which, under the additional hypothesis of (M, ρ) being of determinant class (cf Appendix B), does not vanish. Under this hypothesis, the combinatorial torsion can be defined

log T_{comb}(M, ρ, μ, τ) := 1/2 ∑_{q}(−1)^{q+1}q log det Δ_{q}^{comb}.

It can be shown that the combinatorial torsion is independent of the choice of O_{h}. Given a Riemannian metric g, the metric part T_{met} of the Reidemeister torsion is defined as in [BFKM] and the Reidemeister torsion T_{Reid} is then given by

log T_{Reid} := log T_{comb} + log T_{met}

.

Relative torsion: Notice that the integration on the q-cells of the generalized triangulation τ which are given by the unstable manifolds of −grad_{g′}h, defines a morphism

Int : (Ω, d) → (C, δ),

i.e. Int_{q} : Ω^{q} → C^{q} is an A-linear map so that δ_{q}Int_{q} = Int_{q+1}d_{q} (cf Appendix by F. Laudenbach in [BZ] or [B1]). We would like to define the relative torsion of the system (M, ρ, μ, g, τ) as the torsion of the mapping cone defined by the integration map. Unfortunately, the torsion of this mapping cone cannot be defined (at least not in an obvious way); a first difficulty comes from the fact that the integration maps Int_{k} do not extend to closed maps, defined on a dense domain of H_{0}(Λ^{k}(M; E)). However, for s sufficiently large, the integration morphism has an extension Int_{s}

Int_{s} : H_{s}(Λ(M; E)) → C

to a bounded morphism. We then consider the composition g_{s}

g_{s} : H_{0}(Λ(M; E)) ⟶^{(Δ+Id)^{−s/2}} H_{s}(Λ(M; E)) ⟶^{Int_{s}} C

and prove that g_{s} induces an isomorphism in algebraic cohomology Ker(d_{k})/range(d_{k−1}) as well as in reduced cohomology (cf Proposition 2.5)

Ker(d_{k})/\overline{range(d_{k−1})}.

This implies that the mapping cone C(g_{s}), defined by

C(g_{s})_{k} := C^{k−1} ⊕ H_{0}(Λ^{k}(M; E)); d(g_{s})_{k} := ( \begin{matrix}−δ_{k−1} & g_{k;s} \\ 0 & d_{k} \end{matrix} ) ,

is an algebraically acyclic cochain complex (i.e. Kerd(g_{s})_{k}/range(d(g_{s})_{k−1}) = 0). In particular the complex C(g_{s}) is a cochain complex of A-Hilbert modules (cf Lemma 1.11) whose Laplacians Δ(g_{s})_{k} are unbounded operators which admit a nonvanishing regularized determinant det Δ(g_{s})_{k} in the von Neumann sense. One of the purposes of section 1 is to establish this result.

The relative torsion R_{s} is defined as the torsion T (C(g_{s})) of the mapping cone C(g_{s}),

log R_{s} := log T (C(g_{s})) := \frac{1}{2} ∑_{k}(−1)^{k+1}k log det(Δ(g_{s})_{k}).

In section 2, we show that log R_{s} is independent of s (s sufficiently large) and thus provide a well defined number which we denote by log R.

As first shown in [CMM] in a slightly different and less general version, one verifies that if (M, ρ) is of determinant class, then log R = log T_{an} −log T_{Reid} (cf (2.15)) where T_{an}, resp. T_{Reid}, denotes the analytic torsion resp. Reidemeister torsion. The main result of [BFKM] can thus be stated as follows: When (M, ρ) is of determinant class and μ is parallel then R = 1.

Representations: Let ρ : Γ → Gl_{A}(W) be a representation of Γ on a A−Hilbert module of finite type. The representation ρ^{♯} : Γ → Gl_{A}(W) dual to ρ is defined by ρ^{♯}(γ) := ρ(γ^{−1})^{∗} (γ ∈ Γ) where^{∗} denotes the adjoint in Gl_{A}(W) with respect to the inner product of W. The representation ρ is said to be unitary if ρ^{♯} = ρ. Notice that the flat bundle E → M induced by a unitary representation admits a parallel Hermitian structure. A representation ρ′ : Γ → Gl_{A′}(W′) (W′ a A′−Hilbert module of finite type) is said to be isomorphic to ρ if there exist an isomorphism θ : A → A′ ( with tr_{A′}(θa) = tr_{A}(a), ∀a ∈ A) and an isomorphism Θ : W → W′, such that Θ is θ−linear. As an example, we mention that ρ ⊗ ρ^{♯} and ρ^{♯} ⊗ ρ are isomorphic.

We denote by Rep(Γ, W) the complex analytic space of representations (cf subsection 0.1).

Densities and local quantities: A density α on a manifold M of dimension n is a smooth n−form with coefficients in the orientation bundle O_{M} → M. Densities can be integrated on relative compact open subsets U ⊂ M, and the integral is denoted by ∫_{U} α. In this paper, we use the word local quantity for an assignment which to any set of data G := (E → M, ∇, μ, g, X) consisting of a bundle E → M, a flat connection ∇, a Hermitian structure μ, a Riemannian metric g, and a nonzero vectorfield X on M, provides a density α(G) with the following properties:

(P1) For any open set U ⊂ M, α(G|_{U}) = α(G)|_{U}.

(P2) If (ϕ, Φ) is an isomorphism between G_{1} and G_{2} (i.e. Φ : E_{1} → E_{2} is a bundle isomorphism above the diffeomorphism ϕ : M_{1} → M_{2} such that ϕ^{∗}∇_{2} = ∇_{1}, ϕ^{∗}μ_{2} = μ_{1}, ϕ^{∗}g_{2} = g_{1}, ϕ^{∗}X_{2} = X_{1}) then ϕ^{∗}α(G_{2}) = α(G_{1}).

To a system F = (M, ρ, μ, g, τ) we associate the set of data G_{F} = (E|_{M\Cr(h)} → M \ Cr(h), ∇, μ, X) where (E → M, ∇) is the bundle with flat connection ∇, canonically associated to ρ and X = −grad_{g′}h.

Let us briefly comment on the proofs of the results stated in subsection 0.1: The proof of Proposition 0.1 (cf section 3 and subsection 4.2) uses the Witten deformation of the deRham complex, given by the differentials d_{k}(t) := e^{−th}d_{k}e^{th}. We consider a deformation R(t) of the relative torsion R = R(0), induced by the Witten deformation of the deRham complex. From many different possibilities for defining the deformation R(t) we choose one for which the variation \frac{d}{dt} log R(t) can be computed. We define R(t) as the torsion of the mapping cone associated to

(H_{0}(Λ(M; E)), d(t)) →^{e^{th}} (H_{0}(Λ(M; E)), d) →^{g_{s}} (C, δ),

which will be checked to be independent of s, cf Proposition 2.7 and Definition 2.8. The variation of log R(t) can be computed to be (cf Theorem 2.1)

(0.1) \frac{d}{dt} log R(t) = dim E_{x} · ∫_{M} h e(M, g),

where e(M, g) is the Euler form of the tangent bundle equipped with the Levi-Cività connection induced from g. It follows that log R(t) is given by log R + t ∫_{M} h e(M, g).

To continue our discussion we say that a function G : IR → IR admits an asymptotic expansion for t → ∞ if there exist a sequence i_{1} > i_{2} > … > i_{N} = 0 and constants (a_{k})_{1≤k≤N}, (b_{k})_{1≤k≤N} such that, as t → ∞, (0.2) G(t) = ∑_{1}^{N} a_{k}t^{i_{k}} + ∑_{1}^{N} b_{k}t^{i_{k}} log t + o(1).

For convenience, we denote by F T (G(t)) or F T_{t=∞}(G) the coefficient a_{N} in the asymptotic expansion of G(t) and refer to it as the free term of the expansion. Notice that the equality (0.1) implies that log R(t) admits an asymptotic expansion of the form (0.2) with log R as the free term.

By different considerations outlined below, we derive the existence of an asymptotic expansion for log R(t) of the form (0.2) and calculate the free term of this expansion as the integral on M \ Cr(h) of a local density expressed in terms of g, μ and τ. The proof of Proposition 0.1 will then be derived from the comparison of these two calculations. Using Witten deformation we decompose log R(t) into two parts

log R(t) = log R_{sm}(t) + log T_{la}(t),

and show that each of the two parts admit an asymptotic expansion of the form (0.2) whose free terms can be computed. Precisely, we consider the Witten deformation (Ω(M; E), d(t)) of the deRham complex, d(t) = e^{−th}de^{th}. For t sufficiently large, the deformed deRham complex can be decomposed,

(Ω(M, E), d(t)) = (Ω_{sm}(t), d(t)) ⊕ (Ω_{la}(t), d(t)),

corresponding to the small and the large part of the spectrum of the deformed Laplacians Δ_{k}(t). This decomposition induces the decomposition

(H_{0}(Λ(M; E)), d(t)) = (Ω_{sm}(t), d(t)) ⊕ (H_{0,la}(t), d(t)).

The complex (Ω_{sm}(t), d(t)) is of finite type. Denote by g_{s,sm}(t) the restriction of g_{s} · e^{th} to Ω_{sm}(t), where e^{th} : Ω(M, E) → Ω(M, E) denotes the multiplication by e^{th}. The mapping cone C(g_{s,sm}(t)) is a cochain complex of A-Hilbert modules of finite type which is algebraically acyclic and has a well defined torsion denoted by R_{sm}(t). As in [BFKM, section 6], one verifies that (H_{0,la}(t), d(t)) is algebraically acyclic and has a well defined torsion, denoted by T_{la}(t). We prove that log R(t) = log R_{sm}(t) + log T_{la}(t) (cf subsection 3.1, statement (B)).

To prove that log R_{sm}(t) has an asymptotic expansion, we show that R_{sm}(t) = T (C(f(t))) (cf Proposition 3.1), the torsion of the mapping cone induced by

f(t) : (Ω_{sm}(t), d(t)) →^{e^{th}} (Ω, d) →^{Int} (C, δ)

and use the Witten-Helffer-Sjöstrand theory as presented in [BFKM] to prove that log T (C(f(t))) admits an asymptotic expansion of the type (0.2) and to compute its free term (Proposition 3.1). To analyze log T_{la}(t), we proceed as in [BFK1] or [BFKM]. We derive the existence of an asymptotic expansion for log T_{la}(t) and a closed formula for its free term from Proposition 3.2 iii).

This result states the existence of an asymptotic expansion for the difference log T_{la}(t) − log T̃_{la}(t) where (M, ρ, μ, g, τ) and (M̃, ρ̃, μ̃, g̃, τ̃) are two systems whose Morse functions h and h̃ have the same number of critical points in each index, and provides a formula for its free term. The proof of Theorem 0.1 ( cf subsection 5.1) uses the anomalies of the relative torsion (Propositions 4.1- 4.3) and Proposition 0.1. The locality of the density given Proposition 0.1 is essential for this proof.

The paper is organized in 4 sections and two appendices. Section 1: In subsection 1.1 we single out a class of unbounded A-linear operators on A-Hilbert modules ( A is a von Neumann algebra with finite trace ) for which the regularized determinant can be defined and in subsection 1.2 a class of complexes of A-Hilbert modules for which the Laplacians are operators in the above class. Therefore the torsion of an (algebraically) acyclic complex of such type can be defined. These abstract results are needed because the mapping cone of (a regularized version of) the integration map between the deRham complex and the combinatorial complex has as its components direct sums of A-Hilbert modules of finite type and completions (with respect to a Sobolev norm) of spaces of smooth sections in smooth bundles of A-Hilbert modules of finite type (cf. [BFKM] for definitions). We show that the mapping cone of a regularized version of the integration map is a complex of the type introduced in subsection 1.2. Section 2: Using the results of section 1, we introduce in subsection 2.1 the notion of relative torsion R and in subsection 2.2 the Witten deformation R(t) of the relative torsion R and calculate its variation. Section 3: Proof of Proposition 0.1 in the case g = g′. Section 4: We investigate how the relative torsion R(M, ρ, μ, g, τ) varies with respect to the Riemannian metric g, the Hermitian structure μ and the triangulation τ. Further we prove Proposition 0.1 in full generality. Section 5: Proof of Theorem 0.1 and the construction of an invariant for odd dimensional manifolds. Appendix A: For the convenience of the reader we present a proof of a slightly stronger version of a Lemma due to Carey-Mathai-Mishchenko. Appendix B: We review the concept of determinant class and provide a simple example of a pair (M, ρ), M := S^{1} and ρ a representation of π_{1}(S^{1}) = ZZ on l_{2}(ZZ), which is not of determinant class. The existence of such pairs motivates in part the concept of relative torsion. Throughout the paper the notions of trace (denoted by Tr), dimension, determinant etc. are always understood in the von Neumann sense.

1 Operators and complexes

1.1 Operators

Let W_{1}, W_{2} be A-Hilbert modules and φ : W_{1} → W_{2} an operator. Denote its domain by domain(φ) ⊆ W_{1} and its range by range(φ) = φ(domain(φ)) ⊆ W_{2}.

Introduce the following properties of φ. Op(1) φ is A-linear. Op(2) φ is densely defined, i.e. \overline{domain(φ)} = W_{1}. Op(3) φ is closed (i.e. the graph Γ(φ) of φ is closed in W_{1} × W_{2}).

If, instead of Op(3), φ satisfies

Op(3)’ φ is closable

we consider the minimal extension of φ and denote it again by φ. In the sequel, we will not distinguish between property Op(3) and Op(3)′.

An operator φ satisfying Op(1) − Op(3), admits an adjoint, φ^{∗} (cf. e.g. [RS, p 316]) and we can consider the nonnegative, selfadjoint operator φ^{∗}φ. Functional calculus permits to define the square root |φ| := (φ^{∗}φ)^{1/2}. Then, for any 0 < t < ∞, the heat evolution operator e^{−t|φ|} is bounded, nonnegative and selfadjoint. The operators |φ| and e^{−t|φ|} satisfy the properties Op(1) − Op(3). In the sequel, if not mentioned otherwise, we assume that the operators considered satisfy Op(1) − Op(3).

Denote by dP (λ) ≡ dP_{|φ|}(λ) the (operator valued) spectral measure associated to |φ| defined by the orthogonal projectors P_{|φ|}([0, λ]) (0 ≤ λ ≤ ∞)

P_{|φ|}([0, λ]) = ∫_{0_{−}}^{λ_{+}} dP_{|φ|}(λ).

Most of the operators φ considered in this paper will satisfy the following important property

Op(4) T rP_{|φ|}([a, b]) < ∞ (for any 0 < a ≤ b < ∞)

where

(1.1) P_{|φ|}([a, b]) := ∫_{a_{−}}^{b_{+}} dP_{|φ|}(λ).

If φ satisfies Op(4) one can define the Stiltjes measure dF_{|φ|}(λ) on the half line (0, ∞), given by (0 < a < b < ∞)

(1.2) ∫_{a_{−}}^{b_{+}} dF_{|φ|}(λ) = T rP_{|φ|}([a, b]). The next two properties concern the asymptotic behavior of T rP ([a, b]) for b ↗ ∞ and a ↘ 0 : Op(5) There exists α ≥ 0 such that

T rP_{|φ|}([1, λ]) = 0(λ^{α}) for λ ↗ ∞

and Op(6) T rP_{|φ|}([λ, 1]) = 0(1) for λ ↘ 0. If φ satisfies properties Op(4) and Op(6) one can define the spectral distribution functions, associated to |φ|, (1.3) F_{|φ|}^{+}(λ) = T rP_{|φ|}((0, λ)); F_{|φ|}(λ) := T rP_{|φ|}((−∞, λ)). The spectral distribution function F_{|φ|}^{+}(λ) can be described variationally as follows, (λ > 0) (1.4) F_{|φ|}^{+}(λ) = sup{dim L|L ⊂ domain(|φ|), ⊥ Kerφ, is an A-Hilbert

submodule with |||φ|(x)|| < λ||x|| ∀x ∈ L}

(cf e.g. [GS], [BFKM]). If, in addition, φ satisfies also Op(5), one can define the heat trace θ_{|φ|}(t) associated to φ (excluding zero modes) (1.5) θ_{|φ|}(t) := ∫_{0_{+}}^{∞} e^{−tλ}dF_{|φ|}^{+}(λ) (t > 0). Using integration by parts in the Stiltjes integral (1.5) one concludes from Op(5) that (1.6) θ_{|φ|}(t) = 0(t^{−α}) (t ↘ 0). Next, let us introduce the ‘partial’ zeta function ζ_{|φ|}^{I}(s) associated to the heat trace θ_{|φ|}(t) and defined for s ∈ IC with Res > α (with α given by Op(5)) (1.7) ζ_{|φ|}^{I}(s) := \frac{1}{Γ(s)} ∫_{0}^{1} t^{s−1}θ_{|φ|}(t)dt where Γ(s) denotes the gamma function. Notice that ζ_{|φ|}^{I}(s) is holomorphic in the halfplane Res > α.

The following property is needed to define the notion of regularized determinant:

Op(7) ζ_{|φ|}^{I}(s) has an analytic continuation at s = 0.

A sufficient condition for Op(7) to hold is the existence of an asymptotic expansion of θ_{|φ|}(t) near t = 0 of the form

(Asy) θ_{|φ|} = ∑_{j=0}^{m−1} a_{j}t^{−α_{j}} + a_{m} + R(t) where m ∈ ZZ_{≥0}, 0 < α_{m−1} < … < α_{0}, a_{0}, …, a_{m} ∈ IR and where R(t) is a bounded, continuous

function with R(t) = 0(t^{ρ}) for some ρ ∈ IR_{>0}.

The fact that (Asy) implies Op(7) follows from the following lemma, using integration by parts.

Lemma 1.1 Let f : (0, 1] → IR be a continuous function of the form

f(t) = ∑_{j=0}^{m−1} a_{j}t^{−α_{j}} + a_{m} + R(t)

with a_{0}, …, a_{m}, α_{0}, …, α_{m−1} and R(t) as in (Asy).

Then, the holomorphic function ξ(x) := \frac{1}{Γ(s)} ∫_{0}^{1} t^{s−1}f(t)dt, defined for Res > α_{0}, has a meromorphic continuation to the half plane Res > −ρ with ρ > 0 as in (Asy). It has only simple poles, located at s = α_{j} (0 ≤ j ≤ m − 1). In particular, ξ(s) is regular at s = 0 and ξ(0) = a_{m}.

Some of the operators φ we will consider have the property that 0 is not in the spectrum, i.e.,

Op(8) There exists ε > 0 with the property that T rP_{|φ|}([0, ε]) = 0.

If φ satisfies the properties Op(1) − Op(8), then

(1.8) θ_{|φ|}(t) = 0(e^{−tε/2}) for t → ∞.

Thus ∫_{1}^{∞} t^{s−1}θ_{|φ|}(t)dt is an entire function of s and, as a consequence,

(1.9) ζ_{|φ|}^{II}(s) := \frac{1}{Γ(s)} ∫_{1}^{∞} t^{s−1}θ_{|φ|}(t)dt

is a holomorphic function in s ∈ IC with

(1.10) ζ_{|φ|}^{II}(0) = 0.

(Actually for ζ_{|φ|}^{II}(s), no zeta regularization is needed (cf [BFKM], subsection 6.1).)

For operators φ satisfying properties Op(1) − Op(8), one can introduce the zeta function ζ_{|φ|}(s)

ζ_{|φ|}(s) := ζ_{|φ|}^{I}(s) + ζ_{|φ|}^{II}(s) = \frac{1}{Γ(s)} ∫_{0}^{∞} t^{s−1}θ_{|φ|}(t)dt

for Res > α which has an analytic continuation at s = 0. This allows to define the volume of the operator φ, V olφ, by

(1.11) logV ol(φ) := logdet|φ| := −\frac{d}{ds}|_{s=0} ζ_{|φ|}(s).

Definition 1.2 1. An operator φ : W_{1} → W_{2} is said to be of sF type (strong Fredholm type) if

(i) φ satisfies Op(1) − Op(6) and

(ii) dim(Kerφ) := T rP_{|φ|}({0}) < ∞.

2. An operator φ is a bounded operator of trace class if

(i) φ is bounded (hence satisfies Op(1) − Op(3));

(ii) φ satisfies Op(4);

(iii) ||φ||_{tr} := ∫_{0_{+}}^{∞} λdF_{|φ|}(λ) < ∞.

3. An operator φ is said to be ζ-regular if φ satisfies Op(1) − Op(7).

4. An operator φ is a bounded operator of finite rank if

(i) φ is bounded;

(ii) dim(\overline{range(φ)}) < ∞.

(As a consequence, φ satisfies Op(4) − Op(6) and (Asy) with m = 0 and ρ = 1 and thus, Op(1) − Op(7) hold.)

Notice that an operator φ : W_{1} → W_{2} of sF type might not be bounded in the case where dimW_{1} = ∞.

Proposition 1.3 (cf [Di]) Let u : W_{1} → W_{2} and v : W_{2} → W_{3} be bounded operators with u, respectively v, of trace class. Then the composition vu is a bounded operator of trace class and ||vu||_{tr} ≤ ||v|| ||u||_{tr}, respectively, ||vu|| ≤ ||v||_{tr} ||u||.

As the proof of Proposition 1.3 is fairly standard, we omit it. The following result will be used in the proof of Proposition 1.5, stated below:

Lemma 1.4 Assume that φ : W → W is a nonnegative, selfadjoint operator (i.e φ = |φ|) of sF type. Let 0 < a < b and assume that ε > 0 satisfies ε < F_{φ}(a). Then one can choose a smooth function f ∈ C_{0}^{∞}(IR_{≥0}; IR) such that f(φ) is a bounded operator of finite rank (hence f(φ) is of trace class and commutes with φ) with the following properties:

(i) F_{φ+f(φ)}(λ) = 0 for λ < a (in particular, φ + f(φ) is 1-1);

(ii) F_{φ+f(φ)}(λ) ≥ ε for λ ≥ a;

(iii) F_{φ+f(φ)}(λ) = F_{φ}(λ) for λ ≥ b.

Remark This lemma will be applied in the case when φ is a pseudodifferential operator. Then φ can be chosen so that f(φ) is a smoothing operator.

Proof Let g : [0, ∞) → IR be a smooth, increasing function defined on [0, ∞) with g(λ) = a for λ ≤ a, g(λ) ≤ λ for a ≤ λ ≤ b and g(λ) = λ for λ ≥ b. Let f(λ) := g(λ) − λ and define

f(φ) := ∫_{0}^{∞} f(λ)dP_{φ}(λ).

Proposition 1.5 (A) If φ : W → W is of sF type and u : W → W is a bounded operator of sF type, then φ + u is of sF type. (B) If, in addition, φ and φ + u are selfadjoint, nonnegative operators and u is of trace class, then there exists ε > 0 so that the difference ζ_{φ}^{I}(s) − ζ_{φ+u}^{I}(s), defined for Res ≫ 0,

ζ_{φ}^{I}(s) − ζ_{φ+u}^{I}(s) = \frac{1}{Γ(s)} ∫_{0}^{1} t^{s−1}(θ_{φ}(t) − θ_{φ+u}(t))dt

has an analytic continuation to Res > −ε and its value at s = 0 is equal to dim(Kerφ) − dim(Ker(φ + u)).

Remark If v is a bounded operator and φ and u are operators as in Proposition 1.5(B), one can, instead of θ_{φ}(t) and θ_{φ+v}(t), consider the function θ_{φ,v}(t) := T rve^{−tφ}(Id − P_{φ}({0})) and similarly, θ_{φ+u,v}(t) := T rve^{−t(φ+u)}(Id − P_{φ+u}

({0})). Notice that for v = Id, θ_{φ,Id}(t) = θ_{φ}(t) (t > 0). Both expressions, ζ_{φ,v}^{I}(s) := \frac{1}{Γ(s)} ∫_{0}^{1} t^{s−1}θ_{φ,v}(t)dt and ζ_{φ+u,v}^{I}(s) := \frac{1}{Γ(s)} ∫_{0}^{1} t^{s−1}θ_{φ+u,v}(t)dt, define holomorphic functions for Res ≫ 0 and by the same arguments as in the proof of Proposition 1.5(B) (cf below) their difference is holomorphic for Res > −ε for some ε > 0 and its value at s = 0 is T r(vP_{φ}{0}) − T r(vP_{φ+u}{0}).

Proof of Proposition 1.5 (A) Since φ satisfies Op(1)−Op(3) and u is bounded, φ + u satisfies Op(1) and Op(2) as well. One verifies in a straightforward way that the graph of φ + u is closed (and thus Op(3) is valid) and the null space Ker(φ + u) has finite (von Neumann) dimension. It remains to check that Op(4) and Op(5) hold for φ + u. To see it, notice that, again by the variational characterization of the spectral distribution function, F_{φ+u}(λ) ≤ F_{φ}(λ+||u||) < ∞ ∀λ. Thus

F_{φ+u}(λ) = 0(λ^{α}) for λ ↗ ∞

where α is given by Op(5), satisfied by φ, and

F_{φ+u}(λ) = 0(1) for λ ↘ 0

as φ satisfies Op(4) and Op(5). (B) First we prove (B) in the case where u is of the form u = f(φ) with f ∈ C^{∞}([0, ∞); IR) being of compact support and such that φ + f(φ) is 1-1. We claim that h(t) := θ_{φ+u}(t) − θ_{φ}(t) is real analytic near t = 0 and satisfies h(0) = dim(Kerφ). From Lemma 1.1 one then concludes that the difference ζ_{φ}^{I}(s)−ζ_{φ+f(φ)}^{I}(s) = \frac{1}{Γ(s)} ∫_{0}^{1} t^{s−1}h(t)dt is well defined and holomorphic in s for Res > α, has an analytic continuation to Res > −1 and takes value dim(Kerφ) at s = 0. To prove that h(t) is real analytic in t near t = 0 and satisfies h(0) = dim(Kerφ) we argue as follows: Since φ and f(φ) commute, e^{−t(φ+f(φ))} − e^{−tφ} = e^{−tφ}(e^{−tf(φ)} − 1). By assumption, f has compact support, i.e. suppf ⊆ [0, K] for some K > 0. Therefore

f(φ) = ∫_{0}^{∞} f(λ)dP_{φ}(λ) = ∫_{0}^{K} f(λ)dP_{φ_{K}} (λ) = f(φ_{K})

where φ_{K} := ∫_{0}^{K} λdP_{φ}(λ) is φ restricted to L := \overline{rangeP ([0, K])}. Using that F_{φ_{K}} (λ) is constant for λ > K, we conclude, integrating by parts,

h_{1}(t) := T re^{−tφ_{K}} (e^{−tf(φ_{K})} − 1) = ∫_{0}^{K} e^{−tλ}(e^{−tf(λ)} − 1)dF_{φ_{K}} (λ) =
= e^{−tλ}(e^{−tf(λ)} − 1)F (λ)|_{0}^{K}
−t ∫_{0}^{K} e^{−tλ}(e^{−tf(λ)} − 1 + e^{−tf(λ)}f′(λ))F_{φ_{K}} (λ)dλ

which is real analytic in t and satisfies h_{1}(0) = 0. Further, one obtains

h_{2}(t) := T r(e^{−t(φ_{K}+f(φ_{K}))}P_{φ_{K}+f(φ_{K})}({0}) − e^{−tφ_{K}} P_{φ_{K}} ({0}))
= e^{−tf(0)} dim(Ker(φ + f(φ))) − dim(Kerφ) = − dim(Kerφ)

which is real analytic in t as well and satisfies h_{2}(0) = − dim Ker(φ). We conclude that h(t) = h_{1}(t) − h_{2}(t) is real analytic in t and h(0) = dim(Kerφ). To prove (B) in the general case it suffices, in view of the first step and Lemma 1.4, to consider operators φ and u so that, in addition, φ and φ + u are 1-1 and satisfy Op(8). These additional properties allow to represent ζ_{φ}^{I}(s) and ζ_{φ+u}^{I}(s) by a contour integral as follows: Choose ε > 0 so that F_{φ}(λ) = F_{φ+u}(λ) = 0 for 0 ≤ λ < ε and consider the contour Γ_{ε/2} = Γ_{−} ∪ Γ_{0} ∪ Γ_{+} defined by

Γ_{−} := \{z = re^{−iπ}|∞ > r ≥ 0\}; Γ_{0} := \{z = \frac{ε}{2}e^{iμ}| − π ≤ μ ≤ π\};

Γ_{+} = {z = re^{iπ}|ε/2 ≤ r < ∞}.

For λ ∈ Γ_{ε/2}, λ − φ and λ − φ − u are both invertible and one computes

\begin{aligned}R(λ) & := & (λ − φ − u)^{−1} − (λ − φ)^{−1} = \\ & = & (λ − φ)^{−1}u(λ − φ − u)^{−1}. \end{aligned}

As u is a bounded operator of trace class we conclude by Proposition 1.3 that, for λ ∈ Γ_{ε/2}, R(λ) is a bounded operator of trace class as well. By a standard argument one shows that for λ ∈ Γ_{ε/2},

(1.12) ||(λ − φ)^{−1}|| ≤ \frac{1}{|λ−ε|} (≤ \frac{1}{ε/2})

(1.12’) ||(λ − φ − u)^{−1}|| ≤ \frac{1}{|λ−ε|} (≤ \frac{1}{ε/2}).

These estimates allow to represent the difference A(s) := ζ_{φ}^{I}(s)−ζ_{φ+u}^{I}(s) by the following contour integral

(1.13) A(s) = \frac{1}{2πi} ∫_{Γ_{ε/2}} λ^{−s}T rR(λ)dλ = A_{0}(s) + A_{+}(s) + A_{−}(s)

where

(1.14)_{±} A_{±}(s) = \frac{1}{2πi} ∫_{Γ_{±}} λ^{−s}T rR(λ)dλ.

(1.14)_{0} A_{0}(s) = \frac{1}{2πi} ∫_{Γ_{0}} λ^{−s}T rR(λ)dλ.

Notice that A_{0}(s) is holomorphic function for s ∈ IC and A_{±}(s) are holomorphic functions in a neighborhood of s = 0 due to the fact that ∫_{ε/2}^{∞} \frac{x^{−s}}{(x+ε)^{2}} dx is absolutely convergent for |s| sufficiently small. Moreover, A_{0}(0) = 0 and A_{+}(0)+ A(0) = 0.

Examples 1.6

1. φ : W_{1} → W_{2} bounded and dimW_{1} < ∞: then φ satisfies Op(1) − Op(7) and is of trace class. In fact, Op(6) follows from (Asy) which holds with m = 0 and ρ = 1. Since Kerφ ⊆ W_{1} is of finite dimension, φ is of sF type.

2. φ : W_{1} → W_{2} is bounded and dimW_{2} < ∞ : then the adjoint φ^{∗} of φ is as in Example (1). Therefore φ satisfies Op(1) − Op(6) and (Asy) with m = 0, ρ = 1. Moreover φ is of trace class. However the dimension of the nullspace Kerφ might not be finite. Therefore, it is useful to reduce φ to φ̂ : W_{1}/Kerφ → W_{2}. Then φ̂ is injective and bounded and dim(W_{1}/Kerφ) < ∞.

3. φ of finite rank: then the reduced map φ̂ : W_{1}/Kerφ → W_{2} is bounded and injective and fits either into (1) or (2).

4. Suppose (M,g) is a closed Riemann manifold and E → M is a smooth bundle of A-Hilbert modules of finite type, equipped with a Hermitian structure μ and a smooth connection ∇ so that the A-action is parallel. Using g, μ and ∇ one can define inner products ⟨·, ·⟩_{s} of Sobolev type on the space of smooth sections on E, C^{∞}(E). (For s = 0, the connection ∇ is not needed.) The completion of C^{∞}(E) with respect to ⟨·, ·⟩_{s} is a Hilbert space denoted by H_{s}(E). Different choices of g, μ, ∇ lead to the same topological vector spaces H_{s}(E), with different inner products, but equivalent norms.

A pseudodifferential operator (ΨDO) of order d induces operators A : H_{s}(E) → H_{s′}(E) which are bounded if s′ ≤ s−d. The operator (Δ+id)^{−s′/2}A(Δ+id)^{s/2} : L_{2}(E) → L_{2}(E) has a kernel of class C^{k} if s′ < s−d−k−dimM. If A is an elliptic ΨD0 of order d > 0 it satisfies Op(1) − Op(7) and (Asy) (with m := dimM and ρ = 1/k) cf [BFKM] section 2. Hence A is a ζ-regular operator of sF type.

1.2 Complexes

In this subsection we introduce the class of ζ-regular complexes of sF type and define their algebraic and reduced cohomology. In the case where a ζ-regular complex of sF type is algebraically acyclic or, more generally, of determinant class, one can define its torsion. We recall Milnor’s lemma which relates the torsions of a short exact sequence of complexes of finite (von Neumann) dimension. Further, within our class of complexes, we discuss the notion of a mapping cone which is a complex induced by a morphism f between two complexes and show that if f is of trace class and induces an isomorphism in algebraic cohomology then the mapping cone is algebraically acyclic. Proposition 1.15 relates, under appropriate conditions, the torsions of the mapping cones of two morphisms and their composition. The class of ζ-regular complexes of sF type has been chosen so that it includes the mapping cone of a regularized version of the integration map between the deRham complex and the combinatorial complex, which will be discussed in section 2.

A cochain complex C ≡ (C_{i}, d_{i}) of A-Hilbert modules,

C_{0} →^{d_{0}} C_{1} →^{d_{1}} … → C_{N−1} ⟶^{d_{N−1}} C_{N}

consists of a finite sequence of A-Hilbert modules C_{i}(0 ≤ i ≤ N) and operators d_{i} : C_{i} → C_{i+1}, satisfying Op(1) − Op(3), with the property that range(d_{i}) ⊆ domain(d_{i+1}) and d_{i+1}d_{i} = 0.

Notice that the null space Kerd_{i} is always an A-Hilbert submodule.

Definition 1.7 For 0 ≤ i ≤ N :

algebraic cohomology: H^{i}(C) := Kerd_{i}/range(d_{i−1})

reduced cohomology : \overline{H}^{i}(C) := Kerd_{i}/\overline{range(d_{i−1})}.

We point out that the reduced cohomology \overline{H}^{i}(C) is an A-Hilbert module whereas the algebraic cohomology H^{i}(C) is, in general, not an A-Hilbert module, as range(d_{i−1}) needs not to be closed. H^{i}(C) is always an A- module.

Given a complex (C_{i}, d_{i}) we can define the adjoint operators d_{i}^{∗} : C_{i+1} → C_{i} and, in turn, the Laplacians Δ_{i},

(1.15) Δ_{i} = d_{i}^{∗}d_{i} + d_{i−1}d_{i−1}^{∗}

as well as the Hodge decomposition C_{i} = H_{i} ⊕ C_{i}^{+} ⊕ C_{i}^{−} where

(1.16) H_{i} := Ker(d_{i}) ∩ Ker(d_{i−1}^{∗}).

(1.17) C_{i}^{+} := \overline{range(d_{i−1})}; C_{i}^{−} := \overline{range(d_{i}^{∗})}.

With respect to this decomposition, the operators d_{i}, d_{i}^{∗} and Δ_{i} take the form

(1.18) d_{i} = ( \begin{matrix}0 & 0 & 0 \\ 0 & 0 & \underline{d}_{i} \\ 0 & 0 & 0 \end{matrix} ); d_{i}^{∗} = ( \begin{matrix}0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & \underline{d}_{i}^{∗} & 0 \end{matrix} );

Δ_{i} = ( \begin{matrix}0 & 0 & 0 \\ 0 & \underline{d}_{i−1}\underline{d}_{i−1}^{∗} & 0 \\ 0 & 0 & \underline{d}_{i}^{∗}\underline{d}_{i} \end{matrix} ) .

We write \underline{Δ}_{i}^{+} := \underline{d}_{i−1}\underline{d}_{i−1}^{∗} and \underline{Δ}_{i}^{−} := \underline{d}_{i}^{∗}\underline{d}_{i} Note that \underline{d}_{i} : C_{i}^{−} → C_{i+1}^{+} and \underline{d}_{i}^{∗} : C_{i+1}^{+} → C_{i}^{−} are injective operators with dense image, (hence so are \underline{Δ}_{i}^{+} and \underline{Δ}_{i}^{+},)and H_{i} is isometric to the reduced cohomology \overline{H}^{i}(C).

Definition 1.8

(i) The cochain complex C = (C_{i}, d_{i}) is said to be of sF type if (CX1) the operators \underline{d}_{i} satisfy Op(1)−Op(6) (and hence, in view of the injectivity

of \underline{d}_{i}, are of sF type); (CX2) dimH_{i} < ∞.

(ii) A complex (C_{i}, d_{i}) of sF type is called \underline{ζ-regular} if, in addition, (CX3) the operators \underline{d}_{i} satisfy Op(7).

(iii) A complex (C_{i}, d_{i}) is said to be of finite type (of finite dimension), if the A-Hilbert modules C_{i} are of finite type (of finite dimension) and the operators \underline{d}_{i} are bounded.

Remark 1 The conditions (CX1) and (CX2) are equivalent to (CX_{12}) the Laplacians Δ_{i} are of sF type

and the three properties (CX1)-(CX3) are equivalent to (CX_{123}) the Laplacians Δ_{i} are ζ-regular operators of sF type.

Remark 2 Assume that H_{i_{0}} = 0 for some i_{0}. Using the closed graph theorem one verifies that the following statements are equivalent:

(a) H^{i_{0}}(C) = 0; (b) \underline{d}_{i_{0}} has a bounded inverse.

Given a complex (C_{i}, d_{i}) of sF type, one can define the spectral distribution functions F_{i}(λ) and N_{i}(λ),

(1.19) F_{i}(λ) := F_{\underline{d}_{i}}(λ)( = F_{|\underline{d}_{i}|}(λ)); N_{i}(λ) := F_{Δ_{i}}(λ). Then (1.20) N_{i}(λ) = F_{i}(λ) + F_{i−1}(λ).

A complex (C_{i}, d_{i}) of sF type is said to be \underline{algebraically acyclic} if H^{i}(C) = 0 for 0 ≤ i ≤ N.

Notice that for an acyclic complex of sF type, the spectrum of Δ_{i} does not contain 0. In fact, in view of Remark 2 after Definition 1.9, there exists ε > 0 so that N_{i}(λ) = 0 for 0 ≤ λ < ε and the same is true for the spectral distribution function of \underline{Δ}_{i}^{+} = \underline{d}_{i−1}\underline{d}_{i−1}^{∗} and \underline{Δ}_{i}^{−} = \underline{d}_{i}^{∗}\underline{d}_{i}. This allows to define the torsion T (C) of an algebraically acyclic complex of sF type,

(1.21) \begin{matrix}logT (C) & := & \frac{1}{2} ∑_{q}(−1)^{q+1}q log detΔ_{q} \\ & & ( = \frac{1}{2} ∑_{q}(−1)^{q}log det\underline{Δ}_{q}^{−} = \frac{1}{2} ∑_{q}(−1)^{q+1}log det\underline{Δ}_{q}^{+}). \end{matrix}

More generally, the torsion T (C) can be defined if (C, d) is of determinant class, i.e.

∫_{0^{+}}^{1} logλdN_{i}(λ) > −∞ (0 ≤ i ≤ N).

Definition 1.9 (i) A morphism f : C^{1} → C^{2} between the complexes (C_{i}^{1}, d_{1,i}) and (C_{i}^{2}, d_{2,i}) consists of a collection of bounded operators f_{i} : C_{i}^{1} → C_{i}^{2} so that

• f_{i}(domain(d_{1,i})) ⊆ domain(d_{2,i}); d_{2,i}f_{i} = f_{i+1}d_{1,i} (on domain(d_{1,i})).

(ii) f is said to be of trace class if, in addition,

• the operators f_{i}, d_{2,i}^{∗}f_{i+1} and f_{i}d_{1,i}^{∗} are bounded operators of trace class.

A morphism f : C^{1} → C^{2} induces A-linear maps in algebraic cohomology, H(f); : H^{i}(C^{1}) → H^{i}(C^{2}) and bounded A-linear maps on the reduced cohomology, \overline{H}(f)_{i} : \overline{H}^{i}(C^{1}) → \overline{H}^{i}(C^{2}). Further, with respect to the Hodge decomposition, f_{i} takes the form,

f_{i} = ( \begin{matrix}f_{i,11} & 0 & f_{i,13} \\ f_{i,21} & f_{i,22} & f_{i,23} \\ 0 & 0 & f_{i,33} \end{matrix} )

where f_{i,11} = \overline{H}(f)_{i}.

For the convenience of the reader, we recall Milnor’s lemma which was extended for complexes of A-Hilbert modules in [BFK2]. Assume that

0 → C^{1} →^{f} C^{2} →^{g} C^{3} → 0

is a short exact sequence of complexes of finite dimension. It induces a long weakly exact sequence H in reduced cohomology

… → \overline{H}^{i}(C^{1}) ⟶^{\overline{H}(f_{i})} \overline{H}^{i}(C^{2}) ⟶^{\overline{H}(g_{i})} \overline{H}^{i}(C^{3}) ⟶^{\overline{H}(δ_{i})} \overline{H}^{i+1}(C^{1}) → …

Proposition 1.10 (cf [BFK2, Theorem 1.5]) If three out of the four cochain complexes C^{1}, C^{2}, C^{3}, H are of determinant class, then so is the fourth and one has the following equality

\begin{aligned}logT (C^{2}) & = & logT (C^{1}) + logT (C^{3}) + logT (H)− \\ & − & ∑_{i}(−1)^{i}logT (0 → C_{i}^{1} → C_{i}^{2} → C_{i}^{3} → 0). \end{aligned}

Remark In [BFK2], Lemma B2.8 was only stated for complexes of finite type. By the same proof, the result remains true for complexes of finite dimension.

Given a ζ-regular complex of sF type, 0 → C_{0} →^{d_{0}} … →^{d_{2N}} C_{2N+1} → 0, we can consider its dual. If C_{j}^{♯} := C_{2N+1−j}, and d_{j}^{♯} := d_{2N−j}^{∗} (adjoint of d_{2N−j}), then

0 → C_{0}^{♯} →^{d_{0}^{♯}} … →^{d_{2N}^{♯}} C_{2N+1}^{♯} → 0

is a ζ-regular complex of sF type. Notice that (\underline{d}_{j}^{♯})^{∗}\underline{d}_{i}^{♯} = \underline{d}_{2N−j} \underline{d}_{2N−j}^{∗}. Thus in case C is of determinant class, so is C^{♯} and one obtains

(1.22) T (C^{♯}) = T (C).

We end this subsection with a discussion of the mapping cone C(f) = (C(f)_{i}, d(f)_{i}) associated to a morphism f : C^{1} → C^{2} between ζ-regular complexes (C^{1}, d_{1}) and (C^{2}, d_{2}) of sF type. This is a complex given by

(MC1) C(f)_{i} := C_{i−1}^{2} ⊕ C_{i}^{1};

(MC2) d(f)_{i} = ( \begin{matrix}−d_{2,i−1} & f_{i} \\ 0 & d_{1,i} \end{matrix} ) .

The Laplacians Δ(f)_{i} of C(f) can be computed,

(1.23) Δ(f)_{i} = ( \begin{matrix}Δ_{2,i−1} + f_{i−1}f_{i−1}^{∗} & −d_{2,i−1}^{∗} · f_{i} + f_{i−1} · d_{1,i−1}^{∗} \\ −f_{i}^{∗} · d_{2,i−1} + d_{1,i−1} · f_{i−1}^{∗} & Δ_{1,i} + f_{i}^{∗}f_{i} \end{matrix} )

The following observation, will be used in order to introduce the relative L_{2}- torsion without making the assumption that the manifold is of determinant class:

Lemma 1.11 (i) Let f : C^{1} → C^{2} be a morphism of trace class between complexes C^{1} and C^{2} of sF type. Then the mapping cone C(f) is of sF type.

(ii) If, in addition, C^{1} and C^{2} are ζ-regular, then C(f) is ζ-regular.

(iii) If φ : C^{1} → C^{2} is a morphism of trace class between ζ-regular complexes of sF type which induces an isomorphism in algebraic cohomology, then the mapping cone C(f) is algebraically acyclic and thus has a well defined torsion T (C(f)) (cf (1.21)).

Proof (i) follows from Proposition 1.5 (A) by choosing φ := ( \begin{matrix}Δ_{2,i−1} & 0 \\ 0 & Δ_{1,i} \end{matrix} )

and

u_{i} := Δ(f)_{i} − φ_{i} = ( \begin{matrix}f_{i−1}f_{i−1}^{∗} & −d_{2,i−1}^{∗}f_{i} + f_{i−1}d_{1,i−1}^{∗} \\ −f_{i}^{∗}d_{2,i−1} + d_{1,i−1}f_{i−1}^{∗} & f_{i}^{∗}f_{i} \end{matrix} )

By Remark 1 after Definition 1.8, the φ_{i} are operators of sF type and, by assumption, the u_{i} are bounded operators of trace class. By Proposition 1.5, Δ(f)_{i} is then an operator of sF type. Apply the same Remark 1 once more to conclude that (CX1) holds. Property (CX2) of Definition 1.8 is easily verified.

(ii) follows from Proposition 1.5 (B) and Remark 1 after Definition 1.8.

(iii) In view of (1.21) and the statements (i) and (ii) it remains to verify that C(f) is algebraically acyclic, i.e. that range(d(f)_{i−1}) = Ker(d(f)_{i}). Let (v_{i−1}, u_{i}) ∈ Ker(d(f)_{i}) ⊆ C_{i−1}^{2} ⊕ C_{i}^{1}. Then

−d_{2,i−1}v_{i−1} + f_{i}u_{i} = 0 ; d_{1,i}u_{i} = 0.

In particular, f_{i}u_{i} ∈ range(d_{2,i−1}) and, as f induces an isomorphism in algebraic cohomology, u_{i} ∈ range(d_{1,i−1}), i.e. there exists u_{i−1} ∈ C_{i−1}^{1} with u_{i} = d_{1,i−1} u_{i−1}. As f_{i}d_{1,i−1}u_{i−1} = d_{2,i−1}f_{i−1}u_{i−1}, we conclude that −v_{i−1} + f_{i−1}u_{i−1} ∈ Kerd_{2,i−1}. Using once again the assumption that f induces an isomorphism in algebraic cohomology, one sees that there exists v_{i−2} ∈ C_{i−2}^{2} and ũ_{i−1} ∈ Kerd_{1,i−1} with −v_{i−1} + f_{i−1}u_{i−1} = f_{i−1}ũ_{i−1} + d_{2,i−2}v_{i−2}.

Thus

v_{i−1} = −d_{2,i−2}v_{i−2} + f_{i−1}(u_{i−1} − ũ_{i−1})

and, as ũ_{i−1} ∈ Kerd_{1,i−1}

u_{i} = d_{1,i−1}u_{i−1} = d_{1,i−1}(u_{i−1} − ũ_{i−1})

i.e. we have shown that (v_{i−1}, v_{i}) ∈ ranged(f)_{i−1}, and therefore that Kerd(f)_{i} = ranged(f)_{i−1}.

If f : (C, d_{1}) → (C, d_{2}) is an isomorphism of complexes of finite dimension we can use Lemma 1.11 together with Milnor’s lemma to compute the torsion of the mapping cone C(f). Recall that the suspension ΣC = (ΣC_{i,Σ}d_{i}) of a complex C = (C_{i}, d_{i}) is the complex given by

\begin{matrix}ΣC_{i} ≡ (ΣC)_{i} := C_{i−1}(i ≥ 1); ΣC_{0} := 0; \\ _{Σ}d_{i} := −d_{i−1} (i ≥ 1); d_{0} = 0. \end{matrix}

Proposition 1.12 Assume that C^{i} ≡ (C, d_{j})(j = 1, 2) are ζ-regular complexes of sF type and of finite dimension. If f : C^{1} → C^{2}, is an isomorphism, then the mapping cone C(f) is algebraically acyclic and

logT (C(f)) = ∑(−1)^{j}log vol(f_{j})

Proof By Lemma 1.11, C(f) is an algebraically ζ-regular complex of sF type.

First we treat the case where, in addition, C^{1} (and then C^{2} as well) is algebraically acyclic. Consider the following short exact sequence of cochain complexes

0 → Σ(C, d_{2}) →^{J} C(f) →^{P} (C, d_{1}) → 0

where J (respectively P) is the canonical inclusion (canonical projection).

By Proposition 1.10, the corresponding long weakly exact sequence H in reduced cohomology is of determinant class and

logT (C(f)) = logT (ΣC^{2}) + logT (C^{1}) + logT (H).

Notice that H is given by

… → \overline{H}^{i}(ΣC) → \overline{H}^{i}(C(f)) → \overline{H}^{i}(C) ⟶^{\overline{H}(δ_{i})} \overline{H}^{i+1}(ΣC) → …

where the connecting homomorphism \overline{H}(δ_{i}) is given by the restriction \overline{H}(f_{i}) of f_{i} to \overline{H}^{i+1}(ΣC) = \overline{H}^{i}(C) into \overline{H}^{i}(C). Therefore

logT (H) = ∑(−1)^{i}log vol(\overline{H}(f_{i})).

By Lemma 1.13 below, logT (ΣC^{2}) = −logT (C^{2}). As f_{i+1}d_{1,i} = d_{2,i}f_{i} we conclude that \underline{d}_{1,i} = (f_{i+1}^{+})^{−1}\underline{d}_{2,i}f_{i}^{−} where f_{i}^{±} : C_{i}^{1,±} → C_{i}^{2,±} are the restrictions of f_{i} to C_{i}^{1,±} and are isomorphisms as well.

Thus

logT (C^{1}) = \frac{1}{2} ∑(−1)^{i}log det(\underline{d}_{1,i}^{∗}\underline{d}_{1,i})
= \frac{1}{2} ∑(−1)^{i}log det((f_{i}^{−})^{∗}\underline{d}_{2,i}^{∗}((f_{i+i}^{+})^{−1})^{∗}(f_{i+1}^{+})^{−1}\underline{d}_{2,i}f_{i}^{−})
= \frac{1}{2} ∑(−1)^{i}log det(f_{i+1}^{+}f_{i+1}^{+∗})^{−1}
+ \frac{1}{2} ∑(−1)^{i}log det(\underline{d}_{2,i}^{∗}\underline{d}_{2,i})
+ \frac{1}{2} ∑(−1)^{i}log det(f_{i}^{−}f_{i}^{−∗})
= logT (C^{2}) + ∑_{i}(−1)^{i}(log vol(f_{i}^{+}) + log vol(f_{i}^{−})).

Combining the equality above yields

logT (C(f)) = ∑(−1)^{i}(log vol(f_{i}^{+}) + log vol(f_{i}^{−}))
+ ∑(−1)^{i}log vol(\overline{H}(f_{i}))
= ∑(−1)^{i}log vol(f_{i}).

To prove the result in general we consider a deformation (C, d_{1}(ε)) of the complex (C, d_{1}), depending smoothly on the parameter ε, so that, for ε ≠ 0, (C, d_{1}(ε)) is algebraically acyclic. The operator d_{1}(ε) is constructed as follows: with respect to the Hodge decomposition C_{i} = H_{i} ⊕ C_{i}^{+} ⊕ C_{i}^{−}, d_{1,i} takes the form

d_{1,i} = ( \begin{matrix}0 & 0 & 0 \\ 0 & 0 & \underline{d}_{1,i} \\ 0 & 0 & 0 \end{matrix} ) where \underline{d}_{1,i} : C_{1,i}^{−} → C_{1,i+1}^{+}. Consider the polar decom-

position of \underline{d}_{1,i}, \underline{d}_{1,i} = ξ_{i}, η_{i} where ξ_{i} := \underline{d}_{1,i}(\underline{d}_{1,i}^{∗}\underline{d}_{1,i})^{−1/2} : C_{i}^{−} → C_{i+1}^{+} is an isometry and η_{i} is given by η_{i} := (\underline{d}_{1,i}^{∗}\underline{d}_{1,i})^{1/2}. The operator η_{i} admits a spectral decomposition; η_{i} = ∫_{0^{+}}^{∞} λdP_{i}(λ). Define η_{i}(ε) := ∫_{0^{+}}^{∞}(λ + ε)dP_{i}(λ) and let \underline{d}_{1,i}(ε) := ξ_{i}η_{i}(ε). Then \underline{d}_{1,i}(ε) is an isomorphism and thus of determinant class. Now consider C_{ε}^{1} ≡ (C, \underline{d}_{1}, (ε)) where the operator d_{1,i}(ε), with respect to the

Hodge decomposition of (C, d_{1}), is given by d_{1,i}(ε) = ( \begin{matrix}0 & 0 & 0 \\ 0 & 0 & \underline{d}_{1,i}(ε) \\ 0 & 0 & 0 \end{matrix} ) . Define

(C, d_{2}(ε)) where \underline{d}_{2,i}(ε) := f_{i+1}^{+} \underline{d}_{1,ε}(f_{i}^{−})^{−1}. Then, for any ε, f :(C, d_{1}(ε)) → (C, d_{2}(ε)) is a morphism and, by the argument above, the torsion T (C_{ε}(f)) of the mapping cone C_{ε}(f) induced by f : (C, d_{1}(ε)) → (C, d_{2}(ε)) is given by, for ε ≠ 0,

logT (C_{ε}(f)) = ∑(−1)^{i}log vol(f_{i})

which is independent of ε. As C_{ε}(f) is algebraically acyclic for all values of ε and logT (C_{ε}(f)) depends continuously on ε, we conclude that

logT (C(f)) = ∑_{j}(−1)^{j}log vol(f_{j}).

Lemma 1.13 (i) Let C be a ζ-regular complex of sF type. Then the mapping cone C(id) is an algebraically acyclic ζ-regular complex of sF type and satisfies logT (C(id)) = 0.

(ii) If C is an algebraically acyclic, ζ-regular complex of sF type, then so is ΣC and satisfies

logT (ΣC) = −logT (C).

Proof (i) By Lemma 1.11, C(id) is an algebraically acyclic ζ-regular complex of sF type. Use Lemma 2.4 to conclude that

logT (C(id)) = \frac{1}{2} ∑(−1)^{q+1}log det(Δ_{q} + id) = 0.

(ii) It follows from Definition 1.9 and the assumptions that ΣC is an algebraically acyclic, ζ-regular complex of sF type. Its torsion is therefore well defined, and by (1.21)

\begin{aligned}logT (ΣC) & = & \frac{1}{2} ∑(−1)^{q+1}log det(\underline{Δ(ΣC)}_{q}^{−}) \\ & = & −\frac{1}{2} ∑(−1)^{q+1}log det(\underline{Δ}_{q}^{−}) = −logT (C). \end{aligned}

The following Lemma 1.14 is due to Carey, Mathai and Mishchenko [CMM]. For the convenience of the reader we include its statement. As the proof in the preliminary preprint [CMM]^{2} is somewhat incomplete, we refer the reader to Appendix A for a detailed proof. Let

(1.24) 0 → C^{1} →^{I} C →^{P} C^{2} → 0

be an exact sequence of cochain complexes of sF type where the complex C = (C_{i}, d_{i}) is given by C_{i} = C_{i}^{1} ⊕ C_{i}^{2} and

(1.25) d_{i} = ( \begin{matrix}d_{1,i} & f_{i} \\ 0 & d_{2,i} \end{matrix} )

with f_{i} : C_{i}^{2} → C_{i+1}^{1} satisfying

(1.26) f_{i}(domain(d_{2,i})) ⊆ domain(d_{1,i+1}) and

(1.27) f_{i+1}d_{2,i} + d_{1,i+1}f_{i} = 0 (on domain(d_{2,i})) so that d_{i+1}d_{i} = 0. The morphism I [resp. P] in (1.24) denotes the canonical inclusion [resp. canonical projection].

2The first author thanks Mishchenko for kindly making the preliminary preprint available to him.

Lemma 1.14 ([CMM]) Assume C^{1} and C^{2} are ζ-regular complexes of sF type which are algebraically acyclic and f_{i} : C_{i}^{2} → C_{i+1}^{1} are bounded maps of trace class so that (1.26) and (1.27) are satisfied and d_{1,i}^{∗}f_{i} as well as f_{i}d_{2,i}^{∗} are bounded operators of trace class. Then the complex C, given by (1.24) - (1.25), is an algebraically acyclic, ζ-regular complex of sF type and

(1.28) logT (C) = logT (C^{1}) + logT (C^{2}).

Proof: cf [CMM] or Appendix A.

Proposition 1.15 Suppose that C^{1}, C^{2} and C^{3} are ζ-regular complexes of sF type and f_{1} : C^{1} → C^{2} and f_{2} : C^{2} → C^{3} are morphisms of trace class which induce isomorphism in algebraic cohomology. Then:

(i) f := f_{2} ∘ f_{1} is a morphism of trace class;

(ii) the mapping cones C(f_{1}), C(f_{2}) and C(f) are ζ-regular complexes of sF type;

(iii) C(f_{1}), C(f_{2}) and C(f) are algebraically acyclic (and thus their torsions are well defined).

(iv) If, in addition, either C^{3} (case 1) or C^{1} (case 2) is of finite type, then logT (C(f)) = logT (C(f_{1})) + logT (C(f_{2})).

Proof The proof is an application of Lemma 1.13 and Lemma 1.14.

Case 1 (C^{3} is of finite type): Consider the diagram

(1.29) \begin{matrix}& & ΣC(id_{3}) \\ & & ↓ I_{1} \\ ΣC(−f_{2}) & →^{I_{2}} & C(h) & →^{P_{2}} & C(f) & → & 0 \\ & & ↓ P_{1} \\ & & C(f_{1}) \\ & & ↓ \\ & & 0 \end{matrix}

where C(f_{1}), C(−f_{2}), C(f) and C(id_{3}) are mapping cones, hence C(f_{1}); = C_{i−1}^{2} ⊕ C_{i}^{1}, C(−f_{2})_{i} = C_{i−1}^{3} ⊕ C_{i}^{2}, C(f)_{i} = C_{i−1}^{3} ⊕ C_{i}^{1} and C(id_{3})_{i} = C_{i−1}^{3} ⊕ C_{i}^{3}. The morphism h : C(f_{1}) → C(id_{3}), is given by the operators h_{i} : C_{i−1}^{2} ⊕ C_{i}^{1} → C_{i−1}^{3} ⊕ C_{i}^{3},

(1.30) h_{i} := ( \begin{matrix}f_{2,i−1} & 0 \\ 0 & f_{i} \end{matrix} ) ,

One verifies that h_{i} are bounded maps of trace class. Denote by C(h) the mapping cone of h, hence C(h)_{i} = (C_{i−2}^{3} ⊕ C_{i−1}^{3}) ⊕ (C_{i−1}^{2} ⊕ C_{i}^{1}). Further I_{1}, respectively I_{2}, are the canonical inclusions,

I_{1} : C_{i−2}^{3} ⊕ C_{i−1}^{3} → (C_{i−2}^{3} ⊕ C_{i−1}^{3}) ⊕ (C_{i−1}^{2} ⊕ C_{i}^{1})

and

I_{2} : C_{i−2}^{3} ⊕ C_{i−1}^{2} → (C_{i−2}^{3} ⊕ C_{i−1}^{3}) ⊕ (C_{i−1}^{2} ⊕ C_{i}^{1})

whereas P_{1} and P_{2} denote the canonical projections on the complement of the images of I_{1} resp. I_{2}. One verifies (cf (1.29)) that

(1.31) 0 → ΣC(id_{3}) →^{I_{1}} C(h) →^{P_{1}} C(f_{1}) → 0

and

(1.32) 0 → ΣC(−f_{2}) →^{I_{2}} C(h) →^{P_{2}} C(f) → 0

are short exact sequences of cochain complexes. First notice that by Proposition 1.3, f = f_{2} · f_{1} is a morphism of trace class which proves (i).

By Lemma 1.11, C(f_{1}), C(f_{2}) and C(f) are algebraically acyclic, ζ-regular complexes of sF type and, by Lemma 1.13, C(id_{3}) and ΣC(f_{i}), ΣC(f_{2}), ΣC(id_{3}) are algebraically acyclic, ζ-regular complex of sF type. We would like to apply Lemma 1.14 to (1.31) and (1.32). For this purpose, observe that, given h_{i} : C(f_{1})_{i} = C_{i−1}^{2} ⊕ C_{i} → C(id_{3}) = C_{i−1}^{3} ⊕ C_{i}^{3}, d(h)_{i} admits the following decomposition

(1.33) \begin{matrix}d(h)_{i} & = & ( \begin{matrix}d(ΣC(id_{3}))_{i} & h_{i} \\ 0 & d(C(f_{1})_{i} \end{matrix} ) , \\ & = & ( \begin{matrix}d_{3,i−2} & −id_{3,i−1} & f_{2,i−1} & 0 \\ 0 & −d_{3,i−1} & 0 & f_{i} \\ 0 & 0 & d_{2,i−1} & f_{1,i} \\ 0 & 0 & 0 & d_{1,i} \end{matrix} ) \end{matrix}

Using this decomposition, all assumptions in Lemma 1.15 which have not yet been proved can be verified in a straight forward way.

Therefore, we can apply Lemma 1.14 to (1.31) to conclude that C(h) is an algebraically acyclic, ζ-regular complex of sF type and

(1.34) \begin{matrix}logT (C(h)) & = & logT (ΣC(id_{3})) + logT (C(f_{1})) \\ & = & logT (C(f_{1})) \end{matrix}

where we used that, by Lemma 1.13,

logT (ΣC(id_{3})) = −logT (C(id_{3})) = 0

In order to apply Lemma 1.14 to (1.32), notice that

(1.35) \begin{matrix}( \begin{matrix}d(ΣC(−f_{2}))_{i} & h̃_{i} \\ 0 & d(C(f))_{i} \end{matrix} ) & = \\ ( \begin{matrix}d_{3,i−2} & +f_{2,i−1} & h̃_{i} \\ 0 & −d_{2,i−1} \\ 0 & 0 & −d_{3,i−1} & f_{i} \\ 0 & 0 & 0 & d_{1,i} \end{matrix} ) & = & d(h)_{i} \end{matrix}

where, in view of (1.33), h̃_{i} : C(f)_{i} = C_{i−1}^{3} ⊕ C_{i}^{1} → C_{i−1}^{3} ⊕ C_{i}^{2} = ΣC(−f_{2})_{i+1}

given by h̃_{i} = ( \begin{matrix}−id_{3,i−1} & 0 \\ 0 & f_{1,i} \end{matrix} ) . As C^{3} is of finite type, id_{3,i} : C_{i}^{3} → C_{i}^{3} and

therefore h̃_{i} are bounded maps of trace class. Again one verifies that all other assumption in Lemma 1.14 are satisfied. Thus we can apply Lemma 1.14 to (1.32), to conclude that

(1.36) \begin{matrix}logT (C(h)) & = & logT (ΣC(−f_{2})) + logT (C(f)) \\ & = & −logT (C(−f_{2})) + logT (C(f)) \end{matrix}

where for the last identity we again used Lemma 1.13. Notice that the morphism

Φ : C(f_{2}) → C(−f_{2}), given by Φ_{i} = ( \begin{matrix}Id & 0 \\ 0 & −Id \end{matrix} ) : C_{i−1}^{3} ⊕ C_{i}^{2} → C_{i−1}^{3} ⊕ C_{i}^{2},

is an isometry and therefore logT (C(−f_{2})) = logT (C(f_{2})). Combining this with (1.34) and (1.36) we conclude that statement (iv) in case 1 is proved.

Case 2 (C^{1} of finite type): In view of (1.22) it suffices to consider the dual complexes, to which we can apply case 1.

Proposition 1.16 Suppose that C^{1}, C^{2} and C^{3} are ζ-regular complexes of sF type and f_{1} : C^{1} → C^{2}, f_{2} : C^{2} → C^{3} are morphisms which induce isomorphisms in algebraic cohomology and denote by f the composition f := f_{2} ∘ f_{1}. Then the following statements hold:

(i) If f_{1} is an isometry and f_{2} is of trace class, then the mapping cones C(f_{2}) and C(f) are algebraically acyclic ζ-regular complexes of sF type. Moreover

logT (C(f_{2})) = logT (C(f)).

(ii) If f_{2} is an isometry and f_{1} is of trace class, then C(f_{1}) and C(f) are algebraically acyclic ζ-regular complexes of sF type. Moreover

logT (C(f_{1})) = logT (C(f)).

Proof The assumption imply that in case (i), C(f_{2}) and C(f) are isometric whereas in case (ii), C(f_{1}) and C(f) are isometric. From this observation and Lemma 1.11 the claimed results follow.

2 Relative torsion and its Witten deformation

In subsection 2.1 we introduce the relative torsion (cf [CMM]). As already mentioned in the introduction, the relative torsion cannot be defined as the torsion of the mapping cone associated to the integration map, denoted by Int, as Int cannot be extended to a closed morphism on L_{2}(Λ(M; E)). To circumvent this difficulty we consider s > \frac{n}{2} + 1, and the composition

L_{2}(Λ(M; E)) ≡ H_{0}(Λ(M; E)) ⟶^{(Δ+Id)^{−s/2}} H_{s}(Λ(M; E)) →^{Int_{s}} C

where H_{s}(Λ^{k}(M; E)) denotes the completion of the space Ω^{k}(M; E) of smooth k-forms with respect to the s-Sobolev norm, Δ_{k} denotes the k-Laplacian, Int_{s} is the extension of Int to H_{s}(Λ(M; E)) and C denotes the combinatorial complex associated to E → M, μ and a generalized triangulation τ = (h, g′). It turns out that the composition Int_{s} · (Δ + Id)^{−s/2} is a morphism of trace class which induces an isomorphism in algebraic cohomology and, therefore, the corresponding mapping cone C(Int_{s}(Δ + Id)^{−s/2}) is algebraically acyclic. We show that it admits a torsion, called the relative torsion R associated to E → M, g, μ, τ, and that this torsion is independent of s > \frac{n}{2} + 1. In the case where E → M is of determinant class (cf [BFKM]), one can show that, log R = log T_{an} − log T_{Reid} where T_{an}[T_{Reid}] is the analytic [Reidemeister] torsion.

To prove that log R is a local quantity (in Section 4), we will use the Witten deformation of the deRham complex, given by the differentials d_{k}(t) := e^{−th}d_{k}e^{th}. There are different possibilities for defining the corresponding deformation R(t) of the relative torsion. We chose one for which the variation \frac{d}{dt} log R(t) can be computed. By definition, R(t) is the torsion of the mapping cone associated to

(L_{2}(Λ(M; E)), d(t)) →^{e^{th}} (L_{2}(Λ(M; E)), d) ⟶^{Int_{s}·(Δ+Id)^{−s/2}} C.

We will show, Theorem 2.1 subsection 2.2 that the variation \frac{d}{dt} log R(t) is given by

\frac{d}{dt} log R(t) = dim E_{x} ∫_{M} he(M, g)

where e(M, g) is the Euler form of the tangent bundle equipped with the Levi-Cività connection induced from g.

2.1 Relative torsion

Let M be a closed smooth manifold, E →^{π} M a smooth bundle of A-Hilbert modules of finite type equipped with a smooth flat connection ∇, which makes the fiberwise multiplication with elements of A ∇-parallel. Unlike in [BFKM], or [BFK1] we do not restrict to the case where the Hermitian structure μ provided by the scalar product μ_{x} of the fiber E_{x} of E (x ∈ M) is ∇-parallel.

We denote by E^{♯} → M the dual bundle of E → M. This is a bundle of A-Hilbert modules of finite type with fiber E_{x}^{♯}, the Hilbert space dual to E_{x}.

The flat connection ∇ = ∇_{E} induces a flat connection ∇^{♯} := ∇_{E^{♯}}. E → M and E^{♯} → M are canonically isometric as bundles by an A-linear isometry, but this isometry does not intertwine ∇ and ∇^{♯} unless the Hermitian structure μ is ∇_{E}-parallel. If E → M, ∇ is induced by the representation ρ of Γ = π_{1}(M, x_{0}) then E^{♯} → M, ∇^{♯} is induced by ρ^{♯}. Each of these connections induces a covariant differentiation,

d_{k} ≡ d_{k,∇} : Ω^{k}(M; E) → Ω^{k+1}(M; E)

and

d_{k}^{♯} ≡ d_{k,∇^{♯}} : Ω^{k}(M; E^{♯}) → Ω^{k+1}(M; E^{♯})

where Ω^{k}(M; E) := C^{∞}(Λ^{k}(T^{∗}M) ⊗ E) and where Ω^{k}(M; E^{♯}) is defined similarly. The isomorphism θ induces a canonical isomorphism, again denoted by θ, between Ω^{k}(M; E) and Ω^{k}(M; E^{♯}). To simplify notation, we write Λ^{k}(M; E) for the A-Hilbert module of finite type, Λ^{k}(M; E) := Λ^{k}(T^{∗}M) ⊗ E. Given a Riemannian metric g on M, let J_{k,E} : Λ^{k}(M; E) → Λ^{n−k}(M; E^{♯}) (with n = dim M) be the morphism of A-Hilbert modules defined by J_{k,E} := J_{k} ⊗ θ where J_{k} : Λ^{k}(T^{∗}M) → Λ^{n−k}(T^{∗}M) is the Hodge-star operator induced by g.

Denote by d_{k}^{∗} : Ω^{k+1}(M; E) → Ω^{k}(M; E) the operator defined by the composition

(2.1) d_{k}^{∗} ≡ d_{k,∇}^{∗} := (−1)^{nk−1}J_{n−k,E^{♯}} ∘ d_{n−k−1;∇^{♯}} ∘ J_{k+1,E}.

Further we define a scalar product ≪ ·, · ≫: Ω^{k}(M; E) × Ω^{k}(M; E) → IC given by

(2.2) ≪ w_{1}, w_{2} ≫:= ∫_{M} w_{1} ∧_{E} ∗w_{2} d vol_{g}

where ∗w_{2} ≡ J_{k,E}w_{2} and ∧_{E} is defined by

Ω^{k}(M; E) × Ω^{k}(M; E) →^{Id×J_{k,E}} Ω^{k}(M; E) × Ω^{n−k}(M; E^{♯})

→^{∧} Ω^{n}(M, E ⊗ E^{♯}) →^{ev} Ω^{n}(M; IC),

(w_{1}, w_{2}) ↦ (w_{1}, ∗w_{2}) ↦ w_{1} ∧ ∗w_{2} ↦ w_{1} ∧_{E} ∗w_{2}

and ev is the map induced by the evaluation map E_{x} ⊗ E_{x}^{♯} → IC. Notice that with respect to the inner product ≪ ·, · ≫, d_{k}^{∗} is the formal adjoint of d_{k}, i.e. for w_{1} ∈ Ω^{k}(M, E), w_{2} ∈ Ω^{k+1}(M; E)

≪ w_{1}, d_{k}^{∗}w_{2} ≫=≪ d_{k}w_{1}, w_{2} ≫ .

Introduce the Laplacians

(2.3) Δ_{k} = d_{k}^{∗}d_{k} + d_{k−1}d_{k−1}^{∗} : Ω^{k}(M; E) → Ω^{k}(M; E) which are second order, elliptic essentially selfadjoint nonnegative A-linear operators. In particular, as M is closed, (Id + Δ_{k}) : Ω^{k}(M; E) → Ω^{k}(M; E) is an isomorphism of Fréchet spaces. Using functional calculus, one can define the powers (Id + Δ_{k})^{s} (s ∈ IR arbitrary). They can be used to define a family of scalar products on Ω^{k}(M, E) by setting

(2.4) ≪ w_{1}, w_{2} ≫_{s}:=≪ (Id + Δ_{k})^{s/2}w_{1}, (Id + Δ_{k})^{s/2}w_{2} ≫ .

Clearly, ≪ ·, · ≫_{s} depends on the Hermitian structure μ and the Riemannian metric g. As d_{k}Δ_{k} = Δ_{k+1}d_{k} and Δ_{k}d_{k}^{∗} = d_{k}^{∗}Δ_{k+1}, d_{k}^{∗} is also the adjoint of d_{k} with respect to the inner product ≪ ·, · ≫_{s} . Denote by H_{s}(Λ^{k}(M; E)) the completion of Ω^{k}(M; E) with respect to the norm induced by the scalar product ≪ ·, · ≫_{s} . As M is supposed to be closed, one obtains a family of complexes

0 → … → H_{s}(Λ^{k}(M; E)) →^{d_{k}} H_{s}(Λ^{k+1}(M; E)) → … → 0

which we denote by H_{s}(Λ(M; E)) ≡ (H_{s}(Λ(M; E)), d). Here d_{k} : H_{s}(Λ^{k}(M; E)) → H_{s}(Λ^{k+1}(M; E)) is the maximal closed extension of

d_{k} : Ω^{k}(M; E) → Ω^{k+1}(M; E),

its domain is H_{s+1}(Λ^{k}(M; E)).

As mentioned in Examples 1.6 (4), H_{s}(Λ(M, E)) are ζ-regular complexes of sF type and the morphisms

(1 + Δ_{k})^{t/2} : H_{s}(Λ^{k}(M; E)) → H_{s−t}(Λ^{k}(M; E))

establish an isometry between the complexes H_{s}(Λ(M; E)) and H_{s−t}(Λ(M; E)). The adjoint of d_{k} with respect to the L_{2}-inner product, d_{k}^{∗} : H_{s}(Λ^{k+1}(M; E)) → H_{s}(Λ^{k}(M; E)), is given by the maximal closed extension of d_{k}^{∗} : Ω^{k+1}(M; E) → Ω^{k}(M; E) (which depends on s).

Consider a generalized triangulation τ = (h, g′) of M (cf introduction of [BFK], also [BFKM]) where h : M → IR is a Morse function, g′ is a compatible Riemannian metric on M and O_{h} is a collection of orientations of the unstable manifolds defined by −gradg̃′ h̃, compatible with the deck transformations, where h̃ and g̃′ are lifts of h respectively g′ to the universal cover of M and denote by C ≡ (C(M, τ, T ), δ) the finite dimensional cochain complex of A-Hilbert modules associated with τ and T := (E, ∇, μ). The A-linear map Int_{k} : Ω^{k}(M; E) → C^{k}(M, τ, T ) provided by integration (cf. [BFKM]) are continuous with respect to the Fréchet topology on Ω^{k}(M; E) and extend, for s > n/2, to bounded maps

Int_{k;s} : H_{s}(Λ^{k}(M; E)) → C^{k}(M, τ, T ).

Since the integration map intertwines d and δ, they induce morphisms

(2.6) Int_{k;s} : H_{s}(Λ^{k}(M; E)) → C^{k}(M, τ, T ).

As C is a complex of finite type Hilbert modules and Int_{k;s} is bounded for s > \frac{n}{2}, Int_{k;s} is of trace class (cf Examples 1.6 (2)). For s′ ≤ s, denote by In_{∗;s,s′} the embedding

(2.7) In_{k;s,s′} : H_{s}(Λ^{k}(M; E)) ↪ H_{s′}(Λ^{k}(M; E))

which is a morphism of cochain complexes. For s − s′ > n, In_{k;s,s′} is of trace class.

Proposition 2.1 (cf [CMM, section 4]) The following families of morphisms induce isomorphisms in algebraic cohomology:

(i) (Id + Δ)^{s/2} : H_{r}(Λ(M; E)) → H_{r−s}(Λ(M; E)) (∀r, s);

(ii) In_{s,s′} : H_{s}(Λ(M; E)) → H_{s′}(Λ(M; E)) (s ≥ s′);

(iii) Int_{s} : H_{s}(Λ(M; E)) → C(M, τ, T ) (s > n/2).

Proof (i) is obvious since (Id + Δ)^{s/2} is an isometric morphism.

(ii) For ease of notation, let H_{k;s} := H_{s}(Λ^{k}(M; E)). Denote by Q_{k} : H_{k;0} → H_{k;0} the L_{2}-orthogonal spectral projector corresponding to the interval [0, r], r > 0 and consider its restriction to H_{k;s}, s > 0. Due to the ellipticity of Δ_{k}, Q_{k}ω ∈ Ω^{k}(M; E) and Q_{k}(H_{k;s}) is isomorphic to Q_{k}(Ω^{k}(M; E)). As H_{k,s} = Q_{k}(H_{k;s}) ⊕ (Id − Q_{k})(H_{k;s}), H_{s}(Λ(M; E)) is the sum of two subcomplexes, C^{1} ⊕ C^{2} where C_{k}^{1} := Q_{k}(H_{k;s}) and C_{k}^{2} := (Id−Q_{k})(H_{k,s}). Notice that C^{2} is algebraically acyclic by remark after (1.20). C^{1} has the same algebraic cohomology as H_{s}(Λ(M, E))

and that In_{k;s,s′} = ( \begin{matrix}Id & 0 \\ 0 & In_{k;s,s′|_{C{2 \atop k}}} \end{matrix} ) . Therefore, In_{s,s′}, induces an isomor-

phism in algebraic cohomology.

(iii) We use Witten’s deformation of the deRham complex (cf [BFK2, section 3], also [BFKM, section 5]). For t ∈ IR, let d_{k}(t) := e^{−th}d_{k}e^{th} be the deformed exterior differential where h : M → IR is the Morse function of a generalized triangulation τ = (h, g′) ( h not necessarily selfindexing). For t sufficiently large, the spectrum of the deformed Laplacian Δ_{k}(t) splits into a small part contained in the interval [0,1] and a large part, contained in an interval of the form [Ct, ∞] with C > 0. Denote by (Ω_{sm}(M; E)(t), d(t)) the subcomplex associated to the small part of the spectrum. In particular, Ω_{sm}^{k}(t) ≡ Ω_{sm}^{k}(M; E)(t) = Q_{k}(t)(H_{k;s}) where Q_{k}(t) denotes the orthogonal spectral projector of Δ_{k}(t) corresponding to the interval [0,1]. As Δ_{k}(t) is elliptic, one concludes that Ω_{sm}^{k}(t) ≡ Ω_{sm}^{k}(M; E)(t) ⊆ Ω^{k}(M; E) and, by [BFK2, section 3] (cf also [BFKM, section 5]), is an A-Hilbert module of finite type. Similarly as in the proof of (ii), H_{s}(Λ(M; E)) is the sum of two subcomplexes, Ω_{sm}(t) ⊕ H_{s,la}(t), where H_{k;s,la} := (Id − Q_{k}(t))(H_{k;s}). Introduce

(2.8) f_{k}(t) : Ω_{sm}^{k}(M; E)(t) → C^{k}(M, τ, T )

given by f_{k}(t) := Int_{k}e^{th}. One verifies that f(t) : (Ω_{sm}(t), d(t)) → (C, δ) is a morphism of cochain complexes. By the Helffer-Sjöstrand theory (cf [BFK2, section 3], also [BFKM, section 5]), f(t) is an isomorphism for t sufficiently large. In particular, it induces an isomorphism in algebraic cohomology. For t sufficiently large, H_{s}(Λ (M; E)) can be decomposed into two subcomplexes, C^{1} ⊕C^{2} where C^{1} = (e^{th}Ω_{sm} (t), d) and C^{2} = (e^{th}H_{s,la}(t), d). Notice that C^{2} is algebraically acyclic and that C^{1} has the same algebraic cohomology as H_{s}(Λ(M; E)). For s > n/2,

Int_{s} : H_{s}(Λ(M; E)) = C^{1} ⊕ C^{2} → C

is well defined and takes the form (f(t)e^{−th}, Int_{s}|_{C^{2}}). Therefore Int_{s} induces an isomorphism in algebraic cohomology.

Remark 2.2 For s − s′ > n + 1 the inclusion In_{s,s′} is a morphism of trace class (cf Definition 1.9).

As H_{s}(Λ(M; E)) is a ζ-regular complex of sF type (for any s ∈ IR), we conclude from Proposition 2.1 and Lemma 1.12 that the mapping cone C(In_{s,s′}) is algebraically acyclic and thus has a well defined torsion T (C(In_{s,s′}))(s−s′ > n+1).

Proposition 2.3 For s − s′ > n + 1, log T (C(In_{s,s′})) = 0.

Proof Recall that the mapping cone C(In_{s,s′}) is given by

C(In_{s,s′})_{k} := H_{s′}(Λ^{k−1}(M; E)) ⊕ H_{s}(Λ^{k}(M; E));

d(In_{s,s′})_{k} = ( \begin{matrix}−d_{k−1} & In_{k;s,s′} \\ 0 & d_{k} \end{matrix} ) .

Therefore, by (1.23) and (2.4) ,

(2.9) Δ(In_{s,s′})_{k} = ( \begin{matrix}Δ_{k−1} + (Id + Δ_{k−1})^{s′−s} & 0 \\ 0 & Δ_{k} + (Id + Δ_{k})^{s′−s} \end{matrix} )

Proposition 2.3 then follows from Lemma 2.4 below.

Lemma 2.4 Assume that (C, δ) is a ζ-regular cochain complex of sF type. Then

(i) ∑_{k}(−1)^{k} log det(Δ_{k} + Id) = 0;

(ii) ∑_{k}(−1)^{k} log det(Δ_{k} + (Id + Δ_{k})^{s′−s}) = 0.

Proof As the two statements can be proved in the same way, we consider only the second one. With respect to the Hodge decomposition, Δ_{k} + Id takes the form (cf (1.18))

(2.10) Δ_{k} + (Id + Δ_{k})^{s′−s} =

= diag(Id, \underline{Δ}_{k}^{+} + (Id + \underline{Δ}_{k}^{+})^{s′−s}, \underline{Δ}_{k}^{−} + (Id + \underline{Δ}_{k}^{−})^{s′−s}).

where \underline{Δ}_{k}^{+} := \underline{d}_{k−1}\underline{d}_{k−1}^{∗} and \underline{Δ}_{k}^{−} := \underline{d}_{k}^{∗}\underline{d}_{k} are ζ-regular operators of sF type and

log det(Δ_{k} + (Id + Δ_{k})^{s′−s}) = log det(\underline{Δ}_{k}^{+} + (Id + \underline{Δ}_{k}^{+})^{s′−s})+
+ log det(\underline{Δ}_{k}^{−} + +(Id + \underline{Δ}_{k}^{−})^{s′−s}).

As \underline{Δ}_{k}^{+} and \underline{Δ}_{k−1}^{−} have the same spectral distribution function we conclude that

(2.11) log det(\underline{Δ}_{k}^{+} + (Id + \underline{Δ}_{k}^{+})^{s′−s}) = log det(\underline{Δ}_{k−1}^{−} + (Id + \underline{Δ}_{k−1}^{−})^{s′−s}).

This proves the claimed statement.

Proposition 2.5 Let s > \frac{n}{2} + 1, s̃ > s + n + 1 and p ∈ IR_{>0}. Then

(i) Int_{s} : H_{s}(Λ(M; E)) → C(M, τ, T )

is a morphism of trace class which induces an isomorphism in algebraic cohomology. Moreover

(2.12) log T (C(Ints̃)) = log T (C(Int_{s}))

and

(2.13) log T (C(Int_{s})) = log T (C(Int_{s}(Δ + Id)^{p/2})),

with (Δ + Id)^{p/2} : H_{s+p}(Λ(M; E)) → H_{s}(Λ(M; E))

Proof By Proposition 2.1, Int_{s} is a morphism of cochain complexes which induces an isomorphism in algebraic cohomology. We claim that Int_{s} is a morphism of trace class (cf Definition 1.10). As we have already observed, Int_{k;s} is an operator of trace class for s > n/2 as it is bounded and C^{k} is a Hilbert module of finite type. By the same reason, δ_{k}^{∗}Int_{k+1;s} is of trace class for s > n/2. Finally we have to verify that Int_{k;s}d_{k}^{∗} is of trace class. Use that Int_{k;s} = Int_{k;s−1}In_{k;s,s−1} and that In_{k;s,s−1}d_{k}^{∗} is a bounded operator to deduce from Proposition 1.3 that Int_{k;s}d_{k}^{∗} is of trace class for s > \frac{n}{2} + 1. Further notice that Ints̃ = Int_{s} · In_{s̃,s} for arbitrary s̃ > s + n + 1 where In_{s̃,s} is a morphism of trace class (Remark 2.2) which induces an isomorphism in algebraic cohomology (Proposition 2.1). From Proposition 1.15 and Proposition 2.3 we then conclude that

log T (C(Ints̃)) = log T (C(In_{s̃,s})) + log T (C(Int_{s}))
= log T (C(Int_{s})).

It follows from Proposition 1.3 and the fact that (Δ + Id)^{p/2} is an isometry of cochain complexes that Int_{s}(Δ + Id)^{p/2} is a morphism of trace class which induces an isomorphism in algebraic cohomology. Applying Lemma 1.11 and Proposition 1.16 yields formula (2.13).

We are now ready to define the relative torsion. Our definition differs slightly from the one first introduced by [CMM].

Given M, T , g, τ, define R_{s} = R_{s}(M, T , g, τ),

(2.14) log R_{s} := log T (C(Int_{s}(Δ + Id)^{−s/2})) (s > n/2 + 1).

Notice that, by Proposition 2.5, log R_{s} = log T (C(Int_{s})) and R_{s} is independent of s > \frac{n}{2} + 1.

Definition 2.6 The \underline{relative torsion} is defined by R := R_{s}(M, T , g, τ).

The name of relative torsion is justified by the observation, due to [CMM], that in the case where E → M is of determinant class - and therefore, the analytic torsion T_{an} and the Reidemeister torsion T_{Reid} are well defined (cf [BFKM]) - R is given by

(2.15) log R = log T_{an} − log T_{Reid}

To see that (2.15) hold we would like to apply Proposition 1.10 (Milnor’s Lemma). However H_{s}(Λ(M; ξ)) is not a complex of finite dimension, we therefore decompose the mapping cone C(Int_{s}) as follows: As shown in the proof of Proposition 2.1 (iii), H_{s}(Λ(M; E)) = C^{1} ⊕ C^{2} and f : Int_{s}|_{C^{1}} : C^{1} → C^{2} is a bijective morphism where C^{2} is algebraically acyclic and C^{1} is given by C^{1} := (e^{th}Ω_{sm}(t), d). Notice that C^{1} has the same algebraic cohomology as H_{s}(Λ(M; E)). Denote by C^{3} the mapping cone C(f). As f is a bijective morphism, C^{3} = C(f) is algebraically acyclic. As f satisfies the assumption of Lemma 1.14 one concludes that

log R = log T (C(Int_{s})) = log T (C^{3}) + log T (C^{2}).

The term log T (C^{3}) has to be analyzed further. By assumption, (M, ρ) is of determinant class. Therefore C^{1} is of determinant class as well. We then obtain the following short exact sequence of cochain complexes, each of them being of determinant class,

0 → ΣC → C^{3} → C^{1} → 0.

As ΣC, C^{3} and C^{1} are of finite dimension we can apply Proposition 1.10 to conclude that

log T (C^{3}) = log T (ΣC) + log T (C^{1}) + log T (H)

where H denotes the long weakly exact sequence in reduced cohomology and is of determinant class. By Lemma 1.14 and the definition of the combinatorial torsion T_{comb} (cf [BFKM]), log T (ΣC) = − log T (C) = − log T_{comb}. By the definition of the analytic torsion (cf [BFKM]), log T (C^{1}) + log T (C^{2}) = log T_{an}. Further H is given by

… → H̄^{i}(ΣC) → H̄^{i}(C^{3}) → H̄^{i}(C^{1}) →^{H̄(δ_{i})} H̄^{i+1}(ΣC) → …

Recall that C^{3} is algebraically acyclic and H̄(δ_{i}) : H̄^{i}(C^{1}) → H̄^{i}(C) is given by the restriction of Int_{i;s} to H_{i} = ker(Δ_{i}) and, by the definition of the metric part T_{met} of the torsion

log T (H) = − log T_{met}.

Combining the various equalities above leads to (2.15).

2.2 Witten deformation of the relative torsion

To obtain a local formula for log R we consider the deformed relative torsion R(t) obtained by considering the Witten deformation of the deRham complex.

Using the generalized triangulation τ = (h, g′, O_{h}) we consider for s > n/2 + 1 the following composition g_{s}(t) of morphisms.

(2.16) (L_{2}(Λ(M; E)), d(t)) ⟶^{e^{th}} (L_{2}(Λ(M; E)), d) →^{(Δ+Id)^{−s/2}} (H_{s}(Λ(M, E)), d)

→^{Int_{s}} (C, δ).

where we recall that d_{k}(t) = e^{−th}d_{k}e^{th}. For s > n/2 + 1, Int_{s} is a morphism of trace class (cf Proposition 2.5) and thus, by Proposition 1.3, the composition g_{s}(t) is a morphism of trace class. Due to Proposition 2.1, g_{s}(t) induces an isomorphism in algebraic cohomology, hence log T (C(g_{s}(t))) is well defined.

Proposition 2.7 For s + p > s > 2n + 2 and t ∈ IR,

log T (C(g_{s}(t))) = log T (C(g_{s+p}(t))).

Proof Using that Int_{s} = Int_{s/2}In_{s,s/2}, g_{s}(t) is given by g_{s}(t) = Int_{s/2}α_{s}(t) where α_{s}(t) := In_{s,s/2}(Δ+Id)^{−s/2}e^{th}. As s > 2n+2, α_{s}(t) (cf. Remark 2.2) and Int_{s/2} (cf. Proposition 2.5) are morphisms of trace class. By Proposition 2.1, both Int_{s/2} and α_{s}(t) induce isomorphisms in algebraic cohomology. Therefore we can apply Proposition 1.16 to conclude that

(2.17) log T (C(g_{s}(t))) = log T (C(Int_{s/2})) + log T (C(α_{s}(t))).

Observe that, In_{s+p, \frac{s}{2} +p}(Δ + Id)^{− \frac{s+p}{2}} = (Δ + Id)^{−p/2}In_{s,s/2}(Δ + Id)^{−s/2} to deduce that

g_{s+p}(t) = Int_{p+ \frac{s}{2}} (Δ + Id)^{−p/2}α_{s}(t).

Using the same arguments as above we can apply Proposition 1.16 to obtain

(2.18) \begin{matrix}log T (C(g_{s+p})) & = & log T (C(Int_{p+s/2} · (Δ + Id)^{−p/2}))+ \\ & & + log T (C(α_{s})). \end{matrix}

By Proposition 2.5,

(2.19) log T (C(Int_{\frac{s}{2} +p})) = log T (C(Int_{\frac{s}{2} +p}(Δ + Id)^{−p/2}))

and (cf (2.14))

(2.20) log T (C(Int_{s/2+p})) = log T (C(Int_{s/2})).

Combining (2.17)-(2.20) leads to the claimed result.

In view of Proposition 2.7, log T (C(g_{s}(t))) is independent of s.

Definition 2.8 The Witten deformation R(t) of the relative torsion is defined by

log R(t) := log T (C(g_{s}(t))) (s > n + 2).

Using the same arguments as for the verification of (2.15) one sees that, for E → M of determinant class,

(2.21) log R(t) = log T_{an}(t) − log T_{comb} − log T_{met}(t)

where T_{an}(t) denotes the analytic torsion of the deformed deRham complex (cf [BFKM]), T_{comb} denotes again the combinatorial torsion and log T_{met}(t) = − log T (H(t)) is the torsion of the long weakly exact sequence in cohomology obtained in a similar fashion as in the case of (2.15).

Our next objective is to calculate the variation for log R(t) :

Theorem 2.1

\frac{d}{dt} log R(t) = dim E_{x} ∫_{M} h · e(M, g)

where e(M, g) is the Euler form of the tangent bundle equipped with the Levi-Cività connection induced from g. In particular, \frac{d}{dt} log R(t) is a local quantity, independent of t and the Hermitian structure. Moreover if n = dim M is odd, then

\frac{d}{dt} log R(t) ≡ 0.

Proof For s > n + 2, introduce the morphism g_{s} := Int_{s}(Δ + Id)^{−s/2}. Then g_{s}(t) = g_{s}e^{th}. Observe that for all t, the mapping cone G(t) := C(g_{s}(t)) is a cochain complex whose k′th component G_{k} := C(g_{s}(t))_{k} is independent of t, given by

(2.22) G_{k} = C^{k−1}(M, τ, T ) ⊕ L_{2}(Λ^{k}(M; E)).

With respect to the decomposition (2.22), d_{k}(t) ≡ d_{k}^{G}(t) : G_{k} → G_{k+1} takes the form

(2.23) d_{k}(t) = E_{k+1}(−t)D_{k}E_{k}(t); d_{k}(t)^{∗} = E_{k}(t)D_{k}^{∗}E_{k+1}(−t).

where

E_{k}(t) := ( \begin{matrix}Id & 0 \\ 0 & e^{th} \end{matrix} ) , D_{k} := ( \begin{matrix}−δ_{k−1} & g_{k;s} \\ 0 & d_{k} \end{matrix} )

According to Proposition 2.5 and 1.12, the mapping cone C(g_{s}(t)) is a ζ-regular, algebraically acyclic cochain complex of sF type. Denote by

Δ_{k}(t) ≡ Δ_{k}^{G}(t) the k-Laplacian associated to G ≡ C(g_{s}(t)),

Δ_{k}(t) := Δ_{k}^{+}(t) + Δ_{k}^{−}(t)

where

Δ_{k}^{+}(t) := d_{k−1}(t)d_{k−1}^{∗}(t) , Δ_{k}^{−}(t) := d_{k}^{∗}(t)d_{k}(t).

With respect to the Hodge decomposition G_{k} = G_{k}^{+}(t) ⊕ G_{k}^{−}(t),

(2.24) G_{k}^{+}(t) := \overline{d_{k−1}(t)(G_{k−1})}; G_{k}^{−}(t) := \overline{d_{k}^{∗}(t)(G_{k+1})},

d_{k}(t) is of the form d_{k}(t) = ( \begin{matrix}0 & 0 \\ 0 & \underline{d}_{k}(t) \end{matrix} ) and thus Δ_{k}^{+}(t) = ( \begin{matrix}\underline{Δ}_{k}^{+}(t) & 0 \\ 0 & 0 \end{matrix} )

and Δ_{k}^{−}(t) = ( \begin{matrix}0 & 0 \\ 0 & \underline{Δ}_{k}^{−}(t) \end{matrix} ) where \underline{Δ}_{k}^{+}(t) := \underline{d}_{k−1}(t)\underline{d}_{k−1}^{∗}(t) and \underline{Δ}_{k}^{−}(t) :=

\underline{d}_{k}^{∗}(t)\underline{d}_{k}(t). As C(g_{s}(t)) is a ζ-regular cochain complex, the operators \underline{d}_{k}(t), according to Definition 1.9, satisfy Op(1)-Op(7). As C(g_{s}(t)) is algebraically acyclic, they also satisfy Op(8). (In fact, the operators \underline{d}_{k}(t) satisfy Op(7), because the corresponding heat traces admit an asymptotic expansion of the form (Asy) (cf Lemma 1.1).)

To compute \frac{d}{dt} log R(t) we proceed similarly as in [BFKM] where we compute \frac{d}{dt} log T_{an}(t). According to Definition 2.8, the relative torsion R(t) = T (C(g_{s}(t))) is given by

log R(t) = \frac{1}{2} ∑_{k}(−1)^{k+1} log det \underline{Δ}_{k}^{+}(t)

and

(2.25) \frac{d}{dt} log det(\underline{Δ}_{k}^{+}(t)) = F.p._{z=0}T r(\frac{d}{dt}(\underline{Δ}_{k}^{+}(t))(\underline{Δ}_{k}^{+}(t))^{−z−1})

= F.p._{z=0}T r(\frac{d}{dt}(E_{k}(−t)D_{k−1}E_{k−1}(2t)D_{k−1}^{∗}E_{k}(−t))(Δ_{k}^{+}(t))^{−z−1}).

(By F.p._{z=0} we denote the constant term in the Laurent expansion at z = 0 of the expression which follows it.) To simplify writing we suppress momentarily the subscript k. Then

(2.26) \begin{matrix}\frac{d}{dt}(E(−t)DE(2t)D^{∗}E(−t)) = \\ = ( \begin{matrix}0 & 0 \\ 0 & −h \end{matrix} ) E(−t)DE(2t)D^{∗}E(−t) \\ +E(−t)D ( \begin{matrix}0 & 0 \\ 0 & 2h \end{matrix} ) E(2t)D^{∗}E(−t) \\ +E(−t)DE(2t)D^{∗}E(−t) ( \begin{matrix}0 & 0 \\ 0 & −h \end{matrix} ) . \end{matrix}

Substituting (2.26) into (2.25) we obtain

(2.27) \begin{matrix}\frac{d}{dt} log det(\underline{Δ}^{+}(t)) = \\ = F.p._{z=0}T r( ( \begin{matrix}0 & 0 \\ 0 & −h \end{matrix} ) d(t)d(t)^{∗}Δ^{+}(t)^{−z−1}) \\ +F.p._{z=0}T r(d(t) ( \begin{matrix}0 & 0 \\ 0 & 2h \end{matrix} ) d(t)^{∗}Δ^{+}(t)^{−z−1}) \\ +F.p._{z=0}T r(d(t)d(t)^{∗} ( \begin{matrix}0 & 0 \\ 0 & −h \end{matrix} ) Δ^{+}(t)^{−z−1}). \end{matrix}

Using that d(t)d(t)^{∗}Δ^{+}(t)^{−z−1} = Δ^{+}(t)^{−z} together with the commutativity of the trace we conclude

(2.28) \begin{matrix}\frac{d}{dt} log det(\underline{Δ}_{k}^{+}(t)) = \\ = F.p._{z=0}T r( ( \begin{matrix}0 & 0 \\ 0 & −2h \end{matrix} ) Δ_{k}^{+}(t)^{−z}) \\ +F.p._{z=0}T r( ( \begin{matrix}0 & 0 \\ 0 & 2h \end{matrix} ) d_{k−1}^{∗}(t)Δ_{k}^{+}(t)^{−z−1}d_{k−1}(t)). \end{matrix}

From d_{k−1}^{∗}(t)Δ_{k}^{+}(t) = d_{k−1}^{∗}(t)d_{k−1}(t)d_{k−1}(t)^{∗} = Δ_{k−1}^{−}(t)d_{k−1}(t)^{∗} we conclude that

d_{k−1}^{∗}Δ_{k}^{+}(t)^{−z−1}d_{k−1}(t) = (Δ_{k−1}^{−}(t))^{−z−1}Δ_{k−1}^{−}(t) = Δ_{k−1}^{−}(t)^{−z}

and therefore, after substituting into (2.28)

(2.29) \frac{d}{dt} log det(\underline{Δ}_{k}^{+}(t)) =

= F.p._{z=0}T r( ( \begin{matrix}0 & 0 \\ 0 & −2h \end{matrix} ) (Δ_{k}^{+}(t)^{−z} − Δ_{k−1}^{−}(t)^{−z})).

This leads to

(2.30) \frac{d}{dt} log T (C(g_{s}(t))) = ∑_{k}(−1)^{k}F.p._{z=0}T r( ( \begin{matrix}0 & 0 \\ 0 & h \end{matrix} ) Δ_{k}(t)^{−z}).

As C(g_{s}(t)) is algebraically acyclic, Δ_{k}(t) has no spectrum near 0 and therefore

ζ^{II}(z) := \frac{1}{Γ(z)} ∫_{1}^{∞} λ^{z−1}T r( ( \begin{matrix}0 & 0 \\ 0 & h \end{matrix} ) e^{−λΔ_{k}(t)})dλ is an entire function, with

ζ^{II}(0) = 0.

In view of the fact that the heat trace T r( ( \begin{matrix}0 & 0 \\ 0 & h \end{matrix} ) e^{−λΔ_{k}(t)}) admits an

asymptotic expansion at λ = 0 of the form (Asy) one obtains (cf Lemma 1.1 and Op(7))

(2.31) \begin{matrix}F.p._{z=0} & T r( ( \begin{matrix}0 & 0 \\ 0 & h \end{matrix} ) Δ_{k}(t)^{−z}) = \\ & = F.p._{z=0}(\frac{1}{Γ(z)} ∫_{0}^{∞} λ^{z−1}T r(( \begin{matrix}0 & 0 \\ 0 & h \end{matrix} ) e^{−λΔ_{k}(t)})dλ) \\ & = F.p._{z=0}(\frac{1}{Γ(z)} ∫_{0}^{1} λ^{z−1}T r(( \begin{matrix}0 & 0 \\ 0 & h \end{matrix} ) e^{−λΔ_{k}(t)})dλ). \end{matrix}

With respect to the decomposition (2.22) of the mapping cone C(g_{s}(t)), the Laplacian Δ_{k}(t) ≡ Δ_{k}^{G}(t) takes the form

Δ_{k}^{G}(t) = ( \begin{matrix}0 & 0 \\ 0 & Δ_{k}(t) \end{matrix} ) + Φ_{k}(t)

where Φ_{k}(t) : G_{k} → G_{k} is given by

Φ_{k}(t) := (\begin{matrix}Δ_{k−1}^{comb} + g_{k−1;s}(t)g_{k−1;s}(t)^{∗} & −δ_{k−1}^{∗}g_{k;s}(t) + g_{k−1;s}(t)d_{k−1}^{∗}(t) \\ −g_{k;s}(t)^{∗}δ_{k−1} + d_{k−1}(t)g_{k−1;s}(t)^{∗} & g_{k;s}(t)^{∗}g_{k;s}(t) \end{matrix}) .

As g_{k;s}(t) and g_{k;s}(t)^{∗} are bounded operators of trace class for s > \frac{n}{2} + 1 (cf Proposition 2.5) the maps g_{k;s}(t)d_{k}^{∗}(t) and d_{k}(t)g_{k;s}(t)^{∗} are bounded operators of trace class for s > n/2 + 1 and we conclude that Φ_{k}(t) is a bounded operator of trace class. By Proposition 1.5 and its remark, we conclude that

(2.32) F.p._{z=0}(\frac{1}{Γ(z)} ∫_{0}^{1} λ^{z−1}T r(( \begin{matrix}0 & 0 \\ 0 & h \end{matrix} ) e^{−λΔ_{k}^{G}(t)})dλ) =

= F.p._{z=0}(\frac{1}{Γ(z)} ∫_{0}^{1} λ^{z−1}T r(he^{−λΔ_{k}(t)}(Id − P_{Δ_{k}(t)}(\{0\})))dλ

where P_{Δ_{k}(t)}({0}) denotes the orthogonal projection onto KerΔ_{k}(t).

One verifies (cf [BZ, p 81]) that e^{th}Δ_{k}(t)e^{−th} are equal to the Laplacians Δ′_{k} associated to the undeformed deRham complex, the Riemannian metric g and the Hermitian structure e^{−2th}μ.

Combining this together with (2.30)-(2.32), we obtain

(2.33) \begin{matrix}\frac{d}{dt} log R(t) = \\ = ∑_{k}(−1)^{k}F.p._{z=0}(\frac{1}{Γ(z)} ∫_{0}^{1} λ^{z−1}T r(he^{−λΔ_{k}(t)}(Id − P_{Δ_{k}(t)}(\{0\})))dλ) \\ = ∑_{k}(−1)^{k}F.p._{z=0}(\frac{1}{Γ(z)} ∫_{0}^{1} λ^{z−1}T r(he^{−λΔ′_{k}}(Id − P_{Δ′_{k}}(\{0\})))dλ) \end{matrix}

where we used that, as Δ_{k}(t) = e^{−th}Δ′_{k}e^{th}, e^{−λΔ_{k}(t)} = e^{−th}e^{−λΔ′_{k}}e^{th} and e^{th}(Id − P_{Δ_{k}(t)}({0}))e^{−th} = Id − P_{Δ′_{k}}({0}).

It is known (cf e.g. [BFKM]) that the restriction K_{k}(x; λ, t) of the Schwartz kernel of e^{−λΔ′_{k}}(Id − P_{Δ′_{k}}({0})) to the diagonal has an asymptotic expansion

with respect to λ for λ ↘ 0 of the form ∑_{j≥0} A_{k;n−j}(t)λ^{− \frac{n−j}{2}} where A_{k;n−j}(t) are smooth densities on M with values in End(Λ^{k}(T^{∗}M)⊗E), depending on the parameter t. Substituting into (2.33), and integrating with respect to λ leads to

(2.34) \begin{matrix}\frac{d}{dt} log R(t) = ∑_{k}(−1)^{k}T r(hA_{k;0}(t)) \\ = T r(h(∑_{k}(−1)^{k}A_{k;0}(t))) \\ = dim E_{x}T r(he(M, g)) \end{matrix}

where e(M, g) is the Euler form, which can be proven to be equal to \frac{1}{dim E_{x}} ∑_{k} (−1)^{k}A_{k;0}(t). If dim M is odd, e(M, g) ≡ 0.

3 Proof of Proposition 0.1 in the case g = g′

3.1 Outline of the Proof

In this section we prove Proposition 0.1 in the case where the Riemannian metric g is the same as the metric g′ of the generalized triangulation τ = (h, g′). Throughtout this section we assume that the Hermitian structure μ is τ-admissible, i.e. that there exists a neighborhood U_{h} of the critical points of h so that on U_{h}, μ is parallel with respect to the canonical flat connection on E → M.

From Theorem 2.9 we know that

(A) log R(t) = log R + ∫_{0}^{t} \frac{d}{dt} log R(t)dt is an affine function of t,

hence log R(t) has an asymptotic expansion with log R as the free term.

Recall from the proof of Proposition 2.1 (iii) that, for t sufficiently large, (H_{0}(Λ(M; E)), d(t)) is the direct sum of two subcomplexes,

(H_{0}(Λ(M; E)), d(t)) = (Ω_{sm}(t), d(t)) ⊕ (H_{0,la}(t), d(t))

where

H_{k;0} := H_{0}(Λ^{k}(M; E)),

Ω_{sm}^{k}(t) = Q_{k}(t)(H_{k;0}),

H_{k;0,la}(t) = (Id−Q_{k}(t))(H_{k;0}) and Q_{k}(t) denotes the orthogonal spectral projector Q_{k}(t) : H_{k;0} → H_{k;0} of the k-Laplacian Δ_{k}(t) corresponding to the interval [0,1]. Notice that Ω_{sm}^{k}(t) consists of smooth forms and that (H_{0,la}(t), d(t)) is an algebraically acyclic ζ-regular complex of sF type. Notice also that (Ω_{sm}(t), d(t)) is a ζ-regular complex of finite type for t large enough. The finite type property follows from [BFKM] Theorem 5.5. Denote by g_{s,sm}(t) the restriction of g_{s}(t) to Ω_{sm}(t). This is a morphism of trace class for s > \frac{n}{2} + 1 since g_{s}(t) is of trace class by Proposition 2.5. We have the following short exact sequence of algebraically acyclic, ζ-regular complexes of sF type,

(3.1) 0 → C(g_{s,sm}(t)) →^{I} C(g_{s}(t)) →^{P} H_{0,la}(t) → 0

where I, resp. P, is the obvious inclusion, resp. projection. One verifies in a straightforward way that (3.1) satisfies the assumptions in Lemma 1.14 and therefore, with

(3.2) R_{sm}(t) := T (C(g_{s,sm}(t))),

we obtain the decomposition

(B) log R(t) = log R_{sm}(t) + log T (H_{0,la}(t)).

For consistency with the notation in [BFK2] and [BFKM] we write

log T_{la}(t) := log T (H_{0,la}(t)).

The two terms, log R_{sm}(t) and log T_{la}(t), will each be treated separately. To obtain an asymptotic expansion of log R_{sm}(t) as t → ∞, we proceed in two steps. Recall that f(t) is given by the composition

f(t) : (Ω_{sm}(t), d(t)) →^{e^{th}} (Ω, d) →^{Int} (C, δ).

We show in subsection 3.2, Proposition 3.1 (i), that

R_{sm}(t) = T (C(f(t))).

The morphism f(t), unlike g_{s,sm}(t), can be studied using the Witten-Helffer-Sjöstrand theory (cf [BFK2, section 3], also [BFKM, section 5]). We show in subsection 3.2, Proposition 3.1 (ii) that log T (C(f(t))) has an asymptotic expansion which we calculate up to terms of order 0(\frac{1}{t}). Proposition 3.1 implies that:

(C) log R_{sm}(t) admits an asymptotic expansion for t → ∞ of the form

log R_{sm}(t) = dim E_{x} · (∑_{j}(−1)^{j} ∑_{x∈Cr_{j}(h)} h(x))t
− (∑_{j}(−1)^{j}(\frac{n}{4} − \frac{j}{2})m_{j} · dim E_{x}) log(\frac{π}{t}) + 0(\frac{1}{t}).

Concerning log T_{la}(t), we will establish in subsection 3.3: (D) (i) log T_{la}(t) admits an asymptotic expansion as t → ∞ and

F T_{t=∞}(log T_{la}(t)) = −F T_{t=∞}(log R_{sm}(t)) + ∫_{M} α

where α is a local density on M which vanishes in a neighborhood of the critical points of h.

(ii) If μ is parallel with respect to the canonical flat connection, ∫_{M} α = 0. The results (A)-(D) prove Proposition 0.1 in the case where g = g′.

3.2 Asymptotic expansion of log R_{sm}(t)

In this subsection we prove the statement (C) of subsection 3.1. By the Witten-Helffer-Sjöstrand theory, f(t) := Int·e^{th} is an isomorphism of cochain complexes for t sufficiently large. Therefore the mapping cone C(f(t)) is an algebraically acyclic, ζ-regular complex of sF type and thus admits a torsion, T (C(f(t))).

Proposition 3.1 For t sufficiently large, (i) log R_{sm}(t) = log T (C(f(t))); (ii) log T (C(f(t))) admits an asymptotic expansion for t → ∞ of the form

log T (C(f(t))) = dim E_{x} · (∑_{j}(−1)^{j} ∑_{x∈Cr_{j}(h)} h(x))t
− (∑_{j}(−1)^{j}(\frac{n}{4} − \frac{j}{2})m_{j} dim E_{x}) log(\frac{π}{t}) + 0(\frac{1}{t}).

Proof (i) Notice that the morphism f(t) is equal to the composition (s > n+2)

(3.3) (Ω_{sm}(t), d(t)) →^{e_{s}^{th}} H_{s}(Λ(M; E)) →^{Int_{s}} C(M, τ, T )

whereas g_{s,sm}(t) is given by

(3.4) (Ω_{sm}(t), d(t)) →^{e^{th}} (H_{0}(Λ(M; E)), d) →^{(Δ+Id)^{−s/2}} (H_{s}(Λ(M; E)), d)

→^{Int_{s}} C(M, τ, T ).

By Proposition 2.5, Int_{s} is a morphism of trace class, multiplication by e^{th} is a morphism of trace class as Ω_{sm}(t) is a cochain complex of finite rank (cf Examples 1.6) and (Δ + Id)^{−s/2} is an isometry. Each of them induces an isomorphism in algebraic cohomology (cf Proposition 2.1). Therefore we can apply Proposition 1.15 to (3.3) and (3.4) to obtain

(3.5) log T (C(f(t))) = log T (C(e_{s}^{th})) + log T (C(Int_{s}))

and

(3.6) log T (C(g_{s,sm}(t))) = log T (C(e^{th})) + log T (C(Int_{s}(Δ + Id)^{−s/2})).

In view of Proposition 1.16 and the fact that (Δ + Id)^{−s/2} is an isometry we conclude that

(3.7) log T (C(Int_{s})) = log T (C(Int_{s}(Δ + Id)^{−s/2})).

Further, e^{th} = In_{s,0} e_{s}^{th} is a morphism

(Ω_{sm}(t), d(t)) ⟶^{e_{s}^{th}} H_{s}(Λ(M; E)) ⟶^{In_{s,0}} H_{0}(Λ(M; E))

with In_{s,0} and e_{s}^{th} being both morphisms of trace class and inducing isomorphisms in algebraic cohomology (cf Proposition 2.1). Thus, applying Proposition 1.15 once more, we obtain, in view of Proposition 2.3,

(3.8) log T (C(e^{th})) = log T (C(e_{s}^{th})).

Combining (3.5) - (3.8) leads to the proof of statement (i).

(ii) Recall from [BFK2, section 3.3] the following commutative diagram (for t sufficiently large)

(3.9) \begin{matrix}(Ω_{sm}(t), d(t)) & ⟶^{f(t)} & (C, δ) \\ Φ + 0(\frac{1}{t}) ↘ & & ↓ S(t) \\ & & (C, δ̃) \end{matrix}

where Φ is a morphism which is an isometry, δ̃ = δ̃(t) is chosen so that the diagram above is commutative, δ̃_{k}(t) := S_{k+1}(t)^{−1} · δ_{k} · S_{k}(t), and S(t) is the morphism given by

S_{k}(t) = ( ⨁_{x∈Cr_{k}(h)} S_{x}(t)) : C^{k} ⟶ C^{k}

where C^{k} := ⨁_{x∈Cr_{k}(h)} E_{x} and S_{x}(t) : E_{x} → E_{x} is given by S_{x}(t) := e^{−th(x)}(\frac{π}{t})^{\frac{n}{4} − \frac{k}{2}} Id.

For t sufficiently large, S(t), f(t), Φ + 0(\frac{1}{t}) are isomorphisms of cochain complexes. Thus, by Lemma 1.11, the mapping cones C(Φ + 0(\frac{1}{t})), C(S(t)) and C(f(t)) are algebraically acyclic. By Proposition 1.15,

log T (C(Φ + 0(\frac{1}{t}))) = log T (C(f(t))) + log T (C(S(t))).

Using Proposition 1.12, one verifies that log T (C(Id + 0(\frac{1}{t}))) = 0(\frac{1}{t}).

Therefore

(3.10) log T (C(f(t))) = − log T (C(S(t))) + 0(\frac{1}{t}).

By Proposition 1.12,

log T (C(S(t))) = ∑(−1)^{j} ∑_{x∈Cr_{j}(h)} log(e^{−th(x)}(\frac{π}{t})^{\frac{n}{4} − \frac{j}{2}} ) dim E_{x}.

In view of (3.10) we obtain

log C(f(t)) = (∑(−1)^{j} ∑_{x∈Cr_{j}(h)} h(x) dim E_{x})t
−(∑(−1)^{j}(\frac{n}{4} − \frac{j}{2})m_{j} dim E_{x}) log \frac{π}{t} + 0(\frac{1}{t}).

3.3 Asymptotic expansion of log T_{la}(t)

In this section we prove the statement (D) in subsection 3.1. As log T_{la}(t) = log R(t) − log R_{sm}(t) and in view of Theorem 2.9 and Proposition 3.1, log T_{la}(t) admits an asymptotic expansion for t → ∞.

The arguments to prove (D) follow closely the ones in [BFKM, section 6.2]: we will derive statement (D) from a relative version of this statement , Proposition 3.2, and from the validity of (D) in some simple cases.

We consider systems F := (M, E, ∇, μ, g, τ), where (E, ∇) is a bundle of A -Hilbert modules equipped with a flat connection, μ a Hermitian structure, g a Riemannian metric and τ = (h, g′) a generalized triangulation with the metric g′ = g. We suppose that μ is admissible with respect to τ, i.e there exists an open neighborhood U_{h} of the critical points so that on U_{h}, μ is parallel with respect to the connection ∇. Introduce V_{F}(t, ϵ) := \frac{1}{2} ∑_{q}(−1)^{q+1}q log det(Δ_{q}(t) + ϵ).

Proposition 3.2 For any ϵ > 0 there exists a smooth density α_{F}(ϵ) on M \ Cr(h), which is polynomial in ϵ and a local quantity (cf subsection 0.2 and [BFKM] section 2) so that the following statement hold:

(i) If F′ := (M, E, ∇, μ, g, τ_{D}) denotes the system obtained from F by replacing the triangulation τ = (h, g) by its dual τ_{D} = (n − h, g) (n = dim M), then

(3.11) α_{F}(ϵ) + (−1)^{n+1}α_{F′}(ϵ) = 0.

(ii) If F and F̃ are two systems as above and F ⊗ F̃ denotes the system defined by

F ⊗ F̃ := (M × M̃, E ⊗ Ẽ, \overline{∇}, μ ⊗ μ̃, g × g̃, τ × τ̃)

with \overline{∇} = ∇ ⊗ id + id ⊗ ˜∇, then

(3.12) α_{F⊗F̃}(ϵ) = (α_{F}(ϵ) ⊗ e(M̃, g̃) + e(M, g) ⊗ α̃F̃(ϵ)) dim W,

where e(M, g) denotes the Euler form of (M, g).

(iii) The density α_{F}(ϵ) vanishes on U_{h} \ Cr(h).

(iv) Assume F and F̃ are two systems as above so that ♯Cr_{q}(h) = ♯Cr_{q}(h̃) (0 ≤ q ≤ n), and the bundles E and Ẽ have isomorphic A−Hilbert modules as fibers. Then V_{F}(t, ϵ) − VF̃(t, ϵ) has an asymptotic expansion with

(a) F T_{t=∞}(V_{F}(t, ϵ) − VF̃(t, ϵ)) = ∫_{M\Cr(h)} α_{F}(ϵ) − ∫_{M̃\Cr(h̃)} αF̃(ϵ);

(b) F T_{t=∞}(log T_{la}(t)) − F T_{t=∞}(log T̃_{la}(t)) = ∫_{M\Cr(h)} α_{F}(0) − ∫_{M̃\Cr(h̃)} αF̃(0).

Proof: We begin with the construction of the density α_{F}(ϵ). Away from the critical points of h we choose a coordinate system for E → M. In these coordinates we calculate inductively the quantities r_{−2−j}^{q}(h, ϵ; x, ξ, t, μ), by the formulae 6.59 in [BFKM], from the symbol of the operator with parameter Δ_{q}(t) + ϵ, which is elliptic with parameter t away from Cr(h) (cf [BFKM] (3.2)). We are interested in the quantity r_{−2−n}^{q} only. We use formula (3.4) in [BFKM] to obtain from r_{−2−n}^{q} the density α_{F}^{q}(h, ϵ) on M \ Cr(h). We define

(3.13) α_{F}(h, ϵ) := 1/2 ∑_{q}(−1)^{q+1}qα_{F}^{q}(h, ϵ).

Since by construction r_{−2−n}^{q} is a polynomial in ϵ of degree smaller than n so is α_{F}(h, ϵ).

(i) follows from the following homogeneity property (cf [BFKM] (6.60)):

r_{−2−n}(h, x, −ξ, −t, μ) = (−1)^{n}r_{−2−n}(h, x, ξ, t, μ)

and by a straightforward verification

r_{−2−n}(n − h, x, ξ, t, μ) = r_{−2−n}(h, x, ξ, −t, μ) .

To prove (ii) we consider systems G = (M, E, ∇, μ, g, ω), where M, E, ∇, μ, g are as above and ω is a closed 1-form on M. A system F = (M, E, ∇, μ, g, τ = (g, h)) gives rise to a system G by taking ω = dh. For any such G, define the Witten deformation by taking ω instead of dh and then the Witten Laplacians Δ_{q}(t) for any real number t. If the 1-form ω has no zeros, Δ_{q}(t) is elliptic with parameter and the general theory in [BFKM], section 2 implies that for any fixed ϵ > 0, V_{G}(t, ϵ) := \frac{1}{2} ∑_{q}(−1)^{q+1}q log(Δ_{q}(t) + ϵ) has an asymptotic expansion for t → ∞, whose free term is ∫_{M} α_{G}, with α_{G} a local density on M. Following [BMFK] section 2 this density is calculated in the same way as the density α_{F} (using ω instead of dh). Since F|_{M\Cr(h)} is locally isomorphic to the restriction to an open set of a system G = (M̃, Ẽ, ˜∇, μ̃, g̃, ω̃) such that M̃ is closed and ω̃ has no zeros, it suffices to check (3.11) for α_{G} and αG̃ instead of α_{F} and αF̃.

For a system G and u > 0, t > 0 consider the operator e^{−uΔ_{q}(t)} which is a smoothing operator and denote by λ_{q}(u, t) the pointwise (von Neumann)trace of the restriction of its Schwartz kernel to the diagonal. This provides a two parameter family (in u and t) of densities on M. Denote by

λ_{G}(u, t) := 1/2 ∑_{q}(−1)^{q+1}qλ_{q}(u, t).

Define the smooth three parameter family, η(s, t, ϵ), of densities on M for ℜes sufficiently large,

(3.14) η(s, t, ϵ) = \frac{1}{Γ(s)} ∫_{0}^{∞} u^{s−1}e^{−uϵ}λ_{G}(u, t)du.

which, by analytic continuation, is holomorphic in s near 0 ∈ IC. Denote by θ_{G}(t, ϵ) the density valued function in t and ϵ,

θ_{G}(t, ϵ) := \frac{d}{ds}|_{s=0}η(s, t, ϵ).

For the system G ⊗ G̃, we can prove that

(3.15) θ_{G⊗G̃}(t, ϵ) = θ_{G}(t, ϵ) ⊗ e(M̃, g̃) dim W + e(M, g) ⊗ θG̃(t, ϵ) dim W.

This can be derived as in the proof of Proposition 2.4 in [BFKM] from the following facts:

(i) Δ_{q}^{G⊗G̃}(t) = Δ_{q}^{G} ⊗ Id + Id ⊗ ΔG̃ ; q(t),

whose proof is similar to the proof of Proposition 2.4 in [BFKM] and for any t, (ii) η(0, t, ϵ = 0) = e(M, g) dim W, whose proof follows from the local index theorem of Bismut and Zhang. This takes care of (3.12).

For (iii), we use the locality of the density α_{F}(ϵ) and the explicit formula of Δ_{q}(t). In admissible coordinates in a neighborhood of the critical points Δ_{q}(t), is the same as Δ_{q}(t), for a product of systems G with underlying manifolds of dimension one. Using (3.12) and the vanishing of the Euler form on M = IR, the result follows.

(iv) (a) is actually Proposition 6.6 in [BFKM] and (b) follows from Lemma 6.7 of [BFKM].

Proof of statement (D). We want to apply Proposition 3.2. Set M̃ := M, τ̃ := τ, g = g̃, ρ̃ the trivial representation of Γ ≡ π_{1}(M) on W. Then the associated bundle Ẽ → M is trivial ; we choose μ̃ the trivial Hermitian structure. Then by (2.15) and the equality of analytic and Reidemeister torsion for the trivial representation we have R̃ = 1. Since log R̃(t) admits an asymptotic expansion by (B) so does log T̃_{la}(t). Since the free term of the expansion of log R̃ is zero, one obtains

(3.16) F T_{t=∞} log T̃_{la}(t) = ( ∑_{j}(−1)^{j}(\frac{n}{4} − \frac{j}{2})m_{j} dim E_{x}) log π.

Statement (i) in (D) follows then from Proposition 3.2 (iv) . To prove statement (ii), observe that since μ is parallel with respect to the canonical flat connection ∇, the Hodge ∗ operator induces an isometry between L_{2}(Λ^{k}(M; E)) and L_{2}(Λ^{n−k}(M; E)) and conjugates Δ_{k}(t) with Δ_{n−k}(−t). Thus Δ_{k}(t) and Δ_{n−k}(t) are isospectral and then by the definition of V_{F}(t, ϵ) we have :

V_{F}(t, ϵ) = (−1)^{n+1}V_{F′}(t, ϵ)

and

VF̃(t, ϵ) = (−1)^{n+1}VF̃′(t, ϵ).

Therefore

V_{F}(t, ϵ) − VF̃(t, ϵ) = (−1)^{n+1}V_{F′}(t, ϵ) − (−1)^{n+1}VF̃′(t, ϵ),

and thus by Proposition 3.2 (iv) we have

F T_{t=∞} log T_{la}(F, t) − F T_{t=∞} log T_{la}(F̃, t) =
= (−1)^{n+1}F T_{t=∞} log T_{la}(F′, t) − (−1)^{n+1}F T_{t=∞} log T_{la}(F̃′, t).

Again by Proposition 3.2 (iv)

∫_{M\Cr(h)} α_{F}(0) − ∫_{M\Cr(h)} αF̃(0) =

(−1)^{n+1}(∫_{M\Cr(h)} α_{F′}(0) − ∫_{M\Cr(h)} αF̃′(0)).

In view of (3.12), (D) (i) and

∫_{M\Cr(h)} αF̃(0) = ∫_{M\Cr(h)} αF̃′(0) = 0,

because μ is parallel, one obtains

∫_{M\Cr(h)} α_{F}(0) = (−1)^{n+1} ∫_{M\Cr(h)} α_{F′}(0).

Statement (D)(ii) follows by combining this last formula with Proposition 3.2 (iv)

4 Anomalies of the relative torsion

4.1 Metric and Hermitian anomalies

In this subsection, we investigate how the relative torsion R = R(M, ρ, μ, g, τ) varies with respect to the Riemannian metric g and the Hermitian structure μ. We denote by E → M, ∇ the flat bundle associated to ρ. In order to formulate the results we introduce a number of additional quantities:

The closed 1-form θ = θ(ρ, μ) ∈ Ω^{1}(M): Choose a finite covering M by open sets (U_{j})_{j∈J}, which are simply connected, and points x_{j} ∈ U_{j}. Define v_{j}: U_{j} → IR by

v_{j}(x) := log vol(T_{x,x_{j}}) = \frac{1}{2} log det(T_{x,x_{j}}^{∗}T_{x,x_{j}})

where T_{x,x_{j}}: (E_{x}, μ_{x}) → (E_{x_{j}}, μ_{x_{j}}) denotes the parallel transport from E_{x} to E_{x_{j}} along any curve joining x and x_{j} inside U_{j}. (As the connection ∇ of E → M is flat, the map T_{x,x_{j}} does not depend on the choice of the curve.) Denote by θ = θ(ρ, μ) the smooth 1-form on M defined by θ_{j} := dv_{j}. Notice that if the Hermitian structure μ is parallel with respect to the canonical flat connection ∇, then θ(μ) = 0. We mention that in [BZ] the quantity θ is defined to be 2θ(ρ, μ).

If the representation ρ is unimodular, (i.e. log vol(ρ(g)) = 0 for any g ∈ Γ) then one can find μ so that θ(ρ, μ) = 0. To see this let detIR E → M denote the 1-dimensional real vector bundle obtained by assigning to each point x ∈ M the vector space detIR E_{x}, as described in [CFM], and let det ∇ denote the induced flat connection in detIR E → M. This is the flat bundle associated with the representation detIR ρ. The unimodularity of ρ implies the existence of a parallel section s_{0} in detIR E → M. Take a Hermitian structure μ in E → M, and denote by s the tautological section in detIR E → M induced by μ. Clearly there exists a smooth nonzero function f : M → IR_{+} so that s_{0} = f · s. If μ is parallel above the open set U, then f|_{U} = 1. It is immediate from definition of θ that θ(ρ, f · μ) = 0.

If (ρ_{i}, μ_{i}), i = 1, 2 are two pairs consisting of a representation and a Hermitian structure on the induced flat bundle, then one can verify in a straightforward manner that

(4.1) θ(ρ_{1} ⊗ ρ_{2}, μ_{1} ⊗ μ_{2}) = θ(ρ_{1}, μ_{1}) ⊗ 1 + 1 ⊗ θ(ρ_{2}, μ_{2}).

The function V = V (ρ, μ_{1}, μ_{2}) ∈ Ω^{0}(M): If μ_{j}, j = 1, 2 are two Hermitian structures of E → M, define the smooth function

V (x) := V (ρ, μ_{1}, μ_{2})(x) = log vol(Id_{x} : (E_{x}, μ_{1}(x)) → (E_{x}, μ_{2}(x))).

Note that

(4.1’) θ(ρ, μ_{1}) − θ(ρ, μ_{2}) = dV (ρ, μ_{1}, μ_{2}),

(4.1”) V (ρ, μ_{1}, μ_{2}) = −V (ρ, μ_{2}, μ_{1}),

(4.1”’) V (ρ, μ_{1}, μ_{3}) = V (ρ, μ_{1}, μ_{2}) + V (ρ, μ_{2}, μ_{3}).

The Euler form e(M, g) ∈ Ω^{n}(M; O_{M}): Denote by e(M, g) the Euler form associated to the Levi-Cività connection on the tangent bundle T M. This is a form with coefficients in the orientation bundle of M, a flat, 1-dimensional real vector bundle.

The Chern-Simon element [e_{CS}(M, g_{1}, g_{2})] ∈ Ω^{n−1}(M; O_{M})/dΩ^{n−2}(M; O_{M}): For two Riemannian metrics g_{1} and g_{2} on M denote by [e_{CS} = e_{CS}(M, g_{1}, g_{2})] the Chern-Simon class , cf [BZ], p 46. Recall that

d(e_{CS}(M, g_{1}, g_{2})) = e(M, g_{2}) − e(M, g_{1})

and that there is a canonical construction, due to Chern-Simon, for a representative e_{CS} of [e_{CS}] so that for a smooth 1-parameter family g_{2}(u) of Riemannian metrics on M, e_{CS}(g_{1}, g_{2}(u)) is a smooth 1-parameter family of (n − 1) forms.

The following object appears in Theorem 0.1 and is related to the previous quantities.

The form Ψ(T M, g) ∈ Ω^{n−1}(T M \ M; O_{T M}) : Let π : T M → M be the tangent bundle of M. Clearly, O_{T M} = π^{∗}O_{M}. Following Mathai-Quillen [MQ] Theorem 6.4, or Bismut Zhang [BZ] Theorem 3.4, one can construct a smooth form Ψ(T M, g) ∈ Ω^{n−1}(T M \ M; O_{T M}) with the following properties:

(4.2’) dΨ(T M, g) = π^{∗}(e(M, g))

If g_{1} and g_{2} are two Riemannian metrics on M then (4.2”) Ψ(T M, g_{2}) − Ψ(T M, g_{1}) = π^{∗}e_{CS}(M, g_{1}, g_{2}) (modulo exact forms)

If λ : T M \ M → T M \ M denotes multiplication by the real number λ ≠ 0 then

(4.2”’) λ^{∗}(Ψ(T M, g)) = (λ/|λ|)^{n}Ψ(T M, g).

(4.2^{iv}) For x ∈ M fixed, denote by ω the volume form on T_{x}M and by Y the Euler vectorfield on T_{x}M ( in polar coordinates, Y = r \frac{∂}{∂r} , r the distance to the origin). Then

Ψ(T M, g)|_{T_{x}M} = −\frac{Γ(n/2)}{2π^{n/2}} \frac{ι_{Y} ω}{|Y |^{n}} .

It is easy to check (cf [BZ] ) that for two Riemannian manifolds (M_{i}, g_{i}), i = 1, 2 we have

(4.2^{v}) \begin{matrix}Ψ(T (M_{1} × M_{2}), g_{1} × g_{2}) = Ψ(T (M_{1}), g_{1}) ⊗ e(M_{2}, g_{2})+ \\ +e(M_{1}, g_{1}) ⊗ Ψ(T (M_{2}), g_{2}). \end{matrix}

Proposition 4.1 (Metric anomaly of the relative torsion)

(i) log R(g_{2}) − log R(g_{1}) = − ∫_{M} θ ∧ e_{CS}(g_{1}, g_{2}). (ii) If dim M is odd or μ is a Hermitian structure parallel with respect to the canonical flat connection, the relative torsion R is independent of g.

Proposition 4.2 (Hermitian anomaly of the relative torsion)

log R(μ_{2}) − log R(μ_{1}) = − ∫_{M} V (ρ, μ_{1}, μ_{2})e(M, g)+

+ ∑_{x∈Cr(h)}(−1)^{ind x}V (ρ, μ_{1}, μ_{2})(x).

In the case A = IC these results were established in by Bismut Zhang, cf [BZ]. Propositions 4.1 and 4.2 will be proved at the same time. Their proof is reduced to some local index type results established in [BZ].

Remark Theorem 2.1. can be also derived as a special case of a (slightly more general) version of Proposition 4.2(i).

Proof of Propositions 4.1, 4.2: (ii) follows from (i) by noting that e(M, g) = 0 and e_{CS}(M, g_{1}, g_{2}) = 0 if dimM is odd and that θ(μ) = 0 if the Hermitian structure μ is parallel with respect to the canonical flat connection.

To prove (i), consider a smooth 1-parameter family g(u) of Riemannian metrics and a smooth 1-parameter family μ(u) of Hermitian structures, (−1 < u < +1).

We want to compute \frac{d}{du} log R(g(u), μ_{0}) and \frac{d}{du} log R(g_{0}, μ(u)). We begin by analyzing \frac{d}{du} log R(g(u), μ(u)).

Denote by ⟨⟨·, ·⟩⟩(u) the scalar product defined by g(u) and μ(u) on Ω^{k}(M; E)) (cf. 2.2) and by ⟨⟨·, ·⟩⟩_{s}(u) the one given by (2.4). Clearly ⟨⟨·, ·⟩⟩(u) = ⟨⟨·, ·⟩⟩_{0}(u).

Denote by Δ_{k}(u): Ω^{k}(M; E) → Ω^{k}(M; E) the Laplacian on Ω^{k}(M; E) induced by g(u) and μ(u) and by H_{s}(u)(Λ(M; E)) the completion of Ω(M; E) with respect to ⟨⟨·, ·⟩⟩_{s}(u).

Consider the following commutative diagram

γ_{s+s′}(u)
H_{0}(u)(Λ(M; E)) ⟶ C(τ)(u)
↓ φ_{s,s′}(u)  ↓ ι(u)
γ_{s}(0)
H_{0}(0)(Λ(M; E))  ⟶  C(τ)(0)

where γ_{s}(u) := Int_{s}(Id + Δ(u))^{−s/2}, φ_{s,s′}(u) := (Id + Δ(0))^{+s/2}(Id + Δ(u))^{− \frac{s+s′}{2}} , and ι(u) is the identity map on C(τ) regarded as an isomorphism of C(τ) with the scalar product induced from μ(u) and C(τ) with the scalar product induced by μ(0). We recall from subsection 2.1 that, for s and s′ sufficiently large, γ_{s}(u) and φ_{s,s′}(u) are morphisms of trace class. Since γ_{s+s′}(u) and γ_{s}(0) induce isomorphisms in algebraic cohomology (cf. Proposition 2.5), so does φ_{s,s′}(u). We thus can apply Proposition 1.15 to obtain

(4.3) log T (C(ι(u)) + log R(g(u)) = log R(g(0)) + log T (C(φ_{s,s′}(u))

where C(φ_{s,s′}(u)) denotes the mapping cone associated to φ_{s,s′}(u). Since (Id + Δ(u))^{s/2}: H_{s}(u)(Λ(M; E)) → H_{0}(u)(Λ(M; E)) is an isometry, we conclude from Proposition 1.16 that

(4.4) T (C(φ_{s,s′}(u)) = T (C(In_{s+s′;s}(u))

where

In_{s+s′,s}(u): H_{s+s′}(u)(Λ(M; E)) → H_{s}(0)(Λ(M; E))

denotes the canonical inclusion. To analyze the torsion of the mapping cone (C_{u}, D) := (C(In_{s+s′;s}(u)), d(In_{s+s′,s}(u))), note that

(4.5)  C_{u}_{k} := H_{s}(0)(Λ^{k−1}(M; E)) ⊕ H_{s+s′}(u)(Λ^{k}(M; E)) ;
D_{k} = ( \begin{matrix}−d_{k−1} & In_{s+s′,s}(u) \\ 0 & d_{k} \end{matrix} )

with inner product (ω_{1}, ω′_{1} ∈ Ω^{k−1}, ω_{2}, ω′_{2} ∈ Ω^{k})

(4.6) ⟨⟨(ω_{1}, ω_{2}), (ω′_{1}, ω′_{2})⟩⟩ := ⟨⟨ω_{1}, ω′_{1}⟩⟩_{s}(0) + ⟨⟨ω_{2}, ω′_{2}⟩⟩_{s+s′}(u).

In order to calculate \frac{d}{du} log T (C(u)), introduce the zeroth order differential operators A_{k}(0): Ω^{k}(M; E) → Ω^{k}(M; E) defined by

⟨⟨ω, ω′⟩⟩_{0}(u) = ⟨⟨A_{k}(u)ω, ω′⟩⟩_{0}(0)

and consider the zeroth order pseudo-differential operator

B_{k}(s + s′; u): H_{s+s′}(0)(Λ^{k}(M; E)) → H_{s+s′}(0)(Λ^{k}(M; E))

defined by

B_{k}(u) ≡ B_{k}(s+s′, u) := (Id+Δ_{k}(u))^{−(s+s′)}A_{k}(u)^{−1} \frac{d}{du}(A_{k}(u)(Id+Δ_{k}(u))^{s+s′})

so that the following identity holds

\frac{d}{du}⟨⟨ω, ω′⟩⟩_{s+s′}(u) = ⟨⟨B_{k}(u)ω, ω′⟩⟩_{s+s′}(0) .

Let B_{k}(s + s′, u) be the bounded operator in L(H_{s}(0)(Λ^{k}(M; E)) ⊕ H_{s+s′}(0)(Λ^{k}(M; E))) defined by

B_{k}(u) ≡ B_{k}(s + s′, u) := ( \begin{matrix}0 & 0 \\ 0 & B_{k}(s + s′, u) \end{matrix} ) .

Then we have, in view of (4.6),

\frac{d}{du}⟨⟨(ω_{1}, ω_{2}), (ω′_{1}, ω′_{2})⟩⟩ = ⟨⟨B_{k}(u)(ω_{1}, ω_{2}), (ω′_{1}, ω′_{2})⟩⟩ .

Proceeding as in [BFKM, Appendix 3], one obtains (cf. [BFKM, A3.10]) in view of the fact that the complex C_{u} is algebraically acyclic,

(4.7) \frac{d}{du} log T (C_{u}) = −F p_{z=0} \frac{1}{Γ(z)} ∫_{0}^{1} x^{z−1} \frac{1}{2} ∑_{q=0}^{n}(−1)^{q}T r(B_{q}(u)e^{−xΔ̃_{q}(u)})dx

where Δ̃_{q}(u) denotes the q-Laplacian of C_{u} with respect to the inner product (4.6).

By formula (1.23),

Δ̃_{q}(u) = ( \begin{matrix}Δ_{q−1}(0) + ψ_{q−1}(u) & η_{q}(u) \\ η_{q}(u)^{∗} & Δ_{q}(u) + ψ_{q}(u) \end{matrix} )

where ( \begin{matrix}ψ_{q−1}(u) & η_{q}(u) \\ η_{q}(u)^{∗} & ψ_{q}(u) \end{matrix} ) is a nonnegative selfadjoint operator of trace class

(s, s′ sufficiently large).

By the remark after Proposition 1.5 we conclude, in view of definitions (1.5) and (1.7),

(4.8)  − \frac{1}{2}F p_{z=0} \frac{1}{Γ(z)} ∫_{0}^{1} x^{z−1} ∑_{q=0}^{n}(−1)^{q}T r(B_{q}(u)e^{−xΔ̃_{q}(u)})dx

= − \frac{1}{2}F p_{z=0} \frac{1}{Γ(z)} ∫_{0}^{1} x^{z−1} ∑_{q=0}^{n}(−1)^{q}T r(B_{q}(u)e^{−xΔ_{q}(u)}(Id − P_{q;u}))dx

+ \frac{1}{2} ∑_{q=0}^{n}(−1)^{q}T r(B_{q}(u)P_{q;u})

= − \frac{1}{2}F p_{z=0} \frac{1}{Γ(z)} ∫_{0}^{1} x^{z−1} ∑_{q=0}^{n}(−1)^{q}T r(B_{q}(u)e^{−xΔ_{q}(u)})dx

where P_{q;u} ≡ P_{q;u}({0}) is the orthogonal projector onto the null space of Δ_{q}(u). Introduce

(4.9) a(x; u) := ∑_{q=0}^{n}(−1)^{q}T r(B_{q}(s; u)e^{−xΔ_{q}(u)})

= ∑_{q=0}^{n}(−1)^{q}T r((Id + Δ_{q}(u))^{−s}

A_{q}(u)^{−1} \frac{d}{du}(A_{q}(u)(Id + Δ_{q}(u))^{s})e^{−xΔ_{q}(u)}) .

To investigate the right hand side of (4.8) introduce

(4.10) b(x; u) := ∑_{q=0}^{n}(−1)^{q}T r (A_{q}(u)^{−1} (\frac{d}{du}A_{q}(u)) e^{−xΔ_{q}(u)}) .

According to [BZ],

F p_{z=0} \frac{1}{Γ(z)} ∫_{0}^{1} x^{z−1}b(x; u)dx

is, in the case where μ(u) = μ is constant, given by

(4.11) − ∫_{M} θ(ρ, μ) ∧ \frac{d}{du}e_{CS}(M, g(0), g(u))

and, when g(u) = g is constant by

(4.12) ∫_{M} \frac{d}{du}V (ρ, μ(u), μ(0))e(M, g).

Actually, in [BZ], these formulae are proven only for the case A = IC, but the same arguments work ”word by word” for A arbitrary. In this paper θ(ρ, μ) is defined to be 1/2 of the corresponding quantity in [BZ].

We will show that

a(x; u) = b(x; u) .

From the equality

\frac{d}{du}((Id + Δ_{q}(u))^{s}(Id + Δ_{q}(u))^{−s}) = 0

and the commutativity of the trace, T rAB = T rBA, we obtain

T r((Id + Δ_{q}(u))^{−s}A_{q}(u)^{−1} \frac{d}{du}(A_{q}(u)(Id + Δ_{q}(u))^{s})e^{−xΔ_{q}(u)})

= T r(A_{q}(u)^{−1}(\frac{d}{du}A_{q}(u))e^{−xΔ_{q}(u)})

+ T r((Id + Δ_{q}(u))^{−s} \frac{d}{ds}((Id + Δ_{q}(u))^{−s})e^{−xΔ_{q}(u)})

= T r(A_{q}(u)^{−1}(\frac{d}{du}A_{q}(u))e^{−xΔ_{q}(u)}) +

+ s T r((Id + Δ_{q}(u))^{−1} \frac{d}{du}(Δ_{q}(u))e^{−xΔ_{q}(u)})

= T r(A_{q}(u)^{−1} \frac{dA_{q}(u)}{du}e^{−xΔ_{q}(u)}) + s\frac{d}{du}T rf(Δ_{q}(u))

where f(x, y) := − ∫_{y}^{∞} \frac{e^{−xr}}{(1+r)}dr. Since Δ_{q}(u) is selfadjoint and nonnegative one obtains, for x > 0,

∥f(x, Δ_{q}(u))∥_{tr} := ∫_{0}^{∞} λ|f(x, λ)|dN_{Δ_{q}(u)} ≤ ∫_{0}^{∞} λe^{−λx}dN_{Δ_{q}(u)}(λ) < ∞ .

Therefore, f(x, Δ_{q}(u)) is of trace class for x > 0 and

∑_{q=0}^{n}(−1)^{q}T rf(x, Δ_{q}(u)) = ∑_{q=0}^{n}(−1)^{q}T rf(x, Δ_{q}^{+}(u)) +

+ ∑_{q=0}^{n}(−1)^{q}T rf(x, Δ_{q}^{−}(u))

= ∑_{q=1}^{n}(−1)^{q}(T rf(x, Δ_{q}^{+}(u)) − T rf(x, Δ_{q−1}^{−}))
= 0

where we used that Δ_{q}^{+}(u) and Δ_{q}^{−}(u) are isospectral and, therefore,

T rf(x, Δ_{q}^{+}(u)) = T rf(x, Δ_{q}^{−}(u)) .

Thus, for any x ≥ 0,

a(x; u) = b(x; u) + s\frac{d}{du} ∑_{q=0}^{n}(−1)^{q}T rf(x, Δ_{q}(u))
= b(x; u) .

Note that by Proposition 1.12

log T (C(ι(u)) = ∑_{x∈Crh}(−1)^{ind x}V (ρ, μ(u), μ(0))(x).

Combining these with (4.8)-(4.11), statement (i) in Proposition 4.1 and Proposition 4.2 follow.

4.2 Proof of Proposition 0.1

We now prove Proposition 0.1 in full generality. Let g be an arbitrary Riemannian metric of M and g′ the metric given by the generalized triangulation τ. Denote the corresponding relative torsions by R(g) and R(g′). Proposition 4.1 and Proposition 0.1 in the special case considered in section 3 imply Proposition 0.1: Indeed, denote by α(g′) the density given in subsection 3.1 (D) and define α(g) := α(g′) + θ ∧ e_{CS}(g, g′) Clearly, α is a local quantity and satisfies (cf subsection 3.1 (B) and (D)) log R(g) = ∫_{M\Cr(h)} α(g). Further α(g) vanishes if μ is parallel (cf subsection 3.1 (D) and Proposition 4.1(ii)).

4.3 Anomaly with respect to triangulations

In this subsection we investigate how the relative torsion depends on the triangulation. Suppose τ_{1} = (h_{1}, g_{1}) and τ_{2} = (h_{2}, g_{2}) are two generalized triangulation which admit a common subdivision τ_{0} = (h_{0}, g_{0}). Recall from [BFKM, Subsection 6.3] that τ_{0} is called a subdivision of τ_{1} if :

(i) Cr_{q}(h_{1}) ⊆ Cr_{q}(h_{0}), (0 ≤ q ≤ n), W_{x}^{−}(τ_{0}) ⊆ W_{x}^{−}(τ_{1}), (x ∈ Cr(h_{1}))

and for any y ∈ Cr(h_{1})

(ii) W_{y}^{−}(τ_{1}) = ∪_{x∈Cr(h_{0}),x∈W_{y}^{−}(τ_{1})}W_{x}^{−}(τ_{0}),

where Cr_{q}(h_{i}) denotes the set of critical points of index q of h_{i} (i = 0, 1, 2), W_{x}^{−}(τ_{1}) denotes the unstable manifold of x ∈ Cr(h_{1}) with respect to the gradient flow −grad_{g_{1}}(h_{1}) and g_{0} = g_{1}, h_{0} = h_{1} in a neighborhood of the critical points of h_{1}.

Given x ∈ Cr(h_{0}), there exist unique elements x_{1} ∈ Cr(h_{1}), x_{2} ∈ Cr(h_{2}) so that x ∈ W_{x_{j}}^{−}(τ_{j}) (j = 1, 2).

Introduce the function w = w_{τ_{0}} = w(τ_{1}, τ_{2}; τ_{0}): Cr(h_{0}) → IR by setting

(4.13) w(x) := log vol(T_{x,x_{2}}^{τ_{2}} ∘ (T_{x,x_{1}}^{τ_{1}})^{−1})

where T_{x,x_{j}}^{τ_{j}}: E_{x} → E_{x_{j}} (j = 1, 2) denotes the parallel transport along an arbitrary curve in W_{x_{j}}^{−}(τ_{j}) joining x and x_{j}. Define ω_{τ_{1},τ_{2}} = ω(τ_{1}, τ_{2}; τ_{0}) by

(4.14) ω_{τ_{1},τ_{2}} := ∑_{x∈Cr(h_{0})}(−1)^{index(x)}w(x).

Notice that ω(τ_{1}, τ_{2}; τ_{0}) is independent of the choice of τ_{0}, i.e.,

(4.15) ω(τ_{1}, τ_{2}; τ̃_{0}) = ω(τ_{1}, τ_{2}; τ_{0})

for any generalized triangulation τ̃_{0} = (h̃_{0}, g̃_{0}) which is a subdivision of τ_{0}. To see it, notice that for any y ∈ Cr(h̃_{0}), there exists a unique x ∈ Cr(h_{0}) with y ∈ W_{x}^{−}(τ_{0}). Use that T_{y,x_{j}}^{τ_{j}} = T_{x,x_{j}}^{τ_{j}}T_{y,x}^{τ_{0}} (j = 1, 2) to conclude that wτ̃_{0}(y) = w_{τ_{0}}(x) and thus

∑_{y∈Cr(h̃_{0})∩W_{x}^{−}(τ_{0})}(−1)^{index(y)}wτ̃_{0}(y) = w_{τ_{0}}(x) ∑_{y∈Cr(h̃_{0})∩W_{x}^{−}(τ_{0})}(−1)^{index(y)} .

As τ̃_{0} is a subdivision of τ_{0}

∑ ; \begin{matrix}y ≠ x \\ y ∈ Cr(h̃_{0}) ∩ W_{x}^{−}(τ_{0}) \end{matrix} (−1)^{index(y)} = 0

we conclude that ω(τ_{1}, τ_{2}; τ̃_{0}) = ω(τ_{1}, τ_{2}; τ_{0}). Moreover, if τ_{1}, τ_{2} and τ_{3} are three generalized triangulation which admit a common subdivision τ_{0}, then

(4.16) ω_{τ_{1},τ_{2}} + ω_{τ_{2},τ_{3}} = ω_{τ_{1},τ_{3}}.

Proposition 4.3 Assume that the generalized triangulations τ_{1} and τ_{2} admit a common subdivision τ_{0}. Then

(i) log R(τ_{1}) − log R(τ_{2}) = ω_{τ_{1},τ_{2}} (ii) If μ is parallel with respect to the canonical flat connection of E, then R(τ_{1}) = R(τ_{2}) provided τ_{1} and τ_{2} have a common subdivision.

Proof: Statement (ii) follows from (i) by noticing that w(x) = 0, ∀ x ∈ Cr(h_{0}) and thus ω_{τ_{1},τ_{2}} = 0. To prove (i), notice that

log R(τ_{2}) − log R(τ_{1}) = (log R(τ_{2}) − log R(τ_{0})) + (log R(τ_{0}) − log R(τ_{1})) .

In view of (4.16), it suffices to consider the case where τ_{2} = τ_{0}. The subdivision τ_{2} can be obtained from τ_{1} by a sequence τ_{1} = σ_{1}, …, σ_{N} = τ_{2}, where σ_{j+1} = (h_{j+1}, g′_{j+1}) is a subdivision of σ_{j} = (h_{j}, g′_{j}) with Cr(h_{j+1}) = Cr(h_{j}) ∪ {x_{j+1}, y_{j+1}} so that there exists z_{j} ∈ Cr(h_{j}) with x_{j+1}, y_{j+1} ∈ W_{z_{j}}^{−} and index(x_{j+1}) = index(z_{j}) = index(y_{j+1}) + 1. Recall that W_{x}^{−} ≡ W_{x;h_{j}}^{−} denotes the unstable manifold, associated to x ∈ Cr(h_{j}) and the gradient flow −grad_{g_{j}}h_{j}. In view of (4.15), it suffices to consider the case where τ_{2} = σ_{2}. To ease notation, we write τ := τ_{1} = (h_{1}, g′_{1}), σ := τ_{2} = (h_{2}, g′_{2}) z := z_{1}, x′ := x_{2}, y′ := y_{2}. Then, with q_{0} = index(z)

Cr_{q}(h_{2}) = \{ \begin{matrix}Cr_{q}(h_{1}) & q ≠ q_{0}, q_{0} − 1 \\ Cr_{q}(h_{1}) ∪ \{y′\} & q = q_{0} − 1 \\ Cr_{q}(h_{1}) ∪ \{x′\} & q = q_{0} \end{matrix}

Consider the following commutative diagram

Int_{s}(τ)
H_{s}(Λ(M, E)) ⟶ C(τ)
Int_{s}(σ) ↘  ↗ A
C(σ)

where A_{q} (0 ≤ q ≤ n) is defined as follows: for a section s ∈ C^{∞}(E |_{Cr_{q}h_{2}}) and a critical point x ∈ Cr_{q}(h_{1}), set

A_{q}(s)(x) := ∑_{y∈Cr_{q}(h_{2})∩W_{x}^{−}} T_{yx}(s(y))

where T_{yx}: E_{y} → E_{x} denotes the parallel transport from y to x along a curve in W_{x}^{−} = W_{x}^{−}(τ). The map A is a morphism of cochain complexes of A-Hilbert modules of finite type which induces an isomorphism in algebraic cohomology. As Int_{s} and A are of trace class, we then conclude from Proposition 1.15 that

log R(τ) = log R(σ) + log T (C(A)) .

Thus the claimed result follows once we show that

(4.17) log T (C(A)) = ω_{στ}.

Formula (4.17) can be verified, using a localization argument: Consider first the case where μ is parallel. Clearly, ω_{στ} = 0. On the other hand, log R(τ) = log R(σ) = 0 (Proposition 0.1 (ii)) and thus log T (C(A)) = log R(τ)−log R(σ) = 0 . Hence (4.17) holds in this case. Further, if E admits a Hermitian structure μ_{0} which is parallel, then (4.17) remains valid. Indeed, consider the following commutative diagram

A

C(σ, μ) ⟶ C(τ, μ)
↓Id_{σ,μ,μ_{0}}  ↓Id_{τ,μ,μ_{0}}
A
C(σ, μ_{0}) ⟶ C(τ, μ_{0})

Then, according to Proposition 1.12,

log T (C(Id_{σ,μ,μ_{0}})) + log T (C(A, μ_{0})) = log T (C(A, μ)) + log T (C(Id_{τ,μ,μ_{0}})) . By Proposition 1.14, (j = 1, 2)

log T (C(Id_{σ,μ,μ_{0}})) − log T (C(Id_{τ,μ,μ_{0}}))
= ∑_{q}(−1)^{q} log vol(Id_{q;σ,μ,μ_{0}})

− ∑_{q}(−1)^{q} log vol(Id_{q,τ,μ,μ_{0}})

= ∑_{y∈Cr(h_{2})}(−1)^{index(y)} log vol(Id_{y,μ,μ_{0}})

− ∑_{y∈Cr(h_{1})}(−1)^{index(y)} log vol(Id_{y,μ,μ_{0}})

= (−1)^{index(x′)} log vol(Id_{x′,μ,μ_{0}})

+(−1)^{index(y′)} log vol(Id_{y′,μ,μ_{0}}) .

Combining the above equalities with log T (C(A, μ_{0})) = 0 we obtain

log T (C(A, μ)) = (−1)^{index(x′)} log vol(Id_{x′,μ,μ_{0}})+

+(−1)^{index(y′)} log vol(Id_{y′,μ,μ_{0}}) .

Observe that

log vol(Id_{x′,μ,μ_{0}}) = log vol_{μ}(T_{x′,z′}) + log vol(Id_{z′,μ,μ_{0}})

and, similarly,

log vol(Id_{y′,μ,μ_{0}}) = log vol_{μ}(T_{y′,x_{0}}) + log vol(Id_{z′,μ,μ_{0}}) .

As index(x′) = index(y′) + 1, this leads to

log T (C(A, μ)) = (−1)^{index(x′)} log vol_{μ}(T_{x′z′})
+ (−1)^{index(y′)} log vol_{μ}(T_{y′z′})
= ω_{zτ}

which establishes (4.17) in the case where E admits a Hermitian structure which is parallel. Now use a standard localization to conclude that (4.17) is true in general.

5 Proof of Theorem 0.1

5.1 Proof of Theorem 0.1

Consider the system F := (M, ρ, μ, g, τ) ≡ (M, E, ∇, μ, g, τ), with τ = (h, g′) and denote by X the vector field X := −grad_{g′}h. X defines a smooth map X : M \ Cr(h) → T M \ M. Denote by

β_{F} ≡ β(M, ρ, μ, g, τ) := (−1)^{n+1}θ(ρ, μ)∧X^{∗}(Ψ(T M, g)) ∈ Ω^{n}(M \Cr(h); O_{M}).

Observe that if μ is parallel in a neighborhood of Cr(h) both α ≡ α_{F}, by Proposition 3.2 (iii), and β_{F} vanish in the neighborhood of Cr(h).

Proposition 3.2 (ii) and the equalities (4.1) and (4.2^{v}) imply that for γ(M, ρ, μ, g, τ) = α(M, ρ, μ, g, τ) or β(M, ρ, μ, g, τ) we have

(5.1) γ(M_{1} × M_{2}, ρ_{1} ⊗ ρ_{2}, g_{1} × g_{2}, μ_{1} ⊗ μ_{2}, τ_{1} × τ_{2}) =

γ(M_{1}, ρ_{1}, g_{1}, μ_{1}, τ_{1})e(M_{2}, g_{2})dim(W_{2}) + γ(M_{2}, ρ_{2}, g_{2}, μ_{2}, τ_{2})e(M_{1}, g_{1})dim(W_{1}).

Choose a Hermitian structure μ_{0} which is parallel in a neighborhood of Cr(h). Introduce the quantity

S(M, ρ, μ, μ_{0}, g, τ) := log R(M, ρ, μ, g, τ) − ∫_{M\Cr(h)} β(M, ρ, μ_{0}, g, τ) −

− ∫_{M} V (ρ, μ, μ_{0})e(M, g) + ∑_{x∈Cr(h)}(−1)^{ind(x)}V (ρ, μ, μ_{0})(x)

The proof of Theorem 0.1 uses the following four statements:

(A) S(M, ρ, μ, μ_{0}, g, τ) is independent of μ, μ_{0}, g, τ.

Therefore we write S(M, ρ) instead of S(M, ρ, μ, μ_{0}, g, τ).

(B) If ρ is isomorphic to its dual, then S(M, ρ) + (−1)^{n+1}S(M, ρ) = 0.

(C) S(M, ρ) = (−1)^{n+1}S(M, ρ^{♯}) where ρ^{♯} is the dual representation .

(D) S(M_{1} × M_{2}, ρ_{1} ⊗ ρ_{2}) = S(M_{1}, ρ_{1})χ(M_{2})dimW_{1} + χ(M_{1})S(M_{2}, ρ_{2})dimW_{2}.

(A) follows from Propositions 4.1-4.3. and the formulas 4.1”-4.1”’ and 4.2’-4.2”’. Precisely, the independence of g follows from Proposition 4.1. and formula 4.2”, and the independence of μ from Proposition 4.2 and the formulas 4.1” and 4.1”’. To check the independence on μ_{0} one uses g = g′ and proceeds as follows: Choose coordinates in a neighborhood of the critical points so that (T3) in the definition of a generalized triangulation is satisfied. Denote by D_{x}(ϵ) the disc of radius ϵ and centered at x ∈ Cr(h) and put D(ϵ) := ⨆_{x∈Cr(h)} D_{x}(ϵ). Let μ̃_{0} and μ_{0} be two Hermitian structures parallel in the neighborhood of the critical points. A straightforward application of the definition of S combined with 4.1′ − 4.2^{v} leads to

S(M, ρ, μ, μ̃_{0}, g′, τ) − S(M, ρ, μ, μ_{0}, g′, τ) =

= lim_{ϵ→0}((−1)^{n} ∫_{M\D(ϵ)} d(V (ρ, μ̃_{0}, μ_{0})X^{∗}(Ψ(T M, g′))−

−(−1)^{n} ∫_{M\D(ϵ} V (ρ, μ̃_{0}, μ_{0})e(M, g′))+

+ ∫_{M} V (ρ, μ̃_{0}, μ_{0})e(M, g′) − ∑_{x∈Cr(h)}(−1)^{ind x}(V (ρ, μ̃_{0}, μ_{0})(x).

Using Stokes theorem and, when n is odd, the vanishing of e(M, g′) one obtains

S(M, ρ, μ, μ̃_{0}, g′, τ) − S(M, ρ, μ, μ_{0}, g′, τ) =

= lim_{ϵ→0}((−1)^{n} ∫_{∂(M\D(ϵ))} V (ρ, μ̃_{0}, μ_{0})X^{∗}(Ψ(T M, g′))−

− ∑_{x∈Cr(h)}(−1)^{ind x}(V (ρ, μ̃_{0}, μ_{0})(x) =

= lim_{ϵ→0}((−1)^{n+1} ∫_{∂(D(ϵ))} V (ρ, μ̃_{0}, μ_{0})X^{∗}(Ψ(T M, g′))−

− ∑_{x∈Cr(h)}(−1)^{ind x}V (ρ, μ̃_{0}, μ_{0})(x).

Since V (ρ, μ̃_{0}, μ_{0}) is constant on each D_{x}(ϵ) one obtains

S(M, ρ, μ, μ̃_{0}, g′, τ) − S(M, ρ, μ, μ_{0}, g′, τ) =

= ∑_{x∈Cr(h)} V (ρ, μ̃_{0}, μ_{0})(x)((−1)^{n+1} ∫_{∂D_{x}(ϵ)} X^{∗}(Ψ(T M, g′)) − (−1)^{ind x}).

Since in the coordinates we have chosen

X = ∑_{k=1}^{ind x} x_{k} \frac{∂}{∂x_{k}} − ∑_{k=ind x+1}^{n} x_{k} \frac{∂}{∂x_{k}}

4.2^{iv} implies that

(−1)^{n+1} ∫_{∂D_{x}(ϵ)} X^{∗}(Ψ(T M, g′)) = (−1)^{ind x}

and therefore

S(M, ρ, μ, μ̃_{0}, g′, τ) = S(M, ρ, μ, μ_{0}, g′, τ).

The independence of τ can be verified in the following way: One considers a subdivision τ′ of τ so that only those cells of τ, lying in a contractible open set U, are subdivided. We choose μ_{0} to be parallel on U in addition of being parallel in a neighborhood of the critical points of h and let μ := μ_{0}. Since ω_{τ,τ′} as defined in subsection 4.3 is zero, it follows from Proposition 4.3 that S(M, .., τ) = S(M, .., τ′). If τ_{1} and τ_{2} are two generalized triangulations, one can find a third triangulation, τ_{0}, which is simultaneously a subdivision of τ_{1} and , up to a composition with a diffeomorphism of M isotopic to the identity, a subdivision of τ_{2}. Since one can pass from τ_{i}, i = 1, 2 to τ_{0} by a finite sequence of subdivisions each of them involving subdivisions of cells lying in a contractible open set, the result follows.

To check (B) we set μ := μ_{0} and choose μ_{0} with θ(ρ, μ_{0}) = 0. In view of the assumption that ρ is isomorphic to ρ^{♯} this is possible (cf subsection 4.1). The result follows then from Proposition 3.2 (i).

To check (C ) we choose again μ = μ_{0}. As θ(ρ^{♯}, μ_{0}^{♯}) = −θ(ρ, μ_{0}) and X_{τ_{D}} = −grad_{g′}(n − h) = grad_{g′}h = −X_{τ} we conclude in view of 4.2”’ that

∫_{M} θ(ρ, μ_{0}) ∧ X_{τ}^{∗}Ψ + (−1)^{n} ∫_{M} θ(ρ^{♯}, μ_{0}^{♯}) ∧ X_{D_{τ}}^{∗} Ψ = 0.

Statement (C ) follows once we verify that

∫_{M} α(M, ρ, g, μ, τ, ϵ) + (−1)^{n} ∫_{M} α(M, ρ^{♯}, g, μ^{♯}, τ_{D}, ϵ) = 0.

With the notation from subsection 3.2 and in view of formula 3.13 and the fact that the q-Laplacian corresponding to (ρ, μ, g, h) is conjugated by the Hodge star operator to the (n − q)-Laplacian corresponding to (ρ^{♯}, μ^{♯}, g, n − h), the left hand side of the equality above is ∫_{M} δ_{F}(ϵ) with

(5.2) δ_{F}(ϵ) = δ_{F}(h, ϵ) := n/2 ∑_{q}(−1)^{q+1}α_{F}^{q}(h, ϵ).

We want to check that ∫_{M} δ_{F}(0) = 0 and proceed as in the proof of Proposition 3.2 (iv). Given the system F one chooses a system F̃ with the same underlying Riemannian manifold (M, g), the same triangulation τ, but with ρ̃ the trivial representation over the same Hilbert module and with μ̃ a parallel Hermitian structure.

In view of 5.2, introduce W_{F}(t, ϵ) := \frac{n}{2} ∑_{q}(−1)^{q} log det(Δ_{q}(t)+ϵ). By the same arguments as in the proof of Proposition 3.2 (iv) (Mayer Vietoris arguments), we conclude that

F T_{t=∞}(W_{F}(t, ϵ) − WF̃(t, ϵ)) = ∫_{M\Cr(h)} δ_{F}(ϵ) − ∫_{M̃\Cr(h̃)} δF̃(ϵ).

The left side of the above identity is zero because by a straightforward computation

W_{F}(t, ϵ) = nχ(M)dimW log ϵ = WF̃(t, ϵ).

On the right side of the identity, the term ∫_{M\Cr(h)} δF̃(ϵ) = ∫_{M\Cr(h)}(αF̃ + (−1)^{n}αF̃^{♯}) is zero because in view of Proposition 0.1 both ∫_{M\Cr(h)} αF̃ and ∫_{M\Cr(h)} αF̃^{♯}(ϵ) are zero for the trivial representation and a parallel Hermitian structure. Consequently ∫_{M\Cr(h)} δ_{F}(0) = 0.

( D) Choose μ := μ_{0}. Then the statement follows from formula 5.1 Proposition 0.1 and Proposition 3.2 (ii).

Proof of Theorem 0.1 Step 1: We first show that S(M, ρ) is of the form S(M, ρ) = F (ρ; Γ) · χ(M). Note that when ρ is isomorphic to ρ^{♯}, it follows, by using (B) for n odd and (C) for n even, that S(M, ρ) = 0. If χ(M) = 0, choose an even dimensional manifold M′ with nonzero Euler Poincaré characteristic and with the same fundamental group as M and choose the representation ρ′ := ρ^{♯}. Since ρ ⊗ ρ^{♯} is isomorphic to its dual, and in view of (D)

0 = S(M × M′, ρ ⊗ ρ^{♯}) = S(M, ρ) · χ(M′) dim(W).

Hence the result is true if χ(M) = 0.

Suppose χ(M) ≠ 0. For any M′ with the same fundamental group as M and ρ′ := ρ^{♯}, by the same argument as above for the case χ(M) = 0, one concludes from (D) that

S(M′, ρ^{♯}) = −S(M, ρ) · \frac{χ(M′)}{χ(M)}.

This implies that for any closed manifold with fundamental group Γ, S(M, ρ) = χ(M) · F (ρ; Γ), where F (ρ; Γ) depends only on the representation ρ up to an isomorphism. Notice that if ρ is isomorphic to its dual ρ^{♯}, then by (B) and (C), F (ρ; Γ) · χ(M) = 0.

Step 2: It remains to show that F (ρ; Γ) = 0. Given Γ, choose a closed manifold (M, x_{0}) with dim M = 2n and Γ = π_{1}(M, x_{0}). Let M_{1} := M, M_{2} := M#ICP (n), M_{3} := ICP (n), M_{4} := S^{2n}. Observe that π_{1}(M_{2}, x_{0}) = π_{1}(M_{1}, x_{0}) = Γ and χ(M_{2}) = χ(M_{1}) + (n − 1). Let (E_{i} → M_{i}, ∇_{i}) be the following flat bundles: for i = 1, 2 the ones induced by the representation ρ and for i = 3, 4 the flat bundles induced by ε, the unique representation on W of the group with one element. We show below that one can choose Hermitian structures, Riemannian metrics and generalized triangulations so that, by the locality of the density given in Proposition 0.1 (cf subsection 0.2, (P1),)

(5.3) S(M#ICP (n), ρ) − S(M, ρ) = S(ICP (n), ε) − S(S^{2n}, ε)

On the other hand, by Step 1,

(5.4)S(M#ICP (n), ρ)−S(M, ρ) = F (ρ)·(n−1); S(ICP (n), ε) = 0, S(S^{2n}, ε) = 0

and the claimed result follows. To prove (5.3), decompose M_{i} as a union of compact manifolds with boundary, M_{i} = M_{i}(−) ∪ M_{i}(+) so that

(a) M_{i}(−) ∪ M_{i}(+) = M_{i}; N_{i} := M_{i}(−) ∩ M_{i}(+) ∼ S^{2n−1};

(b1) M_{1}(−) ∼ M_{2}(−) ∼ M \ Int(D^{2n});

(b2) M_{1}(+) ∼ M_{4}(+) ∼ D^{2n};

(b3) M_{2}(+) ∼ M_{3}(+) ∼ ICP (n) \ Int(D^{2n});

(b4) M_{3}(−) ∼ M_{4}(−) ∼ D^{2n}.

Let W_{i}(−) and W_{i}(+) be the bordisms (cf [BFK2]),

W_{i}(−) := (M_{i}(−), ∅, ∂(M_{i}(−))); W_{i}(+) := (M_{i}(+), ∂(M_{i}(+)), ∅).

The isomorphisms in (b1) - (b4) then become isomorphisms of bordisms. Denote by (E_{i}(+), ∇_{i}) respectively (E_{i}(−), ∇_{i}) the restriction of the flat bundles E_{i} to M_{i}(+), respectively to M_{i}(−). Further we have the isomorphisms of flat bundles

(c1) (E_{1}(−), ∇_{1}) ∼ (E_{2}(−), ∇_{2});

(c2) (E_{1}(+), ∇_{1}) ∼ (E_{4}(+), ∇_{4});

(c3) (E_{2}(+), ∇_{2}) ∼ (E_{3}(+), ∇_{3});

(c4) (E_{3}(−), ∇_{3}) ∼ (E_{4}(−), ∇_{4}).

Choose Riemannian metrics g_{i} on M_{i} which are product like near N_{i} = M_{i}(−) ∩ M_{i}(+) and generalized triangulations τ_{i} = (h_{i}, g′_{i}) with g′_{i} product like near N_{i}, 0 a regular value for h_{i}, and h_{i}^{−1}(0) = N_{i}. They induce generalized triangulations on W_{i}(−) and W_{i}(+). Choose Hermitian structures on E_{i}, parallel near the critical points so that (c1)-(c4) are isomorphisms of Hermitian flat bundles. (This is always possible since the space of Hermitian structures which are parallel on a subset is contractible.) Using the locality of the density given in Proposition 0.1 (cf subsection 0.2, (P1)), (5.3) follows.

Remark to Theorem 0.1 Theorem 0.1 answers the question raised by several experts in the field, if the formula of Bismut-Zhang can be extended to L_{2}−torsion. Consider the system (M, E, ∇, μ, g, τ) where (M, g) is a closed Riemannian manifold, (E, ∇) a finite dimensional flat bundle equipped with a Hermitian structure μ and τ a generalized triangulation of M. Suppose that M is connected and let x_{0} ∈ M. Let Γ be the fundamental group of M, Γ = π_{1}(M, x_{0}), N the finite von Neumann algebra associated to Γ, ρ the holonomy representation of Γ on E_{x_{0}} and ρ_{reg} the regular representation of Γ on the N −Hilbert module l_{2}(Γ). Denote by ρ′ the representation of Γ on the N −Hilbert module l_{2}(Γ) ⊗IC E_{x_{0}} given by ρ′ := ρ_{reg} ⊗IC ρ, by M the flat bundle associated with the regular representation ρ_{reg} and by μ_{0} the canonical parallel Hermitian structure on M induced from the scalar product on l_{2}(Γ). The flat bundle induced by the representation ρ′ is M ⊗IC E and is equipped with the Hermitian structure μ′ = μ_{0} ⊗ μ. Under the hypothesis of determinant class for the pair (M, ρ′) (conjecturally always satisfied,) one can use Lott’s approach [L] to define the L_{2}− analytic torsion of (M, E, ∇, μ, g) resp. the L_{2}−Reidemeister torsion of (M, E, ∇, μ, g, τ). It is straightforward to check that these torsions are the analytic torsion of (M, g, ρ′, μ′) resp. the Reidemeister torsion of (M, g, ρ′, μ′, τ) and that the local quantities expressing log R, cf Theorem 0.1, are the same for (M, ρ′, μ′, g, τ) and (M, ρ, μ, g, τ). Therefore, by (2.15), the quotient of the L_{2}−analytic and L_{2}−Reidemeister torsions is the same as the quotient of analytic and Reidemeister torsions .

5.2 An invariant for odd dimensional manifolds.

For Γ a finitely presented group, and denote by Rep(Γ, W) the space of representations of Γ on the A-Hilbert module of finite type W. This space has a structure of a complex analytic space of infinite dimension when A is of infinite dimension.

Let (M, x_{0}) be a base pointed, closed, smooth odd dimensional manifold, with π_{1}(M, x_{0}) = Γ, and let E be an Euler structure in the sense of Turaev cf [Tu]. Recall cf [B2],[Tu] that the set of Euler structures on a base pointed connected manifold (M, x_{0}) can be defined as the set of connected components of the space of continuous vector fields with the only zero at x_{0} equipped with the compact open topology. Recall also that H_{1}(M; ZZ) acts on this set freely and transitively and because χ(M) = 0 there is a canonical identification between sets of Euler structures obtained by using different base points. By using the relative torsion we will associate a (real analytic) function F (M, E) : Rep(Γ, W) → IR which is a smooth invariant for the triple (M, x_{0}, E). We will calculate F (M, E) for M = S^{1} × N, N is a closed simply connected even dimensional manifold and E the canonical Euler structure, defined below. In this case π_{1}(M, x_{0}) = ZZ and the space Rep(ZZ, W) can be identified with Gl_{A}(W). Therefore we will denote a representation ρ by A ∈ Gl_{A}(W), A = ρ(1).

Definition of F(M,E):

First note that an Euler class E can be represented by the following data:

(1) a generalized triangulation τ = (h, g′),

(2) a spray, i.e a collection of disjoint smooth embeddings α_{x} : [0, 1] → M with α_{x}(0) = x_{0}, α_{x}(1) = x, x ∈ Cr(h). Let K := ∪_{x∈Cr(h)}α_{x}([0, 1]) ⊂ M and refer to K as the spray defined by the curves α_{x}.

The data (1) and (2) provide a vector field X by taking X = −grad_{g′}h outside of a contractible smooth regular neighborhood U of K and extending X inside U with the only zero at x_{0}. This vector field determines an Euler structure and any Euler structure can be obtained in this way (cf [B2]). For a representation ρ ∈ Rep(Γ, W) denote by E_{ρ} the flat bundle induced from ρ and choose a Hermitian structure μ in E_{ρ}, so that μ is parallel in the neighborhood of x_{0} and μ(x_{0}) is exactly the scalar product of W ( E_{ρ}(x_{0}) = W).

Define

F (M, E)(ρ) := log R(M, ρ, μ, τ, g).

Using Propositions 4.1-4.3 one verifies that F (M, E)(ρ) is well defined, i.e. is independent of the choice of triangulation, metric and Hermitian structure cf [B2] Main Theorem (1).

Let M = N × S^{1} with N simply connected and of even dimension. Choose the generalized triangulation τ_{S^{1}} of S^{1} = IR/ZZ given by the canonical metric g_{0} and a Morse function with two critical points, a minimum at t_{1} and a maximum at t_{2}, and choose as a base point t_{0} = 1/2. Suppose that 0 < t_{1} < 1/2 < t_{2} < 1. Take K_{S^{1}} = [t_{1}, 1/2] ∪ [1/2, t_{2}]. Choose a generalized triangulation τ_{N} of N, a base point n_{0} and a spray K_{N}. Consider the Euler structure on S^{1} × N represented by τ_{S^{1}} × τ_{N} and spray K = K_{S^{1}} × K_{N} with base point x_{0} = t_{0} × n_{0}. Denote by E_{0} the Euler structure defined by these data. Since N is simply connected this Euler structure is independent of the spray K_{N} and the triangulation τ_{N} and will be refered as the canonical Euler structure. The flat bundle induced by A is actually the tensor product of a flat bundle induced by A on S^{1} for which we choose a Hermitian structure parallel above [t_{1} − ϵ, t_{2} + ϵ], with the trivial line bundle on N, for which we choose a parallel Hermitian structure.

Proposition 5.1 For the canonical Euler structure E_{N} described above F (N × S^{1}, E_{0})(A) = −χ(N)/2 log det(A^{∗}A)^{1/2}.

Proof By the product formula for the relative torsion (use χ(S^{1}) = 0)

log R(N × S^{1}, A, μ, g, τ) = χ(N) log R(S^{1}, A, μ_{S^{1}}, g_{0}, τ_{S^{1}}).

We compute log R(S^{1}, A, ⋯) using Theorem 0.1.

Note that μ_{S^{1}} can be viewed as a smooth family μ(t), t ∈ IR of scalar products in W with μ(t + 1) = A^{−1}^{∗}μ(t)A^{−1} and μ(t) = μ_{0} for t ∈ [t_{1} + k − ϵ, t_{2} + k + ϵ], k ∈ ZZ and μ_{0} the scalar product of W. Recall (subsection 4.1) that θ(μ)(t) = dα(t) where α(t) = log vol(id : (W, μ(t)) → (W, μ_{0})) which in particular says that α(t) = 0 for t ∈ [t_{1} − ϵ, t_{2} + ϵ]. Clearly α(1) − α(0) = log volA^{−1}.

Further, Ψ ∈ Ω^{0}(T S^{1} \ S^{1}) (cf property 4.2^{iv}) is the function given by Ψ(t, ξ) = 1/2 (∀ξ > 0) and Ψ(x, ξ) = −1/2 (∀ξ < 0).

The vectorfield X : S^{1} \ Cr(h) → T S^{1} \ S^{1} is given by −grad_{g′}h and thus the pullback of Ψ by X is

X^{∗}(Ψ)(t) = 1/2 (∀t ∈ [0, 1] \ [t_{1}, t_{2}])

and

X^{∗}(Ψ)(t) = −1/2 (∀t ∈ (t_{1}, t_{2}))

Note that X^{∗}(Ψ)(t) = −1/2 on the points where θ(μ) is zero. Combining the above computations, one obtains

∫_{S^{1}\Cr(h)} θ(μ)X^{∗}(Ψ) = 1/2 ∫_{S^{1}\Cr(h)} θ(μ) = \frac{1}{2} log vol(A^{−1}).

By the formula given in Theorem 0.1, the claimed statement follows.

Appendix A Lemma of Carey-Mathai-Mishchenko

In this Appendix, we prove Lemma 1.14, using a deformation argument.

Introduce f_{q}(t) := tf_{q} and d_{q}(t) := ( \begin{matrix}d_{1,q} & tf_{q} \\ 0 & d_{2,q} \end{matrix} ) and obtain in this way a

complex (C^{∗}, d_{∗}(t)).

By the same arguments as in the proof of Lemma 1.10 one concludes that (C^{∗}, d_{∗}(t)) is an algebraically acyclic, ζ-regular complex of sF type.

Therefore, (C^{∗}, d_{x}(t)) has a well defined torsion T (t) := T (C^{∗}, d_{∗}(t)), given by (cf (1.21))

log T (t) = \frac{1}{2} ∑(−1)^{q} log det(\underline{d}_{q}^{∗}(t)\underline{d}_{q}(t)).

If \frac{d}{dt} log T (t) = 0, then

log T (C) = log T (1) = log T (0) = log T (C^{1}) + log T (C^{2})

where for the last equality we have used that d_{q}(0) = ( \begin{matrix}d_{1,q} & 0 \\ 0 & d_{2,q} \end{matrix} ) and there-

fore Δ_{q}(0) = ( \begin{matrix}Δ_{1,q} & 0 \\ 0 & Δ_{2,q} \end{matrix} ) .

The remaining part of this Appendix is devoted to the proof of the statement

\frac{d}{dt} log T (t) = 0.

Introduce the Hodge decomposition of C^{j}, C_{q}^{j} = C_{q}^{j+} ⊕ C_{q}^{j−}(j = 1, 2). The differential d_{j,q} then have the form

d_{j,q} = ( \begin{matrix}0 & \underline{d}_{j,q} \\ 0 & 0 \end{matrix} ) .

Further, decompose C_{q}, (A.1) C_{q} = C_{q}^{1+} ⊕ C_{q}^{1−} ⊕ C_{q}^{2+} ⊕ C_{q}^{2−}. Then, d_{q}(t) can be written as

d_{q}(t) = ( \begin{matrix}0 & \underline{d}_{1,q} & α_{q} & β_{q} \\ 0 & 0 & ε_{q} & γ_{q} \\ 0 & 0 & 0 & \underline{d}_{2,q} \\ 0 & 0 & 0 & 0 \end{matrix} )

where α_{q} ≡ α_{q}(t) = tα̇_{q}, β_{q} = tβ̇_{q}, ε_{q} = tε̇_{q} and γ_{q} = tγ̇_{q}. From d_{q+1}(t)d_{q}(t) = 0 we deduce (A.2) \underline{d}_{1,q+1}ε_{q} = 0; ε_{q+1}\underline{d}_{2,q} = 0; (A.3) \underline{d}_{1,q+1}γ_{q} + α_{q+1}\underline{d}_{2,q} = 0. As C^{1} and C^{2} are acyclic, \underline{d}_{1,q} and \underline{d}_{2,q} are isomorphisms, and (A.2) and (A.3) imply (A.4) ε_{q} = 0 (A.5) γ_{q} = −\underline{d}_{1,q+1}^{−1}α_{q+1}\underline{d}_{2,q}. Next, let us describe the Hodge decomposition of C_{∗} =: C_{∗}^{+}(t) ⊕ C_{∗}^{−}(t) of (C_{∗}, d_{∗}(t)). Lemma A.1 (i) C_{q}^{+} = Kerd_{q} = \{(x_{+} − \underline{d}_{1,q}^{−1}α_{q}y_{+}, y_{+}, 0)|x_{+} ∈ C_{q}^{1+}; y_{+} ∈ C_{q}^{2+}\}

(ii) C_{q}^{−} = Ker(d_{q−1}^{∗}) = \{(0, x_{−}, (\underline{d}_{1,q}^{−1}α_{q})^{∗}x_{−}, y_{−})|x_{−} ∈ C_{q}^{1−}; y_{−} ∈ C_{q}^{2−}\}.

Proof The statements follow from a straightforward verification.

We want to compute the t-derivative of log T (t) = \frac{1}{2} ∑(−1)^{q} log det(\underline{d}_{q}^{∗}(t)\underline{d}_{q}(t)). Notice that

\frac{d}{dt}(d_{q}^{∗}(t)d_{q}(t)) = ( \begin{matrix}0 & 0 \\ f_{q}^{∗} & 0 \end{matrix} ) ( \begin{matrix}d_{1,q} & tf_{q} \\ 0 & d_{2,q} \end{matrix} ) + ( \begin{matrix}d_{1,q}^{∗} & 0 \\ tf_{q}^{∗} & d_{2,q}^{∗} \end{matrix} ) ( \begin{matrix}0 & f_{q} \\ 0 & 0 \end{matrix} )

= ( \begin{matrix}0 & 0 \\ f_{q}^{∗}d_{1,q} & tf_{q}^{∗}f_{q} \end{matrix} ) + ( \begin{matrix}0 & d_{1,q}^{∗}f_{q} \\ 0 & tf_{q}^{∗}f_{q} \end{matrix} )

= ( \begin{matrix}0 & d_{1,q}^{∗}f_{q} \\ f^{∗}d_{1,q} & 2tf_{q}^{∗}f_{q} \end{matrix} ) .

As f is a morphism of trace class (cf Definition 1.10) we conclude that \frac{d}{dt}d_{q}^{∗}d_{q}(t) and therefore \frac{d}{dt} \underline{d}_{q}^{∗}(t)\underline{d}_{q}(t) are of trace class. In view of Proposition 1.3 and the fact that (\underline{d}_{q}^{∗}(t)\underline{d}_{q}(t))^{−1} is bounded we deduce that \frac{d}{dt}(\underline{d}_{q}^{∗}(t)\underline{d}_{q}(t))(\underline{d}_{q}^{∗}(t)\underline{d}_{q}(t))^{−1} is of trace class. Therefore

\frac{d}{dt} log det(\underline{d}_{q}^{∗}(t)\underline{d}_{q}(t)) = T r(\frac{d}{dt}(\underline{d}_{q}^{∗}(t)\underline{d}_{q}(t))(\underline{d}_{q}^{∗}(t)\underline{d}_{q}(t))^{−1}).

(A.6) \begin{matrix}T r(\frac{d}{dt}(\underline{d}_{q}^{∗}(t)\underline{d}_{q}(t))(\underline{d}_{q}^{∗}(t)\underline{d}_{q}(t))^{−1}) = \\ = T r((\frac{d}{dt} \underline{d}_{q}^{∗}(t))\underline{d}_{q}^{∗}(t)^{−1}) + T r(\underline{d}_{q}(t)^{−1} \frac{d}{dt} \underline{d}_{q}(t)) \\ = 2ReT r((\frac{d}{dt} \underline{d}_{q}(t))\underline{d}_{q}(t)^{−1}). \end{matrix}

It is convenient to introduce D_{q} := d_{q}(t) and denote by D_{q}^{−1} the operator given by D_{q}^{−1}|_{C_{q+1}^{·}} = 0 and D_{q}^{−1}|_{C_{q+1}^{+}} = \underline{d}_{q}(t)^{−1}.

(A.7) T r(D_{q}^{−1} \frac{d}{dt}D_{q}) = T r(\underline{d}_{q}(t)^{−1} \frac{d}{dt} \underline{d}_{q}(t)).

With respect to the decomposition (A.1), D_{q}^{−1} and Ḋ_{q} ≡ \frac{d}{dt}D_{q} (Ḋ_{q} is independent of t) take the form

D_{q}^{−1} = ( \begin{matrix}A_{11} & A_{12} & A_{13} & A_{14} \\ A_{21} & A_{22} & A_{23} & A_{24} \\ A_{31} & A_{32} & A_{33} & A_{34} \\ A_{41} & A_{42} & A_{43} & A_{44} \end{matrix} ); Ḋ_{q} = ( \begin{matrix}0 & 0 & α̇_{q} & β̇_{q} \\ 0 & 0 & 0 & γ̇_{q} \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix} ) .

Ḋ_{q}D_{q}^{−1} = ( \begin{matrix}α̇_{q}A_{31} + β̇_{q}A_{41} & ∗ & ∗ & ∗ \\ ∗ & γ̇_{q}A_{42} & ∗ & ∗ \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix} )

and (A.8) T r(Ḋ_{q}D_{q}^{−1}) = T r(α̇_{q}A_{31} + β̇_{q}A_{41}) + T r(γ̇_{q}A_{42}). Lemma A.2 (i) A_{41} = 0; (ii) (−A_{42} \underline{d}_{1,q+1}^{−1}α_{q+1} + A_{43}) = \underline{d}_{2,q}^{−1}; (iii) A_{42} = −A_{43}(\underline{d}_{1,q+1}^{−1}α_{q+1})^{∗}; (iv) \underline{d}_{2,q}^{−1} = A_{43}((\underline{d}_{1,q+1}^{−1}α_{q+1})^{∗}(\underline{d}_{1,q+1}^{−1}α_{q+1}) + Id);

(v) A_{42} = −\underline{d}_{2,q}^{−1}(Id + (\underline{d}_{1,q+1}^{−1}α_{q+1})^{∗}(\underline{d}_{1,q+1}^{−1}α_{q+1}))^{−1}(\underline{d}_{1,q+1}^{−1}α_{q+1})^{∗}; (vi) γ̇_{q}A_{42} = \underline{d}_{1,q+1}^{−1}α̇_{q+1}(Id + (\underline{d}_{1,q+1}^{−1}α_{q+1})^{∗}(\underline{d}_{1,q+1}^{−1}α_{q+1}))^{−1}(\underline{d}_{1,q+1}^{−1}α_{q+1})^{∗}. Proof (i) Take x = (x_{+}, 0, 0, 0) ∈ C_{q}. By Lemma A.1, x ∈ C_{q+1}^{+}. Then

x = D_{q}D_{q}^{−1}X = ( \begin{matrix}0 & \underline{d}_{1,q} & α_{q} & β_{q} \\ 0 & 0 & 0 & γ_{q} \\ 0 & 0 & 0 & \underline{d}_{2,q} \\ 0 & 0 & 0 & 0 \end{matrix} ) ( \begin{matrix}A_{11} & x_{+} \\ A_{21} & x_{+} \\ A_{31} & x_{+} \\ A_{41} & x_{+} \end{matrix} ) = ( \begin{matrix}∗ \\ ∗ \\ \underline{d}_{2,q} & A_{41} & x_{+} \\ 0 \end{matrix} )

and therefore 0 = \underline{d}_{2,q}A_{41}x_{+}. As \underline{d}_{2,q} is an isomorphism we conclude that A_{41}x_{+} = 0. (ii) Take x(0, 0, 0, y_{−}) ∈ C_{q}. By Lemma A.1, x ∈ C_{q}^{−}. Then

\begin{matrix}x = D_{q}^{−1}D_{q}x = D_{q}^{−1}(β_{q}y_{−}, γ_{−}, \underline{d}_{2,q}y_{−}, 0) = \\ = (∗, ∗, ∗, A_{41}β_{q}y_{−} + A_{42}γ_{q}y_{−} + A_{43}\underline{d}_{2,q}y_{−}). \end{matrix}

In view of (i), A_{41} = 0 and thus (A.9) (A_{42} γ_{q} + A_{43} \underline{d}_{2,q}) y_{−} = y_{−}. Recall that γ_{q} = −\underline{d}_{1,q+1}^{−1}α_{q+1}\underline{d}_{2,q} and substitute into (A.9) to obtain (ii).

(iii) As D_{q}^{−1}|_{C_{q+1}^{−}} = 0 and in view of Lemma A.1,

\begin{matrix}0 = D_{q}^{−1}(0, x_{−}, (\underline{d}_{1,q+1}^{−1}α_{q+1})^{∗}x_{−}, 0) = \\ = (∗, ∗, ∗, A_{42} x_{−} + A_{43} (\underline{d}_{1,q+1}^{−1}α_{q+1})^{∗}x_{−}). \end{matrix}

Thus A_{42} + A_{43}(\underline{d}_{1,q+1}^{−1}α_{q+1})^{∗} = 0 which proves (iii) (iv) follows from substituting (iii) into (ii).

(v) Notice that Id + (\underline{d}_{1,q+1}^{−1}α_{q+1})^{∗}(\underline{d}_{1,q+1}^{−1}α_{q+1}) ≥ Id is invertible and therefore, we can solve (iv) for A_{43}. Substituted into (iii) we obtain (v). (vi) Substitute γ̇_{q} = −\underline{d}_{1,q+1}^{−1}α̇_{q+1}\underline{ḋ}_{2,q} into (v) to get (vi). Lemma A.3

(i) α_{q}A_{31} + \underline{d}_{1,q}A_{21} = Id (on C_{1,q}^{+}), (ii) A_{31} = (\underline{d}_{1,q}^{−1}α_{q})^{∗}A_{21}; (iii) A_{21} = (Id + (\underline{d}_{1,q}^{−1}α_{q})(\underline{d}_{1,q}^{−1}α_{q})^{∗})^{−1}\underline{d}_{1,q}^{−1}; (iv) α̇_{q}A_{31} = α̇_{q}(\underline{d}_{1,q}^{−1}α_{q})^{∗}(Id + (\underline{d}_{1,q}^{−1}α_{q})(\underline{d}_{1,q}^{−1}α_{q})^{∗})^{−1}\underline{d}_{1,q}^{−1}. Proof The proof of Lemma A.3 is similar to the one of Lemma A.2. Lemma A.4

(i) ∑_{q}(−1)^{q}T r(Ḋ_{q}D_{q}^{−1}) = 0; (ii) \frac{d}{dt} log det T (t) = 0. Proof (i) Substituting Lemma A.2 (i) and (vi) as well as Lemma A.3 (iv) into (A.8) leads to

T r(Ḋ_{q}D_{q}^{−1})=T r(α̇_{q}A_{31}) + 0 + T r(γ̇_{q}A_{42})
=T r(α̇_{q}(\underline{d}_{1,q}^{−1}α_{q})^{∗}(Id + (\underline{d}_{1,q}^{−1}α_{q})(\underline{d}_{1,q}^{−1}α_{q})^{∗})^{−1}\underline{d}_{1,q}^{−1})
+T r(\underline{d}_{1,q+1}^{−1}α̇_{q+1}(Id + (\underline{d}_{1,q+1}^{−1}α_{q+1})^{∗}(\underline{d}_{1,q+1}^{−1}α_{q+1}))^{−1}(\underline{d}_{1,q+1}^{−1}α_{q+1})^{∗})
=T r(α̇_{q}(\underline{d}_{1,q}^{−1}α_{q})^{∗}(Id + (\underline{d}_{1,q}^{−1}α_{q})(\underline{d}_{1,q}^{−1}α_{q})^{∗})^{−1}\underline{d}_{1,q}^{−1}
+T r(α̇_{q+1}(\underline{d}_{1,q}^{−1}α_{q})^{∗}(Id + (\underline{d}_{1,q+1}^{−1}α_{q+1})^{∗}(\underline{d}_{1,q+1}^{−1}α_{q+1}))^{−1}\underline{d}_{1,q+1}^{−1})

where for the last equality we have used the fact that T r(AB) = T r(BA) and (Id + AA^{∗})^{−1}A^{∗} = A^{∗}(Id + AA^{∗})^{−1}. Thus with η_{q} := \underline{d}_{1,q}^{−1}α_{q}

∑_{q}(−1)^{q}T r(Ḋ_{q}D_{q}^{−1}) = ∑(−1)^{q}T r(α̇_{q}η_{q}^{∗}(Id + η_{q}η_{q}^{∗})^{−1}\underline{d}_{1,q}^{−1})
− ∑(−1)^{q+1}T r(α̇_{q+1}η_{q+1}^{∗}(Id + η_{q+1}η_{q+1}^{∗})\underline{d}_{1,q+1}^{−1})
= 0.

(ii) follows from (i) and (A.6).

Appendix B Determinant class property

In this appendix, we discuss the concept of determinant class and provide examples of pairs (M, ρ) which are not of determinant class.

Given an A-Hilbert module W and φ ∈ L_{A}(W) an operator which satisfies Op1 − Op6, one says that φ is of determinant class (cf [BFKM]) iff (B.1) ∫_{0+}^{1} log λdF_{|φ|}(λ) > −∞ where |φ| = (φ^{∗} · φ)^{1/2}.

It is not difficult to provide morphisms φ which are not of determinant class. For example for A = N (ZZ) and W = l_{2}(ZZ) = L^{2}(S^{1}; IC), S^{1} = IR/ZZ, the multiplication by a function α ∈ L^{∞}(S^{1}; IC) defines an element φ = M_{α} ∈ L_{A}(W) which satisfies Op1 − Op6. Take α : [0, 1) → IC defined by α(x) = exp(−1/x^{2}). As F_{|M_{α}|}(λ) = μ({x ∈ [0, 1], |α(x)| ≤ λ}), where μ(X) denotes the Lebesgue measure of the set X, one can see that F_{|M_{α}|}(λ) = −(log λ)^{1/2} for 0 < λ < 1/e and the integral (B.1) is not convergent.

Consider (K, τ, ρ, μ) where (K, τ) is a CW complex with finitely many cells in each dimension, ρ is a representation of the group π = π_{1}(K) on the A-Hilbert module of finite type W, and μ is a Hermitian structure in the flat bundle E → K induced by ρ.

We say that (K, τ, ρ, μ) is of c-determinant class (cf [BFKM]) iff the associated cochain complex of A-Hilbert modules of finite type C^{∗}(K, τ, ρ, μ) is of determinant class, i.e. δ_{i}, or equivalently the combinatorial Laplacians Δ_{i}^{comb}, are of determinant class for all i.

We also say that (M, g, ρ, μ), with (M, g) a smooth closed Riemannian manifold and (ρ, μ) as above, is of a- determinant class if the deRham complex Ω^{∗}(M, ρ) of A- Hilbert modules whose Hilbert module structure is given by the scalar products induced by g and μ, is of determinant class, i.e. d_{i}, or equivalently the Laplacians Δ_{i}, are of determinant class.

It was shown in [BFKM] (actually only for μ parallel, but the same arguments remain valid in the generality presented here) that the c-determinant class property is independent of τ and μ, the a-determinant class property is independent of g and μ and both properties are homotopy invariant. Moreover for a compact manifold, possibly with boundary, the a-determinant class property holds iff the c-determinant class property holds. Therefore the determinant class property is a homotopy invariant for a pair (K, ρ) with K a compact space of the homotopy type of a CW complex, which can be defined both analytically and combinatorially.

If ρ is a representation of Γ, induced by a homomorphism π : Γ → G, G a countable discrete group, on the N (G)- Hilbert module W = l_{2}(Γ), then (K, ρ) is of determinant class when G is residually finite or G is amenable. Recently B.Clair [C] has verified the determinant class property when G is a residually amenable group.

A representation ρ: π_{1}(S^{1}) = ZZ → L_{N (ZZ)}(L^{2}(S^{1}; IC)) is determined by ρ(1). Assume that ρ(1) is the operator M_{1+α} given by multiplication by 1 + α where α ∈ L^{∞}(S^{1}; IC). Consider the cochain complex induced by ρ and the generalized triangulation τ = (h, g) with h(t) = \frac{cos(2πt)+1}{2} and g the standard metric. As the triangulation τ has one 0-cell E_{0} and one 1-cell E_{1}, the cochain complex is of the form

0 → W →^{δ} W → 0

where δ(E_{0}) = E_{1} − M_{1+α}E_{1} = −M_{α}E_{1}. Therefore,

log T_{comb} = \frac{1}{2} log det δ^{∗}δ = \frac{1}{2} log det α^{∗}α.

Observe that (M, ρ) is of determinant class iff M_{α} is of determinant class, thus for α(x) = exp(−1/x^{2}), as introduced above, (S^{1}, ρ) is NOT of determinant class. We point out that the regular representation of ZZ corresponds to the function α(t) = exp(2πit) − 1.

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