TORSIONS FOR MANIFOLDS WITH

BOUNDARY AND GLUEING FORMULAS

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

Abstract. We extend the definition of analytic and Reidemeister torsion for closed compact Riemannian manifolds, to compact Riemannian manifolds with boundary (M, ∂M), given a parallel flat bundle F of A-Hilbert modules of finite type and a decomposition of the boundary ∂M = ∂_{−}M ∪ ∂_{+}M into disjoint components. When F is induced from the universal covering of M, and the fundamental group of M is infinite (cf. [BFKM]), these torsions are known as the L_{2}− analytic resp. L_{2}− Reidemeister torsions. If the system (M, ∂_{−}M, ∂_{+}M, F) is of determinant class we compute the quotient of the analytic and the Reidemeister torsion and prove gluing formulas for both of them. In particular we answer positively Conjecture 7.6 in [LL]. If F is induced from a Γ−principal covering where Γ is a residually finite group, we derive from the work of Lück (cf.[Lü3]), that the system (M, ∂_{−}M, ∂_{+}M, F) is of determinant class (cf. Theorem A in Appendix).

Contents

0. Introduction.

1. Linear homological algebra in the von Neumann sense.

2. Torsions for compact manifolds with boundary.

2.1. Reidemeister and analytic torsion.

2.2. Determinant class.

2.3. Witten deformation of the de Rham complex for manifolds

with boundary. 2.4. Asymptotic expansions and a comparison theorem.

3. Applications to torsion.

3.1. Comparison of analytic and Reidemeister torsion.

3.2. Glueing formulas (PartI).

3.3. Glueing formulas (PartII).

3.4. Comparison of the analytic torsion for different boundary conditions.

4. Appendix: On manifolds of determinant class.

All three authors were supported by NSF. The first two authors wish to thank the Erwin-Schrödinger-Institute in Vienna for hospitality and support during the summer of 1995 when part of this work was done.

Typeset by AMS-TEX

0. Introduction.

The purpose of this paper is to extend the study of the analytic and the Reidemeister torsion to compact Riemannian manifolds with boundary. The torsions we will consider will be associated to compact manifolds equipped with a parallel flat bundle of A- Hilbert modules of finite type, F, where A is a finite von Neumann algebra. This torsions involve operators whose spectrum, in general, is not exclusively pure point, but might contain a continuous part as well. In particular, the analytic and Reidemeister L_{2}−torsions are of this type. Among other results we provide a formula for the quotient of the analytic torsion and the Reidemeister torsion. In particular we prove Conjecture 7.6 in [LL]. We also establish glueing formulas for both torsions. These formulas are basic tools for the calculation of these torsions.

In order to formulate our results more precisely, we introduce the following notations. Let M be a compact smooth manifold, not necessarily connected, of dimension d, and with boundary ∂M. Let g be a Riemannian metric on M. Throughout the paper, we will always assume that g has a product structure near the boundary (product-like), i.e. that there exist ϵ > 0 and a diffeomorphism Θ_{∂M} : ∂M × [0, ϵ) → M so that

Θ_{∂M} ↾ _{∂M×{0}} = id (P 1)

Θ_{∂M}^{∗}(g) = g_{∂M} + dt^{2} (P 2)

where g_{∂M} denotes the restriction of g to ∂M and dt^{2} denotes the Euclidean metric on the half open interval [0, ϵ). Suppose ∂M is a union of two disjoint components, not necessarily connected and possibly empty. Let us specify an ordering for them, ∂_{−}M and ∂_{+}M. Then (M, ∂_{−}M, ∂_{+}M) will be referred to as a bordism. A closed manifold M can be regarded as a bordism with ∂_{−}M = ∂_{+}M = ∅. If it is not ambiguous, we will write M instead of (M, ∂_{−}M, ∂_{+}M) and denote by −M the bordism obtained from M by interchanging ∂_{−}M and ∂_{+}M. Next, we introduce the notion of generalized triangulation for a bordism (M, ∂_{−}M, ∂_{+}M).

Definition. A pair τ = (h, g′) consisting of a C^{∞}-function h : M → ℝ with range [a, b] ⊆ ℝ, where a and b are elements in ℤ, and a Riemannian metric g′ is said to be a generalized triangulation for M if the following properties hold:

h(Θ_{∂M}(x, t)) = b − t (x ∈ ∂_{+}M, 0 ≤ t < ϵ) (T1)

h(Θ_{∂M}(x, t)) = a + t (x ∈ ∂_{−}M, 0 ≤ t < ϵ)

where Θ_{∂M} : ∂M × [0, ϵ) → M denotes the exponential map at ∂M, associated to g′ (cf. above).

h(Cr(h)) ⊆ ℤ, (T2)

where Cr(h) denotes the set of critical points of h. (Notice that we do not require that h is self-indexing. This will be convenient for the glueing construction.)

(T3) All critical points of h are nondegenerate and, given any critical point y of index k, y ∈ Cr_{k}(h), there exists a diffeomorphism ϕ_{y} : D_{ϵ_{y}} → M, with D_{ϵ_{y}} being the disc in ℝ^{d} of radius ϵ_{y} > 0, centered at 0, so that ϕ_{y}(0) = y, ϕ_{y}^{#}(g′) is the Euclidean metric on D_{ϵ_{y}} and

h(ϕ_{y}(x_{1}, …, x_{d})) = h(y) − \frac{1}{2} ∑_{j=1}^{k} x_{j}^{2} + \frac{1}{2} ∑_{j=k+1}^{d} x_{j}^{2}.

If for each critical point of h, such a coordinate system exists, the metric g′ is called compatible with h. (T4) The gradient of h with respect to g′, grad_{g′}h, satisfies the Morse-Smale condition (cf. [BFKM]).

A smooth simplicial triangulation t of M with the property that ∂_{±}M become subcomplexes induces generalized triangulations on the bordisms (M, ∂_{−}M, ∂_{+}M), (M, ∂_{+}M, ∂_{−}M), (M, ∅, ∂M), (M, ∂M, ∅) and on the closed manifolds ∂_{±}M.

Assume that the following data is given: (M, ∂_{−}M, ∂_{+}M) a bordism; τ = (h, g′) a generalized triangulation; A a finite von Neumann algebra; F = (E, ∇) a parallel flat bundle where E →^{p} M is a bundle of A-Hilbert modules of finite type and ∇ is a flat connection making the A -module structure and the inner products (·, ·) of E_{y} = p^{−1}(y) parallel with respect to ∇.^{1}

If τ = (h, g′) is a generalized triangulation of (M, ∂_{−}M, ∂_{+}M) then τ_{D} = (−h, g′) is a generalized triangulation of (M, ∂_{+}M, ∂_{−}M). It will be refered to as the dual triangulation.

In section 2 we define, similar as in [BFKM], analytic Laplacians Δ_{q}, acting on q-forms with coefficients in E with relative (or Dirichlet) boundary conditions on ∂_{−}M and absolute (or Neumann) boundary conditions on ∂_{+}M, associated to (M, ∂_{−}M, ∂_{+}M), g and E. Further, we define combinatorial Laplacians Δ_{q}^{comb}, associated to (M, ∂_{−}M), τ and F. We say that the triple {(M, ∂_{−}M, ∂_{+}M), g, F} is of a- determinant class if all the analytic Laplacians Δ_{q} are of determinant class, i.e. log det′_{N}Δ_{q} ∈ ℝ. For {(M, ∂_{−}M, ∂_{+}M), g, F} of a-determinant class the analytic torsion T_{an} = T_{an}(M, ∂_{−}M, g, F) > 0 is defined by

log T_{an} := \frac{1}{2} ∑_{q=0}^{d}(−1)^{q+1}q log det′_{N}Δ_{q}.

Similarly, we say that the triple {(M, ∂_{−}M, ∂_{+}M), τ, F} is of c-determinant class if all the combinatorial Laplacians Δ_{q}^{comb} are of determinant class, i.e. log det′_{N}Δ_{q}^{comb} ∈ ℝ. For {(M, ∂_{−}M, ∂_{+}M), τ, F} of c-determinant class the combinatorial torsion T_{comb} = T_{comb}(M, ∂_{−}M, τ, F) > 0 is defined by

log T_{comb} := \frac{1}{2} ∑_{q=0}^{d}(−1)^{q+1}q log det′_{N}Δ_{q}^{comb}.

1Given a choice of base points (x_{j}) in each of the connected components M_{j} of M(1 ≤ j ≤ k_{M})

F determines and is determined uniquely by the (A, Γ_{j}^{op})−Hilbert modules W_{j} := E_{x_{j}} where the action of Γ_{j} := π_{1}(M_{j}, x_{j}) on W_{j} is given by the parallel transport and where Γ_{j}^{op} denotes the group Γ_{j} with opposite multiplication (cf. [BFKM]).

The Reidemeister torsion T_{Re} = T_{Re}(M, ∂_{−}M, g, τ, F) is then defined by

log T_{Re} = log T_{comb} + log T_{met}

where T_{met} is the metric part of the torsion, defined as in [BFKM] (cf. section 2). T_{met} is equal to 1 if (M, ∂_{−}M) is F- acyclic. It turns out that T_{Re} does not depend on the triangulation τ. The proof of this fact is not difficult and is the same as in the case of finite dimensional unitary representations. This result is also contained in Theorem 3.1. Nevertheless, we will continue to write T_{Re}(M, ∂_{−}M, g, τ, F) in order to emphasize that the Reidemeister torsion is calculated with the triangulation τ.

One can show that the notions of a-determinant class and c-determinant class are equivalent (cf. [BFKM]) which allows us to introduce the notion of a pair {(M, ∂_{−}M, ∂_{+}M), F} being of determinant class. Given a bordism (M, ∂_{−}M, ∂_{+}M) we say that the system (M, ∂_{−}M, ∂_{+}M, F) is of determinant class if the three pairs {(M, ∂_{−}M, ∂_{+}M), F}, {(M, ∅, ∂M), F} and {∂M, F ↾_{∂M}} are of determinant class. As we will see in Proposition 2.5 (4) this is equivalent with {(M, ∅, ∂M), F} and {∂M, F ↾_{∂M}} being of determinant class. If F is a parallel flat bundle over a compact manifold M induced from a Γ−principal covering where Γ is a residually finite group, one can derive from the work of Lück (cf.[Lü3]), that for any bordism (M, ∂_{−}M, ∂_{+}M) the system (M, ∂_{−}M, ∂_{+}M, F) is of determinant class (cf. Theorem A in Appendix). In particular, one concludes that if M is connected and has residually finite fundamental group Γ := π_{1}(M), the system (M, ∂_{−}M, ∂_{+}M, F) is of determinant class if F is the Hilbert bundle induced by the (N (Γ), Γ^{op})−Hilbert module ℓ_{2}(Γ) (cf Appendix).

Given a pair {(M, ∂_{−}M, ∂_{+}M), F} of determinant class, denote by R = R(M, ∂_{−}M, g, τ, F) the quotient of analytic and Reidemeister torsion,

log R = log T_{an}(M, ∂_{−}M, g, F) − log T_{Re}(M, ∂_{−}M, g, τ, F).

In a forthcoming paper [BFK2], we will show that the quantity R is well defined without the hypothesis of the pair being of determinant class, a fact which was proved first in [CMM] in the acyclic case^{2}, and that Theorems 2.10 and 3.1 below hold without this hypothesis.

Extending our earlier results [BFK], [BFKM] and using similar techniques, we prove that log R depends only on the data on the boundary:

Theorem 2.10. Assume that for j = 1, 2, (M_{j}, ∂_{−}M_{j}, ∂_{+}M_{j}) is a bordism equipped with a Riemannian metric g_{j}, a generalized triangulation τ_{j} and that F_{j} = (E_{j}, ∇_{j}) is a parallel flat bundle of A-Hilbert modules of finite type on M_{j}. Assume also that the systems (M_{j}, ∂_{−}M_{j}, ∂_{+}M_{j}, F_{j}) are of determinant class and

(∂_{±}M_{1}, g_{1} ↾_{∂_{±}M_{1}}, F_{1} ↾_{∂_{±}M_{1}}) = (∂_{±}M_{2}, g_{2} ↾_{∂_{±}M_{2}}, F_{2} ↾_{∂_{±}M_{2}}).

Then

R(M_{1}, ∂_{−}M_{1}, g_{1}, τ_{1}, F_{1}) = R(M_{2}, ∂_{−}M_{2}, g_{2}, τ_{2}, F_{2}).

This theorem allows us to extend the comparison theorem (cf. [BFKM] and [BFK]) of analytic and Reidemeister torsions from closed manifolds to bordisms.

2The first author thanks A.Mischenko for providing him with a preliminary version of [CMM].

Theorem 3.1. Assume that the system (M, ∂_{−}M, ∂_{+}M, F) is of determinant class. Then log R is given by

log R(M, ∂_{−}M, g, τ, F) = \frac{1}{4}χ(∂M; F) log 2

where χ(∂M; F) is the Euler characteristic of ∂M with coefficients in F and is equal to χ(∂M; F) = ∑_{k} χ(∂_{k}M) · dim_{N}E ↾_{∂_{k}M} where ∂_{k}M are the connected components of the boundary ∂M, χ(∂_{k}M) is the standard Euler characteristic of ∂_{k}Mand dim_{N}E ↾_{∂_{k}M} is the von Neumann dimension of the fiber of E above ∂_{k}M.

Remark. In the case A is ℝ or ℂ, Theorem 3.1 is due to W. Lück [Lü1] and, independently, Vishik [Vi] (cf. also [Ch]).

The above result shows that R does not depend on the partition ∂M_{−}, ∂M_{+} of the boundary ∂M. However we continue to use the notation R(M, ∂_{−}M, g, τ, F) as the independence of ∂_{−}M will be proven only towards the end of the paper.

Next we present a gluing formula for the analytic torsion. For j = 1, 2, let M_{j} = (M_{j}, ∂_{−}M_{j}, ∂_{+}M_{j}), g_{j}, τ_{j} = (h_{j}, g′_{j}) and F_{j} = (E_{j}, ▽_{j}) be as above and suppose that there exist an isometry

ω : (∂_{+}M_{1}, g_{1} ↾_{∂_{+}M_{1}}) → (∂_{−}M_{2}, g_{2} ↾_{∂_{−}M_{2}})

and a connection preserving bundle isometry Φ above ω which makes the following diagram commutative

E_{1} ↾_{∂_{+}M_{1}}  ⟶^{Φ} E_{2} ↾_{∂_{−}M_{2}}
↓  ↓
∂_{+}M_{1}  ⟶^{ω} ∂_{−}M_{2}.

Then one can form the bordism (M = M_{1} ∪_{ω} M_{2}, ∂_{−}M_{1}, ∂_{+}M_{2}) by glueing ∂_{+}M_{1} to ∂_{−}M_{2} by ω and the parallel flat bundle F by glueing F_{1} and F_{2} by (ω, Φ). The metrics g_{1}, g′_{1} and g_{2}, g′_{2} determine Riemannian metrics g and g′ on M, and the functions h_{1} and h_{2} determine the C^{∞}-function h : M → ℝ given by

h(x) := \{ \begin{matrix}h_{1}(x) & (x ∈ M_{1}) \\ b_{1} − a_{2} + h_{2}(x) & (x ∈ M_{2}) \end{matrix}

where, for j = 1, 2, h_{j}(M_{j}) = [a_{j}, b_{j}].

Definition. Generalized triangulations τ_{1} and τ_{2} as above are said to be compatible if τ = (h, g′) is a generalized triangulation of (M, ∂_{−}M_{1}, ∂_{+}M_{2}).

Notice that by an arbitrary small perturbation, localized in a given neighborhood of ∂_{+}M_{1} one can modify the metric g′_{1} to g^{∼}′_{1} so that the triangulation τ^{∼}_{1} := (h_{1}, g^{∼}′_{1}) is compatible with τ_{2}.

Finally the manifolds M, M_{1}, M_{2} and the metric g induce a cohomology sequence H_{an}(g), which is a long weakly exact sequence of A−Hilbert modules, and hence a cochain complex. Different metrics induce isomorphic, but not necessarily isometric complexes. If H_{an}(g) is of determinant class we denote by T (H_{an}(g)) its torsion.

Similarly, the manifolds M, M_{1}, M_{2} and the generalized triangulation τ induce a cohomology sequence H_{comb}(τ), which is a long weakly exact sequence of A−Hilbert modules and thus, again, a cochain complex. Integration theory (cf section 2) provides an isomorphism (which typically is not an isometry) between the two sequences H_{an}(g) and H_{comb}(τ). Both sequences are of the following form:

0 → H^{0}(M_{2}, ∂_{−}M_{2}, F_{2}) → H^{0}(M, ∂_{−}M, F) → H^{0}(M_{1}, ∂_{−}M_{1}, F_{1}) →

→ H^{1}(M_{2}, ∂_{−}M_{2}, F_{2}) → ⋯ → H^{k}(M_{2}, ∂_{−}M_{2}, F_{2}) → H^{k}(M, ∂_{−}M, F) →

→ H^{k}(M_{1}, ∂_{−}M_{1}, F_{1}) → H^{k+1}(M_{2}, ∂_{−}M_{2}, F_{2}) → …

Theorem 3.2. Assume that for i = 1, 2 the system (M_{i}, ∂_{−}M_{i}, ∂_{+}M_{i}, F_{i}) is of determinant class. Then the following statements hold:

(i) The system (M, ∂_{−}M, ∂_{+}M, F) and the complexes H_{comb}, H_{an} are of determinant class;

(ii)  log T_{Re}(M, ∂_{−}M, g, τ, F) = 2 ; ∑_{j=1} log T_{Re}(M_{j}, ∂_{−}M_{j}, g_{j}, τ_{j}, F_{j})
+ log T (H_{an}).

(iii) log T_{an}(M, ∂_{−}M, g, F) = ∑_{j=1}^{2} log T_{an}(M_{j}, ∂_{−}M_{j}, g_{j}, F_{j}) + log T (H_{an})

− \frac{χ(∂_{+}M_{1}; F_{1})}{2} log 2.

Remark. In the case where A = ℝ this result is due to Vishik [Vi3]. Vishik’s proof is, however, very different from ours.

A sightly more general form of this result is contained in Theorem 3.2′ where only some components of ∂_{+}M_{1} and ∂_{−}M_{2} are glued together. The last result is Theorem 3.3 which compares the analytic torsions of bordisms with the same underlying compact manifold M.

The paper is organized as follows: In section 1 we prove a number of auxiliary results about cochain complexes of determinant class and a generalization of Milnor’s result concerning the torsions of a short exact sequence of finite dimensional complexes to infinite dimension (Proposition 1.13, Theorem 1.14). This section is quite elementary but it is included for the sake of completeness. In section 2 we prove Theorem 2.9 on which all subsequent results depend on. Theorems 3.1, 3.2, 3.2’, 3.3 are proven in section 3. In the appendix, we prove that for any bordism (M, ∂_{−}M, ∂_{+}M) with M a compact connected manifold, M̃ → M a principal covering with residually finite group and F the parallel flat bundle on M, induced by this covering, the system (M, ∂_{−}M, ∂_{+}M, F) is of determinant class. This result is implicit in Lück [Lü3]. Unfortunately, in [Lü3] there are a number of misleading misprints and the definition of the L^{2}-determinant is incorrect. For the convenience of the reader we present an outline of the proof of this result.

By the same arguments as given in [BFKM, Proposition 5.11] it suffices to prove the above Theorems in the case where the fiber W of the bundle E → M is a free A-module. For the rest of the paper the bundle E has as a fiber a free A-module.

The results presented in this paper can be extended to the case where F is a flat bundle with a Hermitian structure which is not necessarily parallel with respect to the connection; in this case the relative torsion R contains an additional ”local quantity” which has been obtained by [BZ] in the case ∂M = ∅ and A = ℂ.

Apart from very minor modifications, this paper coincides with an IHES preprint from January 1996.

Section 1 Linear homological algebra in the von Neumann sense.

In this section we follow, if not stated otherwise, the notations and use the definitions of Section 1 in [BFKM]. In particular, contrary to the standard notations for cochain complexes, the indice for the degree is denoted as a subscript and not as a superscript.

Throughout this section, let A be a von Neumann algebra of finite type. By W, W_{1}, W_{2}, U, … we denote arbitrary A-Hilbert modules of finite type.

Recall that f : W_{1} → W_{2} is said to be a weak isomorphism if f is 1-1 and its range, Im(f), is dense in W_{2}. It admits a polar decomposition f = f_{iso}f_{wiso} where f_{iso} := f(f^{∗}f)^{−1/2} : W_{1} → W_{2} is an isometry and f_{wiso} := (f^{∗}f)^{1/2} : W_{1} → W_{1} is a weak isomorphism which is selfadjoint and positive. Therefore f_{wiso}^{2} + ϵ(ϵ > 0) is an isomorphism and one can define Vol(f) by

(1.1) log Vol(f) := lim_{ϵ→0} \frac{1}{2} log det(f_{wiso}^{2} + ϵ)

where the determinant is the one given by Fuglede-Kadison [FK] and where log denotes the branch of the logarithm with log 1 = 0. Notice that \frac{1}{2} log det(f_{wiso}^{2}) := log Vol(f) is in ℝ ∪ {−∞} and

(1.2) log Vol(f) = lim_{ϵ→0} \frac{1}{2} log det(f^{∗}f + ϵ).

The weak isomorphism f is said to be of \underline{determinant class} iff

−∞ < log Vol(f). (1.3)

Using the spectral distribution function of f,

F_{f}(λ) := sup\{ dim_{N}L; L subspace of W_{1}, ∥f(x))∥ ≤ \sqrt{λ}∥x∥(x ∈ L)\},

one can show that

(1.2′) log Vol(f) = ∫_{0+}^{∞} log(λ)dF_{f}(λ)

and that (1.3) is equivalent to

−∞ < ∫_{0+}^{1} log(λ)dF_{f}(λ). (1.3′)

In case (1.3′) holds, one verifies that

∫_{0}^{1} \frac{F_{f}(λ) − F_{f}(0)}{λ}dλ = − ∫_{0^{+}}^{1} log(λ)dF_{f}(λ),

using that the boundary term obtained, when integrating by parts, vanishes. It will be convenient to extend the concept of determinant class to an arbitrary morphism f ∈ L_{A}(W_{1}, W_{2}). For this purpose we factor f as f = j · f′ · p with p : W_{1} → W′_{1} = W_{1}/Nullf, the canonical projection and j : W′_{2} := \overline{Rangef} → W_{2} the inclusion. Then f′ : W′_{1} → W′_{2} is a weak isomorphism. The morphism f is said to be of determinant class if f′ is of determinant class.

Proposition 1.1. (A) Assume that f ∈ L_{A}(W_{1}, W_{2}) and g ∈ L_{A}(W_{2}, W_{3}) are both weak isomorphisms. Then g · f is a weak isomorphism. Moreover

(1.4) log Vol(g · f) = log Vol(g) + log Vol(f).

In particular g · f is of determinant class iff both g and f are of determinant class. (B) Assume that f ∈ L_{A}(W_{1}, W′_{1}) and g ∈ L_{A}(W_{2}, W′_{2}) are both weak isomor-

phisms and h ∈ L_{A}(W_{2}, W′_{2}). Then f′ = ( \begin{matrix}f & h \\ 0 & g \end{matrix} ) : W_{1} ⊕ W_{2} → W′_{1} ⊕ W′_{2} is a

weak isomorphism and

(1.5) log Vol(f′) = log Vol(f) + log Vol(g).

(C) If f_{i} : W → V_{i}, i = 1, 2 are two morphisms and both are of determinant class then f_{1} ⊕ f_{2} : W → V_{1} ⊕ V_{2} is of determinant class.

Remark. : (a) In the case where W_{1} = W_{2}, statement (A) is verified in [FK].

(b) In the case f, g are isomorphisms and not only weak isomorphisms, statement (B) is proved in [BFKM, Proposition 1.9].

Proof. (A) Consider the polar decomposition f = f_{iso}f_{wiso} and g = g_{iso}g_{wiso}. Then

gf can be written as gf = g^{∼}f^{∼} where

g^{∼} := g_{iso}f_{iso} : W_{1} → W_{3}; f^{∼} := f_{iso}^{−1}g_{wiso}f : W_{1} → W_{1}. (1.6)

Note that g^{∼} is an isometry. Therefore g^{∼}^{∗}g^{∼} = Id_{W_{1}} and (g^{∼}f^{∼})^{∗}(g^{∼}f^{∼}) = f^{∼}^{∗}f^{∼}. From definition (1.1) it then follows that

(1.7) log Vol(gf) = log Vol(f^{∼}).

The map f^{∼} can be decomposed, f^{∼} = (f_{iso}^{−1}g_{wiso}f_{iso})f_{wiso} and we conclude from (1.4), which is applied in the case W_{1} = W_{2}, that

(1.8) log Vol(f^{∼}) = log Vol(f_{iso}^{−1}g_{wiso}f_{iso}) + log Vol(f_{wiso}).

Taking into account (1.2) one concludes that

(1.9) log Vol(f_{iso}^{−1}g_{wiso}f_{iso}) = log Vol(g_{wiso}).

From definition (1.1) we know that log Vol(f) = log Vol(f_{wiso}) and similarly for g. Combining (1.7) - (1.9) leads to

log Vol(gf) = log Vol(g) + log Vol(f).

(B) To verify (B) decompose f′ as follows

f′ = ( \begin{matrix}Id_{W′_{i}} & 0 \\ 0 & g \end{matrix} ) ( \begin{matrix}Id_{W′_{i}} & h \\ 0 & Id_{W_{2}} \end{matrix} ) ( \begin{matrix}f & 0 \\ 0 & Id_{W_{2}} \end{matrix} ) (1.10)

and apply (A) together with Proposition 1.9 in [BFKM].

(C) Denote by F_{1}(λ) := F_{f_{1}}(λ), F_{2}(λ) := F_{f_{2}}(λ) and F (λ) := F_{f_{1}⊕f_{2}}(λ). It is straightforward to verify that F (λ) − F (0) ≤ sup{F_{1}(λ) − F_{1}(0), F_{2}(λ) − F_{2}(0)}. Integration by parts in (1.3′) implies the result. □

Consider a finite A-cochain complex (C, d) of A-Hilbert modules of finite type

0 → C_{0} →^{d_{0}} C_{1} →^{d_{1}} ⋯ → 0. (1.11)

Define the Hodge decomposition C_{i}, C_{i} = H_{i} ⊕ C_{i}^{+} ⊕ C_{i}^{−}, where

C_{i}^{+} := \overline{d_{i−1}(C_{i−1})} ; C_{i}^{−} := \overline{d_{i}^{∗}(C_{i+1})} ; H_{i} := Null(d_{i}) ∩ Null(d_{i−1}^{∗}). (1.12)

Recall that H_{i} is isomorphic to Nulld_{i}/\overline{Range(d_{i−1})} = H^{i}(C). With respect to this decomposition

d_{i} = ( \begin{matrix}0 & 0 & 0 \\ 0 & 0 & \underline{d}_{i} \\ 0 & 0 & 0 \end{matrix} ); \underline{d}_{i} : C_{i}^{−} → C_{i+1}^{+}. (1.13)

Note that \underline{d}_{i} is a weak isomorphism.

Definition. The A-cochain complex C is said to be of determinant class if, for any i, d_{i} (or, equivalently, \underline{d}_{i}) is of determinant class.

If C is of determinant class, its torsion T (C) is a positive real number given by

(1.14) log T (C) = ∑_{j}(−1)^{j} log Vol(\underline{d}_{j}).

For k = 1, 2, let C^{k} be A_{k}-cochain complexes. Denote by χ(C^{k}) the Euler-Poincaré characteristic of C^{k}. Introduce the A_{1} ˆ⊗A_{2}-cochain complex C given by C_{j} = ⊕_{k}(C_{k}^{1} ˆ⊗C_{j−k}^{2}). Here the tensor product is taken in the category of Hilbert spaces.

In [BFKM, Corollary 1.22] we have verified the following Proposition 1.2. Suppose C^{1} and C^{2} are both of determinant class. Then C is of determinant class and

(1.15) log T (C) = χ(C^{2}) log T (C^{1}) + χ(C^{1}) log T (C^{2}).

Suppose f : C^{1} → C^{2} is a morphism of chain complexes. By straightforward inspection (cf. [BFKM, section 1]) one sees that, with respect to the Hodge decompositions of C_{i}^{1} = H_{i}^{1} ⊕ C_{i}^{1,+} ⊕ C_{i}^{1,−} and C_{i}^{2} = H_{i}^{2} ⊕ C_{i}^{2,+} ⊕ C_{i}^{2,−}, the morphisms f_{i} : C_{i}^{1} → C_{i}^{2} are of 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} ) . (1.16)

Moreover

\underline{d}_{i}^{2}f_{i,33} = f_{i+1,22}\underline{d}_{i}^{1}. (1.17)

Further it is straightforward to conclude from Proposition 1.1 that for any i ≥ 0 the following statements hold:

(1.18)  f_{i} is a [weak] isomorphism iff f_{i,11}, f_{i,22} and

f_{i,33} are all [weak] isomorphisms.

(1.19) log Volf_{i} = log Volf_{i,11} + log Volf_{i,22} + log Volf_{i,33}.

(1.20) Assume that f_{i} is a weak isomorphism. Then f_{i} is of determinant class iff f_{i,11}, f_{i,22}, f_{i,33} are all of determinant class.

In the sequel, we will occasionally write H(f_{i}) for f_{i,11} : H_{i}^{1} → H_{i}^{2}.

Proposition 1.3. Let f : C^{1} → C^{2} be a morphism of cochain complexes.

(A) If, for any i, f_{i} is a weak isomorphism and of determinant class, then the following statements hold:

(i) For any i, H(f_{i}) is of determinant class; (ii) C^{1} is of determinant class iff C^{2} is; (iii) if C^{1} is of determinant class (and thus, by (ii), C^{2} is of determinant class as well), then

(1.21)  log T (C^{2}) = log T (C^{1}) − ∑_{i}(−1)^{i} log Vol(f_{i})

+ ∑_{i}(−1)^{i} log Vol(H(f_{i})).

(B) If f is a homotopy equivalence (cf. [BFKM], definition 1.14 ) then C^{1} is of determinant class iff C^{2}is.

Proof. (A) (i) is obtained from (1.20) and (ii) follows from (1.20), (1.17) and Proposition 1.1. To check (iii), apply Proposition 1.1 to \underline{d}_{i}^{2}f_{i,33} = f_{i+1,22}\underline{d}_{i}^{1} to conclude that

(1.22) (−1)^{i} log Vol \underline{d}_{2}^{i}
= (−1)^{i} log Vol\underline{d}_{1}^{i} + (−1)^{i+1} log Volf_{i,33} + (−1)^{i} log Volf_{i+1,22}

which, after summing up and using (1.14), leads to

(1.23)  log T (C^{2}) − log T (C^{1})
= − ∑_{i}(−1)^{i} log Vol(f_{i,22}) − ∑_{i}(−1)^{i} log Vol(f_{i,33})
= − ∑_{i}(−1)^{i} log Vol(f_{i}) + ∑(−1)^{i} log VolH(f_{i}),

where for the last equality, we used (1.19).

(B) This result is due to Gromov-Shubin [GS] ( cf. also Proposition 1.18 in [BFKM]).

□

The following, somewhat tedious considerations (up to the end of (1.69)) will be used for the proof of Lemma 1.11, Lemma 1.12, Proposition 1.13 and Theorem 1.14.

Let C be a cochain complex of the form

0 → C_{0} →^{d_{0}} C_{1} →^{d_{1}} C_{2} → 0. (1.24)

Such a complex is called a three stage complex.

Definition. A three stage complex C is said to be a weakly exact sequence if d_{0} is injective, \overline{Range (d_{0})} = Null(d_{1}) and \overline{Range (d_{1})} = C^{2}.

Note that in this situation \underline{d}_{0} : C_{0} → \overline{Range (d_{0})} and \underline{d}_{1} : Range(d_{0})^{⊥} → C_{2} are weak isomorphisms where \overline{Range (d_{0})}^{⊥} denotes the orthogonal complement of Range(d_{0}) in C_{1}.

In the case where C is of determinant class, its torsion is given by

(1.25) log T (C) = log Vol(\underline{d}_{0}) − log Vol(\underline{d}_{1}).

Lemma 1.4. Suppose that the three stage cochain complex C is a weakly exact sequence of the form

0 → C_{0} →^{d_{0}} C_{1}^{+} ⊕ C_{1}^{−} →^{d_{1}} C_{2} → 0 (1.26)

where d_{0} = \binom{f}{f_{1}} and d_{1} = (g_{1} g). Further assume that d_{1} is onto. Then the following statements hold:

(A) If f : C_{0} → C_{1}^{+} and g : C_{1}^{−} → C_{2} are weak isomorphisms of determinant class, then the complex is of determinant class and

(1.27) log T (C) = log Vol(f) − log Vol(g).

(B) Assume that (1.26) is of determinant class. If f : C_{0} → C_{1}^{+} [resp. g : C_{1}^{−} → C_{2}] is of determinant class, so is g : C_{1}^{−} → C_{2} [resp. f : C_{0} → C_{1}^{+}] and formula (1.27) holds.

Proof. First note that, due to the assumption that d_{1} is onto, \underline{d}_{1} : (Range(d_{0}))^{⊥} → C_{2} is an isomorphism and therefore of determinant class. Moreover, log Vol(\underline{d}_{0}) ∈ ℝ∪\{−∞\}. Therefore, we can define log T (C) = log Vol(\underline{d}_{0})−log Vol(\underline{d}_{1}) ∈ ℝ∪\{−∞\}. Denote the inverse (\underline{d}_{1})^{−1} : C_{2} → Range(d_{0})^{⊥} ⊆ C_{1}^{+} ⊕ C_{1}^{−} by (\underline{d}_{1})^{−1} = \binom{h_{1}}{h} and conclude that

Id_{C_{2}} = (g_{1} g)\binom{h_{1}}{h} = g_{1}h_{1} + gh. (1.28)

Consider the complex C′ given by

0 → C_{0} \underrightarrow{\binom{Id_{C_{0}}}{0}} C_{0} ⊕ C_{2} ⟶^{(0 Id_{C_{2}} )} C_{2} → 0

which is of determinant class and satisfies

log T (C′) = 0. (1.29)

Further introduce the morphism f : C → C′ given by the following diagram

0 →  C_{0}  →  C_{0} ⊕ C_{2}  →  C_{2}  → 0
↓ Id_{C_{0}}  ↓ ( \begin{matrix}f & h_{1} \\ f_{1} & h \end{matrix} )  ↓ Id_{C_{2}}
0 →  C_{0}  \underrightarrow{\binom{f}{f_{1}}}  C_{1}^{+} ⊕ C_{1}^{−}  →^{(g_{1},g)}  C_{2}  → 0.

Using the same arguments as in the proof of Proposition 1.3 one shows that, in view of (1.29),

(1.30) log T (C) = 0 − log Vol ( \begin{matrix}f & h_{1} \\ f_{1} & h \end{matrix} ) + 0.

In particular, one concludes that C is of determinant class iff ( \begin{matrix}f & h_{1} \\ f_{1} & h \end{matrix} ) is of determinant class.

To compute log Vol ( \begin{matrix}f & h_{1} \\ f_{1} & h \end{matrix} ), note that the weak exactness of C implies that g_{1}f + gf_{1} = 0. Therefore

(1.31)  ( \begin{matrix}Id_{C_{1}^{+}} & 0 \\ 0 & g \end{matrix} ) ( \begin{matrix}f & h_{1} \\ f_{1} & h \end{matrix} ) = ( \begin{matrix}f & h_{1} \\ gf_{1} & gh \end{matrix} ) = ( \begin{matrix}f & h_{1} \\ −g_{1}f & gh \end{matrix} )

= ( \begin{matrix}Id_{C_{1}^{+}} & h_{1} \\ −g_{1} & gh \end{matrix} ) ( \begin{matrix}f & 0 \\ 0 & Id_{C_{2}} \end{matrix} )

and, by Proposition 1.1, one obtains (in ℝ ∪ {−∞})

log Vol ( \begin{matrix}f & h_{1} \\ f_{1} & h \end{matrix} ) + log Vol(g) = log Vol(f) + log Vol ( \begin{matrix}Id_{C_{1}^{+}} & h_{1} \\ −g_{1} & gh \end{matrix} ) .

According to (1.28), gh = Id_{C_{2}} − g_{1}h_{1} and therefore

( \begin{matrix}Id_{C_{1}^{+}} & h_{1} \\ −g_{1} & gh \end{matrix} ) = ( \begin{matrix}Id_{C_{1}^{+}} & 0 \\ −g_{1} & Id_{C_{2}} \end{matrix} ) ( \begin{matrix}Id_{C_{1}^{+}} & h_{1} \\ 0 & Id_{C_{2}} \end{matrix} ) . (1.32)

Formula (1.32) implies that ( \begin{matrix}Id_{C_{1}^{+}} & h_{1} \\ −g_{1} & gh \end{matrix} ) is a weak isomorphism of determinant class and

log Vol ( \begin{matrix}Id_{C_{1}^{+}} & h_{1} \\ −g_{1} & gh \end{matrix} ) = 0 + 0. (1.33)

Combining (1.30) - (1.33) statements (A) and (B) follow. □

Lemma 1.5. Let C^{1}, C^{2}, C^{3} be three stage complexes which are short exact sequences, (and thus, in particular, of determinant class)

\begin{aligned}C^{1} & : 0 → C_{0}^{1} →^{d_{0}^{1}} C_{1}^{1} →^{d_{1}^{1}} C_{2}^{1} → 0 \\ C^{2} & : 0 → C_{0}^{1} ⊕ C_{0}^{3} →^{d_{0}^{2}} C_{1}^{1} ⊕ C_{1}^{3} →^{d_{1}^{2}} C_{2}^{1} ⊕ C_{2}^{3} → 0 \\ C^{3} & : 0 → C_{0}^{3} →^{d_{0}^{3}} C_{1}^{3} →^{d_{1}^{3}} C_{2}^{3} → 0 \end{aligned}

Further assume that the following diagram is commutative,

\begin{aligned}0 & → & C_{0}^{1} & → & C_{1}^{1} & → & C_{2}^{1} & → & 0 \\ & & ↓ \binom{Id}{0} & & ↓ \binom{Id}{0} & & ↓ \binom{Id}{0} \\ 0 & → & C_{0}^{1} ⊕ C_{0}^{3} & → & C_{1}^{1} ⊕ C_{1}^{3} & → & C_{2}^{1} ⊕ C_{2}^{3} & → & 0 \\ & & ↓ (0 Id) & & ↓ (0 Id) & & ↓ (0 Id) \\ 0 & → & C_{0}^{3} & → & C_{1}^{3} & → & C_{2}^{3} & → & 0. \end{aligned} (1.34)

Then

(1.35) log T (C^{2}) = log T (C^{1}) + log T (C^{3}).

Proof. We first note that in the Hodge decomposition of C_{1}^{1} = C_{1}^{1,+} ⊕ C_{1}^{1,−}, and C_{1}^{3} = C_{1}^{3,+} ⊕ C_{1}^{3,−} the Hilbert module C_{1}^{j,+} is isometric to C_{0}^{j} and C_{1}^{j,−} is isometric to C_{2}^{j}, (j ∈ \{1, 3\}). Therefore one can write d_{0}^{j} = \binom{\underline{d}_{0}^{j}}{0}; d_{1}^{j} = (0 \underline{d}_{1}^{j}), with \underline{d}_{0}^{j} : C_{0}^{j} → C_{1}^{j,+} and \underline{d}_{1}^{j} : C_{1}^{j,−} → C_{2}^{j}

Taking the above mentioned isometries into account it follows that C^{2} is isometric to the decomposition

0 → C_{0}^{1} ⊕ C_{0}^{3} →^{d_{0}^{2}} C_{0}^{1} ⊕ C_{0}^{3} ⊕ C_{2}^{1} ⊕ C_{2}^{3} →^{d_{1}^{2}} C_{2}^{1} ⊕ C_{2}^{3} → 0

and the commutativity of the diagram (1.34) implies that d_{0}^{2} must be of the form

d_{0}^{2} = ( \begin{matrix}\underline{d}_{0}^{1} & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{matrix} ) + ( \begin{matrix}0 & 0 \\ 0 & \underline{d}_{0}^{3} \\ 0 & 0 \\ 0 & 0 \end{matrix} ) + ( \begin{matrix}0 & h_{1} \\ 0 & 0 \\ 0 & h_{2} \\ 0 & 0 \end{matrix} ) (1.36)

where h_{1} : C_{0}^{3} → C_{0}^{1} and h_{2} : C_{0}^{3} → C_{2}^{1} and d_{1}^{2} must be of the form

d_{1}^{2} = ( \begin{matrix}0 & 0 & \underline{d}_{1}^{1} & 0 \\ 0 & 0 & 0 & 0 \end{matrix} ) + ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \underline{d}_{1}^{3} \end{matrix} ) + ( \begin{matrix}0 & h_{3} & 0 & h_{4} \\ 0 & 0 & 0 & 0 \end{matrix} ) (1.37)

where h_{3} : C_{0}^{3} → C_{2}^{1} and h_{4} : C_{2}^{3} → C_{2}^{1}.

The assumption that C^{2} is an exact sequence implies that

h_{3}\underline{d}_{0}^{3} + \underline{d}_{1}^{1}h_{2} = 0. (1.38)

In order to compute the torsion T (C^{2}) we want to apply Proposition 1.3. First note that \underline{d}_{0}^{1}, \underline{d}_{0}^{3}, \underline{d}_{1}^{1}, \underline{d}_{1}^{3} are all isomorphism as C^{1}, C^{2}, C^{3} are exact sequences. In view of (1.38),

( \begin{matrix}\underline{d}_{0}^{1} & 0 \\ 0 & \underline{d}_{0}^{3} \\ 0 & 0 \\ 0 & 0 \end{matrix} ) ( \begin{matrix}Id_{C_{0}^{1}} & (\underline{d}_{0}^{1})^{−1}h_{1} \\ 0 & Id_{C_{0}^{3}} \end{matrix} ) = ( \begin{matrix}Id & 0 & 0 & 0 \\ 0 & Id & 0 & 0 \\ 0 & (\underline{d}_{1}^{1})^{−1}h_{3} & Id & 0 \\ 0 & 0 & 0 & Id \end{matrix} ) ( \begin{matrix}\underline{d}_{0}^{1} & h_{1} \\ 0 & \underline{d}_{0}^{3} \\ 0 & h_{2} \\ 0 & 0 \end{matrix} ) (1.39)

and

( \begin{matrix}Id & −h_{4}(\underline{d}_{1}^{3})^{−1} \\ 0 & Id \end{matrix} ) ( \begin{matrix}0 & h_{3} & \underline{d}_{1}^{1} & h_{4} \\ 0 & 0 & 0 & \underline{d}_{1}^{3} \end{matrix} ) (1.40)

= ( \begin{matrix}0 & 0 & \underline{d}_{1}^{1} & 0 \\ 0 & 0 & 0 & \underline{d}_{1}^{3} \end{matrix} ) ( \begin{matrix}Id_{C_{0}^{1}} & 0 & 0 & 0 \\ 0 & Id_{C_{0}^{3}} & 0 & 0 \\ 0 & (\underline{d}_{1}^{1})^{−1}h_{3} & Id_{C_{2}^{1}} & 0 \\ 0 & 0 & 0 & Id_{C_{2}^{3}} \end{matrix} ) .

Consider the following isomorphic three stage complexes: C^{2}, C^{21}, C^{22} and the isomorphisms f′ : C^{2} → C^{21}, f′′ : C^{21} → C^{22} given by the following commutative diagram (use equality (1.38))

\begin{matrix}C^{2} : 0 → C_{0}^{1} ⊕ C_{0}^{3} ⟶^{( \begin{matrix}\underline{d}_{0}^{1} & h_{1} \\ 0 & \underline{d}_{0}^{3} \\ 0 & h_{2} \\ 0 & 0 \end{matrix} )} C_{0}^{1} ⊕ C_{0}^{3} & ⊕ C_{2}^{1} ⊕ C_{2}^{3} \underrightarrow{( \begin{matrix}0 & h_{3} & \underline{d}_{1}^{1} & h_{4} \\ 0 & 0 & 0 & \underline{d}_{1}^{3} \end{matrix} )}  C_{2}^{1} ⊕ C_{2}^{3} → 0 \\ ( \begin{matrix}Id & (\underline{d}_{0}^{1})^{−1}h_{1} \\ 0 & Id \end{matrix} ) ↓ & ↓ Id  ↓ ( \begin{matrix}Id & −h_{4}(\underline{d}_{1}^{3})^{−1} \\ 0 & Id \end{matrix} ) \\ C^{21} : 0 → C_{0}^{1} ⊕ C_{0}^{3} ⟶^{( \begin{matrix}\underline{d}_{0}^{1} & 0 \\ 0 & \underline{d}_{0}^{3} \\ 0 & h_{2} \\ 0 & 0 \end{matrix} )} C_{0}^{1} ⊕ C_{0}^{3} & ⊕ C_{2}^{1} ⊕ C_{2}^{3}  \underrightarrow{( \begin{matrix}0 & h_{3} & \underline{d}_{1}^{1} & 0 \\ 0 & 0 & 0 & \underline{d}_{1}^{3} \end{matrix} )} C_{2}^{1} ⊕ C_{2}^{3} → 0 \\ Id ↓ & ↓ ( \begin{matrix}Id & 0 & 0 & 0 \\ 0 & Id & 0 & 0 \\ 0 & (\underline{d}_{1}^{1})^{−1}h_{3} & Id & 0 \\ 0 & 0 & 0 & Id \end{matrix} )  ↓ Id \\ C^{22} : 0 → C_{0}^{1} ⊕ C_{0}^{3} ⟶^{( \begin{matrix}\underline{d}_{0}^{1} & 0 \\ 0 & \underline{d}_{0}^{3} \\ 0 & 0 \\ 0 & 0 \end{matrix} )} C_{0}^{1} ⊕ C_{0}^{3} & ⊕ C_{2}^{1} ⊕ C_{2}^{3}  \underrightarrow{( \begin{matrix}0 & 0 & \underline{d}_{1}^{1} & 0 \\ 0 & 0 & 0 & \underline{d}_{1}^{3} \end{matrix} )}  C_{2}^{1} ⊕ C_{2}^{3} → 0. \end{matrix}

Now apply Proposition 1.3 and use that all three complexes C^{2}, C^{21}, C^{22} are exact sequences to conclude that (1.41) log T (C^{2}) = log T (C^{21}) = log T (C^{22})

Moreover,

(1.42) log T (C^{22}) = log T (C^{1}) + log T (C^{3}).

Combining (1.41) and (1.42) leads to (1.35).

□

The proof of Lemma 1.5 actually leads to a slightly stronger result:

Lemma 1.5′. Assume that the three stage complexes C^{1}, C^{2}, C^{3} are weakly exact sequences of determinant class and, in addition, assume that the morphisms \underline{d}_{0}^{1}, \underline{d}_{1}^{1} and \underline{d}_{1}^{3} (or \underline{d}_{0}^{1}, \underline{d}_{0}^{3} and \underline{d}_{1}^{3}) are isomorphisms. Then

log T (C^{2}) = log T (C^{1}) + log T (C^{3}).

Remark. To conclude the result of Lemma 1.5′ in the case where \underline{d}_{0}^{1}, \underline{d}_{0}^{3} and \underline{d}_{1}^{3} are isomorphisms one replaces the entry (\underline{d}_{1}^{1})^{−1}h_{3} in the 4 x 4 matrix which appears in the diagram above by −h_{2}(\underline{d}_{0}^{3})^{−1}.

The following considerations are a preparation for the proof of Theorem 1.14. Assume that

0 → C^{1} →^{f} C^{2} →^{g} C^{3} → 0

is an exact sequence of A-cochain complexes. Consider the Hodge decomposition of each of the three complexes, k = 1, 2, 3,

C_{i}^{k} = H_{i}^{k} ⊕ C_{i}^{k,+} ⊕ C_{i}^{k,−}.

With respect to these decompositions, f_{i} : C_{i}^{1} → C_{i}^{2} and g_{i} : C_{i}^{2} → C_{i}^{3} take 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} ) , g_{i} = ( \begin{matrix}g_{i,11} & 0 & g_{i,13} \\ g_{i,21} & g_{i,22} & g_{i,23} \\ 0 & 0 & g_{i,33} \end{matrix} ) . (1.43)

Moreover

\underline{d}_{i}^{2}f_{i,33} = f_{i+1,22}\underline{d}_{i}^{1}; \underline{d}_{i}^{3}g_{i,33} = g_{i+1,22}\underline{d}_{i}^{2}. (1.44)

The exactness of the sequences 0 → C_{k}^{1} → C_{k}^{2} → C_{k}^{3} → 0 imply that the following sequences

0 → C_{i}^{1,+} →^{f_{i,22}} C_{i}^{2,+} →^{g_{i,22}} C_{i}^{3,+} → 0 (1.45)

0 → C_{i}^{1,−} →^{f_{i,33}} C_{i}^{2,−} →^{g_{i,33}} C_{i}^{3,−} → 0 (1.46)

are three stage complexes with the property that f_{i,22} is 1-1 and g_{i,33} is onto. Using (1.44) one verifies that f_{i,22} has closed image, the range of g_{i,22} is dense in C_{i}^{3,+} and f_{i,33} is 1-1. Therefore, the cochain complexes (1.45) and (1.46) have a Hodge decomposition

0 → C_{i}^{1,+} →^{f_{i,22}} C_{i}^{2,+0} ⊕ C_{i}^{2,++} ⊕ C_{i}^{2,+−} →^{g_{i,22}} C_{i}^{3,+} → 0 (1.47)

and

0 → C_{i}^{1,−} →^{f_{i,33}} C_{i}^{2,−0} ⊕ C_{i}^{2,−+} ⊕ C_{i}^{2,−−} →^{g_{i,33}} C_{i}^{3,−} → 0. (1.48)

With respect to this decomposition,

f_{i,22} = ( \begin{matrix}0 \\ α_{i,43} \\ 0 \end{matrix} ); g_{i,22} = ( 0 0 β_{i,35} ) (1.50)

f_{i,33} = ( \begin{matrix}0 \\ α_{i,74} \\ 0 \end{matrix} ); g_{i,33} = ( 0 0 β_{i,48} ) . (1.51)

Notice that α_{i,43} and β_{i,48} are isomorphisms and β_{i,35} and α_{i,74} are weak isomorphisms.

The exact sequence of A-cochain complexes of finite type

0 → C^{1} →^{f} C^{2} →^{g} C^{3} → 0

induces a long weakly exact sequence H in cohomology (cf. [CG, p. 10, Thm 2.1])

⋯ → H^{i}(C^{1}) →^{H(f_{i})} H^{i}(C^{2}) →^{H(g_{i})} H^{i}(C^{3}) →^{H(δ_{i})} H^{i+1}(C^{1}) → … (1.52)

or, equivalently, in terms of the harmonic spaces H_{i}^{k} := Null(d_{i}^{k}) ∩ Null(d_{i−1}^{k∗})

⋯ → H_{i}^{1} →^{f_{i,11}} H_{i}^{2} →^{g_{i,11}} H_{i}^{3} →^{δ_{i}} H_{i+1}^{1} →^{f_{i+1,11}} H_{i+1}^{2} → … (1.52′)

In particular the cochain complex H is acyclic. Consider its Hodge decomposition

⋯ → H_{i}^{1,+} ⊕ H_{i}^{1,−} ⟶^{( \begin{matrix}0 & α_{i,12} \\ 0 & 0 \end{matrix} )} H_{i}^{2,+} ⊕ H_{i}^{2,−} ⟶^{( \begin{matrix}0 & β_{i,12} \\ 0 & 0 \end{matrix} )} (1.53)

H_{i}^{3,+} ⊕ H_{i}^{3,−} ⟶^{( \begin{matrix}0 & γ_{i} \\ 0 & 0 \end{matrix} )} H_{i+1}^{1,+} ⊕ H_{i+1}^{1,−} → …

Notice that α_{i,12}, β_{i,12} and γ_{i} are weak isomorphisms.

In view of (1.43) - (1.53) f_{i} and g_{i} can be written as

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} ) = ( \begin{matrix}( \begin{matrix}0 & α_{i,12} \\ 0 & 0 \end{matrix} ) & ( \begin{matrix}0 \\ 0 \end{matrix} ) & ( \begin{matrix}α_{i,14} \\ α_{i,24} \end{matrix} ) \\ ( \begin{matrix}α_{i,31} & α_{i,32} \\ α_{i,41} & α_{i,42} \\ α_{i,51} & α_{i,52} \end{matrix} ) & ( \begin{matrix}0 \\ α_{i,43} \\ 0 \end{matrix} ) & ( \begin{matrix}α_{i,34} \\ α_{i,44} \\ α_{i,54} \end{matrix} ) \\ ( \begin{matrix}0 & 0 \\ 0 & 0 \\ 0 & 0 \end{matrix} ) & ( \begin{matrix}0 \\ 0 \\ 0 \end{matrix} ) & ( \begin{matrix}0 \\ α_{i,74} \\ 0 \end{matrix} ) \end{matrix} ) (1.54)

and

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

= ( \begin{matrix}( \begin{matrix}0 & β_{i,12} \\ 0 & 0 \end{matrix} ) & ( \begin{matrix}0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} ) & ( \begin{matrix}β_{i,16} & β_{i,17} & β_{i,18} \\ β_{i,26} & β_{i,27} & β_{i,28} \end{matrix} ) \\ ( β_{i,31} β_{i,32} ) & ( 0 0 β_{i,35} ) & ( β_{i,36} β_{i,37} β_{i,38} ) \\ ( 0 0 ) & ( 0 0 0 ) & ( 0  0  β_{i,48} ) \end{matrix} )

Recall that f_{i} : H_{i}^{1,+} ⊕ H_{i}^{1,−} ⊕ C_{i}^{1,+} ⊕ C_{i}^{2,−} → H_{i}^{2,+} ⊕ H_{i}^{2,−} ⊕ C_{i}^{2,+0} ⊕ C_{i}^{2,++} ⊕ C_{i}^{2,+−} ⊕ C_{i}^{2,−0} ⊕ C_{i}^{2,−+} ⊕ C_{i}^{2,−−} and g_{i} : H_{i}^{2,+} ⊕ H_{i}^{2,−} ⊕ C_{i}^{2,+0} ⊕ C_{i}^{2,++} ⊕ C_{i}^{2,+−} ⊕ C_{i}^{2,−0} ⊕ C_{i}^{2,−+} ⊕ C_{i}^{2,−−} → H_{i}^{3,+} ⊕ H_{i}^{3,−} ⊕ C_{i}^{3,+} ⊕ C_{i}^{3,−}.

We have already noticed that α_{i,12}, α_{i,74}, β_{i,12}, β_{i,35} are weak isomorphisms, while α_{i,43} and β_{i,48} are isomorphisms. Using that g_{i} · f_{i} = 0 and the fact that α_{i,74} and β_{i,35} are weak isomorphisms one verifies that

α_{i,51} = 0; β_{i,27} = 0 (1.56)

(1.56′)  β_{i,12}α_{i,24} + β_{i,17}α_{i,74} = 0; β_{i,31}α_{i,12} + β_{i,35}α_{i,52} = 0
β_{i,31}α_{i,14} + β_{i,32}α_{i,24} + β_{i,35}α_{i,54} + β_{i,37}α_{i,74} = 0.

20 TORSIONS FOR MANIFOLDS WITH BOUNDARY...

Lemma 1.6. For any i, (A) α_{i,31} : H_{i}^{1,+} → C_{i}^{2,+0} is an isomorphism; (B) β_{i,26} : C_{i}^{2,−0} → H_{i}^{3,−} is an isomorphism. Proof. (A) Inspecting (1.54) and in view of α_{i,51} = 0 (cf. 1.56) one concludes that h_{i} := f_{i} ↾_{H_{i}^{1,+}⊕C_{i}^{1,+}} has a range which is contained in C_{i}^{2,+0} ⊕ C_{i}^{2,++} and admits a representation of the form

h_{i} := ( \begin{matrix}α_{i,31} & 0 \\ α_{i,41} & α_{i,43} \end{matrix} ) : H_{i}^{1,+} ⊕ C_{i}^{1,+} → C_{i}^{2,+0} ⊕ C_{i}^{2,++}.

From (1.55) one sees that C_{i}^{2,+0} ⊕ C_{i}^{2,++} ⊆ Null g_{i}. Due to the exactness 0 → C_{i}^{1} →^{f_{i}} C_{i}^{2} →^{g_{i}} C_{i}^{3} → 0, the map h_{i} is an isomorphism. Further we already know that α_{i,43} is an isomorphism and therefore conclude that α_{i,31} : H_{i}^{1,+} → C_{i}^{2,+0} is an isomorphism as well. (B) From (1.54) one sees that C_{i}^{2,−0} ⊕ C_{i}^{2,−−} ⊆ (Range f_{i})^{⊥}. Therefore, using that

0 → C_{i}^{1} →^{f_{i}} C_{i}^{2} →^{g_{i}} C_{i}^{3} → 0 is exact, h′_{i} := g_{i} ↾_{C_{i}^{2,−0}⊕C_{i}^{2,−−}} is 1-1 and its image is

given by H_{i}^{3,−} ⊕ C_{i}^{3,−}, i.e. h′_{i} := ( \begin{matrix}β_{i,26} & β_{i,28} \\ 0 & β_{i,48} \end{matrix} ) : C_{i}^{2,−0} ⊕ C_{i}^{2,−−} → H_{i}^{3,−} ⊕ C_{i}^{3,−} is

bijective. We already know that β_{i,48} is an isomorphism and (B) follows. □ Further notice that d_{i}^{2} takes the following form

d_{i}^{2} = ( \begin{matrix}0 & 0 & 0 \\ 0 & 0 & \underline{d}_{i}^{2} \\ 0 & 0 & 0 \end{matrix} ) = ( \begin{matrix}0 & 0 & 0 \\ 0 & 0 & ( \begin{matrix}d_{i,36}^{2} & 0 & d_{i,38}^{2} \\ d_{i,46}^{2} & d_{i,47}^{2} & d_{i,48}^{2} \\ 0 & 0 & d_{i,58}^{2} \end{matrix} ) \\ 0 & 0 & 0 \end{matrix} ) (1.57)

where we used again (1.44) to verify that d_{i,56}^{2}, d_{i,57}^{2} and d_{i,37}^{2} all vanish and that, for any i,

(1.58) \underline{d}_{i}^{3}β_{i,48} = β_{i+1,35}d_{i,58}^{2} is a weak isomorphism and

(1.59) d_{i,47}^{2}α_{i,74} = α_{i+1,43}\underline{d}_{i}^{1} is a weak isomorphism. Further, considering C_{i}^{2,−} ⟶^{g_{i,13}} H_{i}^{3} ⟶^{δ_{i}} H_{i+1}^{1} ⟶^{f_{i+1,21}} C_{i+1}^{2,+} and taking into account the definition of δ_{i} = ( \begin{matrix}0 & γ_{i} \\ 0 & 0 \end{matrix} ) one obtains d_{i,36}^{2} = α_{i+1,31}γ_{i}β_{i,26}. (1.60)

In a straightforward way one verifies the following Lemma 1.7. Assume that F : W_{1} ⊕ W_{2} → V_{1} ⊕ V_{2} is an A-linear map of the form

F = ( \begin{matrix}f & h \\ 0 & g \end{matrix} ).

(A) If F is 1-1 and has closed range and f is an isomorphism, then g is 1-1 and has closed range. (B) If F is onto and g is an isomorphism, then f is onto.

Lemma 1.8. For any i, α_{i,12} and α_{i,74} are of determinant class iff β_{i,12} and β_{i,35} are of determinant class.

Proof. Introduce φ_{i} : H_{i}^{1,−} ⊕ C_{i}^{1,−} → H_{i}^{2,+} ⊕ C_{i}^{2,−+} ⊕ H_{i}^{2,−} ⊕ C_{i}^{2,+−} and ψ_{i} : H_{i}^{2,+} ⊕ C_{i}^{2,−+} ⊕ H_{i}^{2,−} ⊕ C_{i}^{2,+−} → H_{i}^{3,+} ⊕ C_{i}^{3,+} given as follows

ψ_{i} = ( \begin{matrix}0 & β_{i,17} & β_{i,12} & 0 \\ β_{i,31} & β_{i,37} & β_{i,32} & β_{i,35} \end{matrix} ); φ_{i} = ( \begin{matrix}α_{i,12} & α_{i,14} \\ 0 & α_{i,74} \\ 0 & α_{i,24} \\ α_{i,52} & α_{i,54} \end{matrix} ) . (1.61)

Due to (1.56) - (1.56′) we have ψ_{i} · φ_{i} = 0. Moreover α_{i,12}, α_{i,74}, β_{i,12} and β_{i,35} are all weak isomorphisms. Therefore, for any i,

0 → H_{i}^{1,−} ⊕ C_{i}^{1,−} ⟶^{φ_{i}} H_{i}^{2,+} ⊕ C_{i}^{2,−+} ⊕ H_{i}^{2,−} ⊕ C_{i}^{2,+−} ⟶^{ψ_{i}} H_{i}^{3,+} ⊕ C_{i}^{3,+} → 0 (1.62)

is a weak exact sequence.

Using that ( \begin{matrix}α_{i,31} & 0 \\ α_{i,41} & α_{i,43} \end{matrix} ) is an isomorphism and applying Lemma 1.7 (A) twice

to f_{i} one concludes that φ_{i} has closed image. Similarly, using that ( \begin{matrix}β_{i,26} & β_{i,28} \\ 0 & β_{i,48} \end{matrix} )

is an isomorphism and applying Lemma 1.7 (B) one concludes that ψ_{i} is onto. Therefore, the sequence (1.62) is, in fact, an exact sequence and thus of determinant class. The statement then follows from Lemma 1.4(B) and Proposition 1.1(B).

□

Lemma 1.9. For any i, d_{i,36}^{2}, d_{i,47}^{2}, d_{i,58}^{2} are weak isomorphisms and satisfy: (A) d_{i,36}^{2} is of determinant class iff γ_{i} is of determinant class. (B) d_{i,47}^{2} and α_{i,74} are of determinant class iff \underline{d_{i}^{1}} is of determinant class.

(C) d_{i,58}^{2} and β_{i+1,35} are of determinant class iff \underline{d_{i}^{3}} is of determinant class.

Proof. The fact that d_{i,36}^{2}, d_{i,47}^{2} and d_{i,58}^{2} are weak isomorphisms follows from inspection of (1.57). To prove (A), (B) and (C) one applies Proposition 1.1 and uses formula (1.60) (in case of (A)), formula (1.59) (in case of (B)) and formula (1.58) (in case of (C)) together with the properties that α_{i+1,31} and β_{i,26} are isomorphisms (Lemma 1.6) and therefore of determinant class (in case (A)), α_{i+1,43} is an isomorphism (in case (B)) and β_{i,48} is an isomorphism (in case (C)).

□

Lemma 1.10. Assume that C^{1} and C^{3} are of determinant class. Then, for any i, the following statements hold: (A) the three stage complexes D_{±}^{i} given by 0 → C_{i}^{1,±} → C_{i}^{2,±} → C_{i}^{3,±} → 0 are of determinant class and

(1.63) log T (D_{+}^{i}) = log Vol α_{i,43} − log Vol β_{i,35}

(1.64) log T (D_{−}^{i}) = log Vol α_{i,74} − log Vol β_{i,48}.

(B) Assume, in addition, that the cochain complex H (cf (1.52)) is of determinant class. Then, for any i, the three stage complex D^{i}, given by 0 → C_{i}^{1} ⟶^{f_{i}} C_{i}^{2} ⟶^{g_{i}} C_{i}^{3} → 0, is of determinant class and

(1.65) log T (D^{i}) = log Vol α_{i,12} + log Vol α_{i,31} + log Vol α_{i,43} + log Vol α_{i,74}
− log Vol β_{i,12} − log Vol β_{i,26} − log Vol β_{i,35} − log Vol β_{i,48}.

Proof. (A) Recall that D_{+}^{i} is given by

0 → C_{i}^{1,+} ⟶^{f_{i,22}} C_{i}^{2,+} ⟶^{g_{i,22}} C_{i}^{3,+} → 0

where, according to (1.50), f_{i,22} = ( \begin{matrix}0 \\ α_{i,43} \\ 0 \end{matrix} ) and g_{i,22} = (0 0 β_{i},_{35} ). The map α_{i,43}

is an isomorphism and therefore of determinant class. Due to Lemma 1.9, the weak isomorphism β_{i,35} is of determinant class as well and (1.63) follows.

Similarly one argues for D_{−}^{i} given by

0 → C_{i}^{1,−} ⟶^{f_{i,33}} C_{i}^{2,−} ⟶^{g_{i,33}} C_{i}^{3,−} → 0

where, according to (1.50),

f_{i,33} = ( \begin{matrix}0 \\ α_{i,74} \\ 0 \end{matrix} ) , g_{i,33} = ( 0 0 β_{i,48} ) .

Notice that β_{i,48} is an isomorphism and therefore of determinant class. According to Lemma 1.9, the weak isomorphism α_{i,74} is of determinant class and therefore (1.64) is proved.

(B) The complexes D^{i}, given by 0 → C_{i}^{1} ⟶^{f_{i}} C_{i}^{2} ⟶^{g_{i}} C_{i}^{3} → 0 are exact and therefore of determinant class. To verify formula (1.65) we want to apply Lemma 1.4.

Decompose C_{i}^{2} = W_{i}^{2} ⊕ V_{i}^{2} where

W_{i}^{2} := H_{i}^{2,+} ⊕ C_{i}^{2,+0} ⊕ C_{i}^{2,++} ⊕ C_{i}^{2,−+}

and

V_{i}^{2} := (W_{i}^{2})^{⊥} = H_{i}^{2,−} ⊕ C_{i}^{2,+−} ⊕ C_{i}^{2,−0} ⊕ C_{i}^{2,−−}.

Then f_{i} = ( \begin{matrix}f′_{i} \\ f′′_{i} \end{matrix} ) and g_{i} = (g′_{i} g′′_{i}) have the following representations

f′_{i} = ( \begin{matrix}0 & α_{i,12} & 0 & α_{i,14} \\ α_{i,31} & α_{i,32} & 0 & α_{i,34} \\ α_{i,41} & α_{i,42} & α_{i,43} & α_{i,44} \\ 0 & 0 & 0 & α_{i,74} \end{matrix} ) (1.66)

g′′_{i} = ( \begin{matrix}β_{i,12} & 0 & β_{i,16} & β_{i,18} \\ 0 & 0 & β_{i,26} & β_{1,28} \\ β_{i,32} & β_{i,35} & β_{i,36} & β_{i,38} \\ 0 & 0 & 0 & β_{i,48} \end{matrix} ) . (1.67)

Due to the assumption that H is of determinant class and due to Lemma 1.6 and Lemma 1.9, the maps α_{i,31}, α_{i,12}, α_{i,43}, α_{i,74}, β_{i,12}, β_{i,35}, β_{i,26} and β_{i,48} are weak isomorphisms of determinant class. Therefore f′_{i} and g′′_{i} are weak isomorphisms of determinant class. As g_{i} is onto, we can apply Proposition 1.4 to conclude that

(1.68) log T (0 → C_{i}^{1} → C_{i}^{2} → C_{i}^{3} → 0) = log Volf′_{i} − log Volg′′_{i}.

Applying Proposition 1.1, we conclude from (1.66) - (1.67)

(1.69) log Volf′_{i} = log Volα_{i,31} + log Volα_{i,12} + log Volα_{i,43} + log Volα_{i,74}
log Volg′′_{i} = log Volβ_{i,12} + log Volβ_{i,35} + log Volβ_{i,26} + log Volβ_{i,48}.

Substituting (1.69) into (1.68) leads to the claimed result (1.65). □

Let f : C^{1} → C^{2} be a morphism of cochain complexes. Define the mapping cone of f to be the cochain complex C(f) := (C_{i}(f), d_{i}(f)) where C_{i}(f) = C_{i}^{2} ⊕ C_{i+1}^{1},

d_{i}(f) = ( \begin{matrix}d_{i}^{2} & f_{i+1} \\ 0 & −d_{i+1}^{1} \end{matrix} ) . Denote by SC = (SC_{i}, Sd_{i}) the suspension of a cochain

complex C, defined by SC_{i} = C_{i+1}, Sd_{i} = −d_{i+1}. Consider the suspension SC^{1} of C^{1} and the morphisms p(f) : C(f) → SC^{1} and

j(f) : C^{2} → C(f) given by p_{i}(f) = (0 Id) and j_{i}(f) = ( \begin{matrix}Id \\ 0 \end{matrix} ) . Then

0 → C^{2} →^{j(f)} C(f) →^{p(f)} SC^{1} → 0

is a short exact sequence of cochain complexes. The induced long weakly exact sequence in cohomology (cf. 1.52) becomes in this case

⋯ → H^{i}(C^{2}) → H^{i}(C(f)) → H^{i}(SC^{1}) →^{H(δ_{i})} H^{i+1}(C^{2}) → … (1.70)

Notice that H^{i}(SC^{1}) is equal to H^{i+1}(C^{1}) and the connecting homomorphism H(δ_{i}) : H^{i}(SC^{1}) → H^{i+1}(C^{2}) is given by the map H(f_{i+1}) : H^{i+1}(C^{1}) → H^{i+1}(C^{2}), induced by f in cohomology.

Lemma 1.11 (A). Let f : C^{1} → C^{2} be a morphism of cochain complexes. Assume that the cochain complexes C^{k}, (k = 1, 2) are of determinant class as well as the morphisms H(f_{i}) (i ≥ 0). Then the cochain complex C(f) is of determinant class.

Proof. The claimed result follows from applying Proposition 1.1 and Lemma 1.9 (cf the proof of Proposition 1.13(i)). □

Let

0 → C^{1} →^{f} C^{2} →^{g} C^{3} → 0

be a short exact sequence of cochain complexes. Let π : C(f) → C^{3}, ρ : SC^{1} → C(g)

and S(f) : SC^{1} → SC^{2} be the morphisms defined by π_{i} := (g_{i} 0), ρ_{i} := ( \begin{matrix}0 \\ f_{i+1} \end{matrix} ) ,

S_{i}(f) := f_{i+1.} Clearly π_{i} · j_{i}(f) = g_{i} and p_{i}(g) · ρ_{i} = S_{i}(f).

Lemma 1.11 (B). Both π and ρ are homotopy equivalences.

Proof. Without loss of generality one can assume that C_{i}^{2} = C_{i}^{1} ⊕ C_{i}^{3}, f_{i} = ( \begin{matrix}Id \\ 0 \end{matrix} ),

g_{i} = (0 Id) and d_{i}^{2} = ( \begin{matrix}d_{i}^{1} & θ_{i} \\ 0 & d_{i}^{3} \end{matrix} ) with θ_{i} : C_{i}^{3} → C_{i+1}^{1}, satisfying

d_{i+1}^{1}θ_{i} + θ_{i+1}d_{i}^{3} = 0. (1.71)

Then ρ_{i} = ( \begin{matrix}0 \\ Id \\ 0 \end{matrix} ) and π_{i} = (0 Id 0), as well as C_{i}(f) = C_{i}^{1} ⊕ C_{i}^{3} ⊕ C_{i+1}^{1} and

C_{i}(g) = C_{i}^{3} ⊕ C_{i+1}^{1} ⊕ C_{i+1}^{3}. Notice that d_{i}(f), resp. d_{i}(g), are given by

d_{i}(f) = ( \begin{matrix}d_{i}^{1} & θ_{i} & Id \\ 0 & d_{i}^{3} & 0 \\ 0 & 0 & −d_{i+1}^{1} \end{matrix} ) . (1.72)

d_{i}(g) = ( \begin{matrix}d_{i}^{3} & 0 & Id \\ 0 & −d_{i+1}^{1} & −θ_{i+1} \\ 0 & 0 & −d_{i+1}^{3} \end{matrix} ) . (1.72′)

Define the morphisms of cochain complexes σ : C^{3} → C(f) and ω : C(g) → SC^{1}

by σ_{i} = ( \begin{matrix}0 \\ Id \\ −θ_{i} \end{matrix} ) respectively ω_{i} = (−θ_{i} − Id 0) and notice that π_{i} · σ_{i} = Id and

ω_{i}·ρ_{i} = −Id. It remains to check that σ·π and −ρ·ω are both homotopic to Id. Since the cochain complexes Null(π) and Null(ω) are, as is verified easily, exact sequences this will follow from Lemma 1.12 below (applied to 0 → Null(π) → C(f) → C^{3} → 0, respectively 0 → Null(ω) → C(g) → SC^{1} → 0). □

Lemma 1.12. Let

0 → C^{1} →^{f} C^{2} →^{g} C^{3} → 0

be a short exact sequence of cochain complexes and let s : C^{3} → C^{2} be a morphism so that g · s = Id. If C^{1} is an exact sequence, then s · g and Id are homotopic.

Proof. It is to prove (cf. Definition 1.14 in [BFKM]) that there exist morphisms t_{i} : C_{i}^{2} → C_{i−1}^{2} such that Id−s_{i}g_{i} = d_{i−1}^{2}t_{i} +t_{i+1}d_{i}^{2}. Consider the Hodge decomposition

of C^{1}, C_{i}^{1} = C_{i}^{1,+} ⊕ C_{i}^{1,−} with d_{i}^{1} = ( \begin{matrix}0 & \underline{d}_{i}^{1} \\ 0 & 0 \end{matrix} ) . In view of the exactness of C^{1}, \underline{d}_{i}^{1} are

isomorphisms. Without loss of generality one can assume that C_{i}^{2} = C_{i}^{1,+}⊕C_{i}^{1,−}⊕C_{i}^{3},

f_{i} = ( \begin{matrix}Id & 0 \\ 0 & Id \\ 0 & 0 \end{matrix} ) , g_{i} = (0 0 Id), s_{i} = ( \begin{matrix}s_{i}^{+} \\ s_{i}^{−} \\ Id \end{matrix} ) and d_{i}^{2} = ( \begin{matrix}0 & \underline{d}_{i}^{1} & θ_{i}^{+} \\ 0 & 0 & θ_{i}^{−} \\ 0 & 0 & d_{i}^{3} \end{matrix} ) . Note that

d_{i+1}^{2} · d_{i}^{2} = 0 implies that

\underline{d}_{i+1}^{1}θ_{i}^{−} + θ_{i+1}^{+}d_{i}^{3} = 0, θ_{i+1}^{−}d_{i}^{3} = 0. (1.73)

The condition that s is a morphism of cochain complexes implies that

\underline{d}_{i}^{1}s_{i}^{−} + θ_{i}^{+} = s_{i+1}^{+}d_{i}^{3}, θ_{i}^{−} = s_{i+1}^{−}d_{i}^{3}. (1.74)

Define t_{i} : C_{i}^{1,+} ⊕ C_{i}^{1,−} ⊕ C_{i}^{3} → C_{i−1}^{1,+} ⊕ C_{i−1}^{1,−} ⊕ C_{i−1}^{3} by

t_{i+1} = ( \begin{matrix}0 & 0 & 0 \\ (\underline{d}_{i}^{1})^{−1} & 0 & −(\underline{d}_{i}^{1})^{−1} · s_{i+1}^{+} \\ 0 & 0 & 0 \end{matrix} ) . (1.75)

One verifies that Id − s_{i}g_{i} = t_{i+1}d_{i}^{2} + d_{i−1}^{2}t_{i}. □

Let 0 → C^{1} →^{f} C^{2} →^{g} C^{3} → 0 be an exact sequence of A-cochain complexes of finite type. Recall that it induces the long weakly exact sequence in cohomology (1.52)

⋯ → H^{i}(C^{1}) →^{H(f_{i})} H^{i}(C^{2}) →^{H(g_{i})} H^{i}(C^{3}) →^{H(δ_{i})} H^{i+1}(C^{1}) → … (1.52)

which will be also viewed as a cochain complex and denoted by H.

Proposition 1.13. Let 0 → C^{1} →^{f} C^{2} →^{g} C^{3} → 0 be an exact sequence of A-cochain complexes of finite type.

(i) If C^{1}, C^{3} is of determinant class and H(δ_{i}) : H^{i}(C^{3}) → H^{i+1}(C^{1}) is of determinant class for any i, then C^{2} is of determinant class. (ii) If C^{1}, C^{2} are of determinant class and H(f_{i}) : H^{i}(C^{1}) → H^{i}(C^{2}) is of determinant class for any i, then C^{3} is of determinant class.

(iii) If C^{2}, C^{3} are of determinant class and H(g_{i}) : H^{i}(C^{2}) → H^{i}(C^{3}) is of determinant class for any i, then C^{1} is of determinant class.

Proof. (i) Recall that

\underline{d}_{i}^{2} = ( \begin{matrix}( \begin{matrix}d_{i,36}^{2} & 0 \\ d_{i,46} & d_{i,47} \end{matrix} ) & ( \begin{matrix}d_{i,38}^{2} \\ d_{i,48}^{2} \end{matrix} ) \\ ( 0  0 ) & d_{i,58}^{2} \end{matrix}  ) .

¿From Proposition 1.1 we conclude that \underline{d}_{i}^{2} is of determinant class iff d_{i,36}^{2}, d_{i,47}^{2} and d_{i,58}^{2} are of determinant class. In view of Lemma 1.9 it then follows that C^{2} is of determinant class, using that \underline{d}_{i}^{1}, \underline{d}_{i}^{3} and γ_{i} are of determinant class for any i. (ii) By Lemma 1.11(A) (or statement (i) above), the mapping cone C(f) is of determinant class. As C(f) is homotopy equivalent to C^{3} (Lemma 1.11(A)) it follows therefore from Proposition 1.3(B) that C^{3} is of determinant class. (iii) Applying again Lemma 1.11(A) and (B) one concludes that the mapping cone C(g) is of determinant class and homotopy equivalent to SC^{1}. Thus Proposition 1.3(B) implies that C^{1} is of determinant class. □

Theorem 1.14. 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

(1.75)  log T (C^{2}) = log T (C^{1}) + log T (C^{3}) + log T (H)

− ∑_{i}(−1)^{i} log T (0 → C_{i}^{1} → C_{i}^{2} → C_{i}^{3} → 0).

Proof. In view of Proposition 1.13 it remains to prove, apart from formula (1.75), that H is of determinant class if C^{1}, C^{2} and C^{3} are of determinant class. Notice that the following statements hold:

(i) If C^{2} is of determinant class then d_{i,36}^{2} is of determinant class which implies that H(δ_{i}) is of determinant class for any i (cf Lemma 1.9(A)). (ii) If C^{3} is of determinant class then C(f) is of determinant class (Lemma 1.11(B), Proposition 1.3(B)). Moreover, by Proposition 1.13(i), the morphisms H^{i}(SC^{1}) → H^{i+1}(C^{2}) are of determinant class which is equivalent to H(f_{i+1}) being of determinant class (cf (1.70)). (iii) If C^{1} is of determinant class then C(g) is of determinant class (Lemma 1.11(B), Proposition 1.3(B)) and, by Proposition 1.13(i), H(g_{i}) : H^{i}(C^{2}) → H^{i}(C^{3}) is of determinant class for any i. Combining (i)-(iii) one concludes that if C^{1}, C^{2}, C^{3} are of determinant class, then H is of determinant class.

To prove formula (1.75) consider the commutative diagram (1.76)

0 →  C_{i}^{1,−}  \underrightarrow{( \begin{matrix}0 \\ α_{i,74} \\ 0 \end{matrix} )}  C_{i}^{2,−0} ⊕ C_{i}^{2,−+} ⊕ C_{i}^{2,−−}  ⟶^{(0 0 β_{i,48})}  C_{i}^{3,−} →  0

↓ \underline{d_{i}^{1}}  ↓ \underline{d_{i}^{2}}  ↓ \underline{d_{i}^{3}}

(1.77)

0 → C_{i+1}^{1,+} \underrightarrow{( \begin{matrix}0 \\ α_{i+1,43} \\ 0 \end{matrix} )} C_{i+1}^{2,+0} ⊕ C_{i+1}^{2,++} ⊕ C_{i+1}^{2,+−} ⟶^{(0 0 β_{i+1,35})} C_{i+1}^{3,+} → 0.

Notice that \underline{d}_{i}^{1}, \underline{d}_{i}^{2} and \underline{d}_{i}^{3} establish a weak isomorphism of determinant class between the complexes (1.76) and (1.77). By Lemma 1.9 both complexes (1.76) and (1.77) are of determinant class. By Proposition 1.3, for any i

(1.78) logT ( 0 → C_{i+1}^{1,+} → C_{i+1}^{2,+} → C_{i+1}^{3,+} → 0 )
= log T ( 0 → C_{i}^{1,−} → C_{i}^{2,−} → C_{i}^{3,−} → 0 )
− log Vol(\underline{d}_{i}^{1}) + log Vol(\underline{d}_{i}^{2}) − log Vol(\underline{d}_{i}^{3}) − log Vol(d_{i,36}^{2}),

where we used that H(\underline{d_{i}^{1}}) = 0 = H(\underline{d}_{i}^{3}) and H(\underline{d}_{i}^{2}) = d_{i,36}^{2}. Rearrange terms in (1.78), multiply by (−1)^{i+1} and sum over i to obtain

(1.79) ∑_{i}(−1)^{i} log Vol(\underline{d_{i}^{1}}) − ∑_{i}(−1)^{i} log Vol(\underline{d_{i}^{2}}) + ∑_{i}(−1)^{i} log Vol(\underline{d_{i}^{3}})

= − ∑_{i}(−1)^{i} log T (0 → C_{i+1}^{1,+} → C_{i+1}^{2,+} → C_{i+1}^{3,+} → 0)

+ ∑_{i}(−1)^{i} log T (0 → C_{i}^{1,−} → C_{i}^{2,−} → C_{i}^{3,−} → 0)

− ∑_{i}(−1)^{i} log Vol(d_{i,36}^{2}).

Recall formula (1.60), d_{i,36}^{2} = α_{i+1,31}γ_{i}β_{i,26}. Using that both d_{i,36}^{2} and γ_{i} are weak isomorphisms of determinant class and, according to Lemma 1.6 which states that α_{i+1,31} and β_{i,26} are isomorphisms we conclude from (1.4) that

(1.80) log Vold_{i,36}^{2} = log Volα_{i+1,31} + log Volγ_{i} + log Volβ_{i,26}.

Substituting (1.80) into (1.79) and taking into account that

log T (H) = ∑_{i}(−1)^{i} log Vol(α_{i,12}) − ∑(−1)^{i} log Vol(β_{i,12}) + ∑(−1)^{i} log Vol(γ_{i}),

28 TORSIONS FOR MANIFOLDS WITH BOUNDARY...

one obtains

(1.81) logT (C^{1}) − log T (C^{2}) + log T (C^{3})

= − log T (H) − ∑_{i}(−1)^{i} log Volα_{i+1,31} − ∑_{i}(−1)^{i} log Volβ_{i,26}

− ∑_{i}(−1)^{i} log Volβ_{i,12} + ∑_{i}(−1)^{i} log Volα_{i,12}

+ ∑_{i}(−1)^{i} log T (0 → C_{i}^{1,−} → C_{i}^{2,−} → C_{i}^{3,−} → 0)

− ∑_{i}(−1)^{i} log T (0 → C_{i+1}^{1,+} → C_{i+1}^{2,+} → C_{i+1}^{3,+} → 0).

By Lemma 1.10,

(1.82)  logT (0 → C_{i+1}^{1,+} → C_{i+1}^{2,+} → C_{i+1}^{3,+} → 0)
= log Vol(α_{i+1,43}) − log Vol(β_{i+1,35})

and

(1.83) log T (0 → C_{i}^{1,−} → C_{i}^{2,−} → C_{i}^{3,−} → 0) = log Volα_{i,74} − log Volβ_{i,48}. Changing the index of summation from i + 1 to i where necessary, one obtains

logT (C^{2}) = log T (C^{1}) + log T (C^{3}) + log T (H)

− ∑_{i}(−1)^{i} log Volα_{i,31} − ∑(−1)^{i} log Volα_{i,12}

− ∑(−1)^{i} log Volα_{i,74} − ∑(−1)^{i} log Volα_{i,43} + ∑_{i}(−1)^{i} log Volβ_{i,12} + ∑_{i}(−1)^{i} log Volβ_{i,26} + ∑_{i}(−1)^{i} log Volβ_{i,35} + ∑_{i}(−1)^{i} log Volβ_{i,48} = log T (C^{1}) + log T (C^{3}) + log T (H) − ∑(−1)^{i} log T (0 → C^{1} → C^{2} → C^{3} → 0)

where, for the last equality, we used again Lemma 1.10. This proves (1.75). □

2. Torsions for compact manifolds with boundary.

2.1 Reidemeister and analytic torsion.

Let (M, g) be a compact Riemannian manifold with boundary and (M, ∂_{−}M, ∂_{+}M) a bordism. Further let A be a unital von Neumann algebra and E →^{p} M be a bundle of A-Hilbert modules of finite type equipped with a flat connection ∇ with the property that the inner products (·, ·) of E_{y} = p^{−1}(y) are parallel. Such a pair (E, ∇) will be denoted by F and refered to as a parallel flat bundle of A-Hilbert modules.

Given a generalized triangulation τ = (h, g′) of M, the collection of the unstable manifolds in M\∂M, W_{y}^{−}, associated to the critical points y ∈ Cr(h), and the flow, induced by grad_{g′}h, provides a \underline{relative} CW-complex structure on the pair of spaces (S, ∂_{−}M), S = ∂_{−}M ∪_{y∈Cr(h)} W_{y}^{−} whose open cells are given by W_{y}^{−}. The inclusion of pairs (S, ∂_{−}M) ⊂ (M, ∂_{−}M) is a homotopy equivalence. Notice that if τ = (h, g′) is a generalized triangulation for M = (M, ∂_{−}M, ∂_{+}M), then τ_{D} = (−h, g′) is a triangulation for −M. Further if M, resp. N, are two closed manifolds and τ′ = (h′, g′), resp. τ′′ = (h′′, g′′), are generalized triangulations, then τ_{0} := (h′ + h′′, g′ ⊕ g′′) provides a generalized triangulation on M × N. The CW-complex structure induced by τ_{0} is the product of the CW-complex structures induced by τ′ and τ′′. Notice that if (M, ∂_{−}M, ∂_{+}M) is a bordism and ∂M ≠ ∅, then h_{0} := h′ +h′′ is not constant on the boundary components ∂_{±}M ×N, and therefore τ_{0} is not a generalized triangulation as defined in the introduction. However it is possible to modify h_{0} in an arbitrary small neighborhood of ∂M × N, so that the new function h has the same critical points as h_{0}, τ := (h, g′ ⊕ g′′) is a generalized triangulation and the relative CW-complex structure induced by τ is the product of the relative CW complex structure induced by τ′ and the CW complex structure induced by τ′′. The generalized triangulation τ will be refered to as a product triangulation τ = τ′ × τ′′.

Denote by M = ∪M_{j} the disjoint union of the connected components M_{j} of a

manifold M. Let M^{∼}_{j} be the universal cover of M_{j} and introduce M^{∼} = ∪M^{∼}_{j}. Let h^{∼}

be the lift of h to M^{∼}, and g^{∼}′ be the pull back of the Riemannian metric g′ on M^{∼}. Then the gradient vector field grad_{g^{∼}}_{′} h^{∼} satisfies the Morse-Smale condition as well.

For an arbitrary critical point y^{∼} ∈ Cr(h^{∼}), choose an orientation O_{y^{∼}} of the de-

scending manifold W − ; y^{∼} and let O_{h} := {O_{y^{∼}}; y^{∼} ∈ Cr(h)}. Given this choice one con-

structs, as in [BFKM, section 4], the cochain complex (C^{q}(M, ∂_{−}M, τ, O_{h}, F), δ_{q}) of A-Hilbert modules of finite type.

Denote by Δ_{q}^{comb} the combinatorial Laplacians

Δ_{q}^{comb} := δ_{q}^{∗}δ_{q} + δ_{q−1}δ_{q−1}^{∗}.

Here δ_{q}^{∗} denotes the adjoint of δ_{q}. Observe that Δ_{q}^{comb} is a bounded, nonnegative selfadjoint A-linear operator on C^{q} and for any ϵ > 0 one can define its regularized determinant in the von Neumann sense, by the following Stieltjes integral

log det_{N}(Δ_{q}^{comb} + ϵ) := ∫_{0^{+}}^{∞} log(λ + ϵ)dN_{Δ_{q}^{comb}}(λ)

where N_{Δ_{q}^{comb}}(λ) is the spectral distribution function associated to Δ_{q}^{comb}.

The function log det_{N}(Δ_{q}^{comb} + ·) is an element in the vector space 𝔻 consisting of equivalence classes [f] of real analytic functions f : (0, ∞) → ℝ with f ∼ g iff lim_{λ→0}(f(λ) − g(λ)) = 0. (The elements of 𝔻 represented by the constant functions

form a subspace of 𝔻 which can be identified with ℝ, the space of real numbers.) We then define T_{comb} = T_{comb}(M, ∂_{−}M, τ, F) as the following element in 𝔻

log T_{comb} = \frac{1}{2}∑_{q}(−1)^{q+1}q log det(Δ_{q}^{comb} + ·).

One can show that log T_{comb} is independent of the choice of orientations O_{h}. To define the analytic torsion we first have to introduce some more notation. Denote by Λ^{q}(M; E) the A-module of all smooth q-forms and let d_{q} : Λ^{q}(M; E) →

Λ^{q+1}(M; E) be the exterior differential induced by the exterior differential on scalar valued q-forms and the covariant differentiation given by the connection ∇ on E. Then (Λ^{∗}(M; E), d_{∗}) is a cochain complex of A-modules. The Riemannian metric g together with the connection ∇ induce the Hodge ∗-operators J_{q} : Λ^{q}(M; E) → Λ^{d−q}(M; E), denoted by ∗. The operator J_{q} induces a scalar product ⟨·, ·⟩ on Λ^{q}(M; E), with ⟨ω′, ω′′⟩ given by (ω′, ω′′ in Λ^{q}(M; E))

⟨ω′, ω′′⟩ = ∫_{M} ω′ ∧_{E} ∗ω′′ (2.1)

where ∧ ≡ ∧_{E} is defined by the composition

Λ^{q}(M; E) × Λ^{k}(M; E) → Λ^{q+k}(M; E ⊗ E) → Λ^{q+k}(M).

Denote by d_{q−1}^{∗} : Λ^{q}(M; E) → Λ^{q−1}(M; E) the formal adjoint of d_{q−1}, i.e

d_{q−1}^{∗} = (−1)^{d(q−1)+1} ∗ d_{d−q} ∗ .

Define the submodules Λ_{1}^{q}(M; E) and Λ_{2}^{q}(M; E) of Λ^{q}(M; E)

Λ_{1}^{q}(M; E) := {ω ∈ Λ^{q}(M; E); i_{∂_{−}M}^{#}ω = 0; i_{∂_{+}M}^{#}(∗ω) = 0} (2.2)

and

(2.3)  Λ_{2}^{q}(M; E) := {ω ∈ Λ^{q}(M; E); i_{∂_{−}M}^{#}ω = 0; i_{∂_{+}M}^{#}(∗ω) = 0;
i_{∂_{−}M}^{#}(d_{q−1}^{∗}ω) = 0; i_{∂_{+}M}^{#}(d_{d−q−1}^{∗}(∗ω)) = 0}

where i_{∂_{±}M}^{#} : Λ^{∗}(M; E) → Λ^{∗}(∂M_{±}; E) are the pullbacks induced by the embeddings i_{∂_{±}M} : ∂_{±}M ↪ M.

Consider the Laplacians acting on q-forms in Λ_{2}^{q}(M; E),

Δ_{q} := d_{q}^{∗}d_{q} + d_{q−1}d_{q−1}^{∗}.

The operators Δ_{q} are essentially self-adjoint, nonnegative, elliptic and A-linear. For any ϵ > 0 one can define its regularized determinant in the von Neumann sense,

log det_{N}(Δ_{q} + ϵ) = −\frac{d}{ds}ζ_{Δ_{q}+ϵ}(s)|_{s=0} − ϵ log(tr_{N}P_{q})

where P_{q} denotes the orthogonal projection onto the null space of Δ_{q} and ζ_{Δ_{q}+ϵ}(s) is the meromorphic continuation of the function defined for Re(s) > d/2 by the formula (cf. [BFKM])

ζ_{Δ_{q}+ϵ}(s) = \frac{1}{Γ(s)} ∫_{0}^{∞} t^{s−1}e^{−tϵ}tr_{N}e^{−tΔ_{q}}dt.

Then log det_{N}(Δ_{q} + ·) is an element in the vector space 𝔻 introduced above and the analytic torsion T_{an} = T_{an}(M, ∂_{−}M, g, F) is defined as the following element in 𝔻

log T_{an} = \frac{1}{2} ∑_{q}(−1)^{q+1}q log det(Δ_{q} + ·).

Denote by L_{2}(Λ^{q}(M; E)) the completion of Λ^{q}(M; E) with respect to the scalar product ⟨·, ·⟩. L_{2}(Λ^{q}(M; E)) is an A-Hilbert module. Notice that the completions of (Λ_{j}^{q}(M; E)) (j = 1, 2) give the same space L_{2}(Λ^{q}(M; E)).

Introduce the space of harmonic forms,

H_{q}(M; E) := {ω ∈ Λ_{2}^{q}(M; E); d_{q}ω = 0; d_{q−1}^{∗}ω = 0}.

It is an immediate consequence of the definition of Λ_{j}^{q}(M; E), j = 1, 2, that the Hodge ∗ operators induce isometries (Poincaré duality)

Λ_{j}^{q}(M; E) → Λ_{j}^{d−q}(−M; E)
H_{q}(M; E) → H_{d−q}(−M; E).

(Recall that −M is obtained from the bordism (M, ∂_{−}M, ∂_{+}M) by interchanging the role of ∂_{+}M and ∂_{−}M.)

By the theory of elliptic differential A-operators one concludes that, for any q, H_{q}(M; E) is an A-Hilbert module of finite von Neumann dimension (in fact it is of finite type). Denote by P_{q}(λ) : Λ^{q}(M; E) → Λ_{2}^{q}(M; E) the spectral projections corresponding to the Laplacian Δ_{q} and the boundary operator

(2.4) B_{q} : Λ^{q}(M; E) → Λ^{q}(∂_{−}M; E) × Λ^{q−1}(∂_{−}M; E) × Λ^{d−q}(∂_{+}M; E) × Λ^{d−q−1}(∂_{+}M; E)
ω ↦ (i_{∂_{−}M}^{#} (ω), i_{∂_{−}M}^{#} (d_{q−1}^{∗}ω), i_{∂_{+}M}^{#} (∗ω), i_{∂_{+}M}^{#} (d_{d−q−1}^{∗}(∗ω)).

Proposition 2.1. (Hodge decomposition) For any 0 ≤ q ≤ d, there exist orthogonal decompositions of A-pre Hilbert modules

(HD) Λ^{q}(M; E) = H_{q}(M; E) ⊕ closure d_{q−1}(Λ_{1}^{q−1}(M; E)) ⊕ closure d_{q}^{∗}(Λ_{1}^{q+1}(M; E))

where the word closure refers to the closure with respect to the usual C^{∞}−topology.

(HD)_{1} Λ_{1}^{q}(M; E) = H_{q}(M; E) ⊕ closure d_{q−1}(Λ_{2}^{q−1}(M; E)) ⊕ closure d_{q}^{∗}(Λ_{2}^{q+1}(M; E))

and

\overline{(HD)} L_{2}(Λ_{1}^{q}(M; E)) = H_{q}(M; E) ⊕ L_{2}(d_{q−1}(Λ_{2}^{q−1}(M; E))) ⊕ L_{2}(d_{q}^{∗}(Λ_{2}^{q+1}(M; E))).

Proof. Firs observe that H_{q}(M; E), d_{q−1}(Λ_{2}^{q−1}(M; E)) and d_{q}^{∗}(Λ_{2}^{q+1}(M; E)) are Asubmodules of Λ_{1}^{q}(M; E) and H_{q}(M; E), d_{q−1}(Λ_{1}^{q−1}(M; E)) and d_{q}^{∗}(Λ_{1}^{q+1}(M; E)) are A-submodules of Λ^{q}(M; E). In each of the two cases the three spaces are, due to Stokes’ theorem, pairwise orthogonal with respect to ⟨·, ·⟩. Moreover \overline{(HD)} follows from (HD) or (HD)_{1}.

Thus it remains to check that an arbitrary element ω in Λ^{q}(M; E) or Λ_{1}^{q}(M; E) can be decomposed as stated. With this in mind, we introduce for ϵ > 0 the operator G_{q;ϵ} : Λ^{q}(M; E) → Λ_{2}^{q}(M; E), which vanishes on the subspace P_{q}(ϵ)Λ^{q}(M; E) and is equal to Δ_{q}^{−1} (considered with boundary conditions defined by the boundary operator (2.4)) on (Id − P_{q}(ϵ))Λ^{q}(M; E).

Notice that Δ_{q}G_{q;ϵ} = Id − P_{q}(ϵ). Thus, for ω ∈ Λ^{q}(M; E), one has

ω = P_{q}(ϵ)ω + d_{q−1}d_{q−1}^{∗}G_{q;ϵ}ω + d_{q}^{∗}d_{q}G_{q;ϵ}ω

and therefore

ω − (P_{q}(ϵ) − P_{q}(0))ω = P_{q}(0)ω + d_{q−1}d_{q−1}^{∗}G_{q;ϵ}ω + d_{q}^{∗}d_{q}G_{q;ϵ}ω. (2.5)

Since G_{q;ϵ}ω ∈ Λ_{2}^{q}(M; E)

d_{q−1}^{∗}G_{q;ϵ}ω ∈ Λ_{1}^{q−1}(M; E) and d_{q}G_{q;ϵ}ω ∈ Λ_{1}^{q+1}(M; E). (2.5′)

One concludes that (2.5) is a decomposition of type (HD) for ω_{ϵ} = ω − (P_{q}(ϵ) − P_{q}(0))ω. Let us show that if ω ∈ Λ_{1}^{q}(M; E) then

d_{q−1}^{∗}G_{q;ϵ}ω ∈ Λ_{2}^{q−1}(M; E) and d_{q}G_{q;ϵ}ω ∈ Λ_{2}^{q+1}(M; E).

In view of (2.3),(2.5’) and d · d = 0 it remains to check that

i_{∂_{+}M}^{#}( ∗ d_{q−1}d_{q−1}^{∗}G_{q;ϵ}ω) = 0 and i_{∂_{−}M}^{#}(d_{q}^{∗}d_{q}G_{q;ϵ}ω) = 0. (2.6)

Observe that

∗d_{q−1}d_{q−1}^{∗}G_{q;ϵ}ω = ∗(Δ_{q} − d_{q}^{∗}d_{q})G_{q;ϵ}ω = ∗ω − ∗P_{q}(ϵ)ω ± d_{d−q−1} ∗ d_{q}G_{q;ϵ}ω

and

d_{q}^{∗}d_{q}G_{q;ϵ}ω = (Δ_{q} − d_{q−1}d_{q−1}^{∗})G_{q;ϵ}ω = ω − P_{q}(ϵ)ω − d_{q−1}d_{q−1}^{∗}G_{q;ϵ}ω.

But i_{∂_{+}M}^{#} ∗ ω = 0 and i_{∂_{−}M}^{#}ω = 0 (as ω ∈ Λ_{1}^{q}(M; E)), i_{∂_{+}M}^{#} ∗ P_{q}(ϵ)ω = 0 and i_{∂_{−}M}^{#}P_{q}(ϵ)ω = 0 (as P_{q}(ϵ)ω ∈ Λ_{2}^{q}(M; E)) and i_{∂_{+}M}^{#}d_{d−q−1}∗d_{q}G_{q;ϵ}ω = d_{d−q−1}i_{∂_{+}M}^{#} ∗ d_{q}G_{q;ϵ}ω = 0 and i_{∂_{−}M}^{#}d_{q−1}d_{q−1}^{∗}G_{q;ϵ}ω = d_{q−1}i_{∂_{−}M}^{#}d_{q−1}^{∗}G_{q;ϵ} ∗ ω = 0 (as G_{q;ϵ}ω ∈ Λ_{2}^{q}(M; E)).

Consequently the decompositions (HD) and (HD)_{1} hold for ω_{ϵ} = ω − (P_{q}(ϵ) − P_{q}(0))ω. Note that lim_{ϵ→0} ω_{ϵ} = ω in Λ^{q}(M; E) with respect to the C^{∞}−topology and all spaces in (HD) and (HD)_{1} are closed in this topology. □

Define

Λ^{q}(M, ∂_{−}M; E) := {ω ∈ Λ^{q}(M; E); i_{∂_{−}M}^{#}ω = 0}. (2.7)

Notice that (Λ^{q}(M, ∂_{−}M; E), d_{q}) is a subcomplex. The integration Int^{(q)} on the q-cells of a generalized triangulation τ = (h, g′), which is given by the unstable manifolds of grad_{g′}h, defines an A-linear map

Int^{(q)} : Λ^{q}(M, ∂_{−}M; E) → C^{q}(M, ∂_{−}M, τ, O_{h}; F)

so that δ_{q}Int^{(q)} = Int^{(q+1)}d_{q}. Denote by π_{q} the canonical projection π_{q} : C^{q} → Null (Δ_{q}^{comb}). By a theorem of Dodziuk [Do] of de Rham type, generalized for manifolds with boundary, the map π_{q} Int^{(q)}, restricted to H_{q}, is an isomorphism of Hilbert modules. Denote its inverse by θ_{q}. Define the metric part T_{met} = T_{met}(M, ∂_{−}M, g, τ, F) of the Reidemeister torsion by

log T_{met} := \frac{1}{2} ∑_{q}(−1)^{q} log det_{N}(θ_{q}^{∗}θ_{q}).

log T_{met} is independent of the choice of the orientations O_{h}. The Reidemeister torsion T_{Re} = T_{Re}(M, ∂_{−}M, g, τ, F) is defined as an element in 𝔻 by

log T_{Re} = log T_{comb} + log T_{met}.

In the remaining of this subsection we will discuss the behavior of the torsions with respect to the Poincaré duality and the product formula for torsions.

Suppose M is oriented. Then the orientations O_{h} of the unstable manifolds corresponding to the gradient vector field grad_{g′}h induce orientations on the stable manifolds of grad_{g′}h. These manifolds can be identified with the stable manifolds of grad_{g′}(−h) with the orientations O_{−h}. Since the critical points of index k of h are the critical points of index d − k of −h one obtains isometries of Hilbert modules J_{q}^{comb} : C^{q}(M, ∂_{−}M, τ; F) → C^{d−q}(M, ∂_{+}M, τ_{D}; F) ( which induce Poincaré duality) and verifies that

J_{q+1}^{comb} · δ_{q;τ} = (−1)^{q(d−q)}δ_{d−q−1;τ_{D}}^{∗} · J_{q}^{comb} (2.8)

where δ_{q;τ} resp. δ_{q;τ_{D}} are the differentials in the cochain complexes C(M, ∂_{−}M, τ; F) resp. C(M, ∂_{+}M, τ_{D}; F) and δ_{q;τ}^{∗} denotes the adjoint of δ_{q;τ} . As in section 4 of

[BFKM] denote by \overline{Int}_{τ}^{(q)} the restriction of π_{q,τ} Int_{τ}^{(q)} to H_{q}(M; E), where π_{q,τ} is the orthogonal projection on Null(Δ_{q,τ}^{comb}), and by (\overline{Int}_{τ}^{(q)})^{∗} the adjoint of \overline{Int}_{τ}^{(q)}. Formula (2.8) implies that J_{q}^{comb}(Null(Δ_{q,τ}^{comb})) = Null(Δ_{d−q,τ_{D}}^{comb}) The same arguments as in [BFKM] Proposition 5.15 lead to

(2.9)  J_{q} = (\overline{Int}_{τ_{D}}^{(d−q)})^{∗}J_{q}^{comb}\overline{Int}_{τ}^{(q)}

By using 2.8 and 2.9 one obtains

Proposition 2.2. The following identities hold in 𝔻 :

(i) log T_{comb}(M, ∂_{−}M, τ; F) = (−1)^{d+1} log T_{comb}(M, ∂_{+}M, τ_{D}; F);

(ii) log T_{an}(M, ∂_{−}M, g; F) = (−1)^{d+1} log T_{an}(M, ∂_{+}M, g; F);

(iii) log T_{met}(M, ∂_{−}M, g, τ; F) = (−1)^{d+1} log T_{met}(M, ∂_{+}M, g, τ_{D}; F).

Proposition 2.3. Let (M, ∂_{−}M, ∂_{+}M) be a bordism equipped with a Riemannian metric g, a generalized triangulation τ and a parallel flat bundle F of A-Hilbert modules of finite type. Let N be a closed manifold equipped with a Riemannian metric g_{N}, a generalized triangulation τ_{N} and a parallel flat bundle F_{N} of A_{N}−Hilbert modules of finite type. Denote by \underline{τ} a product triangulation and by \underline{A} the von Neumann algebra \underline{A} = A ˆ⊗A_{N}. Then F ˆ⊗F_{N} is a parallel flat bundle of \underline{A}−Hilbert modules of finite type. If the pairs {(M, ∂_{−}M, ∂_{+}M), F} and {N, F_{N}} are of determinant class, so is the pair {(M × N, ∂_{−}M × N, ∂_{+}M × N), F ˆ⊗F_{N}} and the following equalities hold:

log T_{an}(M × N, (∂_{−}M) × N, g ⊕ g_{N}, F ˆ⊗F_{N}) = χ(M, ∂_{−}M; F)
(2.10)  logT_{an}(N, g; F_{N}) + χ(N; F_{N}) log T_{an}(M, ∂_{−}M, g; F)

log T_{Re}(M × N, (∂_{−}M) × N, g ⊕ g_{N}, \underline{τ}, F ˆ⊗F_{N}) = χ(M, ∂_{−}M; F)
(2.10′)  log T_{Re}(N, g, τ_{N}; F_{N}) + χ(N; F_{N}) log T_{Re}(M, ∂_{−}M, g, τ; F).

The proof of Proposition 2.3 is identical to the one of Proposition 4.1 in [BFKM].

2.2 Determinant class.

Following [BFKM, section 4.1] we introduce the following Definition. (1) The triple {(M, ∂_{−}M, ∂_{+}M), τ, F} is said to be of c-determinant class iff for 0 ≤ k ≤ d

−∞ < ∫_{0^{+}}^{1} log λdN_{Δ_{k}^{comb}}(λ).

(2) The triple {(M, ∂_{−}M, ∂_{+}M), g, F} is said to be of a-determinant class iff for 0 ≤ k ≤ d

−∞ < ∫_{0^{+}}^{1} log λdN_{Δ_{k}}(λ).

As in [BFKM] Proposition 5.6, one can conclude from work of Gromov-Shubin [GS] and using Hellfer Sjöstrand theory (cf. also [Ef1],[Ef2]) that the following result holds:

Proposition 2.4. ([GS]) The triple {(M, ∂_{−}M, ∂_{+}M), g, F} is of a-determinant class iff {(M, ∂_{−}M, ∂_{+}M), τ, F} is of c-determinant class.

Proposition 2.4 allows us to introduce the following

Definition. (i) The pair {(M, ∂_{−}M, ∂_{+}M), F} is said to be of determinant class if there exists a generalized triangulation τ such that {(M, ∂_{−}M, ∂_{+}M), τ, F} is of c-determinant class. (ii) The system (M, ∂_{−}M, ∂_{+}M, F) is said to be of determinant class if the three pairs {(M, ∂_{−}M, ∂_{+}M), F}, {(M, ∅, ∂M), F}, {∂M, F ↾_{∂M}} are of determinant class.

Notice that a system (M, ∂_{−}M, ∂_{+}M, F) being of determinant class implies that the two pairs {∂_{±}M, F ↾_{∂_{±}M}} are of determinant class as well as, by Poincaré duality, that the two pairs {(M, ∂_{+}M, ∂_{−}M), F} and {(M, ∂M, ∅), F} are of determinant class.

Proposition 2.5. (1) If the system (M, ∂_{−}M, ∂_{+}M, F) is of determinant class then the long weakly exact sequences in cohomology with coefficients in F associated to (M, ∂M) and (M, ∂_{±}M) are of determinant class.

(2) Suppose that the systems (M_{i}, ∂_{−}M_{i}, ∂_{+}M_{i}, F_{i}) (i = 1, 2) have the property that (∂_{+}M_{1}, F_{1} ↾_{∂_{+}M_{1}}) = (∂_{−}M_{2}, F_{2} ↾_{∂_{−}M_{2}}). Let (M, ∂_{−}M, ∂_{+}M, F) be the system where M is obtained by glueing M_{1} to M_{2} along ∂_{+}M_{1} = ∂_{−}M_{2} and F defined by F_{i} on M_{i}. If the systems (M_{i}, ∂_{−}M_{i}, ∂_{+}M_{i}, F_{i}) are of determinant class then so is the system (M, ∂_{−}M, ∂_{+}M, F).

(3) Suppose that each connected component of a closed Riemannian manifold N is simply connected and let F̃ denote the pullback of F by the canonical projection of M × N on M. Then the pair {(M, ∂_{−}M, ∂_{+}M), F} is of determinant class iff the pair {(M × N, ∂_{−}M × N, ∂_{+}M × N), F̃} is.

(4) The system (M, ∂_{−}M, ∂_{+}M, F) is of determinant class iff the system (M, ∅, ∂M, F) is of determinant class.

Proof. (1) We verify the statement for (M, ∂_{−}M). Similar arguments can be used for (M, ∂M) and (M, ∂_{+}M). Choose generalized triangulations τ_{−} = (h_{−}, g′_{−}) of ∂_{−}M and τ = (h, g′) of (M, ∂_{−}M, ∂_{+}M). The descending manifolds of grad_{g′_{−}}h_{−} in ∂_{−}M and of grad_{g′}h in M\∂_{−}M provide a cell structure for the space X = S∪∂_{−}M, where S is the union of all open cells of τ (i.e. of all descending manifolds of grad_{g′}h in M\∂_{−}M). By an arbitrary small change of the metric g′_{−} which keeps the CWcomplex structure of ∂_{−}M isomorphic to the unperturbed one, one can arrange that the cell structure is in fact a CW-complex structure, denoted by τ̃. Choose orientations O_{h} resp. O_{h_{−}} for the cells of τ resp. τ_{−}. These orientations define orientations O for the cells of τ̃. Consider the cochain complex (C^{q}(X, τ̃, O; F), δ_{q}) of A -Hilbert modules of finite type associated to the CW-complex (X, τ̃) and F. The short exact sequence of cochain complexes

0 → C^{∗}(M, ∂_{−}M, τ, O_{h}; F) → C^{∗}(X, τ̃, O; F) → C^{∗}(∂_{−}M, τ_{−}, O_{h_{−}}; F ↾_{∂_{−}M}) → 0

induces a long cohomology sequence which is weakly exact, hence a cochain complex. This complex will be called the cohomology sequence of the pair (M, ∂_{−}M). Different generalized triangulations give rise to isomorphic (but not necessarily isometric) cochain complexes. Using subdivisions, one can show that any generalized triangulation τ′ = (h′, g′′) of the bordism (M, ∅, ∂M) produces a cochain complex C^{∗}(M, τ′, O_{h′}; F) of A-Hilbert modules which is homotopy equivalent to C^{∗}(X, τ̃, O; F). By hypotheses, C^{∗}(M, τ′, O_{h′}; F) is of determinant class and therefore, by Proposition 1.3(B), C^{∗}(X, τ̃, O; F) is of determinant class as well. Thus, by Theorem 1.14, the cohomology sequence of the pair (M, ∂_{−}M) is of determinant class.

(2) We verify that the pair {(M, ∅, ∂M), F} is of determinant class. Similar arguments can be applied to the pair {(M, ∂_{−}M, ∂_{+}M), F}. (Notice that, by assumption, {∂M, F ↾_{∂M}} is of determinant class.) Choose generalized triangulations τ_{1} = (h_{1}, g′_{1}) for (M_{1}, ∂_{+}M_{1}, ∂_{−}M_{1}), τ_{2} = (h_{2}, g′_{2}) for (M_{2}, ∂_{−}M_{2}, ∂_{+}M_{2}) and τ_{0} = (h_{0}, g′_{0}) for ∂_{+}M_{1} = ∂_{−}M_{2}. The open cells of τ_{0}, τ_{1}, τ_{2} provide a CW-complex structure τ̃ (if necessary, modify g′_{0} slightly) for the space Y = ∂_{+}M_{1} ∪ S_{1} ∪ S_{2} where S_{1} resp. S_{2} are the union of the open cells of τ_{1} resp. τ_{2}. Choose orientations O_{h_{0}}, O_{h_{1}}, O_{h_{2}} for the cells of τ_{0}, τ_{1}, τ_{2} which define orientations O for the cells of τ̃, and consider the associated cochain complex (C^{q}(Y, τ̃, O; F), δ_{q}) of A−Hilbert modules of finite type. Consider the short exact sequence of cochain complexes

0 → C^{q}(M_{1}, ∂_{+}M_{1}, τ_{1}, O_{h_{1}}; F_{1})⊕C^{q}(M_{2}, ∂_{−}M_{2}, τ_{2}, O_{h_{2}}; F_{2}) → C^{q}(Y, τ̃, O; F) →
→C^{q}(∂_{+}M_{1}, τ_{0}, O_{h_{0}}; F ↾_{∂_{+}M_{1}}) → 0.

Notice that the connecting homomorphisms in the induced cohomology sequence, H^{q}(∂_{+}M_{1}, τ_{0}; F ↾_{∂_{+}M_{1}}) → H^{q+1}(M_{1}, ∂_{+}M_{1}, τ_{1}; F_{1}) ⊕ H^{q+1}(M_{2}, ∂_{−}M_{2}, τ_{2}; F_{2}), are the sum of the connecting homomorphisms in the cohomology sequences of the pairs (M_{1}, ∂_{+}M_{1}) and (M_{2}, ∂_{−}M_{2}) and hence of determinant class by (1) and Proposition 1.1 (C). Therefore, Proposition 1.13 (i) together with the made assumptions implies that the complex C^{∗}(Y, τ̃, O; F) is of determinant class. By Propostion 1.3(B), the homotopy equivalence of C^{∗}(Y, τ̃, O; F) with the cochain complex associated to an arbitrary generalized triangulation of the bordism (M, ∅, ∂M) implies that the pair {(M, ∂_{−}M, ∂_{+}M), F} is of determinant class.

(3) follows from Proposition 1.2.

(4) In view of the definition of a system being of determinant class it suffices to prove that the pair {(M, ∂_{−}M, ∂_{+}M); F} is of determinant class if the system (M, ∅, ∂M, F) is of determinant class. First notice the following two facts:

(i) The disjoint union of two systems of determinant class is a system of determinant class and (ii) if {N, F} is a pair of determinant class (N a closed manifold) and F_{[a,b]} denotes the pullback of F by the projection p : N × [a, b] → N ([a, b] a compact interval on ℝ) then (N × [a, b], N × a, N × b; F_{[a,b]}) is a system of determinant class.

For i = 1, 2 define bordisms (N_{i}, ∂_{−}N_{i}, ∂_{+}N_{i})

(N_{1}, ∂_{−}N_{1}, ∂_{+}N_{1}) := (M, ∅, ∂M) ⊔ (∂_{−}M × [−1, 0], ∂_{−}M × {−1}, ∂_{−}M × {0})

(N_{2}, ∂_{−}N_{2}, ∂_{+}N_{2}) := (∂_{−}M × [0, 1], ∂_{−}M × {0, 1}, ∅)
⊔ (∂_{+}M × [0, 1], ∂_{+}M × {0}, ∂_{+}M × {1})

and whose parallel flat bundle G_{i} of A−Hilbert modules are equal to F on M and to (F ↾_{∂_{±}M})_{[a,b]} on the other components. (Recall that A ⊔ B denotes the disjoint union of two sets A and B.) The statements (i) and (ii) above imply that the systems (N_{i}, ∂_{−}N_{i}, ∂_{+}N_{i}; G_{i}) are of determinant class. Since (∂_{+}N_{1}, G_{1} ↾_{∂_{+}N_{1}}) = (∂_{−}N_{2}, G_{2} ↾_{∂_{−}N_{2}}) one can glue these two systems to obtain a system (N, ∂_{−}N, ∂_{+}N, G) which by (2) is of determinant class. The result follows once we observe that (N, ∂_{−}N, ∂_{+}N, G) is isomorphic to (M, ∂_{−}M, ∂_{+}M, F). □

2.3 Witten deformation of the de Rham complex for manifolds with boundary.

Given a triangulation τ = (h, g′) of M, consider the Witten deformation d_{q}(t) of d_{q},

d_{q}(t) = e^{−th}d_{q}e^{th} = d_{q} + tdh ∧ . (2.11)

Noticing that d_{q−1}^{∗}(t) = (−1)^{d(q−1)+1} ∗ d_{d−q}(−t)∗ we define the spaces Λ_{j;t}^{∗}(M; E) ≡ Λ_{j}^{∗}(M; E)(t) (j = 1, 2) and H_{∗;t}(M; E) by replacing in the definition (2.2) - (2.3) the exterior derivative d_{q} and its adjoint d_{q}^{∗} with d_{q}(t) respectively d_{q}^{∗}(t). We point out that Λ_{1}^{q}(M; E) = Λ_{1;t}^{q}(M; E) is independent of t.

Using the same proof, the above Proposition 2.1 can be generalized as follows:

Proposition 2.1’. For any t ∈ ℝ and 0 ≤ q ≤ d, there exists an orthogonal decomposition of A-pre Hilbert modules

(HD)^{t}Λ^{q}(M; E) = H_{q;t}(M; E) ⊕ closure d_{q−1}(t)(Λ_{1;t}^{q−1}(M; E)) ⊕ closure d_{q}^{∗}(t)(Λ_{1;t}^{q+1}(M; E)),

(HD)_{1}^{t}Λ_{1}^{q}(M; E) = H_{q;t}(M; E) ⊕ closure d_{q−1}(t)(Λ_{2;t}^{q−1}(M; E)) ⊕ closure d_{q}^{∗}(t)(Λ_{2;t}^{q+1}(M; E))

(where the word closure refers to closure with respect to the C^{∞}−topology) and

(\overline{HD})^{t} L_{2}(Λ^{q}(M; E)) = H_{q;t}(M; E) ⊕ L_{2}(d_{q−1}(t)(Λ_{2;t}^{q−1}(M; E))) ⊕ L_{2}(d_{q}^{∗}(t)(Λ_{2;t}^{q+1}(M; E))).

Consider the differential boundary operator

(2.4) B_{q}(t) : Λ^{q}(M; E) → Λ^{q}(∂_{−}M; E) × Λ^{q−1}(∂_{−}M; E) × Λ^{d−q}(∂_{+}M; E) × Λ^{d−q−1}(∂_{+}M; E)
ω ↦ (i_{∂_{−}M}^{#} (ω), i_{∂_{−}M}^{#} (d_{q−1}^{∗}(t)ω), i_{∂_{+}M}^{#} (∗ω), i_{∂_{+}M}^{#} (d_{d−q−1}^{∗}(t)(∗ω))).

One verifies that, for any t ∈ ℝ, (Δ_{q}(t), B_{q}(t)) is an elliptic boundary value problem which is essentially selfadjoint and nonnegative. Hence the spectrum σ_{q}(t) := Spec (Δ_{q}(t), B_{q}(t)) of (Δ_{q}(t), B_{q}(t)) is contained in [0, ∞). Following the proof of Proposition 5.2 in [BFKM] one concludes that there exist constants C > 0, C′ > 0 and t_{0} > 0 so that for t ≥ t_{0} and 0 ≤ q ≤ d,

σ_{q}(t) ∩ (e^{−tC}, C′t) = ∅.

Choose t_{0} so large that

e^{−t_{0}C} < 1 < C′t_{0}

and introduce the orthogonal projections Q_{q}(t) : Λ^{q}(M; E) → Λ_{2;t}^{q}(M; E),

Q_{q}(t) := \frac{1}{2πi} ∫_{S^{1}} (λ − Δ_{q}(t))^{−1}dλ

for t ≥ t_{0}, where (λ − Δ_{q}(t))^{−1} is viewed as a map

(λ − Δ_{q}(t))^{−1} : Λ^{q}(M; E) → Λ_{2;t}^{q}(M; E)

and S^{1} is the circle in the complex λ-plane, centered at 0, with radius 1. Introduce the following A-submodules of Λ_{1}^{q}(M; E),

Λ_{t}^{q}(M; E)_{sm} := Q_{q}(t)(Λ^{q}(M; E))

and

Λ_{t}^{q}(M; E)_{la} := (Id − Q_{q}(t))(Λ_{1}^{q}(M; E)).

Notice that Λ_{t}^{q}(M; E)_{sm} ⊆ Λ_{2;t}^{q}(M; E) is a A-Hilbert submodule of finite von Neumann dimension. Further (Λ_{t}^{q}(M; E)_{sm}, d_{q}(t)) and (Λ_{t}^{q}(M; E)_{la}, d_{q}(t)) is a subcomplex of (Λ^{q}(M, ∂_{−}M; E), d_{q}(t)).

Let us now describe (Λ_{t}^{q}(M; E)_{sm}, d_{q}(t)) in more detail.

Recall that the cochain complex (C^{q} = C^{q}(M, ∂_{−}M, τ, O_{h}; F), δ_{q}) is given by

C^{q} = ⊕_{x∈Cr_{q}(h)}E_{x}

where Cr_{q}(h) = {x_{q;j}; 1 ≤ j ≤ m_{q}} denotes the set of critical points of h which are of index q. Given q and 1 ≤ j ≤ m_{q} := #Cr_{q}(h), denote by e_{q;j,i}(1 ≤ i ≤ ℓ) an orthonormal basis of E_{x_{q;j}} (x_{q;j} ∈ Cr_{q}(h)). (Recall that E_{x} is assumed to be a free A−Hilbert module (cf introduction).) Define E_{q;j,i} ∈ C^{q} by

E_{q;j,i}(x_{q;j′}) = δ_{j′j}e_{q;j,i}

where δ_{j′j} denotes the Kronecker delta. With respect to this basis, δ_{q} is given by

δ_{q}(E_{q;i,j}) = _{∑_{1≤i′≤m_{q+1}} ; 1≤j′≤ℓ}γ_{q;ij,i′j′}E_{q+1;i′j′}.

Generalizing results of Helffer-Sjöstrand, one constructs for t ≥ t_{0} an orthonormal base φ_{q;i,j}(t) of Λ_{t}^{q}(M; E)_{sm} to conclude that Λ_{t}^{q}(M; E)_{sm} is a free A-Hilbert module of finite type.

Let us recall from [BFKM] the notion of a H-neighborhood U_{qj} ≡ U_{x_{q;j}} of a critical point x_{q;j} ∈ Cr_{q}(h):

Definition. U_{qj} ⊆ M\∂M is said to be an H-neighborhood of x_{q;j} if there exist a disc B_{2α} := {x ∈ ℝ^{d}; |x| < 2α} and diffeomorphisms φ : B_{2α} → U_{qj} and Φ : B_{2α} × W → E ↾_{U_{qj}} with the following properties:

(i) φ(0) = x_{q;j};

(ii) when expressed in the coordinates induced by φ, h is of the form

h(x) = h(x_{q;j}) − \frac{1}{2} ∑_{k=1}^{q} x_{k}^{2} + \frac{1}{2} ∑_{k=q+1}^{d} x_{k}^{2};

(iii) the pullback φ^{∗}(g′) of the Riemannian metric g′ is the Euclidean metric; (iv) Φ is a trivialization of E ↾_{U_{qj}}.

For later use, we call the coordinates provided by φ H-coordinates and define U′_{qj} := φ(B_{α}).

A collection (U_{x})_{x∈Cr(h)} of H-neighborhoods is called a system of H-neighborhoods if, in addition, they are pairwise disjoint.

Taking into account that h is not necessarily a self indexing Morse function one obtains the following version of Theorem 5.7 in [BFKM]. (We recall that we assume throughout this section that the W_{j}’s are free A-Hilbert modules.)

Theorem 2.6. Assume (U_{x})_{x∈Cr(h)} is a system of H-neighborhoods such that, for any q and j ≠ j′

U_{x_{q;j}} ∩ W_{q;j′}^{−} = ∅

where W_{q;j′}^{−} denotes the descending manifold associated to the critical point x_{q;j′} and the gradient flow grad_{g′}h. Then there exists t_{1} ≥ t_{0} such that for t > t_{1}, the elements φ_{q;j,i}(t), constructed in [BFKM] with 1 ≤ j ≤ m_{q}, 1 ≤ i ≤ ℓ form an orthonormal basis of the A-Hilbert module Λ_{t}^{q}(M; E)_{sm} with the following properties:

(i) There exist C > 0, η > 0 so that for t ≥ t_{1}, 1 ≤ r ≤ ℓ,

sup_{x∈M\U_{x_{q,j}}}∥φ_{q;j,r}(t)(x)∥ ≤ Ce^{−tη}.

(ii) When expressed in H-coordinates on U_{x_{q;j}} ∩W_{q,j}^{−}, the q-forms φ_{q;j,r}(t) satisfy the following estimate

φ_{q;j,r}(t)(x) = (\frac{t}{π} )^{d/4} e^{−t|x|^{2}/2} (dx_{1} ∧ ⋯ ∧ dx_{q} ⊗ e_{q;j,r} + 0(\frac{1}{t})) .

(iii) With respect to this basis,

Int^{(q)}(e^{th}φ_{q;j,r}(t)) = (\frac{t}{π} )^{\frac{d−2q}{4}} e^{th(x_{q;j})}(E_{q;j,r} + 0(\frac{1}{t})).

(iv) With respect to this basis

d_{q}(t)φ_{q;j,r}(t) = _{∑_{1≤j′≤m_{q+1}} ; 1≤r′≤ℓ}η_{q;jr,j′r′}(t)φ_{q+1;j′,r′}(t)

and the coefficients η_{q;jr,j′r′} satisfy

η_{q;jr,j′r′}(t) = ((\frac{t}{π} )^{1/2}γ_{q;jr,j′r′} + 0(1)) e^{−t(h(x_{q+1,i′})−h(x_{q;i}))}.

Consider the isomorphism of A-Hilbert modules

f_{k}(t) : Λ_{t}^{k}(M; E)_{sm} → C^{k}(M, ∂_{−}M, τ, O_{h}; F)

given by

f_{k}(t)(φ_{k;jr}(t)) := ((\frac{π}{t})^{\frac{d−2k}{4}} e^{−th(x_{q;j})})Int^{(k)}(e^{th}φ_{k;jr}(t)). (2.12)

Then the maps f_{k}(t) provide an isomorphism between (Λ_{t}^{k}(M; E)_{sm}, d^{∼}_{k}(t)) and (C^{k}(M, ∂_{−}M, τ, O_{h}; F), δ_{k}). As a consequence of Theorem 2.6 (iv), f_{k}(t), when expressed with respect to the basis (φ_{k;ir})_{i,r} of Λ_{t}^{k}(M; E)_{sm} and the basis (E_{k;ir})_{i,r} of C^{k}(M, ∂_{−}M, τ, O_{h}; F), is of the form

f_{k}(t) = Id + 0(\frac{1}{t}) (t → ∞).

2.4 Asymptotic expansions and a comparison theorem.

Suppose that (M, ∂_{−}M, ∂_{+}M) is a bordism, g a Riemannian metric, τ = (h, g′) a generalized triangulation and F a parallel flat bundle. Denote by log T (M, ∂_{−}M, g, h, F)(t) the analytic torsion defined by the Laplace operators Δ_{q}(t) associated to d_{q}(t) and the metric g. For t large enough let log T_{sm}(M, ∂_{−}M, g, h, F)(t) and log T_{la}(M, ∂_{−}M, g, h, F)(t) be the torsion of the subcomplex (Λ_{t}^{q}(M; E)_{sm}, d_{q}(t)) respectively (Λ_{t}^{q}(M; E)_{la}, d_{q}(t)). Both, log T (M, ∂_{−}M, g, h, F)(t) and log T_{sm}(M, ∂_{−}M, g, h, F)(t) are elements in the vector space 𝔻 and log T_{la}(M, ∂_{−}M, g, h, F)(t) is a real number. In the case the pair {(M, ∂_{−}M, ∂_{+}M), F} is of determinant class the Laplace operators Δ_{q}(t) associated to d_{q}(t) and the metric g are of determinant class. As in [BFKM] one proves the following two theorems:

Theorem A. Let M = (M, ∂_{−}M, ∂_{+}M) be a bordism, g a Riemannian metric, τ = (h, g) a generalized triangulation and F a parallel flat bundle of A−Hilbert modules so that the pair {(M, ∂_{−}M, ∂_{+}M), F} is of determinant class. Then the following statements hold:

(i) The functions log T (M, ∂_{−}M, g, h, F)(t), log T_{sm}(M, ∂_{−}M, g, h, F)(t) and log T_{la}(M, ∂_{−}M, g, h, F)(t) admit asymptotic expansions for t → ∞ of the form

a_{0} + ∑_{j=1}^{d+1} a_{j}t^{j} + b log t + o(1).

(ii) The asymptotic expansion of log T (M, ∂_{−}M, g, h, F)(t) is of the form

log T (M, ∂_{−}M, g, h, F)(0) − log T_{met}(M, ∂_{−}M, τ, g, F)

+ (log π)( ∑_{q=0}^{d}(−1)^{q} \frac{d − 2q}{4} β_{q}) + ( ∑_{q=0}^{d}(−1)^{q+1} \frac{d − 2q}{4} β_{q}) log t

+ ( ∑_{q=0}^{d}(−1)^{q+1}qβ_{q})t + ∑_{j=1}^{d+1} ( ∑_{q=0}^{d}(−1)^{q}p_{q,j})t^{j} + o(1)

where p_{q,j} can be written as a sum, p_{q,j} = p_{q,j}^{I} + p_{q,j}^{II} with p_{q,j}^{I} being a local term on M and p_{q,j}^{II} being a local term on ∂M.

(iii) The asymptotic expansion of log T_{sm}(M, ∂_{−}M, g, h, F)(t) is of the form

log T_{comb}(M, ∂_{−}M, τ, F) + \frac{1}{2} ∑_{q=0}^{d}(−1)^{q}q(m_{q}ℓ − β_{q})(2t − log t + log π) + o(1).

Theorem B. Assume that the pairs {(M_{j}, ∂_{−}M_{j}, ∂_{+}M_{j}), F_{j}} (j = 1, 2) are of determinant class and τ_{j} = (h_{j}, g_{j}) are generalized triangulations so that there exist neighborhoods U_{j} of ∂M_{j} ∪ Cr(h_{j}) in M_{j} and a diffeomorphism Ψ : U_{1} → U_{2} with Ψ(∂_{±}M_{1}) = ∂_{±}M_{2}, Ψ(Cr_{q}(h_{1})) = Cr_{q}(h_{2}) for 0 ≤ q ≤ d, as well as Ψ^{∗}g_{2} = g_{1} and h_{2} ∘ Ψ = h_{1} on U_{1}.

Then the free term

F T ( log T_{la}(M_{1}, ∂_{−}M_{1}, g_{1}, h_{1}, F_{1})(t) − log T_{la}(M_{2}, ∂_{−}M_{2}, g_{2}, h_{2}, F_{2})(t))

of the asymptotic expansion of

log T_{la}(M_{1}, ∂_{−}M_{1}, g_{1}, h_{1}, F_{1})(t) − log T_{la}(M_{2}, ∂_{−}M_{2}, g_{2}, h_{2}, F_{2})(t)

for t → ∞ is given by

∫_{M_{1}\Cr(h_{1})} a_{0}(h_{1}, ϵ = 0, x_{1}) − ∫_{M_{2}\Cr(h_{2})} a_{0}(h_{2}, ϵ = 0, x_{2})

where a_{0}(h_{1}, ϵ, x_{1}) and a_{0}(h_{2}, ϵ, x_{2}) are densities (forms of degree d) and are given by explicit local formulas; the difference of the two integrals has to be taken in the same way as in [BFK, section 0, Remarks after Theorem 3].

Introduce

\begin{matrix}A & (M_{1}, M_{2}, τ_{1}, τ_{2}, F_{1}, F_{2}) \\ := ∫_{M_{1}\Cr(h_{1})} a_{0}(h_{1}; ϵ = 0, x_{1}) − ∫_{M_{2}\Cr(h_{2})} a_{0}(h_{2}; ϵ = 0, x_{2}). \end{matrix}

Corollary C. Assume in addition to the hypotheses in Theorem B that there exists a system (M_{3}, ∂_{−}M_{3}, ∂_{+}M_{3}, F_{3}) with the following properties (j = 1, 2):

(i) ∂_{±}M_{3} = ∂_{±}M_{j} and F_{3} ↾_{∂_{±}M_{3}}= F_{j} ↾_{∂_{±}M_{j}}; (ii) the pair {N_{j}, F̃_{j}} is of determinant class where N_{j} denotes the closed manifold obtained by gluing M_{j} to M_{3} and F̃_{j} is given by F_{j} on M_{j} and by F_{3} on M_{3}. Then

A(M_{1}, M_{2}, τ_{1}, τ_{2}, F_{1}, F_{2}) = 0

Proof. Let a be the minimum of h_{1} (which is also the minimum of h_{2}) and let b be its maximum. Choose a smooth function h_{3} : M_{3} → ℝ with h_{3}(M_{3}) ⊂ [a − 1, b + 1] as well as h_{3} ↾_{∂_{−}M_{3}}= a, h_{3} ↾_{∂_{+}M_{3}}= b (cf. Figure 1), and a metric g_{3} on M_{3} so that, for i = 1, 2, h_{i} together with h_{3} defines a Morse function h̃_{i} on N_{i} and g_{i} together with g_{3} defines a metric g̃_{i} on N_{i}. Further choose g_{3} (if necessary, modify g_{3} in an arbitrary small neighborhood of ∂M_{3}) so that (h̃_{i}, g̃_{i}) is a generalized triangulation for N_{i}. Since {N_{i}, F_{i}}, (i = 1, 2) are, by assumption, of determinant class, one concludes from Theorem A

F T ( log T (N_{1}, g̃_{1}, h̃_{1}, F̃_{1})(t) − log T (N_{2}, g̃_{2}, h̃_{2}, F̃_{2})(t))
= log T_{an}(N_{1}, g̃_{1}, h̃_{1}, F̃_{1}) − log T_{an}(N_{2}, g̃_{2}, h̃_{2}, F̃_{2})
− log T_{met}(N_{1}, τ̃_{1}, g̃_{1}, F̃_{1}) + log T_{met}(N_{2}, τ̃_{2}, g̃_{2}, F̃_{2})

D. BURGHELEA, L. FRIEDLANDER, T. KAPPELER 43

and

F T ( log T_{sm}(N_{1}, g̃_{1}, h̃_{1}, F̃_{1})(t) − log T_{sm}(N_{2}, g̃_{2}, h̃_{2}, F̃_{2})(t))
= log T_{comb}(N_{1}, g̃_{1}, F̃_{1}) − log T_{comb}(N_{2}, τ̃_{2}, F̃_{2}).

As N_{1} and N_{2} are both closed manifolds of determinant class we conclude from [BFKM] that

log T_{an}(N_{j}, g̃_{j}, F̃_{j}) = log T_{comb}(N_{j}, τ̃_{j}, F̃_{j})
+ log T_{met}(N_{j}, τ̃_{j}, g̃_{j}, F̃_{j}).

Combining the above equalities with the identity

F T ( log T (N_{j}, g̃_{j}, h̃_{j}, F̃_{j})(t)) = F T ( log T_{sm}(N_{j}, g̃_{j}, h̃_{j}, F̃_{j})(t))

+ F T ( log T_{la}(N_{j}, g̃_{j}, h̃_{j}, F̃_{j})(t))

one obtains, with A defined as above,

A(N_{1}, N_{2}, τ̃_{1}, τ̃_{2}, F̃_{1}, F̃_{2}) = 0.

In view of the fact that A is local, one concludes that

A(M_{1}, M_{2}, τ_{1}, τ_{2}, F_{1}, F_{2}) = 0.

□

44 TORSIONS FOR MANIFOLDS WITH BOUNDARY...

Consider the pair {(M, ∂_{−}M, ∂_{+}M), F} and assume that it is of determinant class. Define R(M, ∂_{−}M, g, τ, F) by

log R(M, ∂_{−}M, g, τ, F) := log T_{an}(M, ∂_{−}M, g, F) − log T_{Re}(M, ∂_{−}M, τ, g, F).

Our aim is to show that the ratio R depends only on the data on the boundary at least in the case when the system (M, ∂_{−}M, ∂_{+}M, F) is of determinant class.

First observe that the result concerning the metric anomaly of the analytic torsion [BFKM, Lemma 6.11] extends to manifolds with boundary:

Proposition 2.8. Assume that the pair {(M, ∂_{−}M, ∂_{+}M), F} is of determinant class and that g(s), a ≤ s ≤ b, is a smooth 1-parameter family of Riemannian metrics on M so that g(s) is independent of s in a collar neighborhood of ∂M. Then

\frac{d}{ds} log T_{an}(M, ∂_{−}M, g(s), F) = \frac{d}{ds} log T_{met}(M, ∂_{−}M, g(s), τ, F).

Proof. Arguing as in [RS] one has

\begin{aligned}\frac{d}{ds} & log T_{an}(M, ∂_{−}M, g(s), F) \\ & = \frac{d}{ds} log T_{met}(M, ∂_{−}M, g(s), τ, F) + ∑_{q=0}^{d}(−1)^{q}q̂ρ_{q} \end{aligned}

where ρ̂_{q} = ∫_{M}ρ_{q}(x, s) and ρ_{q}(x, s) is a density (i.e. a d-form) on M, which can

be calculated in terms of the symbol of Δ_{q}(s) acting on Λ^{q}(M; E). Due to our assumption that g(s) does not depend on s near the boundary, ρ_{q}(x, s) vanishes for x near the boundary. If d = dim M is odd one can then use a parity argument to

conclude that ρ_{q}(x, s) ≡ 0 for 0 ≤ q ≤ d. If d = dim M is even, ∑^{d} ; q=0(−1)^{q}qρ̂_{q} = 0.

Indeed take the double of M and use that log T_{an} and log T_{met} are 0 for closed manifolds of even dimension.

Corollary 2.9. Suppose that the pair {(M, ∂_{−}M, ∂_{+}M), F}) is of determinant class and that τ = (h, g′) is a generalized triangulation. If g_{1} and g_{2} are two Riemannian metrics on M which agree on ∂M, then

R(M, ∂_{−}M, g_{1}, τ, F) = R(M, ∂_{−}M, g_{2}, τ, F).

Theorem 2.10. Assume that, for j = 1, 2, M_{j} = (M_{j}, ∂_{−}M_{j}, ∂_{+}M_{j}) is a bordism equipped with a Riemannian metric g_{j}, a generalized triangulation τ_{j} and a parallel flat bundle F_{j} = (E_{j}, ∇_{j}) of A-Hilbert modules of finite type on M_{j}.Further assume that the systems (M_{j}, ∂_{−}M_{j}, ∂_{+}M_{j}, F_{j}) are of determinant class and

(∂_{±}M_{1}, g_{1} ↾_{∂_{±}M_{1}}, F_{1} ↾_{∂_{±}M_{1}}) = (∂_{±}M_{2}, g_{2} ↾_{∂_{±}M_{2}}, F_{2} ↾_{∂_{±}M_{2}}).

Then

R(M_{1}, ∂_{−}M_{1}, g_{1}, τ_{1}, F_{1}) = R(M_{2}, ∂_{−}M_{2}, g_{2}, τ_{2}, F_{2}).

Proof. The proof proceeds in several steps.

(I) In the case where the hypotheses of Corollary C are satisfied the claim follows from Theorem A.

(II) Next assume that only the hypotheses of Theorem B are satisfied. Consider S^{2} equipped with a generalized triangulation τ_{S^{2}} = (h_{0}, g_{0}) and denote by \underline{τ}_{j} = (\underline{h}_{j}, \underline{g}′_{j}) a product triangulation on M_{j} × S^{2}. We claim that, for j = 1, 2, (M_{j} × S^{2}, g_{j} ×g_{0}, \underline{τ}_{j}, \underline{F_{j}}), where \underline{F_{j}} is the pullback of F_{j} by the projection M_{j} ×S^{2} → M_{j}, satisfies the assumptions of Corollary C. Indeed denote by D^{3} the unit disc in ℝ^{3} and choose for M_{3} the disjoint union of ∂_{−}M_{1} × D^{3} with ∂_{+}M_{1} × D^{3}. Then ∂_{±}M_{3} = ∂_{±}M_{1} × S^{2}. Let F_{3} be the pull back of F_{1} ↾_{∂M_{1}} by the projection ∂M_{1} × D^{3} → ∂M_{1}. Since the pairs {∂_{±}M_{i}, F_{i} ↾_{∂_{±}M_{i}}} are of determinant class, the system (M_{3}, ∂_{−}M_{3}, ∂_{+}M_{3}, F_{3}) is of determinant class and therefore, by Proposition 2.3, the pairs {N_{i}, F̃_{i}} (as defined in Corollary C) are of determinant class. One concludes then by (I) that

R(M_{1} × S^{2}, ∂_{−}M_{1} × S^{2}, g_{1} × g_{0}, \underline{τ}_{1}, \underline{F_{1}}) = R(M_{2} × S^{2}, ∂_{−}M_{2} × S^{2}, g_{2} × g_{0}, \underline{τ}_{2}, \underline{F_{2}}).

By applying Proposition 2.5 (3) and Proposition 2.3 (formulas (2.10)- (2.10’)), one obtains

R(M_{1}, ∂_{−}M_{1}, g_{1}, τ_{1}, F_{1}) = R(M_{2}, ∂_{−}M_{2}, g_{2}, τ_{2}, F_{2}).

(III) Applying Corollary 2.9 it suffices to prove the statement in the case where g_{j} and τ_{j} = (h_{j}, g′_{j}) (j = 1, 2) have the additional property that g_{j} = g′_{j}.

(IV) (A) It suffices to prove the result under the additional hypothesis that χ(M_{1}, ∂_{−}M_{1}) = χ(M_{2}, ∂_{−}M_{2}). Indeed, if dimM_{1} is odd, χ(M_{1}, ∂_{−}M_{1}) = χ(M_{2}, ∂_{−}M_{2}). If dimM_{1} is even, then

χ(M_{1}, ∂_{−}M_{1}) = χ(M_{2}, ∂_{−}M_{2}) + 2k.

If, in addition, k > 0 we replace, without loss of generality, M_{1} by the disjoint union of M_{2} with k copies of (S^{d}, F), where F is the trivial parallel flat bundle with the same fiber as F_{1}. (Observe that R = 0 for (S^{d}, F). ) In the case when k < 0 interchange the role of M_{1} and M_{2}.

(B) Under the additional hypotheses χ(M_{1}, ∂_{−}M_{1}) = χ(M_{2}, ∂_{−}M_{2}) the result can be proved as follows: By the invariance of the Reidemeister torsion under a subdivision of a generalized triangulation (cf. Lemma 6.12 in [BFKM] for the case of a closed manifold) it suffices to prove that there exist subdivisions τ̂_{j} of τ_{j} so that the assumptions of Theorem B are satisfied. But such subdivisions exist if χ(M_{1}, ∂_{−}M_{1}) = χ(M_{2}, ∂_{−}M_{2}) (cf. [BFKM]). □

3. Applications to torsions.

3.1 Comparison of analytic and Reidemeister torsion.

In this subsection we provide a formula for the ratio of analytic and Reidemeister torsion of a compact Riemannian manifold with boundary as introduced at the end of section 2.

We begin by some auxiliary considerations. Assume that (M, g) is a closed Riemannian manifold, τ = (h, g′) a generalized triangulation, and F = (E, ∇) a parallel flat bundle on M. Consider the cylinder M_{I} = M × I where I is a compact interval [a, b] in ℝ with a, b ∈ 4ℤ. Denote by g_{0} the standard metric on [a, b] and by τ_{i} = (h_{i}, g_{0}), (i = 1, 2), the generalized triangulations of the bordisms ([a, b], \{a\}, \{b\}) resp. ([a, b], ∅, \{a, b\}) with h_{1}(x) = x resp. h_{2} = \frac{1}{2}(x − (b + a)/2)^{2}. ( To satisfy condition (T1) in the definition of a generalized triangulation, one can perturb h_{2} slightly so that it is linear near the boundary of the interval.) Notice that the generalized triangulation τ_{1} has no cells. Denote by T the trivial 1-dimensional complex vector bundle with trivial connection. T is a parallel flat bundle over [a, b]. An easy calculation gives

(3.1)  log T_{comb}(I, \{a\}, τ_{1}, T ) = 0, log T_{an}(I, \{a\}, g_{0}, T ) = \frac{1}{2} log 2,
log T_{met}(I, {a}, τ_{1}, g_{0}, T ) = 0,

(3.1′) log T_{comb}(I, ∅, τ_{2}, T ) = 0, log T_{an}(I, ∅, g_{0}, T ) = \frac{1}{2}(log 2 + log(b − a)),

log T_{met}(I, ∅, τ_{2}, g_{0}, T ) = \frac{1}{2} log(b − a).

Assume that the pair {M, F} is of determinant class. Then, by Proposition 2.3, the systems (M_{I}, ∅, M ×∂I, F_{I}) and (M_{I}, M ×{a}, M ×{b}, F_{I}) are of determinant class (where F_{I} is the pull back of F by the projection M × I → M) and

(3.2)  log T_{comb}(M_{I}, M × {a}, τ ⊕ τ_{1}, F_{I}) = 0,

log T_{an}(M_{I}, M × \{a\}, g ⊕ g_{0}, F_{I}) = χ(M; F)\frac{log 2}{2}

log T_{met}(M_{I}, M × {a}, g ⊕ g_{0}, τ ⊕ τ_{2}, F_{I}) = 0,

(3.2′) log T_{comb}(M_{I}, ∅, τ ⊕ τ_{2}, F_{I}) = log T_{comb}(M, τ, F),
log T_{an}(M_{I}, ∅, g ⊕ g_{0}, F_{I}) = \frac{χ(M; F)}{2}(log 2 + log(b − a)) + log T_{an}(M, g, F),
log T_{met}(M_{I},∅, g ⊕ g_{0}, τ ⊕ τ_{2}, F_{I}) = \frac{χ(M; F)}{2} log(b − a) + log T_{met}(M, g, τ, F)

(where we have used that χ(I, ∅) = 1 and χ(I, {a}) = 0).

Since M is closed and the pair {M, F}, is of determinant class one obtains, by [BFKM],

log T_{an}(M, g, F) = log T_{Re}(M, g, τ, F).

Combining with the formulas (3.2)-(3.2’), one obtains

log R(M_{I}, M × \{a\}, g ⊗ g_{0}, τ ⊕ τ_{1}, F_{I}) = \frac{χ(M; F) log 2}{2}. (3.3)

log R(M_{I}, ∅, g ⊗ g_{0}, τ ⊕ τ_{2}, F_{I}) = \frac{χ(M; F) log 2}{2}. (3.3′)

Theorem 3.1. Assume that the system (M, ∂_{−}M, ∂_{+}M, F) is of determinant class. Then

log R(M, ∂_{−}M, g, τ, F) = \frac{1}{4}χ(∂M; F) log 2. (3.4)

Remark. In the case where A = ℝ, the result, as it stands, is due independently to Lück ([Lü1]) and Vishik [Vi] (cf. also [Ch]). However, even in the case A = ℝ, the proof presented here is new, elementary and short.

Proof. The proof proceeds in two steps.

(A) We first consider the case where M = (M, ∅, ∂M). Take a \underline{disjoint} union M ⊔ M of two copies of M and consider the bordism (M ⊔M, ∅, (∂M)⊔(∂M)). Further introduce (∂M)_{I} := ∂M×I and consider the bordism ((∂M)_{I}, ∅, ∂((∂M)_{I})). Notice that ∂((∂M)_{I}) = (∂M × {0}) ⊔ (∂M × {1}) = (∂M) ⊔ (∂M). Since by hypothesis (∂M, F ↾_{∂M}) is of determinant class so is (∂M × I, ∅, (F ↾_{∂M})_{I}) (cf Proposition 2.5(3)). Therefore, by Theorem 2.10,

log R(∂M × I, ∅, g_{I}, τ_{I}, (F ↾_{∂M})_{I}) = log R(M ⊔ M, ∅, g ⊔ g, τ ⊔ τ, F ⊔ F)

where (F ↾_{∂M})_{I} is defined as above, g_{I} := g ↾_{∂M} ⊕g_{0} and τ_{I} is a product triangulation. Notice that

log R(M ⊔ M, ∅, g ⊔ g, τ ⊔ τ, F ⊔ F) = 2 log R(M, ∅, g, τ, F),

and, by (3.3’),

log R(∂M × I, ∅, g_{I}, τ_{I}, F_{I} ↾_{∂M}) = \frac{χ(∂M; F)}{2} log 2.

Combining the three equalities above, we obtain

log R(M, ∅, g, τ, F) = \frac{χ(∂M; F)}{4} log 2

and statement (3.4) is proved in the case where ∂_{−}M = ∅.

(B) In view of Proposition 2.3 it suffices to check (3.4) for the system (M × S^{2}, ∂_{−}M × S^{2}, ∂_{+}M × S^{2}, F̃) which is, by Proposition 2.5 (3), of determinant class because (M, ∂_{−}M, ∂_{+}M, F) is. Here F̃ denotes the pullback of F by the projection M × S^{2} → M. In view of Theorem 2.10 it suffices to prove the statement (3.4) for the two systems (∂_{+}M × D^{3}, ∅, ∂_{+}M × S^{2}, (F ↾_{∂_{+}M})˜ ) and (∂_{−}M × D^{3}, ∂_{−}M × S^{2}, ∅, (F ↾_{∂_{−}M})˜ ) where (F ↾_{∂_{±}M})˜ denotes the pullback of F ↾_{∂_{±}M} by the projections ∂_{±}M × D^{3} → ∂_{±}M. For the first system it suffices to apply (A) and for the second system, (3.4) follows from Proposition 2.2 (Poincaré duality) and (A). □

3.2. Glueing formulas (Part I).

In this subsection we present a gluing formula for the analytic and Reidemeister torsions. Let (M_{j}, ∂_{−}M_{j}, ∂_{+}M_{j}), be two bordisms equipped with Riemannian metrics g_{j}, generalized triangulations τ_{j} = (h_{j}, g′_{j}) and parallel flat bundles F_{j} = (E_{j}, ∇_{j}). Suppose that there exists an isometry

ω : (∂_{+}M_{1}, g_{1} ↾_{∂_{−}M_{1}}) → (∂_{−}M_{2}, g_{2} ↾_{∂_{+}M_{2}}) (3.5)

and a connection preserving bundle isomorphism Φ above ω which makes the following diagram commutative

\begin{matrix}E_{1} ↾_{∂_{−}M_{1}} & ⟶^{Φ} & E_{2} ↾_{∂_{+}M_{2}} \\ ↓ & & ↓ \\ ∂_{−}M_{1} & ⟶^{ω} & ∂_{+}M_{2}. \end{matrix} (3.6)

Then one can form a bordism (M := M_{1} ∪_{ω} M_{2}, ∂_{−}M_{1}, ∂_{+}M_{2}) by gluing ∂_{+}M_{1} to ∂_{−}M_{2} by ω and a parallel flat bundle F by gluing F_{1} and F_{2} by (ω, Φ). The metrics g_{1}, g′_{1} and g_{2}, g′_{2} determine Riemannian metrics g, g′ on M, and the functions h_{1} and h_{2} determine the C^{∞}-function h : M → ℝ given by

h(x) := \{ \begin{matrix}h_{1}(x) & (x ∈ M_{1}) \\ b_{1} − a_{2} + h_{2}(x) & (x ∈ M_{2}) \end{matrix} (3.7)

where, for j=1,2, h_{j}(M_{j}) = [a_{j}, b_{j}]. Notice that if τ = (h, g′) is not a generalized triangulation (i.e. violates (T4)) one can modify, by an arbitrary small perturbation and localized in a given neighborhood of ∂_{+}M_{1}, the metric g′_{1} to g^{∼}′_{1} so that the triangulation τ^{∼}_{1} := (h_{1}, g^{∼}′_{1}) is compatible with τ_{2} and τ^{∼}_{1} and τ_{1} provide the same relative CW-complex structure. Therefore, without loss of generality, we may assume that τ = (h, g′) is a generalized triangulation of (M, ∂_{−}M_{1}, ∂_{+}M_{2}).

Choose orientations O_{h} for M. They induce orientations O_{h_{i}} for M_{i}. Now consider the short exact sequence of cochain complexes

0 → C^{∗}(M_{2}, ∂_{−}M_{2}, τ_{2}, O_{h_{2}}; F_{2}) →^{i_{∗}} C^{∗}(M, ∂_{−}M_{1}, τ, O_{h}; F)

→^{r_{∗}} C^{∗}(M_{1}, ∂_{−}M_{1}, τ_{1}, O_{h_{1}}; F_{1}) → 0, (3.8)

where i_{∗} is the map which extends the cochains, defined on the cells of M_{2}, to cochains defined on all cells of M by assigning the value zero on the cells of M_{1} and where r_{∗} is the map which restricts the cochains, defined on the cells of M, to the cells of M_{1}. The sequence (3.8) induces a long weakly exact sequence H_{comb}(τ) in cohomology of A- Hilbert modules (cf (1.52))

⋯ → H^{q}(M_{2}, ∂_{−}M_{2}, τ_{2}; F_{2}) → H^{q}(M, ∂_{−}M_{1}, τ; F) → H^{q}(M_{1}, ∂_{−}M_{1}, τ_{1}; F_{1}) →

H^{q+1}(M_{2}, ∂_{−}M_{2}, τ_{2}; F_{2}) → ⋯ (3.9)

A similar sequence (depending on the Riemannian metric g), denoted by H_{an}(g), can be obtained using de Rham cohomology H^{q}(M, ∂_{−}M, g; F) instead of the combinatorial one,

⋯ → H^{q}(M_{2}, ∂_{−}M_{2}, g_{2}; F_{2}) → H^{q}(M, ∂_{−}M, g; F) → H^{q}(M_{1}, ∂_{−}M_{1}, g_{1}; F_{1}) →

H^{q+1}(M_{2}, ∂_{−}M_{2}, g_{2}; F_{2}) → ⋯ (3.10)

As we have seen in section 2, the integration theory provides isomorphisms (θ_{j})^{−1} of A-Hilbert modules from de Rham cohomology to the combinatorial cohomology. Moreover the (θ_{j}) define an isomorphism of cochain complexes from H_{comb}(τ) to H_{an}(g).

Theorem 3.2. Assume that, for i=1,2, the system (M_{i}, ∂_{−}M_{i}, ∂_{+}M_{i}, F_{i}) is of determinant class. Then the following statements hold:

The system(M, ∂_{−}M, ∂_{+}M, F) and the complexes H_{comb}, H_{an}

(i) are of determinant class;

(ii)  log T_{Re}(M, ∂_{−}M, g, τ, F) = 2 ; ∑_{j=1} log T_{Re}(M_{j}, ∂_{−}M_{j}, g_{j}, τ_{j}, F_{j})
+ log T (H_{an}).

(iii) log T_{an}(M, ∂_{−}M, g, F) = ∑_{j=1}^{2} log T_{an}(M_{j}, ∂_{−}M_{j}, g_{j}, F_{j}) + log T (H_{an})

− \frac{χ(∂_{+}M_{1}; F_{1} ↾_{∂_{+}M_{1}})}{2} log 2.

Remark. In the case where A = ℝ, the result (ii) is due to Milnor [Mi]. Work related to (i) can be found in [LL]. In the case where A = ℝ, (iii) is due to Vishik [Vi] . Vishik’s proof, however, is completely different than ours.

Proof. (i) Proposition 2.5 implies that the system (M, ∂_{−}M, ∂_{+}M, F) is of determinant class. Theorem 1.14 applied to the short exact sequence of cochain complexes (3.8) implies that H_{comb} is of determinant class. In view of the fact that θ defines an isomorphism of cochain complexes, Proposition 1.3 implies that H_{an} is of determinant class as well.

(ii) Concerning the torsion of the sequence (3.8), notice that

log T (0 → C^{q}(M_{2}, ∂_{−}M_{2},τ_{2}, O_{h_{2}}; F) →^{i_{q}} C^{q}(M, ∂_{−}M, τ, O_{h}; F)

→^{r_{q}} C^{q}(M_{1}, ∂_{−}M_{1}, τ_{1}, O_{h_{1}}; F) → 0) = 0

since log V ol(i_{q}) = 0 and log V ol(r_{q} ↾ (Null(r_{q})^{⊥}) = 0. Theorem 1.14 implies that

(3.11) log T_{comb}(M, ∂_{−}M,τ, F) =

2 ; ∑_{j=1} log T_{comb}(M_{j}, ∂_{−}M_{j}, τ_{j}, F_{j}) + log T (H_{comb}).

From Proposition 1.3 (C) and in view of the fact that θ is an isomorphism of cochain complexes of A−Hilbert modules one obtains

(3.12)  log T_{met}(M, ∂_{−}M, g, τ, F) = 2 ; ∑_{j=1} log T_{met}(M_{j}, ∂_{−}M_{j}, g_{j}, τ_{j}, F_{j})
+ log T (H_{an}) − log T (H_{comb}).

Combining (3.11) and (3.12) one obtains (ii).

(iii) follows from (ii) and Theorem 3.1. □

3.3. Glueing formulas (Part II).

In this subsection we extend Theorem 3.2 for partial glueing, i.e. to a situation where not necessarily all of the components of ∂_{+}M_{1} and ∂_{−}M_{2} are glued together.

Suppose that (M_{i}, ∂_{−}M_{i}, ∂_{+}M_{i}; F_{i}), i = 1, 2 are two systems with ∂_{+}M_{1} resp. ∂_{−}M_{2} consisting of two disjoint components ∂_{+}M_{1} = V_{0} ⊔ V_{1}, resp. ∂_{−}M_{2} = W_{0} ⊔W_{1}. Assume that there exist a diffeomorphism ω_{0} : V_{0} → W_{0} and a connection preserving isomorphism above ω_{0}, Φ_{0} : F_{1} ↾_{V_{0}}→ F_{2} ↾_{W_{0}} . Then M_{1} and M_{2} can be glued together to give rise to the bordism (M, ∂_{−}M, ∂_{+}M) with M := M_{1} ∪_{ω} M_{2} and boundary components ∂_{−}M = ∂_{−}M_{1} ⊔ W_{1} and ∂_{+}M = V_{1} ⊔ ∂_{+}M_{2}. Using Φ_{0}, F_{1} can be glued with F_{2} to obtain a parallel flat bundle F on M. Suppose, in addition, that Riemannian metrics g_{1} on M_{1} and g_{2} on M_{2} are given so that ω^{#}(g_{2} ↾_{W_{0}}) = g_{1} ↾_{V_{0}} . The metrics g_{1} and g_{2} define a Riemannian metric g on M. One then obtains the following (weakly exact) sequence H in de Rham cohomology

⋯ →H^{q}(M_{2}, ∂_{−}M_{2}, g_{2}; F_{2}) → H^{q}(M, ∂_{−}M, g; F) →

→ H^{q}(M_{1}, ∂_{−}M_{1}, g_{1}; F_{1}) → H^{q+1}(M_{2}, ∂_{−}M_{2}, g_{2}; F_{2}) → ⋯ (3.19)

This sequence is the cohomology sequence induced by the exact sequence of cochain complexes (cf (2.7))

0 → Λ^{∗}(M_{2}, ∂_{−}M_{2}; F_{2}) → Λ^{∗}(M, ∂_{−}M; F) → Λ^{∗}(M_{1}, ∂_{−}M_{1}; F_{1}) → 0. (3.20)

(To be completely correct, Λ^{q}(M_{2}, ∂_{−}M_{2}; F_{2}) should be the smaller space of smooth q−forms which are the restriction to M_{2} of q−forms which are defined on all of M, but vanish on M_{1}.)

Theorem 3.2′. If the systems (M_{i}, ∂_{−}M_{i}, ∂_{+}M_{i}, F_{i}) (i=1,2) are of determinant class then so are the system (M, ∂_{−}M, ∂_{+}M, F) and the cohomology sequence H, given by (3.19). Moreover

log T_{an}(M, ∂_{−}M, g, F) = ∑_{j=1}^{2} log T_{an}(M_{j}, ∂_{−}M_{j}, g_{j}, F_{j}) + log T (H)

− χ(V_{0}; F_{1} ↾_{V_{0}})\frac{log 2}{2}.

Proof. We will derive the stated results from Theorem 3.2. For that purpose introduce, for ϵ > 0, the systems (N_{i,ϵ}, ∂_{−}N_{i,ϵ}, ∂_{+}N_{i,ϵ}, G_{i,ϵ}), i = 1, 2, defined by

N_{1,ϵ} := W_{1} × [−ϵ, 0], ∂_{−}N_{1,ϵ} := W_{1} × {−ϵ}, ∂_{+}N_{1,ϵ} := W_{1} × {0};
N_{2,ϵ} := V_{1} × [0, ϵ], ∂_{−}N_{2,ϵ} := V_{1} × {0}, ∂_{+}N_{2,ϵ} := V_{1} × {ϵ};
G_{1,ϵ} := (F_{2} ↾_{W_{1}})_{[−ϵ,0]}, G_{2,ϵ} := (F_{1} ↾_{V_{1}})_{[0,ϵ]},

and let g_{i,ϵ} be the metrics on N_{1,ϵ} defined by g_{1,ϵ} := g_{2} ↾_{W_{1}} ⊕g_{0} and g_{2,ϵ} := g_{1} ↾_{V_{1}} ⊕g_{0} (where g_{0} is the standard Euclidean metric) (see Figure 2). Denote by (M_{i,ϵ}, ∂_{−}M_{i,ϵ}, ∂_{+}M_{i,ϵ}, F_{i,ϵ}) the disjoint union of the systems (M_{i}, ∂_{−}M_{i}, ∂_{+}M_{i}, F_{i}) and (N_{i,ϵ}, ∂_{−}N_{i,ϵ}, ∂_{+}N_{i,ϵ}, G_{i,ϵ}) and let ω_{ϵ} : ∂_{+}M_{1,ϵ} := V_{0} ⊔ V_{1} ⊔ W_{1} → ∂_{−}M_{2,ϵ} := W_{0} ⊔ V_{1} ⊔ W_{1} be the diffeomorphism which is equal to ω_{0} on V_{0} and to the identity on V_{1} ⊔ W_{1}. Moreover ω_{0} and Φ_{0} induce an isomorphism of parallel flat bundles

Φ_{ϵ} : F_{1,ϵ} ↾_{∂_{+}M_{1,ϵ}}→ F_{2,ϵ} ↾_{∂_{−}M_{2,ϵ}} .

Following the glueing instructions described in subsection 3.2, one forms the system (M_{ϵ}, ∂_{−}M_{ϵ}, ∂_{+}M_{ϵ}, F_{ϵ}) and defines the Riemannian metric g_{ϵ} given by g_{i} on M_{i} and g_{i,ϵ} on N_{i,ϵ}. There exists a commutative diagram connecting the cohomology sequence H, defined in (3.19), with the sequence (3.10) for the bordism (M_{ϵ}, ∂_{−}M_{ϵ}, ∂_{+}M_{ϵ}), which we denote by H_{ϵ},

H^{q}(M_{2}, ∂_{−}M_{2}, g_{2}; F_{2})  → H^{q}(M, ∂_{−}M, g; F) →  H^{q}(M_{1}, ∂_{−}M_{1}, g_{1}; F_{1})
↓ (i_{2,ϵ})_{q}  ↓ (i_{ϵ})_{q}  ↓ (i_{1,ϵ})_{q}
H^{q}(M_{2,ϵ}, ∂_{−}M_{2,ϵ}, g_{2,ϵ}; F_{2,ϵ}) → H^{q}(M_{ϵ}, ∂_{−}M_{ϵ}, g_{ϵ}; F_{ϵ}) → H^{q}(M_{1,ϵ}, ∂_{−}M_{1,ϵ}, g_{1,ϵ}; F_{1,ϵ}).

In this diagram, the horizontal lines represent the cohomology sequence (3.19) and (3.10) respectively and the vertical arrows are isomorphisms of Hilbert modules of finite type. The isomorphisms (i_{1,ϵ})_{q}, (i_{ϵ})_{q} and (i_{2,ϵ})_{q} can be described as the composition

(i_{1,ϵ})_{q} := (i′_{1,ϵ})_{q} · ((i′′_{1,ϵ})_{q})^{−1}, (i_{2,ϵ})_{q} := (i′_{2,ϵ})_{q} · ((i′′_{2,ϵ})_{q})^{−1},

(i_{ϵ})_{q} = (i′_{ϵ})_{q} · ((i′′_{ϵ})_{q})^{−1}

where (i′_{k,ϵ})_{q} and (i′′_{k,ϵ})_{q} are induced by the inclusions (k=1,2)

i′_{k,ϵ} : (M_{k}, ∂_{−}M_{k}) → (M_{k,ϵ}, ∂_{−}M_{k,ϵ} ∪ N_{k,ϵ}) and
i′′_{k,ϵ} : (M_{k,ϵ}, ∂_{−}M_{k,ϵ}) → (M_{k,ϵ}, ∂_{−}M_{k,ϵ} ∪ N_{k,ϵ})

and where (i′_{ϵ})_{q} and (i′′_{ϵ})_{q} are induced by the inclusions

i′_{ϵ} :(M_{,}∂_{−}M) → (M_{ϵ}, ∂_{−}M_{ϵ} ∪ N_{1,ϵ}) and
i′′_{ϵ} : (M_{ϵ}, ∂_{−}M_{ϵ}) → (M_{ϵ}, ∂_{−}M_{ϵ} ∪ N_{1,ϵ}).

Since the harmonic forms, which represent the cohomology classes in H^{∗}(M_{k}, ∂_{−}M_{k}, g_{k}; F_{k}) and H^{∗}(M_{k,ϵ}, ∂_{−}M_{k,ϵ}, g_{k,ϵ}; F_{k,ϵ}) are in fact the same, the homomorphisms (i_{k,ϵ})_{q} are isometries for any ϵ (k=1,2). Notice that one can provide a smooth family of diffeomorphisms φ_{ϵ} : M → M_{ϵ}, with φ_{0} = id so that φ_{ϵ}^{#}(g_{ϵ}) = g̃_{ϵ} is a smooth family of Riemannian metrics on M with g̃_{0} = g. This implies that

lim_{ϵ→0} log T_{an}(M_{ϵ}, ∂_{−}M_{ϵ}, g_{ϵ}, F_{ϵ}) = log T_{an}(M, ∂_{−}M, g, F).

Using φ_{ϵ}, one can also show that

lim_{ϵ→0} log V ol((i_{ϵ})_{q}) = 0.

In view of Proposition 1.3 A(iii) H is of determinant class iff H_{ϵ} is of determinant class (for one and then for any ϵ > 0) and if so

log T (H) = log T (H_{ϵ}) + ∑_{q}(−1)^{q} log V ol(i_{ϵ})_{q}.

Note that if the systems (M_{i}, ∂_{−}M_{i}, ∂_{+}M_{i}, F_{i}), i = 1, 2, are of determinant class, then the systems (M_{i,ϵ}, ∂_{−}M_{i,ϵ}, ∂_{+}M_{i,ϵ}, F_{i,ϵ}), i = 1, 2, are of determinant class as well. From (3.2) one obtains

log T_{an}(M_{i,ϵ}, ∂_{−}M_{i,ϵ}, g_{i,ϵ}, F_{i,ϵ}) = log T_{an}(M_{i}, ∂_{−}M_{i}, g_{i}, F_{i}) + χ_{i} · \frac{log 2}{2}

with χ_{1} = χ(W_{1}; F_{2} ↾_{W_{1}}) and χ_{2} = χ(V_{1}; F_{1} ↾_{V_{1}}). Then by Theorem 3.2

log T_{an}(M_{ϵ}, ∂_{−}M_{ϵ}, g_{ϵ},F_{ϵ}) = ∑_{i=1,2} log T_{an}(M_{i}, ∂_{−}M_{i}, g_{i}, F_{i})+

+ log T (H_{ϵ}) − χ(V_{0}, F_{1} ↾_{V_{0}}) · \frac{log 2}{2}

The result follows by passing to the limit ϵ → 0. □

3.4. Comparison of the analytic torsion for different boundary conditions.

Suppose (M, g) is a Riemannian manifold whose boundary ∂M is a disjoint union of three components ∂_{1}M, ∂_{2}M, and ∂_{3}M and F is a parallel flat bundle of A−Hilbert modules on M. In this subsection we will compare the analytic torsions of (M, ∂_{1}M, g, F) and (M, ∂_{1}M ∪ ∂_{3}M, g, F).

Introduce g_{3} := g ↾_{∂_{3}M} and F_{3} := F ↾_{∂_{3}M} and denote by H′ the cohomology sequence

⋯ → H^{q}(M, ∂_{1}M ∪ ∂_{3}M, g; F) → H^{q}(M, ∂_{1}M, g; F) →

H^{q}(∂_{3}M, g_{3}; F_{3}) → H^{q+1}(M, ∂_{1}M ∪ ∂_{3}M, g; F) → ⋯ (3.21)

which is induced by the short exact sequence

0 → Λ^{∗}(M, ∂_{1}M ∪ ∂_{3}M; F) → Λ^{∗}(M, ∂_{1}M; F) → Λ^{∗}(∂_{3}M; F_{3}) → 0.

Theorem 3.3. The system (M, ∂_{1}M, ∂_{2}M ∪ ∂_{3}M, F) is of determinant class iff the system (M, ∂_{1}M ∪ ∂_{3}M, ∂_{2}M, F) is of determinant class and if so

log T_{an}(M,∂_{1}M, g, F) = log T_{an}(M, ∂_{1}M ∪ ∂_{3}M, g, F) + log T (H′)+
(3.22)  logT_{an}(∂_{3}M, g_{3}, F_{3}).

Remark. For A = ℂ, the result, as it stands, is implicit in Vishik [Vi3]. Again his proof is very different from ours.

Proof. The first statement follows from Proposition 2.5 (4). To prove (3.22), we apply Theorem 3.2′ (partial gluing theorem) to the systems

(M_{1}, ∂_{−}M_{1}, ∂_{+}M_{1}, F_{1}) := (M, ∂_{1}M, ∂_{2}M ∪ ∂_{3}M, F)

and, for ϵ > 0,

(M_{2}, ∂_{−}M_{2}, ∂_{+}M_{2}, F_{2}) := (∂_{3}M × [0, ϵ], ∂_{3}M × {0, ϵ}, ∅, F_{[0,ϵ]})

where F_{[0,ϵ]} is the pull back of F ↾_{∂_{3}M} by the projection ∂_{3}M × [0, ϵ] → ∂_{3}M. We regard M_{1} equipped with the metric g and M_{2} with the metric g_{3} ⊕ g_{0} (g_{0} the Euclidean metric). Further, using the same notation as in subsection 3.3,

V_{0} = W_{0} := ∂_{3}M, V_{1} := ∂_{2}M and W_{1} := ∂_{3}M.

In view of Proposition 2.3 and (3.1′) we have

log T_{an}(M_{2}, ∂_{−}M_{2}, g_{2}, F_{2}) = − log T_{an}(∂_{3}M, g_{3}, F_{3})
(3.23)  +\frac{1}{2}χ(∂_{3}M; F_{3}) · (log 2 + log ϵ).

Denote the system obtained by partial gluing by (N_{ϵ}, ∂_{−}N_{ϵ}, ∂_{+}N_{ϵ}, F_{ϵ}) and the resulting metric by g_{ϵ}. Then, by Theorem 3.2’ and (3.23),

(3.24)  log T_{an}(N_{ϵ}, ∂_{−}N_{ϵ}, g_{ϵ}, F_{ϵ}) = log T_{an}(M, ∂_{1}M, g, F) + log T ((H(ϵ))
− log T_{an}(∂_{3}M,g_{3}, F_{3}) + \frac{1}{2}χ(∂_{3}M, F_{3}) · log ϵ,

where H(ϵ) is the cohomology sequence (3.19) for the partial gluing of (M_{1}, ∂_{−}M_{1}, ∂_{+}M_{1}, F_{1}) and (M_{2}, ∂_{−}M_{2}, ∂_{+}M_{2}, F_{2}). Notice that there exists a commutative diagram connecting the cohomology sequence H′ (cf (3.21)) and the cohomology sequence H(ϵ)

H^{q−1}(∂_{3}M, g_{3}; F_{3})  → H^{q}(M, ∂_{1}M ∪ ∂_{3}M, g; F) → H^{q}(M, ∂_{1}M, g; F) ⋯
↓ (r_{2,ϵ})_{q}  ↓ (r_{ϵ})_{q}  ↓ Id
H^{q}(M_{2}, ∂_{−}M_{2}, g_{3} ⊕ g_{0}; F_{[0,ϵ]}) →  H^{q}(N_{ϵ}, ∂_{−}N_{ϵ}, g_{ϵ}; F_{ϵ})  → H^{q}(M, ∂_{1}M, g; F) · · · .

In this commutative diagram the vertical arrows are isomorphisms of A−Hilbert modules. The maps (r_{ϵ})_{q} are induced by a family of smooth retractions r_{ϵ} : (N_{ϵ}, ∂_{−}N_{ϵ}) → (M, ∂_{1}M ∪ ∂_{3}M) with r_{ϵ=0} = id which depends smoothly on ϵ and has the property that the restriction of r_{ϵ} to M \U_{ϵ} is the identity where U_{ϵ} is a collar neighborhood of ∂_{3}M of size ϵ. The maps ((r_{2,ϵ})_{q}), (q ≥ 1), are isomorphisms and induced by the maps, assigning to ω ∈ Λ^{q−1}(∂_{3}M; F_{3}) the wedge product dt ∧ ω. One can easily see that

lim_{ϵ→0} log V ol(r_{ϵ})_{q} = 0 (3.25)

and

log V ol((r_{2,ϵ})_{q}) = \frac{1}{2} log ϵ · dim_{N}H^{q−1}(∂_{3}M, g_{3}; F_{3}). (3.26)

From Proposition 1.3 A(iii), using (3.26), we conclude that

− log T (H(ϵ)) = log T (H′) + \frac{1}{2}χ(∂_{3}M; F_{3}) log ϵ + ∑(−1)^{q} log V ol(r_{ϵ})_{q}.

(3.27)

Notice that one can provide a smooth family of diffeomorphisms φ_{ϵ} : M_{ϵ} → M, φ_{0} = id so that φ_{ϵ}^{#}(g_{ϵ}) = g̃_{ϵ} is a smooth family of Riemannian metrics with g̃_{0} = g. This implies that

lim_{ϵ→0} log T_{an}(N_{ϵ}, ∂_{−}N_{ϵ}, g_{ϵ}, F_{ϵ}) = log T_{an}(M, ∂_{1}M ∪ ∂_{3}M, g, F)

The result then follows from (3.24), (3.25) and (3.27) by passing to the limit ϵ → 0. □ Appendix: On manifolds of determinant class.

To state our result let us introduce the following notation. Let M be a compact, connected manifold (possibly with boundary) with fundamental group π_{1}(M). Let Γ be a group and θ an arbitrary group homomorphism, θ : π_{1}(M) → Γ. Denote by W_{0} the (N (Γ), π_{1}(M)^{op})−Hilbert module ℓ_{2}(Γ) with the π_{1}(M)^{op}−action induced by θ and the right translation of Γ. Denote by F_{θ} the parallel flat bundle of N (Γ)−Hilbert modules of finite type, induced by W_{0}. Notice that this is the parallel flat bundle induced by the principal covering M̃ → M, defined by θ.

Theorem A. If Γ is residually finite then for any bordism (M, ∂_{−}M, ∂_{+}M), the pair {(M, ∂_{−}M, ∂_{+}M), F_{θ}} is of determinant class. As a consequence, the system (M, ∂_{−}M, ∂_{+}M, F_{θ}) is of determinant class.

Although Lück does not use the notion of determinant class, Theorem A is implicit in his paper [Lü3]. Unfortunately, there are a number of misleading misprints and his definition of the L^{2}-determinant [Lü3, Definition, p. 471] is incorrect. For the convenience of the reader we present an outline of Lück’s arguments to prove Theorem A.

We recall that a group Γ is residually finite if there exists a sequence (Γ_{m})_{m≥1} of nested normal subgroups of Γ, Γ_{0} = Γ ⊇ Γ_{1} ⊇ Γ_{2} ⊇ …, such that

(i) ∩_{m≥1}Γ_{m} = {e}; (ii) γ_{m} := #(Γ : Γ_{m}) < ∞ (m ≥ 1).

Further recall that (C^{∗}(M, τ, O_{h}, F_{θ}), δ_{∗}) is of determinant class if, for 0 ≤ q ≤ d,

−∞ < ∫_{0^{+}}^{1} log λdN_{q}(λ),

where N_{q}(λ) = N_{q}^{comb}(λ) denotes the L_{2}-combinatorial spectral distribution function of the combinatorial q-Laplacian Δ_{q} = Δ_{q}^{comb}.

Let M_{m} be the principal (Γ/Γ_{m})- finite cover of M induced by the composition of θ with the projection Γ → Γ/Γ_{m} and denote by Δ_{m;q} ≡ Δ_{m;q}^{comb} the combinatorial q-Laplacians on M_{m}. Further let P_{m;q}(λ) ≡ P_{m;q}^{comb}(λ) be the spectral family associated to Δ_{m;q}, defined in such a way that it is right continuous, and introduce the corresponding normalized spectral distribution function

N_{m;q}(λ) = \frac{1}{γ_{m}} trP_{m;q}(λ).

We point out N_{m;q}(λ) are step functions. Denote by det′Δ_{m;q} the modified determinant of Δ_{m,q}, i.e. the product of all \underline{non zero} eigenvalues of Δ_{m;q}. Let a_{m;q} be the smallest non zero eigenvalue and b_{m;q} the largest eigenvalue of Δ_{m;q}. Then, for any a and b, such that 0 < a < a_{m;q} and b > b_{m;q},

\frac{1}{γ_{m}} log det′Δ_{m;q} = ∫_{a}^{b} log λdN_{m;q}(λ). (A.1)

Integrating by parts, the Stieltjes integral ∫_{a}^{b} log λdN_{m;q}(λ) can be written as

(A.2)  ∫_{a}^{b} log λdN_{m;q}(λ) =(log b)(N_{m;q}(b) − N_{m;q}(0))

− ∫_{a}^{b} \frac{N_{m;q}(λ) − N_{m;q}(0)}{λ}dλ.

Recall that the spectral distribution function N_{q}(λ) ≡ N_{q}^{comb}(λ) is given by

N_{q}(λ) := tr_{N (Γ)}P_{q}(λ)

with tr_{N (Γ)}P_{q}(λ) denoting the von Neumann trace where P_{q}(λ) is the spectral family corresponding to Δ_{q} defined in such a way that it is right continuous with respect to λ. Notice that N_{q}(λ) is continuous to the right in λ. Denote by det′Δ_{q} the modified determinant of Δ_{q}, given by the following Stieltjes integral,

log det′Δ_{q} = ∫_{0^{+}}^{b} log λdN_{q}(λ) := lim_{ϵ→0^{+}} ∫_{ϵ}^{b} log λdN_{q}(λ)

with 1 ≤ b < ∞ chosen in such a way that |||Δ_{q}||| < b (operator norm).

Notice that log det′(Δ_{q}) ∈ [−∞, ∞) and recall that Δ_{q} is said to be of determinant class if ∫_{0^{+}}^{b} log λdN_{q}(λ) ∈ (−∞, ∞).

Integrating by parts, one obtains

(A.3) log det′(Δ_{q}) = log b(N_{q}(b) − N_{q}(0))

+ lim_{ϵ→0^{+}}\{(− log ϵ)(N_{q}(ϵ) − N_{q}(0)) − ∫_{ϵ}^{b} \frac{N_{q}(λ) − N_{q}(0)}{λ}dλ\}.

Using that lim inf_{ϵ→0^{+}}(− log ϵ)(N_{q}(ϵ) − N_{q}(0)) ≥ 0 (in fact, this limit exists and is

zero) and \frac{N_{q}(λ)−N_{q}(0)}{λ} ≥ 0 for λ > 0, one sees that

log det′(Δ_{q}) ≥ (log b)(N_{q}(b) − N_{q}(0)) − ∫_{0}^{b} \frac{N_{q}(λ) − N_{q}(0)}{λ}dλ. (A.4)

Proof of Theorem A.2.

The main ingredient is Lück’s estimate of log det′(Δ_{q}) in terms of log det′Δ_{m,q} combined with the fact that log det′Δ_{m,q} ≥ 0 as the determinant det′Δ_{m,q} is integer valued. By [Lü, Lemma 2.5], there exists 1 ≤ b < ∞ such that, for m ≥ 1,

|||Δ_{m,q}||| ≤ b; |||Δ_{q}||| ≤ b.

By [Lü, Lemma 3.3.1],

∫_{0}^{b} \frac{N_{q}(λ) − N_{q}(0)}{λ}dλ ≤ lim inf_{m→∞} ∫_{0}^{b} \frac{N_{m;q}(λ) − N_{m;q}(0)}{λ}dλ. (A.5)

Combining (A.1) and (A.2) with the inequalities log det′Δ_{m;q} ≥ 0, it follows that

∫_{0}^{b} \frac{N_{m;q}(λ) − N_{m;q}(0)}{λ}dλ ≤ (log b)(N_{m;q}(b) − N_{m;q}(0)). (A.6)

¿From (A.4) - (A.6) we then conclude that

(A.7)  log det′Δ_{q} ≥(log b)(N_{q}(b) − N_{q}(0))

− lim inf_{m→∞}(log b)(N_{m;q}(b) − N_{m;q}(0)).

The estimate (A.7) together with the identities ([Lü3, Theorem 2.3.1])

N_{q}(λ) = lim_{ϵ→0^{+}} lim inf_{m→∞} N_{m;q}(λ + ϵ) ([Lü3, Theorem 2.3.1])

and

N_{q}(0) = lim_{m→∞} N_{m;q}(0) ([Lü3, Theorem 2.3.2])

yield that log det′Δ_{q} ≥ 0 which proves that Δ_{q} is of determinant class. Since q is arbitrary this shows that the pair {(M, ∂M_{−}, ∂M_{+}), F_{θ}} is of determinant class.

□

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