“Torsion and fibrations” by

Wolfgang Lück, Thomas Schick and Thomas Thielmann

Abstract

We study the behaviour of analytic torsion under smooth fibrations. Namely, let F → E ⟶^{f} B be a smooth fiber bundle of connected closed oriented smooth manifolds and let V be a flat vector bundle over E. Assume that E and B come with Riemannian metrics and V comes with a unimodular (not necessarily flat) Riemannian metric. Let ρ_{an}(E; V) be the analytic torsion of E with coefficients in V, let ρ_{an}(F_{b}; V) be the analytic torsion of the fiber over b with coefficients in V restricted to F_{b} and let Pf_{B} be the Pfaffian dim(B)-form. Let H_{dR}^{q}(F; V) be the flat vector bundle over B whose fiber over b ∈ B is H_{dR}^{q}(F_{b}; V) with the Riemannian metric which comes from the HodgedeRham decomposition and the Hilbert space structure on the space of harmonic forms induced by the Riemannian metrics. Let ρ_{an}(B; H_{dR}^{q}(F; V)) be the analytic torsion of B with coefficients in this bundle. The Leray-Serre spectral sequence for deRham cohomology determines a certain correction term ρ_{dR}^{LS}(f). We prove

ρ_{an}(E; V) = ∫_{B} ρ_{an}(F_{b}; V) · Pf_{B} + ∑_{q}(−1)^{q} · ρ_{an}(B; H_{dR}^{q}(F; V)) + ρ_{dR}^{LS}(f).

This formula simplifies in special cases such as bundles with S^{n} as fiber or base, in which case the correction terms ρ_{dR}^{LS}(f) reduces to the torsion of the associated Gysin or Wang sequence, resp.

AMS-classification number: 57Q10, 55R20, 55T10

0. Introduction

Let M be a connected closed smooth manifold with Riemannian metric and V be a flat vector bundle with a not necessarily flat Riemannian metric. The definition of analytic torsion due to Ray and Singer [20] for an orthogonal representation V, or, equivalently, for a flat Riemannan metric on V, still makes sense in the setting above [1, page 35] and [18, page 730]. Namely, let ζ_{p}(s) be the zeta-function of the Laplace operator Δ^{p} : Ω^{p}(E; V) ⟶ Ω^{p}(E; V) which is for Re(s) >> 0 the holomorphic function ∑_{λ>0} λ^{−s} where λ runs over the positive eigenvalues of Δ^{p}. It has a meromorphic extension to the complex plane which is analytic in zero. Define

ρ_{an}(E; V) := \frac{1}{2} · ∑_{q≥0}(−1)^{q} · q · ζ′_{q}(0) ∈ ℝ.  (0.1)

We want to study it for smooth fiber bundles. The main result of this paper is Theorem 0.2 Let F → E ⟶^{f} B be a smooth fiber bundle of connected closed oriented smooth manifolds and let V be a flat vector bundle over E. Assume that E and B come with Riemannian metrics. Suppose that dim(E) is odd and V is unimodular and comes with an arbitrary Riemannian metric or that dim(E) is even and V comes with a unimodular Riemannian metric. Then

ρ_{an}(E; V) = ∫_{B} ρ_{an}(F_{b}; V) · Pf_{B} + ∑_{q}(−1)^{q} · ρ_{an}(B; H_{dR}^{q}(F; V)) + ρ_{dR}^{LS}(f).

Here are some explanations of the assumptions and the formula in Theorem 0.2.

For a path w in E the fiber transport gives a linear isomorphism V_{w} : V_{w(0)} ⟶ V_{w(1)} which depends only on the homotopy class relative endpoints of w since V is flat. We call V unimodular if for one (and hence all) e ∈ E and all loops w with base point e we get | det(V_{w} : V_{e} ⟶ V_{e})| = 1. We call a Riemannian metric on V unimodular if for any path w in E we get det(V_{w}^{∗} ∘ V_{w} : V_{w(0)} ⟶ V_{w(0)}) = 1 where V_{w}^{∗} is the adjoint of V_{w} with respect to the Hilbert space structure on the fibers of V given by the Riemannian metric. This is a weaker condition than being a flat Riemannian metric what would mean that V_{w} is always an isometry. Notice that V is unimodular if and only if it carries a unimodular Riemannian metric.

Of course ρ_{an}(E; V) is just the analytic torsion with respect to the given Riemannian metrics on E and V. These induce also a metric on H_{dR}^{p}(E, V). Each fiber F_{b} = p^{−1}(b) inherits a Riemannian metric from E by restriction. We denote the restriction of V to F_{b} again by V. Hence ρ_{an}(F_{b}; V) is defined and is a smooth function in b ∈ B.

Let Pf_{B} be the Pfaffian dim(B)-form on the oriented Riemannian manifold B. It is a representative of the Euler class of B in Chern-Weil theory and satisfies by the Gauss-Bonnet theorem

∫_{B} Pf_{B} = χ(B)

where χ(B) is the Euler characteristic. If dim(B) is odd, then Pf_{B} is defined to be zero.

Let H^{q}(F_{b}; V) be the space of harmonic q-forms, i.e. the kernel of the Laplace operator Δ^{q} : Ω^{q}(F_{b}; V) ⟶ Ω^{q}(F_{b}; V). It inherits an inner product from the Riemannian metrics on F_{b} and V. The harmonic Hilbert structure on the deRham cohomology H_{dR}^{q}(F_{b}; V) is the Hilbert space structure for which the canonical Hodge isomorphism

H^{q}(F_{b}; V) ⟶ H_{dR}^{q}(F_{b}; V)

is isometric. Thus we get a Riemannian metric on the canonical flat vector bundle H_{dR}^{q}(F; V) whose fiber over b ∈ B is H_{dR}^{q}(F_{b}; V). Hence the analytic torsion ρ_{an}(B; H_{dR}^{q}(F; V)) is defined, and H_{dR}^{p}(B, H_{dR}^{q}(F, V)) inherits a natural Hilbert space structure.

There is the following natural descending filtration of the deRham complex Ω^{∗}(E; V). Define F_{p}Ω^{n}(E; V) to be those n-forms with coefficients in V which can be written as finite sums of n-forms on E with coefficients in V of the shape ω ∧ f^{∗}η for ω ∈ Ω^{n−k}(E; V) and η ∈ Ω^{k}(B) for some k ≥ p. This filtration is compatible with the differential since d(ω ∧ f^{∗}η) = d(ω) ∧ f^{∗}η ± ω ∧ f^{∗}d(η). The associated spectral cohomology sequence is the Leray-Serre spectral sequence for deRham cohomology, which we recall in Section 4.

Part of the Leray-Serre spectral sequence for deRham cohomology is the filtration of the cohomology H^{n}(E; V)

{0} = F^{n+1,−1} ⊂ ⋯ ⊂ F^{p+1,n−p−1} ⊂ F^{p,n−p} ⊂ ⋯ ⊂ F^{0,n} = H_{dR}^{n}(E; V),

the natural identification of the E_{2}-term

V_{2}^{p,q} : H_{dR}^{p}(B; H_{dR}^{q}(F; V)) ⟶^{≅} E_{2}^{p,q},  (0.3)

the identification of the cohomology of the r-th term of the spectral sequence with the (r + 1)-th term and the identification of the E_{∞}-term with the filtration quotients

\begin{aligned}ϕ_{r}^{p,q} : & H^{0}(E_{r}^{p+r∗,q−(r−1)∗}) & ⟶^{≅} & E_{r+1}^{p,q}, \\ ψ^{p,q} : & F^{p,q}/F^{p+1,q−1} & ⟶^{≅} & E_{∞}^{p,q}. \end{aligned}

For r sufficiently large, the differentials in E_{r}^{∗,∗} are trivial.

Next we explain the term ρ_{dR}^{LS}(f) appearing in Theorem 0.2.

For a linear isomorphism f : V → W of finite-dimensional real Hilbert spaces, set

[[f]] := \frac{1}{2} ln(| det(f^{∗}f)|) ∈ ℝ  (0.4)

Let C = C^{∗} be an acyclic finite Hilbert cochain complex. Define

ρ(C) := [[(c^{∗} + γ^{∗}) : C^{ev} ⟶ C^{odd}]] ∈ ℝ  (0.5)

where c^{∗} is the differential and γ^{∗} a chain contraction. If f : C ⟶ D is a chain homotopy equivalence of finite Hilbert cochain complexes, cone(f) is the cochain complex with n-th differential

( \begin{matrix}c^{n} & 0 \\ f^{n} & −d^{n−1} \end{matrix} ) : C^{n} ⊕ D^{n−1} ⟶ C^{n+1} ⊕ D^{n}.

It is acyclic and we define

t(f) := ρ(cone(f)).  (0.6)

Let C be a finite Hilbert cochain complex such that H(C^{∗}) carries a Hilbert structure. There is up to homotopy precisely one chain map i : H(C) ⟶ C whith H(i) = id, where we consider H(C) as a cochain complex with the trivial differential. Define

ρ(C) := −t(i) ∈ ℝ.  (0.7)

In the Leray-Serre spectral sequence, equip E_{2}^{p,q} with the Hilbert space structure which makes V_{2}^{p,q} to an isometry. Equip inductively E_{r}^{p+r∗,q−(r−1)∗}, H(E_{r}^{p+r∗,q−(r−1)∗}) (r ≥ 2) and E_{∞}^{p,q} with the Hilbert sub- and quotient structures. In particular, ϕ_{r}^{p,q} become isometries. Equip F^{p,q} ⊂ H_{dR}^{p+q}(E, V) and F^{p,q}/F^{p+1,q−1} with the Hilbert sub- and quotient structures Now define

ρ_{dR}^{LS}(f) := ∑_{r≥2} ∑_{p=0}^{r−1} ∑_{q}(−1)^{p+q} · ρ(E_{r}^{p+r∗,q−(r−1)∗}) − ∑_{p,q}(−1)^{p+q} · [[ψ^{p,q}]]

This number depends only on f and the Riemannian metrics on E, B and V.

In general the correction term ρ_{dR}^{LS}(f) is very involved and is as complicated as the Leray-Serre spectral cohomology sequence is. However, there are cases where ρ_{dR}^{LS}(f) and the whole formula in Theorem 0.2 are easy to understand. Namely, we will prove under the assumptions of Theorem 0.2 the following three corollaries. The first one generalizes a result of Fried [7] for orthogonal V.

Corollary 0.8 Suppose that H_{dR}^{q}(F; V) vanishes for all q. Then χ(B) · ρ_{an}(F_{b}; V) is independent of b and

ρ_{an}(E; V) = χ(B) · ρ_{an}(F_{b}; V).

Corollary 0.9 Suppose that F is S^{n} and V = f^{∗}W for a flat vector bundle with Riemannian metric over B. Let G^{∗} be the acyclic cochain complex of finite-dimensional Hilbert spaces given by the Gysin sequence

… \underrightarrow{∫} H_{dR}^{p}(B; W) ⟶^{∧e(f)} H_{dR}^{p+n+1}(B; W) ⟶^{f^{∗}} H_{dR}^{p+n+1}(E; V) \underrightarrow{∫} H_{dR}^{p+1}(B; W) ⟶^{∧e(f)} …

where ∧e(f) is the product with the Euler class e(f) ∈ H_{dR}^{n+1}(B) of the sphere bundle, ∫ is integration over the fiber and G^{1} = H_{dR}^{0}(E; V). Then the torsion ρ(G^{∗}) ∈ ℝ is defined and we get:

ρ_{an}(E; V) = χ(S^{n}) · ρ_{an}(B; W) + ρ(G^{∗}).

The condition V = f^{∗}W is no loss of generality, provided that n ≥ 2 or that f induces an isomorphism π_{1}(E) ⟶ π_{1}(B).

Corollary 0.10 Suppose that B = S^{n}. Let W^{∗} be the acyclic cochain complex of finitedimensional Hilbert spaces given by the Wang sequence and the harmonic structures for some b ∈ B:

… ⟶ H_{dR}^{q−1}(E; V) ⟶ H_{dR}^{q−1}(F_{b}; V) ⟶ H_{dR}^{q−n}(F_{b}; V) ⟶ H_{dR}^{q}(E; V) ⟶ …

where W^{1} = H_{dR}^{0}(E; V). Then

ρ_{an}(E; V) = χ(S^{n}) · ρ_{an}(F_{b}; V) + ρ(W^{∗}).

We make some remarks about the proof of Theorem 0.2. It will depend on the following deep results of Bismut-Zhang [1] and Müller [18]. In the sequel M is a connected closed oriented Riemannian manifold and V is a flat vector bundle over M with Riemannian metric. We have introduced the analytic torsion ρ_{an}(M; V) above. Its topological counterpart

ρ_{top}(M; V) ∈ ℝ  (0.11)

is the Milnor torsion of M with respect to some triangulation and the harmonic Hilbert structure on cohomology which we will recall in Definition 3.3.

Theorem 0.12 If the Riemanniann metric on V is unimodular, then

ρ_{an}(M; V) = ρ_{top}(M; V).

Let g_{M} and \overline{g_{M}} be two Riemannian metrics on M and g_{V} and \overline{g_{V}} be two arbitrary Riemannian metrics on V. Let ρ_{an}(M; V) and \overline{ρ_{an}(M; V)} be the analytic torsion with respect to (g_{M}, g_{V}) and (\overline{g_{M}}, \overline{g_{V}}). Analogously we denote the Hilbert spaces H_{dR}^{p}(M; V) and \overline{H_{dR}^{p}(M; V)} equipped with the harmonic Hilbert structures with respect to (g_{M}, g_{V}) and (\overline{g_{M}}, \overline{g_{V}}) and the Hilbert spaces V_{x} and \overline{V_{x}} equipped with the Hilbert structure with respect to g_{V} and \overline{g_{V}}. Denote by Pf_{M} the Pfaffian with respect to g_{M}. Let P̃f(M, g_{M}, \overline{g_{M}}) be the Chern-Simons n − 1-form. Its image under the differential is the difference of the two Pfaffians of M with respect to g_{M} and \overline{g_{M}}. Let θ(V, \overline{g_{V}}) be the closed 1-form defined in [1, Definition 4.5]. It measures the deviation of \overline{g_{V}} from being unimodular and vanishes if \overline{g_{V}} is unimodular.

Theorem 0.13 We get under the conditions and in the notations above.

1. If dim(M) is odd, then

ρ_{an}(M; V) − \overline{ρ_{an}(M; V)} = − ∑_{p}(−1)^{p} · [[H_{dR}^{p}(M; V) ⟶^{id} \overline{H_{dR}^{p}(M; V)}]];

2. If dim(M) is even, then

ρ_{an}(M; V) − \overline{ρ_{an}(M; V)} = − ∑_{p}(−1)^{p} · [[H_{dR}^{p}(M; V) ⟶^{id} \overline{H_{dR}^{p}(M; V)}]]

+ ∫_{M}[[V_{x} ⟶^{id} \overline{V_{x}}]] · Pf_{M} − ∫_{M} θ(V, \overline{g_{V}}) · P̃f(M, g_{M}, \overline{g_{M}}).

Theorem 0.12 and Theorem 0.13 for odd-dimensional M are due to Müller [18, Theorem 1 and Theorem 2.6], who generalizes Cheeger’s and his proof [3] and [17] of the Ray-Singer Conjecture ρ_{an}(M; V) = ρ_{top}(M; V) for orthogonal representations V to the unimodular setting. Bismut and Zhang [1, Theorem 0.1 and Theorem 0.2] have generalized Müller’s work to all dimensions and to a setting where V is not necessarily unimodular. We say more about their version of Theorem 0.12 in Section 6.

Theorem 0.12 and Theorem 0.13 enable us to show that Theorem 0.2 follows from its topological version. This will be done in Section 6 where we also explain how Corollaries 0.8, 0.9 and 0.10 follow from Theorem 0.2 and its topological version. In the topological case we can treat a more general setting. Namely, we consider a fibration f : E ⟶ B such that B is a connected finite CW -complex, the homotopy fiber has the homotopy type of a finite CW -complex and a certain cohomology class θ_{f} ∈ H^{1}(B; Wh(E)) vanishes. Then we get for a local coefficient system V with unimodular Hilbert structure, and fixed Hilbert structures on the relevant singular cohomology groups:

Theorem 5.4

ρ(E; V) = χ(B) · ρ(F; V) + ρ(B; ∑_{q}(−1)^{q} · H_{sing}^{q}(F; V)) + ρ_{sing}^{LS}(f)

Freed [6] obtained results about torsion and spectral sequences similar to those of section 4. The latter are used to deduce the topological fibration formula. There is also work of Maumary [13] about Whitehead torsion and spectral sequences.

The topological version includes manifolds with boundary and the question whether Theorem 0.2 generalizes to manifolds with boundary comes down to the question whether Theorem 0.12 and Theorem 0.13 generalize to manifolds with boundary. At least under the condition that V is an orthogonal representation Cheeger and Müller’s results for closed manifolds have been extended to manifolds with boundary (see [8], [9] and [11]).

There are also interesting generalizations of the topological versions to other settings. For example one can consider an L^{2}-version. Or one can substitute V by a local coefficient system of finitely generated projective modules over a ring R. Then the torsion takes value in the algebraic K-theory of R.

Instead of explicitely choosing Hilbert structures to define torsion as a real number, one can consider it as an element in certain determinant spaces. The latter approach has the advantage not to depend on artificial choices. However, it does not allow the generalizations we have mentioned. In particular, one is bound to the finite dimensional setting (i.e. even the singular cochain complex of a finite CW-complex is not allowed). Also, if one carries out explicit computations, in most cases it is preferable to deal with real numbers. Therefore we used the first method. In our context, both languages are completely equivalent and we supply a dictionary to translate between then in appendix A. One should also mention that there is a third equivalent approach which uses norms constructed on determinant lines (compare [1]).

Finally we mention that in the case where V is orthogonal Dai and Melrose [5] have proven the formula of Theorem 0.2 in the adiabatic limit by completely different methods, namely, by a careful analysis of the heat kernel in the adiabatic limit and the adiabatic version of the Leray-Serre spectral sequence [14]. We remark that in the special case where V is an orthogonal representation ρ_{an}(E; V) vanishes if dim(E) is even but this is not true in general for a non-flat Riemannian metric on V.

The first two authors decidate this paper to their friend and colleague Thomas Thielmann who died in a car accident in November 1994.

The paper is organized as follows

1. Simple structures on spaces
2. Milnor torsion for cochain complexes
3. Milnor torsion for spaces and local coefficient systems
4. Torsion and spectral sequences
5. The fibration formula for Milnor torsion
6. The fiber bundle formula for analytic torsion
A Torsion and determinants

References

1. Simple structures on spaces

In this section we explain additional structures on arbitrary topological spaces and fibrations which allow the definition of torsion invariants on them. This extension from the category of finite CW-complexes serves two purposes: on the one hand there are interesting spaces which are not finite CW-complexes, f.i. classifiying spaces of discrete groups. On the other hand, our approach singles out what exactly is used in the definition of torsion invariants, and this definitely clarifies the exposition.

We will always assume for a pair of spaces (X, A) that the inclusion of A into X is a cofibration. This condition is satisfied if (X, A) is a pair of CW -complexes or if X is a manifold with submanifold A. A map (F, f) : (X, A) ⟶ (Y, B) of pairs is a relative homotopy equivalence if F ∪ id : X ∪_{f} B ⟶ Y is a homotopy equivalence. Here X ∪_{f} B is obtained from X ∐ B by identifying a ∈ A with f(a) ∈ B. Notice that (F, f) is a homotopy equivalence of pairs if and only if f is a homotopy equivalence and (F, f) is a relative homotopy equivalence. Given a cellular relative homotopy equivalence (F, f) : (X, A) ⟶ (Y, B) of pairs of finite CW -complexes, define its Whitehead torsion as the Whitehead torsion of the homotopy equivalence F ∪ id : X ∪_{f} B ⟶ Y of finite CW -complexes

τ(F, f) := τ(F ∪ id)  ∈ Wh(Y).  (1.1)

We refer to [4, §6, §21 and §22] for the definition of the geometric Whitehead group and Whitehead torsion and their identifications with the algebraic Whitehead group Wh(π_{1}(Y)) and Whitehead torsion. Notice that τ(F, f) depends only on the homotopy class of (F, f) and satisfies the composition formula, the formula for pairs and the product formula as stated in 1.6. This follows from the special case A = ∅ in [4, §22 and §23]. Given a pair (Y, B), we call two relative homotopy equivalences (F_{i}, f_{i}) : (X_{i}, A_{i}) ⟶ (Y, B) with pairs of finite CW -complexes as source for i = 0, 1 equivalent if τ((G, g) ∘ (F_{0}, f_{0})) = τ((G, g) ∘ (F_{1}, f_{1})) holds for any relative homotopy equivalence (G, g) : (Y, B) ⟶ (Y′, B′) into a pair of finite CW -complexes.

Definition 1.2 A relative simple structure on a pair (Y, B) is an equivalence class of relative homotopy equivalences (F, f) : (X, A) ⟶ (Y, B) for a pair of finite CW -complexes as source.

1.3 Let (X, A) be a pair with a relative simple structure and g : A′′ ⟶ A be a homotopy equivalence with a finite CW -complex as source. Then we can extend g to a representative for the simple structure (G, g) : (X′′, A′′) ⟶ (X, A) which is a homotopy equivalence of pairs as follows. Choose some representative (F, f) : (X′, A′) ⟶ (X, A) for the given simple structure. Furthermore choose a homotopy inverse g^{−1} : A ⟶ A′′, a homotopy ϕ from g^{−1} ∘f to a cellular map A′ ⟶ A′′ and a homotopy ψ from g ∘ g^{−1} ∘ f to f. Let X′′ be the finite CW -complex X′ ∪_{ϕ_{1}} A′′. Define (G, g) : (X′′, A′′) ⟶ (X, A) by the following composition of (relative) homotopy equivalences or their homotopy inverses

X′ ∪_{ϕ_{1}} A′′ ⟶ X′ × [0, 1] ∪_{ϕ} A′′ ⟵ X′ ∪_{g^{−1}∘f} A′′ ⟶ X′ ∪_{g∘g^{−1}∘f} A
⟶ X′ × [0, 1] ∪_{ψ} A ⟵ X′ ∪_{f} A ⟶ X.

The proof that (G, g) represents the given simple structure is done by the results of [4, §5].

1.4 Given a (relative) simple structure on (X, A) and on A, we construct a preferred simple structure on X as follows. Because of 1.3 there is a homotopy equivalence of pairs (G, g) : (X′′, A′′) ⟶ (X, A) such that it represents the given relative simple structure on (X, A) and g represents the given simple structure on A. The preferred simple structure on X is then represented by G. This is independent of the choice of (G, g) by 1.6.

Given a relative homotopy equivalence (F, f) : (X, A) ⟶ (Y, B) of pairs with relative simple structures, we still can define its (relative) Whitehead torsion

τ(F, f) ∈ Wh(Y)  (1.5)

as follows. Choose representatives (G, g) : (X′, A′) ⟶ (X, A) and (H, h) : (Y′, B′) ⟶ (Y, B) for the relative structures. Because of 1.3 one can arrange that (H, h) is a homotopy equivalence of pairs. Define τ(F, f) as the image of Whitehead torsion τ((H, h)^{−1} ∘ (F, f) ∘ (G, g)) defined in 1.1 under the map H_{∗} : Wh(Y′) ⟶ Wh(Y) induced by H.

1.6 We have already mentioned homotopy invariance, the composition formula

τ((G, g) ∘ (F, f)) = τ(G, g) + G_{∗}τ(F, f)

the formula for pairs

τ(F) = τ(F, f) + i_{∗}τ(f)

and the product formula

τ((F, f) × id_{Y}) = χ(Y) · i_{∗}τ(F, f)

where i denotes the obvious inclusions. One easily checks that they remain true in the more general case that (X, A) is not necessarily a pair of finite CW -complexes, but carries a relative simple structure.

1.7 Let f : E ⟶ B be a fibration such that B is a finite CW -complex and the fiber has the homotopy type of a finite CW -complex. Suppose that we are given a cellular base point system {b_{c} | c ∈ I} for B, i.e. a choice of points b_{c} in the interior c^{∘} for each c ∈ I where here and elsewhere I_{n} is the set of n-cells and I is the disjoint union of the I_{n}-s. Furthermore suppose that we have specified a simple structure on each fiber F_{b_{c}} = f^{−1}(b_{c}). We want to define a simple structure on E depending only on these choices as follows. Recall that any homotopy class of paths w from b_{0} to b_{1} defines a homotopy class of homotopy equivalences t_{w} : F_{b_{0}} ⟶ F_{b_{1}} by the fiber transport [21, 15.12].

Let E_{n} be f^{−1}(B_{n}). As B_{n−1} ⟶ B_{n} is a cofibration, the same is true for E_{n−1} ⟶ E_{n} [22, I.7.14]. Because of the construction 1.4 it suffices to specify a relative simple structure for (E_{n}, E_{n−1}) for all n ≥ 0. This will be done by the next Lemma 1.8 taking into account 1.6 and that the Whitehead torsion of any homotopy equivalence (D^{n}, S^{n−1}) ⟶ (D^{n}, S^{n−1}) is trivial since D^{n} is simply-connected and hence Wh(D^{n}) is trivial.

Lemma 1.8 Suppose we have specified a cellular base point system for B and for each element in I_{n} an orientation. Then there is a relative homotopy equivalence which is uniquely defined up to homotopy

∐_{c∈I_{n}} F_{b_{c}} × (D^{n}, S^{n−1}) ⟶ (E_{n}, E_{n−1}).

If we change the orientation of the cell c, then the map is changed by the selfhomotopy equivalence id ×s : F_{b_{c}} × (D^{n}, S^{n−1}) ⟶ F_{b_{c}} × (D^{n}, S^{n−1}) where s is a map of degree −1. If we change the base point b_{c} of c to b′_{c}, then the map is changed by the homotopy equivalence t × id : F_{b_{c}} × (D^{n}, S^{n−1}) ⟶ F_{b′_{c}} × (D^{n}, S^{n−1}) where t : F_{b_{c}} ⟶ F_{b′_{c}} is given by the fiber transport along any path in c^{∘} connecting b_{c} and b′_{c}.

\underline{\underline{Proof}} : The map will be constructed as the composition of the following four maps or their homotopy inverses.

There is up to homotopy one orientation preserving homotopy equivalence of pairs (D^{n}, S^{n−1}) ⟶ (c^{∘}, c^{∘} − b_{c}). This gives the first map

∐_{c∈I_{n}} F_{b_{c}} × (D^{n}, S^{n−1}) ⟶ ∐_{c∈I_{n}} F_{b_{c}} × (c^{∘}, c^{∘} − b_{c}).

Choose a homotopy ϕ_{c} : c^{∘} × [0, 1] ⟶ B from the canonical inclusion c^{∘} ⟶ B to the constant map with value b_{c} such that its evaluation at b_{c} gives a path within c^{∘}. By the homotopy lifting property we obtain a strong fiber homotopy equivalence which is unique up to fiber homotopy [21, Proposition 15.11]

F_{b_{c}} × (c^{∘}, c^{∘} − b_{c}) ⟶ (E|_{c^{∘}}, E|_{c^{∘}−b_{c}}).

The second map is the disjoint union of these maps over I_{n}. The third map is the relative homotopy equivalence given by the inclusion

∐_{c∈I_{n}}(E|_{c^{∘}}, E|_{c^{∘}−b_{c}}) ⟶ (E_{n}, E|_{B_{n}−{∐_{c∈In} b_{c}}}).

The fourth map is the homotopy equivalence of pairs given by the inclusion

(E_{n}, E_{n−1}) ⟶ (E_{n}, E|_{B_{n}−{∐_{c∈In} b_{c}}}).

This finishes the proof of Lemma 1.8.

1.9 Let {b_{c} | c ∈ I} and {b′_{c} | c ∈ I} be two base point systems and suppose F_{b_{c}} and F_{b′_{c}} come with simple structures. Let σ and σ′ be the simple structures on E given by the construction 1.7 for these two choices. Let t_{c} : F_{b_{c}} ⟶ F_{b′_{c}} be the homotopy equivalence which is given by the fiber transport along any path in c^{∘} from b_{c} to b′_{c}. Denote by i(b′_{c}) : F_{b′_{c}} ⟶ E the inclusion. Then we get from 1.6 and Lemma 1.8

τ(id : (E, σ) ⟶ (E, σ′)) = ∑_{n≥0}(−1)^{n} · ∑_{c∈I_{n}} i(b′_{c})_{∗}τ(t_{c} : F_{b_{c}} ⟶ F_{b′_{c}}).

Next we give a criterion when the choice of base point system does not affect the simple structure on E. Fix a base point b ∈ B. Given an element w in π_{1}(B, b), define

θ_{f}(w) = i(b)_{∗}τ(t_{w} : F_{b} ⟶ F_{b}) ∈ Wh(E)

where t_{w} is the fiber transport and i(b) : F_{b} ⟶ E is the inclusion. As i(w(1)) ∘ t_{w} ≃ i(w(0)) holds for any path w in B, one easily checks using 1.6 that this defines a homomorphism from π_{1}(B, b) to Wh(E) and thus a cohomology class which is independent of the choice of b ∈ B

θ_{f} ∈ H^{1}(B; Wh(E)).  (1.10)

Definition 1.11 A choice of simple structures on the fibers is a choice of simple structure on each fiber F_{b}. It is called coherent if for any path w in B we get

i(w(1))_{∗}τ(t_{w} : F_{w(0)} ⟶ F_{w(1)}) = 0 ∈ Wh(E).

1.12 Notice that a coherent choice of simple structures on the fibers exists if and only if θ_{f} is trivial. If we fix a coherent choice of simple structures {σ(F_{b}) | b ∈ B} on the fibers, the induced simple structure σ on E of 1.7 is independent of the cellular base point system by 1.9. Moreover, if we have a different choice of coherent structures on the fibers, then

τ(id : (E, σ) ⟶ (E, σ′)) = χ(B) · i(b)_{∗}(τ(id : (F_{b}, σ(F_{b})) ⟶ (F_{b}, σ′(F_{b}))).

In particular we see for a fibration p : E ⟶ B over a finite CW -complex B such that the homotopy fiber has the homotopy type of a finite CW -complex and χ(B) = 0 and θ_{f} = 0 holds that E has a preferred simple structure.

Lemma 1.13 Let F ⟶ E ⟶^{f} B be a smooth bundle of compact smooth manifolds. Equip B and each fiber with the simple structure given by a smooth triangulation. This gives a coherent choice of simple structures on the fibers. Then the simple structure on E given by a smooth triangulation agrees with the one given by 1.7.

Remark 1.14 The construction 1.7 can be extended to the case where B is not necessarily a CW -complex but carries a simple structure. Namely, choose a representative g : X ⟶ B of the simple structure. The pull back construction yields a fibration \overline{f} : g^{∗}E ⟶ X and a fiber homotopy equivalence \overline{g} : g^{∗}E ⟶ E. Equip \overline{f} with the coherent choice of simple structures on the fibers induced by \overline{g} and the given one on f. Construction 1.12 applies to \overline{f} and gives a simple structure on g^{∗}E. Equip E with the simple structure for which τ(\overline{g}) vanishes. It is not hard to check that this is independent of the choice of the representative g. The main step is to show in the case where X and B are finite CW -complexes and g is an elementary expansion that τ(\overline{g}) vanishes with respect to the simple structures on E and g^{∗}E given by construction 1.12.

2. Milnor torsion for cochain complexes

In this section we give a brief introduction to torsion invariants of cochain complexes as defined in the introduction. We give no proofs but refer to [16] and [12].

Given linear isomorphism f, g of finite-dimensional real Hilbert spaces, the number [[·]] of 0.4 has the following properties:

[[f]] = ln(| det(A)|) ∈ ℝ  (2.1)

where A is the matrix describing f with respect to some choice of orthonormal basis for source and range.

[[f ∘ g]] = [[f]] + [[g]];
[[( \begin{matrix}f & h \\ 0 & g \end{matrix} )]] = [[f]] + [[g]];
[[f^{∗}]] = [[f]].

Let C = C^{∗} be a finite Hilbert cochain complex, i.e. a cochain complex of finite-dimensional Hilbert spaces such that C^{i} = 0 for |i| ≥ N for some natural number N. If C is acyclic, we defined in 0.5 ρ(C) := [[(c^{∗} + γ^{∗}) : C^{ev} ⟶ C^{odd}]] ∈ ℝ, where c^{∗} is the differential and γ^{∗} a chain contraction. This is independent of the choice of γ^{∗}. For f : C ⟶ D a chain homotopy equivalence of finite Hilbert cochain complexes, t(f) was defined in 0.6. It turns out that t(f) depends only on the homotopy class of f and t(f ∘ g) = t(f) + t(g). Notice that with these conventions we get for an isomorphisms f : C ⟶ D of finite Hilbert cochain complexes

t(f) = ∑_{n}(−1)^{n} · [[f^{n}]].

Let C be a finite Hilbert cochain complex such that H(C^{∗}) carries a Hilbert structure, i.e. H^{n}(C^{∗}) is equipped with a Hilbert space structure for each n ∈ ℤ. If i : H(C) ⟶ C is the chain map which induces on cohomology the indentity, we defined ρ(C) := −t(i) ∈ ℝ. The minus sign ensures that this definition coincides with the one in 0.5 in the acyclic case. If we fix an orthonormal basis for each C^{n} and each H^{n}(C), then the logarithm of the torsion defined by Milnor [16, page 365] is ρ(C).

Let C be a finite Hilbert cochain complex and equip ker(c^{p}) and im(c^{p−1}) with the Hilbert substructures and H^{p}(C) = ker(c^{p})/ im(c^{p−1}) and C^{p}/ ker(c^{p}) with the quotient Hilbert structures. If \overline{c^{p}} : C^{p}/ ker(c^{p}) ⟶ im(c^{p}) is the obvious isomorphism induced by c^{p} and \overline{Δ^{p}} : C^{p}/ ker(Δ^{p}) ⟶ C^{p}/ ker(Δ^{p}) is the automorphism induced by the endomorphism Δ^{p} = (c^{p})^{∗} ∘ c^{p} + c^{p−1} ∘ (c^{p−1})^{∗} : C^{p} ⟶ C^{p}, then we get

ρ(C) = ∑_{p}(−1)^{p} · [[\overline{c^{p}}]] = − ∑_{p}(−1)^{p} · p · \frac{1}{2} · ln (det(\overline{Δ^{p}})) .  (2.2)

A simple structure on a (real) cochain complex C is an equivalence class of chain homotopy equivalences u : \overline{C} ⟶ C with a finite Hilbert cochain complex as source where u and v : \overline{\overline{C}} ⟶ C are equivalent if t(v^{−1} ∘ u) vanishes. Let f : C ⟶ D be a chain homotopy equivalence of cochain complexes with simple structure. Define

t(f) = t(v^{−1} ∘ f ∘ u) ∈ ℝ  (2.3)

for any representatives u : \overline{C} ⟶ C and v : \overline{D} ⟶ D of the simple structures. Let C be a cochain complex with simple structure such that H(C) has a Hilbert structure. Define

ρ(C) := ρ(\overline{C}) ∈ ℝ  (2.4)

for any representative u : \overline{C} ⟶ C where we use the Hilbert structure on H(\overline{C}) for which H(u) is an isometry.

Next we collect the basic properties of these invariants. We mention that one firstly verifies them for finite Hilbert cochain complexes and uses this to show that the definitions and results extend to cochain complexes with simple structures.

If 0 ⟶C ⟶D ⟶E ⟶0 is an exact sequence of finite Hilbert cochain complexes, we can view C^{n} ⟶ D^{n} ⟶ E^{n} as an acyclic finite Hilbert cochain complex concentrated in dimension 0, 1 and 2 and define

ρ(C ⟶ D ⟶ E) := ∑_{n∈ℤ}(−1)^{n} · ρ(C^{n} ⟶ D^{n} ⟶ E^{n}).

This extends to an exact sequence 0 ⟶C ⟶D ⟶E ⟶0 of cochain complexes with simple structures as follows. Construct a commutative diagram of cochain complexes

0 ⟶ C ⟶ D ⟶ E ⟶ 0
f↑  ^{g}↑  ^{h}↑
0 ⟶ \overline{C} ⟶ \overline{D} ⟶ \overline{E} ⟶ 0

with the property that the rows are exact and the vertical arrows are homotopy equivalences which represent the given simple structures. In particular the lower row is an exact sequence of finite Hilbert cochain complexes and we put

ρ(C ⟶ D ⟶ E) := ρ(\overline{C} ⟶ \overline{D} ⟶ \overline{E}).  (2.5)

2.6 In the following list of basic properties all cochain complexes come with simple structures.

1. homotopy invariance f ≃ g ⇒ t(f) = t(g);

2. composition formula

t(f ∘ g) = t(f) + t(g);

3. exactness Given a commutative diagram of the shape above with exact rows and homotopy equivalences as vertical arrows, then

t(f) − t(g) + t(h) = ρ(\overline{C} ⟶ \overline{D} ⟶ \overline{E}) − ρ(C ⟶ D ⟶ E);

4. sum formula Let 0 ⟶C ⟶D ⟶E ⟶0 be exact and the cohomology of C, D and E come with Hilbert structures. Let LHS be the acyclic finite Hilbert cochain complex given by the long cohomology sequence where LHS^{0} = H^{0}(C). Then:

ρ(D) − ρ(C) − ρ(E) = ρ(LHS) − ρ(C ⟶ D ⟶ E);

5. transformation formula If f : C ⟶ D is a homotopy equivalence and the cohomology of C and D come with Hilbert structures, then:

ρ(C) − ρ(D) = t(f) − ∑_{n∈ℤ}(−1)^{n} · [[H^{n}(f)]].

3. Milnor torsion for spaces and local coefficient systems

The fundamental groupoid Π(X) of a space X has as objects points in X and a morphism w : x ⟶ y is a homotopy class relative end points of paths in X from y to x. For x ∈ X let X̃(x) be the set of all morphisms w : x ⟶ y with x as source. The projection p(x) : X̃(x) ⟶ X sends w to y. Assume that X is locally path-connected and semi-locally simply connected. This condition is always satisfied if X is a CW -complex or manifold. It ensures that there is precisely one topology on X̃(x) such that p(x) is a model for the universal covering of the path component of X containing x. Thus we get a contravariant functor

X̃ : Π(X) ⟶ SPACES .

Composing it with the covariant functor singular chain complex with real coefficients yields the contravariant functor

C_{∗}^{sing}(X̃) : Π(X) ⟶ ℝ − CHAIN

A local coefficient system V on X is a contravariant functor V : Π(X) ⟶ ℝ − VECTOR into the category of finite-dimensional real vector spaces. For instance a flat vector bundle over X defines a local coefficient system. Define the singular cochain complex and the singular cohomology of X with coefficient in V by

C_{sing}^{∗}(X; V) := hom(C_{∗}^{sing}(X̃), V)  (3.1)
H_{sing}^{∗}(X; V) := H^{∗}(C_{sing}^{∗}(X; V))

where hom denotes the real vector space of natural transformations.

Suppose that X is a CW -complex. Then X̃ becomes a functor from Π(X) into the category of CW -complexes and we can define the cellular versions C_{cell}^{∗}(X; V) and H_{cell}^{∗}(X; V) of the definitions above by using the cellular chain complex instead of the singular one. There is a homotopy equivalence

C_{cell}^{∗}(X; V) ⟶ C_{sing}^{∗}(X; V)  (3.2)

which is unique up to homotopy and natural in X and V [10, page 263]. It induces a natural isomorphism H_{cell}^{∗}(X; V) ⟶ H_{sing}^{∗}(X; V).

Let X be a space and V^{0}, V^{1}, …, V^{r} local coefficient systems. We say ∑_{q=0}^{r}(−1)^{q} · V^{q} is unimodular if we have for each automorphism w : x ⟶ x in Π(X)

∏_{q=0}^{r} |det(V_{w}^{q} : V_{x}^{q} ⟶ V_{x}^{q})|^{(−1)^{q}} = 1.

A Hilbert structure on ∑_{q=0}^{r}(−1)^{q} · V^{q} is a choice of Hilbert space structure on V_{x}^{q} for each q ∈ {1, 2 … r} and x ∈ X. It is called unimodular if we have for each morphism w : y ⟶ x in Π(X)

∑_{q=0}^{r}(−1)^{q} · [[V_{w}^{q} : V_{x}^{q} ⟶ V_{y}^{q}]] = 0

in the notation of 0.4. Notice that ∑_{q=0}^{r}(−1)^{q} · V^{q} is unimodular if and only if admits a unimodular Hilbert structure.

Definition 3.3 Let X be a space and V^{0}, V^{1}, …, V^{r} be local coefficient systems. Assume that X comes with a simple structure, ∑_{q=0}^{r}(−1)^{q} · V^{q} with a unimodular Hilbert structure and H_{sing}^{∗}(X; V^{q}) with a Hilbert structure. Define the Milnor torsion

ρ(X; ∑_{q=0}^{r}(−1)^{q} · V^{q}) := ∑_{q=0}^{r}(−1)^{q} · ρ(C_{sing}^{∗}(X; V^{q}))

where ρ(C_{sing}^{∗}(X; V^{q})) was defined in 2.4 and we use the simple structure on C_{sing}^{∗}(X; V^{q}) defined below.

Let f : Y ⟶ X be a representative for the simple structure on X. Fix a cellular base point system {b_{c} | c ∈ I}, i.e. a choice of base points b_{c} ∈ c^{∘} for each cell c of Y, and an orientation for each cell c. For each cell c there is precisely one lift c̃ ⊂ Ỹ(b_{c}) which contains the canonical base point in Ỹ(b_{c}) given by the constant path at b_{c}. The orientation of c induces an orientation of c̃. Thus we get an element [c̃] ∈ C_{dim(c)}^{cell}(Ỹ(b_{c})). Define an isomorphism

C_{cell}^{n}(Y, f^{∗}V^{q}) ⟶ ⊕_{c∈I_{n}}V_{f(b_{c})}^{q}  t ↦ (t(b_{c})([c̃]))_{c∈I_{n}} .  (3.4)

Equip C_{cell}^{n}(Y, f^{∗}V^{q}) with the Hilbert structure induced by the isomorphism 3.4 above and the given Hilbert space structure on the various V_{f(b_{c})}^{q}-s. The desired simple structure on C_{sing}^{∗}(X; V^{q}) is represented by the composition of the chain homotopy equivalences given by 3.2 and f

C_{cell}^{∗}(Y; f^{∗}V^{q}) ⟶ C_{sing}^{∗}(Y; f^{∗}V^{q}) ⟶^{(f^{∗})^{−1}} C_{sing}^{∗}(X; V^{q}).

The choice of the orientations of the cells does not affect this simple structure. If we change the base point system, the simple structure and hence ρ(C_{sing}^{∗}(X; V^{q})) changes. However, ∑_{q=0}^{r}(−1)^{q} · ρ(C_{sing}(X; V)) does not change because of the formulas 2.6 and the assumption that the Hilbert structure on ∑_{q=0}^{r}(−1)^{q} · V^{q} is unimodular.

If ∑_{q=0}^{r}(−1)^{q} · V^{q} is unimodular, we can define a homomorphism

Φ(∑_{q=0}^{r}(−1)^{q} · V^{q}) : Wh(X) ⟶ ℝ  (3.5)

as follows. An element [u] ∈ Wh(X) can be represented by an automorphism u : M ⟶ M of a finitely generated free ℤΠ(X)-module M. Composition with u defines an automorphism u^{q} : hom(M, V^{p}) ⟶ hom(M, V^{q}) of a finite-dimensional real vector space. Put

Φ(∑_{q=0}^{r}(−1)^{q} · V^{q})([u]) := ∑_{q=0}^{r}(−1)^{q} · ln | det(u^{q})|.

The unimodularity condition on ∑_{q=0}^{r}(−1)^{q} · V^{q} ensures that trivial units are send to 0. The following lemma is a consequence of the definitions and the formulas 2.6

Lemma 3.6 Let f : X ⟶ Y be a homotopy equivalence of spaces with simple structures. Let V^{0}, V^{1}, …, V^{r} be local coefficient systems on Y. Assume that V := ∑_{q=0}^{r}(−1)^{q} · V^{q} comes with a unimodular Hilbert structure and H_{sing}^{∗}(X; f^{∗}V^{q}) and H_{sing}^{∗}(Y; V^{q}) come with Hilbert structures. Then:

ρ(Y; V) − ρ(X; f^{∗}V) = Φ(V)(τ(f))

− ∑_{q=0}^{r}(−1)^{q} · ∑_{p≥0}(−1)^{p} · [[f^{p} : H_{sing}^{p}(Y; V^{q}) ⟶ H_{sing}^{p}(X; f^{∗}V^{q})]].

4. Torsion and spectral sequences

Let C be a cochain complex with finite descending filtration by cochain complexes F_{p}C

C = F_{0}C ⊃ F_{1}C ⊃ F_{2}C ⊃ ⋯ ⊃ F_{p}C ⊃ F_{p+1}C ⊃ ….

Finite means that there is a natural number l with F_{l}C = {0}. Put F_{p}C = C for p ≤ −1. We recall the construction of the associated spectral cohomology sequence (E_{∗}^{∗,∗}, d_{∗}^{∗,∗}) converging to H^{∗}(C) since we will need it explicitely (see [2] or [15]). In the sequel ∂ denotes the boundary operator of the long exact cohomology sequence associated to a short exact sequence of cochain complexes. We will abbreviate F_{p}C by F_{p}. Define for r ≥ 1:

Z_{r}^{p,q} := im(H^{p+q}(F_{p}/F_{p+r}) ⟶ H^{p+q}(F_{p}/F_{p+1}))
B_{r}^{p,q} := im(H^{p+q−1}(F_{p−r+1}/F_{p}) ⟶^{∂} H^{p+q}(F_{p}/F_{p+1}))
Z_{∞}^{p,q} := im(H^{p+q}(F_{p}) ⟶ H^{p+q}(F_{p}/F_{p+1}))
B_{∞}^{p,q} := im(H^{p+q−1}(C/F_{p}) ⟶^{∂} H^{p+q}(F_{p}/F_{p+1}))
E_{r}^{p,q} := Z_{r}^{p,q}/B_{r}^{p,q}
E_{∞}^{p,q} := Z_{∞}^{p,q}/B_{∞}^{p,q}
F^{p,q} := im(H^{p+q}(F_{p}) ⟶ H^{p+q}(C))

We have the inclusions:

{0} = B_{1}^{p,q} ⊂ ⋯ ⊂ B_{r}^{p,q} ⊂ B_{∞}^{p,q} ⊂ Z_{∞}^{p,q} ⊂ Z_{r}^{p,q} ⊂ ⋯ ⊂ Z_{1}^{p,q} = H^{p+q}(F_{p}/F_{p+1})

The map H^{p+q}(F_{p}/F_{p+r}) ⟶ H^{p+q}(F_{p}/F_{p+1}) resp. H^{p+q}(F_{p}/F_{p+r}) ⟶ H^{p+q+1}(F_{p+r}/F_{p+r+1}) is induced by the inclusion resp. is a boundary operator. We get epimorphisms

H^{p+q}(F_{p}/F_{p+r}) ⟶ Z_{r}^{p,q}/Z_{r+1}^{p,q}

and

H^{p+q}(F_{p}/F_{p+r}) ⟶ B_{r+1}^{p+r,q−r+1}/B_{r}^{p+r,q−r+1}.

Now the standard diagram chase shows that these maps have the same kernel. Hence we obtain canonical isomorphisms

γ_{r}^{p,q} : Z_{r}^{p,q}/Z_{r+1}^{p,q} ⟶ B_{r+1}^{p+r,q−r+1}/B_{r}^{p+r,q−r+1} for r ≥ 1.

Analogously one gets a natural isomorphism

ψ^{p,q} : F^{p,q}/F^{p+1,q−1} ⟶ E_{∞}^{p,q}.

We define the differential

d_{r}^{p,q} : E_{r}^{p,q} = Z_{r}^{p,q}/B_{r}^{p,q} ⟶ E_{r}^{p+r,q−r+1} = Z_{r}^{p+r,q−r+1}/B_{r}^{p+r,q−r+1}

by the composition:

Z_{r}^{p,q}/B_{r}^{p,q} ⟶ Z_{r}^{p,q}/Z_{r+1}^{p,q} ⟶^{γ_{r}^{p,q}} B_{r+1}^{p+r,q−r+1}/B_{r}^{p+r,q−r+1} ⟶ Z_{r}^{p+r,q−r+1}/B_{r}^{p+r,q−r+1}

We obtain cochain complexes

(E_{r}^{p+r∗,q−(r−1)∗}, d_{r}^{p+r∗,q−(r−1)∗}) r ≥ 0

if we use for r ≥ 1 the definition above and define E_{0}^{p,∗} to be the (−p)-th suspension Σ^{−p}F_{p}/F_{p+1} of F_{p}/F_{p+1}. Since ker(d_{r}^{p,q}) is Z_{r+1}^{p,q}/B_{r}^{p,q} and im(d_{r}^{p,q}) is B_{r+1}^{p+r,q−r+1}/B_{r}^{p+r,q−r+1}, we obtain a canonical isomorphism

ϕ_{r}^{p,q} : H^{0}(E_{r}^{p+r∗,q−(r−1)∗}) ⟶ E_{r+1}^{p,q}

in the case r ≥ 1; in the case r = 0 we use the identification H^{0}(E_{0}^{p,q+∗}) = H^{p+q}(F_{p}/F_{p+1}).

4.1 Now suppose we have fixed the following data

1. Simple structures on F_{p}/F_{p+1};

2. a Hilbert structure on H(C).

Equip F_{p} inductively with the simple structure for which the number defined in 2.5 satisfies

ρ(F_{p+1} ⟶ F_{p} ⟶ F_{p}/F_{p+1}) = 0.

In particular we get a preferred simple structure on C = F_{0} and ρ(C) is defined. Equip F^{p,q} ⊂ H^{p+q}(C) with the Hilbert substructure and F^{p,q}/F^{p+1,q−1} with the Hilbert quotient structure. Do iteratively the same for E_{r}^{p,q} and H(E_{r}^{p+r∗,q−(r−1)∗}). Observe that ϕ_{r}^{p,q} become isometries. Notice that 4.1.1 implies that E_{r}^{p,q} is finite-dimensional for r ≥ 1.

Definition 4.2 Define for the finite descending filtration F_{∗}C with respect to the data 4.1 and choices above :

ρ_{fil}^{≥2}(F_{∗}C) := ∑_{r≥2} ∑_{p=0}^{r−1} ∑_{q}(−1)^{p+q} · ρ(E_{r}^{p+r∗,q−(r−1)∗}) − ∑_{p,q}(−1)^{p+q} · [[ψ^{p,q}]];

ρ_{fil}(F_{∗}C) := ∑_{p} ρ(F_{p}/F_{p+1}) + ∑_{r≥1} ∑_{p=0}^{r−1} ∑_{q}(−1)^{p+q} · ρ(E_{r}^{p+r∗,q−(r−1)∗})

− ∑_{p,q}(−1)^{p+q} · [[ψ^{p,q}]];

where ρ was defined in 2.4 and [[ ]] in 0.4.

Remark 4.3 ρ_{fil}(F_{∗}C) depends on the choices 4.1. For the definition of ρ_{fil}^{≥2}(F_{∗}C) one can fix Hilbert structures on E_{2}^{p,q} instead of data 4.1.1 and equip E_{r}^{p,q} and H(E_{r}^{p+r∗,q−(r−1)∗}) (r ≥ 2) with the corresponding Hilbert sub- and quotient structures. Then, ρ_{fil}^{≥2}(F_{∗}C) depends not on 4.1.1 but on these choices as follows: if U_{2}^{p,q} : \overline{E}_{2}^{p,q} → E_{2}^{p,q} is the identity on E_{2} equipped with two different Hilbert structures, then

\overline{ρ_{fil}^{≥2}(F_{∗}C)} − ρ_{fil}^{≥2}(F_{∗}C) = ∑_{p,q}(−1)^{p+q}[[U_{2}^{p,q}]]

(torsion computed using these two Hilbert structures). This follows from the transformation formula 2.6.

The main result of this section is:

Theorem 4.4 We get with respect to the data 4.1 and the conventions above

ρ(C) = ρ_{fil}(F_{∗}C).

It will follow from the next three lemmas.

Lemma 4.5 It suffices to treat the following special case: C itself is a finite Hilbert cochain complex with the simple structure represented by id : C ⟶ C and the Hilbert structures on F_{p} and F_{p}/F_{p+1}, are obtained by the given one on C by taking Hilbert sub- and quotient structures.

\underline{\underline{Proof}} : One easily constructs a finite Hilbert cochain complex D with finite cofiltration F_{∗}D together with a chain homotopy equivalence f : D ⟶ C with the following property: f induces chain homotopy equivalences F_{p}f : F_{p}D ⟶ F_{p}C for all p such that the given simple structure on F_{p}C/F_{p+1}C is represented by F_{p}f/F_{p+1}f : F_{p}D/F_{p+1}D ⟶ F_{p}C/F_{p+1}C. Equip H(D) with the Hilbert structure for which H(f) : H(D) ⟶ H(C) becomes an isometry. We have by assumption and construction for all p that

ρ(F_{p+1}D ⟶ F_{p}D ⟶ F_{p}D/F_{p+1}D) = 0 = ρ(F_{p+1}C ⟶ F_{p}C ⟶ F_{p}C/F_{p+1}C).

Hence we conclude from 2.6

ρ(C) = ρ(D).

The maps F_{p}f induce isometries between the E_{r}^{p+r∗,q−(r−1)∗} for r ≥ 1 and H(E_{r}^{p+r∗,q−(r−1)∗}) for r ≥ 0 associated to F_{∗}D and F_{∗}C. Now one easily verifies

ρ_{fil}(F_{∗}C) = ρ_{fil}(F_{∗}D).

This finishes the proof of Lemma 4.5.

From now on we will only consider the special case described in Lemma 4.5. The proof of the next lemma is similiar to the proof in [19, Theorem 2.2.]

Lemma 4.6

ρ(C) = ∑_{p} ρ(F_{p}/F_{p+1}) + ∑_{r≥1} ∑_{p,q}(−1)^{p+q} · [[γ_{r}^{p,q}]] − ∑_{p,q}(−1)^{p+q} · [[ψ^{p,q}]]

\underline{\underline{Proof}} : We do induction over l with F_{l}C = \{0\}. The begin of induction l = 0 is trivial, the induction step from l − 1 to l ≥ 1 done as follows.

Let \overline{C} be F_{1} = F_{1}C with the cofiltration
\overline{F}_{p} = F_{p}\overline{C} := F_{p+1}C.

We will denote the various data coming from the spectral sequence associated to \overline{C} as the ones for C decorated with an additional bar. Let LHS be the acyclic finite Hilbert cochain complex given by the long cohomology sequence associated to 0 ⟶ \overline{C} ⟶ C ⟶ C/\overline{C} ⟶ 0. It induces for n ≥ 0 an acyclic finite Hilbert cochain complex D(n) concentrated in dimensions 0, 1, 2 and 3 by

H^{n}(C)/F^{1,n−1} ⟶ H^{n}(C/\overline{C}) ⟶ H^{n+1}(\overline{C}) ⟶ F^{1,n}

where we equip F^{1,n−1} = im(H^{n}(\overline{C}) ⟶ H^{n}(C)) respectively H^{n+1}(C)/F^{1,n} with the Hilbert substructure respectively quotient structure. Define an acyclic Hilbert subcochain complex D(n, r) for r ≥ 1 of D(n) by

H^{n}(C)/F^{1,n−1} ⟶ Z_{r}^{0,n} ⟶ \overline{F}^{r−1,n+2−r} ⟶ F^{r,n+1−r}

Notice that D(n, 1) = D(n). The following diagram of acyclic finite Hilbert cochain complexes concentrated in dimensions 1, 2 and 3 commutes:

Z_{r}^{0,n}/Z_{r+1}^{0,n}  ⟶ \overline{F}^{r−1,n+2−r}/\overline{F}^{r,n+1−r} ⟶ F^{r,n+1−r}/F^{r+1,n−r}
γ_{r}^{0,n}↓  \overline{ψ}^{r−1,n+2−r}↓  ^{ψ^{r,n+1−r}}↓
B_{r+1}^{r,n+1−r}/B_{r}^{r,n+1−r} ⟶ \overline{Z}_{∞}^{r−1,n+2−r}/\overline{B}_{∞}^{r−1,n+2−r} ⟶ Z_{∞}^{r,n+1−r}/B_{∞}^{r,n+1−r}

where the upper row is the quotient D(n, r)/D(n, r + 1) and the lower row is induced by the obvious inclusions and projections if one takes the following identities into account

\overline{B}_{∞}^{r−1,n+2−r} = B_{r}^{r,n+1−r};
B_{r+1}^{r,n+1−r} = B_{∞}^{r,n+1−r};
\overline{Z}_{∞}^{r−1,n+2−r} = Z_{∞}^{r,n+1−r}.

We conclude from the sum formula and transformation formula 2.6

ρ(C) = ρ(\overline{C}) + ρ(C/\overline{C}) + ρ(LHS);
ρ(LHS) = ∑_{n≥−1}(−1)^{n+1}ρ(D(n));

ρ(D(n)) = ρ(D(n, l)) + ∑_{r=1}^{l−1} ρ(D(n, r)/D(n, r + 1));
ρ(D(n, r)/D(n, r + 1)) = −[[γ_{r}^{0,n}]] + [[\overline{ψ}^{r−1,n+2−r}]] − [[ψ^{r,n+1−r}]]  for 1 ≤ r ≤ l.

Since D(n, l) is concentrated in dimensions 0 and 1 and its zero-th differential is ψ^{0,n}, we get

ρ(D(n, l)) = [[ψ^{0,n}]].

We compute using the fact

\overline{γ}_{r}^{p,q} = γ_{r}^{p+1,q−1} for p ≥ 1

and the induction hypothesis applied to \overline{C}.

ρ(C) = ρ(\overline{C}) + ρ(F_{0}/F_{1}) +

+ ∑_{n≥−1}(−1)^{n+1} · ([[ψ^{0,n}]] + ∑_{r=1}^{l−1} (−[[γ_{r}^{0,n}]] + [[\overline{ψ}^{r−1,n+2−r}]] − [[ψ^{r,n+1−r}]]))

= ∑_{p≥1} ρ(F_{p}/F_{p+1}) + ∑_{r≥1} ∑_{p≥1,q}(−1)^{p+q−1}[[γ_{r}^{p,q−1}]] − ∑_{p,q}(−1)^{p+q} · [[\overline{ψ}^{p,q}]] + ρ(F_{0}/F_{1})

+ ∑_{n≥−1}(−1)^{n+1} · ([[ψ^{0,n}]] + ∑_{r=1}^{l−1} −[[γ_{r}^{0,n}]] + [[\overline{ψ}^{r−1,n+2−r}]] − [[ψ^{r,n+1−r}]])

= ∑_{p} ρ(F_{p}/F_{p+1}) + ∑_{r≥1} ∑_{p,q}(−1)^{p+q} · [[γ_{r}^{p,q}]] − ∑_{p,q}(−1)^{p+q} · [[ψ^{p,q}]].

This finishes the proof of Lemma 4.6.

Lemma 4.7

ρ_{fil}(F_{∗}C) = ∑_{p} ρ(F_{p}/F_{p+1}) + ∑_{r≥1} ∑_{p,q}(−1)^{p+q} · [[γ_{r}^{p,q}]] − ∑_{p,q}(−1)^{p+q} · [[ψ^{p,q}]]

\underline{\underline{Proof}} : We get from 2.2

ρ(E_{r}^{p+r∗,q−(r−1)∗}) = ∑_{k}(−1)^{k} · [[γ_{r}^{p+rk,q−(r−1)k}]].

This implies

∑_{r≥1} ∑_{p,q}(−1)^{p+q} · [[γ_{r}^{p,q}]] = ∑_{r≥1} ∑_{p=0}^{r−1} ∑_{q}(−1)^{p+q} · ∑_{k}(−1)^{k} · [[γ_{r}^{p+rk,q−(r−1)k}]]

= ∑_{r≥1} ∑_{p=0}^{r−1} ∑_{q}(−1)^{p+q} · ρ(E_{r}^{p+r∗,q−(r−1)∗})

This finishes the proof of Lemma 4.7 and hence of Theorem 4.4.

5. The fibration formula for Milnor torsion

For the remainder of this section we suppose that we have a fibration f : E ⟶ B so that B is a finite connected CW -complex, the homotopy fiber has the homotopy type of a finite CW -complex, the class θ_{f} ∈ H^{1}(B; Wh(E)) defined in 1.10 is trivial and E is locally path-connected.

5.1 Moreover, we assume that we are also given the following data:

1. A local coefficient system V on E with a unimodular Hilbert structure;

2. a Hilbert structure on H_{sing}^{∗}(E; V);

3. a coherent choice of simple structures on the fibers;

4. a unimodular Hilbert structure on ∑_{q}(−1)^{q}·H_{sing}^{q}(F; V) for the local coefficient systems H_{sing}^{q}(F; V) over B;

5. a Hilbert structure on H_{cell}^{∗}(B; H_{sing}^{q}(F; V)).

We mention that a choice of a unimodular Hilbert structure on ∑_{q}(−1)^{q} · H_{sing}^{q}(F; V) is possible because of

Lemma 5.2 If f : E ⟶ B is a fibration as desribed above, then ∑_{q}(−1)^{q} · H_{sing}^{q}(F; V) is unimodular.

\underline{\underline{Proof}} : We get from Lemma 3.6 for a loop w in B with base point b if t_{w} : F_{b} ⟶ F_{b} is given by the fiber transport along w:

∑_{q}(−1)^{q} · [[(t_{w})^{∗} : H_{sing}^{q}(F_{b}; V) ⟶ H_{sing}^{q}(F_{b}; V)]] = ρ(F_{b}; V) − ρ(F_{b}; V) + Φ(V)(θ_{f}(w)) = 0

where we think of θ_{f} as homomorphism π_{1}(B, b) ⟶ Wh(E) and Φ(V) was introduced in 3.5.

Notice that ∑_{q}(−1)^{q} · H_{sing}^{q}(F; V) is unimodular but not necessarily each H_{sing}^{q}(F; V).

5.3 We will consider the following real numbers:

1. ρ(E; V)

We get by construction 1.7 a simple structure on E from the data 5.1.3 if we specify a cellular base point system on B. However, by 1.9 the choice of cellular base point system does not matter. Now we use this simple structure and data 5.1.1 and 5.1.2 to define the Milnor torsion ρ(E; V) according to Definition 3.3. Notice that it depends only on 5.1.1, 5.1.2 and 5.1.3. This is the invariant we want to compute;

2. ρ(F; V)

Choose b ∈ B. Then we get from Definition 3.3 applied to data 5.1.1, 5.1.3 and 5.1.4 the Milnor torsion ρ(F_{b}; V). Here and elsewhere we supress in the notation that we view V as a local coefficient system over F_{b} by the inclusion of F_{b} into E. We get from Lemma 3.6 that ρ(F_{b}; V) is independent of b since B is path connected. We abbreviate ρ(F; V) = ρ(F_{b}; V). This number depends only on 5.1.1, 5.1.3 and 5.1.4;

3. ρ(B; ∑_{q}(−1)^{q} · H_{sing}^{q}(F; V))

This is the Milnor torsion and depends on the data 5.1.4 and 5.1.5; 4. ρ_{sing}^{LS}(f)

We have the Leray-Serre spectral sequence for singular cohomology associated to the fibration f : E ⟶ B. Namely, the skeletal filtration of B induces a filtration E_{p} = f^{−1}(B_{p}). It yields a cofiltration on C_{sing}^{∗}(E; V) by putting

F_{p}C_{sing}^{∗}(E; V) = C_{sing}^{∗}(E, E_{p−1}; V).

Recall that C_{sing}^{∗}(E, E_{p−1}; V) is the kernel of the canonical map C_{sing}^{∗}(E; V) ⟶ C_{sing}^{∗}(E_{p−1}; V) induced from the inclusion. We will later recall in 5.5 the isomorphism which identifies the E_{2}-term of the associated spectral sequence

U_{2}^{p,q} : H_{cell}^{p}(B; H_{sing}^{q}(F; V)) ⟶ E_{2}^{p,q}.

Equip E_{2}^{p,q} with the Hilbert structure so that U_{2}^{p,q} becomes an isometry. Using Definition 4.2 put

ρ_{sing}^{LS}(f) := ρ_{fil}^{≥2}(F_{∗}C_{sing}^{∗}(X; V)) + ∑_{p,q}(−1)^{p+q}[[U_{2}^{p,q}]].

Using Remark 4.3, ρ_{sing}^{LS}(f) depends only on the data 5.1.2 and 5.1.5.

The main result of this section is

Theorem 5.4 If F is a fibration as described above with data 5.1 then

ρ(E; V) = χ(B) · ρ(F; V) + ρ(B; ∑_{q}(−1)^{q} · H_{sing}^{q}(F; V)) + ρ_{sing}^{LS}(f).

\underline{\underline{Proof}} :

Fix a cellular base point system {b_{c} | c ∈ I} for B and an orientation for each cell c ∈ I. The homotopy equivalence of Lemma 1.8 together with the suspension cochain homotopy equivalence and the obvious identification of F_{p}/F_{p+1} with C_{sing}^{∗}(E_{p}, E_{p−1}; V) yields a cochain homotopy equivalence unique up to homotopy

U_{0}^{p,∗} : ⊕_{c∈I_{p}}Σ^{p}C_{sing}^{∗}(F_{b_{c}}; V) ⟶ F_{p}/F_{p+1}.

We have defined an isomorphism in 3.4

μ^{p,q} : C_{cell}^{p}(B; H_{sing}^{q}(F; V)) ⟶ ⊕_{c∈I_{p}}H_{sing}^{q}(F_{b_{c}}; V).

Recall that we equip C_{cell}^{p}(B; H_{sing}^{q}(F; V)) with the Hilbert structure for which μ^{p,q} becomes an isometry. Define an isomorphism by the composition

U_{1}^{p,q} : C_{cell}^{p}(B; H_{sing}^{q}(F; V)) ⟶^{μ^{p,q}} ⊕_{c∈I_{p}}H_{sing}^{q}(F_{b_{c}}; V) = ⊕_{c∈I_{p}}H^{p+q}(Σ^{p}C_{sing}^{∗}(F_{b_{c}}; V))

⟶^{H^{p+q}(U_{0}^{p,∗})} H^{p+q}(F_{p}/F_{p+1}) = H^{0}(E_{0}^{p,q+∗}) ⟶^{ϕ_{0}^{p,q}} E_{1}^{p,q}

This isomorphism is independent of the choice of base point system and orientation of the cells and is compatible with the differentials on C_{cell}^{∗}(B; H_{sing}^{q}(F; V)) and E_{1}^{∗,q}. Hence we get a canonical isomorphism

U_{2}^{p,q} : H_{cell}^{p}(B; H_{sing}^{q}(F; V)) ⟶^{H^{p}(U_{1}^{∗,q})} H^{p}(E_{1}^{∗,q}) = H^{0}(E_{1}^{p+∗,q}) ⟶^{ϕ_{1}^{p,q}} E_{2}^{p,q}.  (5.5)

The simple structure on F_{b_{c}} induces one on C_{sing}^{∗}(F_{b_{c}}; V). Put on F_{p}/F_{p+1} the simple structure for which the torsion t(U_{0}^{p,∗} : ⊕_{c∈I_{p}}Σ^{p}C_{sing}^{∗}(F_{b_{c}}; V) ⟶ F_{p}/F_{p+1}) defined in 2.3 vanishes. Equip F_{p} inductively with the simple structure for which ρ(F_{p+1} ⟶ F_{p} ⟶ F_{p}/F_{p+1}) defined in 2.5 vanishes. Then this simple structure on F_{0} = C_{sing}^{∗}(E; V) agrees with the one we get from the simple structure on E which we have defined in 1.7 by an inductive procedure over the E_{n}-s and Lemma 1.8. We conclude from Theorem 4.4:

ρ(E; V) = ρ_{fil}(F_{∗}C_{sing}^{∗}(E; V)).

Because ϕ_{r}^{p,q} are isometries, we get from the transformation formula 2.6:

∑_{p} ρ(F_{p}/F_{p+1}) = ∑_{p} ∑_{c∈I_{p}} ρ(Σ^{p}C_{sing}^{∗}(F_{b_{c}}; V)) + ∑_{p} ∑_{q}(−1)^{q} · [[H^{q}(U_{0}^{p,∗})]]
= ∑_{p}(−1)^{p} · ∑_{c∈I_{p}} ρ(F; V) + ∑_{p,q}(−1)^{p+q} · [[H^{p+q}(U_{0}^{p,∗})]]
= χ(B) · ρ(F; V) + ∑_{p,q}(−1)^{p+q} · [[U_{1}^{p,q}]]
∑_{q}(−1)^{q} · ρ(E_{1}^{∗,q}) = ∑_{q}(−1)^{q} · ρ(C_{cell}^{∗}(B; H_{sing}^{q}(F; V)) − ∑_{q}(−1)^{q} · ∑_{p}(−1)^{p} · [[U_{1}^{p,q}]]

+ ∑_{q}(−1)^{q} · ∑_{p}(−1)^{p} · [[H^{p}(U_{1}^{∗,q})]]

= ρ(B; ∑_{q}(−1)^{q} · H^{q}(F; V)) − ∑_{p,q}(−1)^{p+q} · [[U_{1}^{p,q}]] + ∑_{p,q}(−1)^{p+q} · [[U_{2}^{p,q}]]

We get from 5.3.4 and Definition 4.2:

ρ_{fil}(F_{∗}C_{sing}^{∗}(E; V)) = ∑_{p} ρ(F_{p}/F_{p+1}) + ∑_{q}(−1)^{q} · ρ(E_{1}^{∗,q})

+ρ_{sing}^{LS}(f) − ∑_{p,q}(−1)^{p+q} · [[U_{2}^{p,q}]].

Now Theorem 5.4 follows.

Remark 5.6 One can extend Theorem 5.4 to the case where B is not necessarily a finite CW -complex but carries a simple structure and one requires in data 5.1.5 a Hilbert structure on H_{sing}^{∗}(B; H^{q}(F; V)) instead of H_{cell}^{∗}(B; H^{q}(F; V)). Then we still get a simple structure on E by Remark 1.14. It remains to define ρ_{sing}^{LS}(f). Choose an arbitrary homotopy equivalence h : Y ⟶ B for a finite CW -complex Y. Let \overline{f} : h^{∗}E ⟶ Y be the pull back fibration and \overline{h} : h^{∗}E ⟶ E be the canonical fiber homotopy equivalence. Equip H_{sing}^{∗}(h^{∗}E; \overline{h}^{∗}V) and H_{cell}^{∗}(Y; h^{∗}H_{sing}^{q}(F; V)) with the Hilbert structures for which the following isomorphisms become isometries (see 3.2)

H_{sing}^{∗}(h^{∗}E; \overline{h}^{∗}V) ⟶^{(\overline{h}^{∗})^{−1}} H_{sing}^{∗}(E; V)

and

H_{cell}^{∗}(Y; h^{∗}H_{sing}^{q}(F; V)) ⟶ H_{sing}^{∗}(Y; h^{∗}H_{sing}^{q}(F; V)) ⟶^{(h^{∗})^{−1}} H_{sing}^{∗}(B; H_{sing}^{q}(F; V)).

Then ρ_{sing}^{LS}(\overline{f}) is defined and we put

ρ_{sing}^{LS}(f) = ρ_{sing}^{LS}(\overline{f}).

This is independent of the choice of h and Theorem 5.4 remains true.

6. The fiber bundle formula for analytic torsion

In this section we give the proof of Theorem 0.2 and Corollaries 0.8, 0.9 and 0.10.

We begin with the proof of Theorem 0.2. In the first step we reduce the claim to the case where the Riemannian metric on V is unimodular. By assumption we only have to treat the case where dim(E) is odd. We will vary the Riemannian metric on V, but fix the Riemannian metrics on E and B. Denote by V the flat bundle V with the given Riemannian metric and by V̂ the flat vector bundle V with some unimodular Riemannian metric. Let ρ_{dR}^{LS}(f) and { \atop ρ_{dR}^{LS}(f)} be the two values of the correction term for the choice of these two Riemannian metrics on V. Notice that either Pf_{B} is zero or dim(F_{b}) is odd. We get from Theorem 0.13 and Definition A.7

ρ_{an}(E; V) − ρ_{an}(E; V̂) = − ∑_{p}(−1)^{p} · [[H_{dR}^{p}(E; V) ⟶^{id} H_{dR}^{p}(E; V̂)]];

∫_{B} ρ_{an}(F_{b}; V) · Pf_{B} − ∫_{B} ρ_{an}(F_{b}; V̂) · Pf_{B}

= − ∫_{B} (∑_{q}(−1)^{q} · [[H_{dR}^{q}(F_{b}; V) ⟶^{id} H_{dR}^{q}(F_{b}; V̂)]]) · Pf_{B};

ρ_{an}(B; H_{dR}^{q}(F; V)) − ρ_{an}(B; H_{dR}^{q}(F; V̂))

= − ∑_{p}(−1)^{p} · [[H_{dR}^{p}(B; H_{dR}^{q}(F; V)) ⟶^{id} H_{dR}^{p}(B; H_{dR}^{q}(F; V̂))]]

+ ∫_{B}[[H_{dR}^{q}(F_{b}; V) ⟶^{id} H_{dR}^{q}(F_{b}; V̂)]] · Pf_{B};

ρ_{dR}^{LS}(f) − { \atop ρ_{dR}^{LS}(f)} = ∑_{p,q}(−1)^{p+q} · [[H_{dR}^{p}(B; H_{dR}^{q}(F; V)) ⟶^{id} H_{dR}^{p}(B; H_{dR}^{q}(F; V̂))]]

− ∑_{p}(−1)^{p} · [[H_{dR}^{p}(E; V) ⟶^{id} H_{dR}^{p}(E; V̂)]]

We conclude from the equations above:

ρ_{an}(E; V) − (∫_{B} ρ_{an}(F_{b}; V) · Pf_{B} + ∑_{q}(−1)^{q} · ρ_{an}(B; H_{dR}^{q}(F; V)) + ρ_{dR}^{LS}(f))

= ρ_{an}(E; V̂) − (∫_{B} ρ_{an}(F_{b}; V̂) · Pf_{B} + ∑_{q}(−1)^{q} · ρ_{an}(B; H_{dR}^{q}(F; V̂)) + { \atop ρ_{dR}^{LS}(f)}) .

Hence we can assume in the sequel without loss of generality that the Riemannian metric on V is unimodular.

Given a closed smooth manifold M and a flat vector bundle W over M with Riemannian metrics on M and W, the harmonic Hilbert structure on the singular cohomology H_{sing}^{∗}(M; W) is given by the deRham isomorphism

H_{dR}^{∗}(M; W) ⟶ H_{sing}^{∗}(M; W)

and the harmonic Hilbert structure on H_{dR}^{∗}(M; W) coming from the Hodge decomposition and the Hilbert space structure on the space of harmonic forms as explained in the introduction. If we have choosen a smooth triangulation on M, the harmonic Hilbert structure on H_{cell}^{∗}(M; W) is induced by the canonical isomorphism

H_{cell}^{∗}(M; W) ⟶ H_{sing}^{∗}(M; W)

induced from 3.2. Denote by ρ_{top}(M; W) the Milnor torsion of M defined in 3.3 with respect to the simple structure represented by a smooth triangulation and the harmonic Hilbert structure on H_{sing}^{∗}(M; W).

Denote by \overline{H_{sing}^{q}(F; V)} the flat bundle over B with fibers H_{sing}^{q}(F_{b}; V) equipped with some Riemannian metric such that the induced Hilbert structure on ∑_{q}(−1)^{q} · \overline{H_{sing}^{q}(F; V)} is unimodular. Recall that this can be done by Lemma 5.2. Equip B and F_{b} with the CW - structure given by some smooth triangulation. Let \overline{ρ(F_{b}; V)} be the Milnor torsion of F_{b} with respect to \overline{H_{sing}^{q}(F; V)} (see Definition 3.3). We have seen in 5.3.2 that it is independent of b ∈ B and abbreviate it by \overline{ρ(F; V)}. Let \overline{ρ_{sing}^{LS}(f)} be the correction term of 5.3 with respect to the harmonic Hilbert structures on H_{sing}^{∗}(E; V) and H_{sing}^{p}(B; \overline{H_{sing}^{q}(F; V)}). We conclude from Lemma 1.13 and Theorem 5.4:

ρ_{top}(E; V) = χ(B) · \overline{ρ(F; V)} + ρ(B; ∑_{q}(−1)^{q} · \overline{H_{sing}^{q}(F; V)}) + \overline{ρ_{sing}^{LS}(f)}.  (6.1)

Next we want to show:

ρ_{an}(E; V) = ρ_{top}(E; V);  (6.2)
ρ_{an}(F_{b}; V) = ρ_{top}(F_{b}; V);  (6.3)
∑_{q}(−1)^{q} · ρ_{an}(B; \overline{H_{sing}^{q}(F; V)}) = ρ(B; ∑_{q}(−1)^{q} · \overline{H_{sing}^{q}(F; V)}).  (6.4)

The first two equations follow directly from Theorem 0.12. For the proof of the last equation we must take a closer look at the result of Bismut and Zhang [1, Theorem 0.2]. The problem is that the Riemannian metric on ∑_{q}(−1)^{q} · \overline{H_{sing}^{q}(F; V)} is unimodular but it is not true in general that each of the flat bundles H_{sing}^{q}(F; V) is unimodular or, equivalently, admits some unimodular Riemannian metric.

Choose a Morse function f on B. Let X be its gradient vector field with respect to some Riemannian metric on B which can be different from the given Riemannian metric such that X satisfies the Smale transversality conditions. We obtain a CW -structure on B, denoted by B′, and a cellular base point system {b_{c} | c ∈ I} given by the critical points of f. The simple structure represented by id : B′ ⟶ B is the same as the one represented by any smooth triangulation. Let C_{cell}^{∗}(B′, \{b_{c}\}; \overline{H_{sing}^{q}(F; V)}) be the cellular cochain complex with the Hilbert structure defined in 3.4. We equip H^{p}(C_{cell}^{∗}(B′, \{b_{c}\}; \overline{H_{sing}^{q}(F; V)})) with the harmonic Hilbert structure. We get by inspecting Definition 3.3

ρ(B, ∑_{q}(−1)^{q} · \overline{H_{sing}^{q}(F; V)}) = ∑_{q}(−1)^{q} · ρ(C_{cell}^{∗}(B′, \{b_{c}\}; \overline{H_{sing}^{q}(F; V)})).

But C_{cell}^{∗}(B′, \{b_{c}\}; \overline{H_{sing}^{q}(F; V)}) is isometrically isomorphic to the Thom-Smale complex. Let vol_{harm} be the volume form associated to the harmonic Hilbert structure. Hence we get from the definitions (cf. [1, Remark 2.3 and Remark 1.10])

ρ_{an}(B; \overline{H_{sing}^{q}(F; V)}) = − ln (|| vol_{harm} ||_{det H_{dR}^{•}(B;\overline{H_{sing}^{q}(F;V)})}^{RS});
ρ(C_{cell}^{∗}(B′, \{b_{c}\}; \overline{H_{sing}^{q}(F; V)}) = − ln (|| vol_{harm} ||_{det H_{cell}^{•}(B;\overline{H_{sing}^{q}(F;V)})}^{M,X});

where the terms on the right side are the invariants appearing in [1]. We get from [1, Theorem 0.2])

ρ_{an}(B; \overline{H_{sing}^{q}(F; V)}) − ρ(C_{cell}^{∗}(B′, \{b_{c}\}; \overline{H_{sing}^{q}(F; V)})) = ∫_{B} θ(\overline{H_{sing}^{q}(F; V)}) · X^{∗}ψ(B)

for a certain (dim(B) − 1)-form ψ(B). We conclude

∑_{q}(−1)^{q} · ρ_{an}(B; \overline{H_{sing}^{q}(F; V)}) − ρ(B, ∑_{q}(−1)^{q} · \overline{H_{sing}^{q}(F; V)})

= ∑_{q}(−1)^{q} · ρ_{an}(B; \overline{H_{sing}^{q}(F; V)}) − ∑_{q}(−1)^{q} · ρ(C_{cell}^{∗}(B′, \{b_{c}\}; \overline{H_{sing}^{q}(F; V)}))

= ∫_{B} ∑_{q}(−1)^{q} · θ(\overline{H_{sing}^{q}(F; V)}) · X^{∗}ψ(B).

Now one easily checks that ∑_{q}(−1)^{q} · θ(\overline{H_{sing}^{q}(F; V)}) vanishes as the Riemannian metric on ∑_{q}(−1)^{q} · \overline{H_{sing}^{q}(F; V)}) is unimodular. This finishes the proof of 6.4.

Denote by H_{dR}^{q}(F; V) and H_{sing}^{q}(F; V) the flat bundles with the harmonic Riemannian metrics (in contrast to \overline{H_{sing}^{q}(F; V)}). We get from the transformation formula 2.6 and 6.3

χ(B) · \overline{ρ(F; V)} − ∫_{B} ρ_{an}(F_{b}; V) · Pf_{B}  (6.5)
= ∫_{B} (\overline{ρ(F_{b}; V)} − ρ_{top}(F_{b}; V)) · Pf_{B}
= − ∫_{B} ∑_{q}(−1)^{q} · [[\overline{H_{sing}^{q}(F_{b}; V)} ⟶^{id} H_{sing}^{q}(F_{b}; V)]] · Pf_{B} .

We conclude from 6.4 and Theorem 0.13:

ρ(B; ∑_{q}(−1)^{q} · \overline{H_{sing}^{q}(F; V)}) − ∑_{q}(−1)^{q} · ρ_{an}(B; H_{dR}^{q}(F; V))  (6.6)
= ∑_{q}(−1)^{q} · (ρ_{an}(B; \overline{H_{sing}^{q}(F; V)}) − ρ_{an}(B; H_{sing}^{q}(F; V)))

= ∑_{q}(−1)^{q} · (− ∑_{p}(−1)^{p} · [[H_{dR}^{p}(B, \overline{H_{sing}^{q}(F; V)}) ⟶^{id} H_{dR}^{p}(B, H_{sing}^{q}(F; V))]]

+ ∫_{B}[[\overline{H_{sing}^{q}(F; V)} ⟶^{id} H_{sing}^{q}(F; V)]] · Pf_{B})

= − ∑_{p,q}(−1)^{p+q} · [[H_{dR}^{p}(B, \overline{H_{sing}^{q}(F; V)}) ⟶^{id} H_{dR}^{p}(B, H_{sing}^{q}(F; V))]]

+ ∑_{q}(−1)^{q} · ∫_{B}[[\overline{H_{sing}^{q}(F_{b}; V)} ⟶^{id} H_{sing}^{q}(F_{b}; V)]] · Pf_{B} .

The deRham isomorphism H_{dR}^{n}(E; V) ⟶ H_{sing}^{n}(E; V) is compatible with the two filtrations and the identifications of the E_{2}-term in the Leray-Serre spectral sequence for deRham and singular cohomology 0.3 and 5.5. It yields isomorphisms on E_{r}^{p,q} for r ≥ 2. This implies using Remark 4.3

\overline{ρ_{sing}^{LS}(f)} − ρ_{dR}^{LS}(f) = ∑_{p,q}(−1)^{p+q}[[H_{dR}^{p}(B, \overline{H_{sing}^{q}(F; V)}) ⟶^{id} H_{dR}^{p}(B, H_{sing}^{q}(F; V))]]. (6.7)

Now Theorem 0.2 follows from 6.1, 6.2, 6.5, 6.6 and 6.7.

Next we prove Corollary 0.8. Recall that H_{dR}^{q}(F; V) vanishes by assumption. In particular the E_{2}-term of the Leray-Serre spectral sequences for cohomology vanishes. Hence Theorem 0.2 implies

ρ_{an}(E; V) = ∫_{B} ρ_{an}(F_{b}; V) · Pf_{B} .

If dim(B) is odd, the right hand side of the equation above and χ(B) · ρ_{an}(F_{b}; V) vanish and the claim follows. Suppose that dim(B) is even. Then dim(F_{b}) is even and the Riemannian metric on V is unimodular by assumption or dim(F_{b}) is odd. In both cases ρ_{an}(F_{b}; V) is independent of b ∈ B by Theorem 0.13 and the claim follows. This finishes the proof of Corollary 0.8.

Next we outline the proof of Corollary 0.9. Analogously to the first step in the proof of Theorem 0.2 we can show that we can assume without loss of generality that the Riemannian metric on W is unimodular. Put V = f^{∗}W. There is a canonical isomorphism of local coefficient systems over B

H_{sing}^{q}(F; ℤ) ⊗_{ℤ} W ⟶ H_{sing}^{q}(F; V).

Recall that we assume that E and B are oriented. (One can drop this assumption if one allows an additional twist for W.) This implies that π_{1}(B) acts orientation preserving on the homology of the fiber. Hence we can choose an isomorphism of local coefficient systems of ℤ-modules from H_{sing}^{q}(F; ℤ) to the trivial coefficient system with value ℤ. Hence we obtain identifications

H_{sing}^{q}(F; V) = \{ \begin{matrix}W & q = 0, n \\ 0 & q ≠ 0, n \end{matrix} .

We conclude from Theorem 5.4

ρ_{top}(E; V) = χ(B) · ρ(F_{b}; V) + ρ(B; ∑_{q}(−1)^{q} · H_{sing}^{q}(F; V)) + ρ_{sing}^{LS}(f)

where we use on H_{sing}^{q}(F_{b}; V) the unimodular Hilbert structure given by the one on W and the identification above and on H_{sing}^{q}(B; ∑_{q}(−1)^{q} · H_{sing}^{q}(F; V)) the harmonic one. We get

ρ(F; V) = 0;
ρ(B; ∑_{q}(−1)^{q} · H_{sing}^{q}(F; V)) = χ(S^{n}) · ρ_{top}(B; W);
ρ_{sing}^{LS}(f) = ρ_{dR}^{LS}(f);
ρ_{an}(E; V) = ρ_{top}(E; V);
ρ_{an}(B; W) = ρ_{top}(B; W);

since there is a canonical isomorphism

C_{sing}^{∗}(F_{b}; ℤ) ⊗_{ℤ} W ⟶ C_{sing}^{∗}(F_{b}; V)

and H_{sing}^{∗}(F_{b}; ℤ) is free as ℤ-module, the deRham isomorphism is compatible with the cofiltrations on the de Rham complex and the singular cochain complex and with the identifications of the E_{2}-terms of the associated Leray-Serre spectral sequences 0.3 and 5.5 and we have Theorem 0.12. Hence it remains to show

ρ_{dR}^{LS}(f) = ρ(G^{∗}).

Notice that the E^{2}-term of the Leray-Serre spectral sequence is trivial except for the 0-th and n-th row. Hence we can splice the spectral sequence together to one exact sequence

… ⟶ E_{n}^{p,n} ⟶^{d_{n+1}^{p,n}} E_{n}^{p+n+1,0} ⟶ H^{p+n+1}(E; V) ⟶ E_{n}^{p+1,n} ⟶^{d_{n+1}^{p+1,n}} …

If one takes the identification of the E_{2}-term 0.3 and the obvious identification E_{2}^{p,q} = E_{n}^{p,q} into account, one gets an exact sequence whose torsion is precisely ρ_{dR}^{LS}(f). Moreover, it can be identified with the Gysin sequence up to sign. This finishes the proof of Corollary 0.9. The proof of Corollary 0.10 is similar.

A Torsion and determinants

We start with a description of the determinant of a finite dimensional vector space. Let V be a n-dimension (real) vector space. Define its determinant det(V) to be the 1-dimensional vector space ∧^{n}V given by the n-th exterior power. There are canonical isomorphisms

det(V ⊕ W)  ⟶^{≅}  det(V) ⊗ det(W);
det(V^{∗})  ⟶^{≅}  det(V)^{∗};
det(V)^{∗} ⊗ det(W)  ⟶^{≅}  hom_{ℝ}(det(V), det(W));
det(U^{∗}) ⊗ det(V) ⊗ det(V^{∗}) ⊗ det(W)  ⟶^{≅}  det(U^{∗}) ⊗ det(W);
det(V) ⟶ (det(V)^{∗})^{∗}.

Given a homomorphisms f : V ⟶ W, we obtain a well-defined element

det(f) ∈ det(V)^{∗} ⊗ det(W).

Under the identifications above

det(f ∘ g) = det(f) ⊗ det(g);
det ( \begin{matrix}f & h \\ 0 & g \end{matrix} ) = det(f) ⊗ det(g).

If (V_{p})_{p} is a graded vector space graded by nonnegative integers p ≥ 0 and almost all V_{p} are zero, then we define

det((V_{p})_{p}) := ⊗_{p≥0} det(V_{p})^{(−1)^{p}}

where det(V)^{1} is det(V) and det(V)^{−1} is defined by det(V)^{∗}. If (V_{p,q})_{p,q} is bigraded by non-negative integers p, q ≥ 0 and almost all V_{p,q} are zero, then we define

det((V_{p,q})_{p,q}) = det((W_{n})_{n})

where (W_{n})_{n} is the associated graded vector space with W_{n} = ⊕_{p=0}^{n}V_{p,n−p}.

Now we will introduce algebraic torsion invariants in the language of determinants.. Given an acyclic finite cochain complex, we define

\overline{ρ}(C) ∈ det((C^{p})_{p})^{−1}  (A.1)

by the determinant of the isomorphism c^{∗} + γ^{∗} : C^{ev} ⟶ C^{odd}. For a homotopy equivalence f : C ⟶ D of finite cochain complexes we define

\overline{ρ}(f) := \overline{ρ}(cone(f)) ∈ det((C^{p})_{p})^{−1} ⊗ det((D^{p})_{p}) ≅ hom_{ℝ}(det(C^{p})_{p}, det(D^{p})_{p}). (A.2)

Given a finite cochain complex C, we use the cochain map i : H(C) ⟶ C to define the isomorphism

\overline{ρ}(C) := (\overline{ρ}(i))^{−1}  : det((C^{p})_{p}) ⟶ det((H^{p}(C))_{p}).  (A.3)

The topological torsion \overline{ρ}(X) of a finite CW-complex X is defined as the torsion of the cellular cochain complex.

The definition of the torsion of a filtration/spectral sequence is a little bit more elaborate.

Let F_{∗}W be a fibration of a finite-dimensional vector space W

{0} = F_{−1} ⊂ ⋯ ⊂ F_{p} ⊂ F_{p+1} ⊂ ⋯ ⊂ F_{n} = W.

This determines an isomorphism

\overline{ρ}^{fil}(F_{∗}W) : ⊗_{p≥0} det(F_{p}/F_{p−1})) ⟶^{≅} det(W).  (A.4)

which is defined inductively over n. The begin of induction n = 1 is given by the Milnor torsion of the acyclic 2-dimensional cochain complex F_{0} ⟶ W ⟶ W/F_{0}. For the step n−1 to n observe that

\overline{ρ}^{fil}(F_{∗}F_{n−1} : ⨁_{p=0}^{n−1} det(F_{p}/F_{p−1}) ⟶^{≅} det(F_{n−1}

is defined by induction, and we simply compose (after tensoring with det(W/F_{n−1})) with the Milnor torsion of F_{n−1} → W → W/F_{n−1}.

In particular, suppose f : E → B is a fibration of closed manifolds as in the introduction. This yields a filtration of H_{dR}^{n}(E; V) for n ≥ 0. Take the tensor product of the corresponding torsion isomorphisms resp. their inverses depending on the parity of n to obtain an isomorphism

\overline{ρ}^{fil} : det((F^{p,q}/F^{p+1,q−1})_{p,q}) ⟶ det((H^{n}(E; V))_{n}).

In the de Rham spectral sequence, we can compute the torsions of the cochain complex (E_{r}^{p+r∗,q−(r−1)∗}, d_{r}^{p+r∗,q−(r−1)∗}). We obtain an isomorphism

\overline{ρ}(E_{r}^{p+r∗,q−(r−1)∗}) : ⊗_{n} det(H^{n}(E_{r}^{p+r∗,q−(r−1)∗})^{(−1)^{n}} ⟶ ⊗_{n} det(E_{r}^{p+rn,q−(r−1)n})^{−1)^{n}}.

Taking ⊗_{p=0}^{r−1} ⊗_{q≥0} (\overline{ρ}(E_{r}^{p+r∗,q−(r−1)∗}))^{(−1)^{p+q}} yields an isomorphism for r ≥ 2

\overline{ρ} : det((H^{0}(E_{r}^{p+r∗,q−(r−1)∗}))_{p,q}) ⟶ det((E_{r+1}^{p,q})_{p,q}).

Together with the natural isomorphisms V^{p,q} : H_{dR}^{p}(B, H_{dR}^{q}(F, V) → E_{2}^{p,q} and ϕ_{r}^{p,q} between the homology of E_{r} and E_{r+1}, this definies an isomorphism

\overline{ρ}_{dR}^{LS}(f) : det((H_{dR}^{p}(B; H_{dR}^{q}(F; V))_{p,q}) ⟶^{≅} det((H_{dR}^{n}(E; V))_{n})  (A.5)

as the composition of isomorphisms (use the fact that E_{r} and E_{∞} are equal for r large enough)

det((H_{dR}^{p}(B; H_{dR}^{q}(F; V))_{p,q}) ⟶^{(V^{p,q})_{p,q}} det((E_{2}^{p,q})_{p,q}) ⟶^{\overline{ρ}}det((H^{0}(E_{2}^{p+2∗,q−∗}))_{p,q}) ⟶^{(ϕ_{2}^{p,q})_{p,q}}

det((E_{3}^{p,q})_{p,q}) ⟶^{\overline{ρ}}det((H^{0}(E_{3}^{p+3∗,q−2∗}))_{p,q}) ⟶^{(ϕ_{3}^{p,q})_{p,q}} det((E_{4}^{p,q})_{p,q}) ⟶ … ⟶

det((E_{∞}^{p,q})_{p,q}) ⟶^{((ψ^{p,q})^{−1})_{p,q}} det((F^{p,q}/F^{p+1,q−1})_{p,q}) ⟶^{\overline{ρ}^{fil}} det((H_{dR}^{n}(E; V))_{n})).

For the spectral sequence associated to an arbitrary filtration F_{∗}W of a finite-dimensional vector space W one can define \overline{ρ}_{fil}(F_{∗}W) and \overline{ρ}_{fil}^{≥2}(F_{∗}W) in the obvious analogous way.

Now, we have to relate the invariants \overline{ρ} to the previously defined real numbers ρ —using given inner products:.

If the finite-dimensional vector space V has a Hilbert structure, it induces in a canonical way a Hilbert space structure on det(V). In particular the norm ∥v∥ ∈ ℝ^{≥0} is defined for any element v ∈ V. If V and W come with Hilbert space structures, for any element

u ∈ hom_{ℝ}(det(U), det(V)) = det(V)^{∗} ⊗ det(W)

we get its norm

∥u∥ ∈ ℝ^{≥0}.  (A.6)

It turns out that for a finite Hilbert cochain complex C with given Hilbert structures on H(C)

ρ(C) = ln(∥\overline{ρ}(C)∥)

In the case of the fibration f : E → B of the introduction, the Riemannian metrics on E, B and V induce harmonic Hilbert structures on H_{dR}^{n}(E; V) and H_{dR}^{p}(B; H_{dR}^{q}(F; V)) as described there. Then the norm of the element \overline{ρ}_{dR}^{LS}(f) defined in A.5 is just

ρ_{dR}^{LS}(f) = ln (∥\overline{ρ}_{dR}^{LS}(f)∥) .  (A.7)

Of course, with similar definitions a similar result can be obtained for ρ_{sing}^{LS}(f). Also, the corresponding equation for ρ_{fil} of a filtration is true.

References

[1] Bismut, J.-M. and Zhang, W.: “An extension of a theorem by Cheeger and Müller”, Astérisque 205 (1992)

[2] Cartan, H. and Eilenberg, S.: “Homological Algebra”, Princeton University Press, Princeton (1956)

[3] Cheeger, J.: “Analytic torsion and the heat equation”, Ann. of Math. 109, 259-322 (1979)

[4] Cohen, M.M.: “A course in simple homotopy theory”, Graduate Texts in Mathematics 10, Springer (1973)

[5] Dai, X. and Melrose, R.B.: “Adiabatic limit of the analytic torsion”, preprint

[6] Freed, D.: “Reidemeister torsion, spectral sequences, and Brieskorn spheres”, J. reine angew. Math. 429, 75–89

[7] Fried, D.: “Lefschetz formulas for flows”, Contemp. Math. 58 III, Lefschetz Centennial Conference, 19 -69 (1987)

[8] Hassell, A.: “Analytic surgery and analytic torsion”, Thesis, MIT (1994)

[9] Lott, J. and Rothenberg, M.: “Analytic torsion for group actions”, J. of Diff. Geom. 34, 431 - 481 (1991)

[10] Lück, W.: “Transformation groups and algebraic K-theory”, Lecture Notes in Mathematics 1408, Springer (1989)

[11] Lück, W.: “Analytic and topological torsion for manifolds with boundary and symmetries”, J. of Diff. Geom. 37, 263 - 322 (1993)

[12] Lück, W. and Rothenberg, M.: “Reidemeister torsion and the K-theory of von Neumann algebras”, K-theory 5, 213 - 264 (1991)

[13] Maumary, S.: “Contributions a la theorie du type simple d’homotopy”, Comm. Math. Helv. 44, 410–437 (1969)

[14] Mazzeo, R.R. and Melrose, R.B.: “The adiabatic limit, Hodge cohomology and Leray’s spectral sequence for a fibration”, J. of Diff. Geom. 31, 185 - 213 (1990)

[15] McCleary, J.: “User’s guide to spectral sequences”, Mathematics Lecture Notes Series 12, Publish or Perish (1985)

[16] Milnor, J.: “Whitehead torsion”, Bull. of the AMS 72, 358 - 426 (1966)

[17] Müller, W.: “Analytic torsion and R-torsion of Riemannian manifolds”, Adv. in Math. 28, 233-305 (1978)

[18] Müller, W.: “Analytic torsion and R-torsion for unimodular representations”, J. of AMS 6, 721 - 753 (1993)

[19] Munkholm, H.J.: “Transfer on algebraic K-theory and Whitehead torsion for PLfibrations”, J. of Pure and Applied Algebra 20, 195 - 225 (1981)

[20] Ray, D.B. and Singer, I.: “R-torsion and the Laplacian on Riemannian manifolds”, Adv. in Math. 7, 145 - 210 (1971)

[21] Switzer, R.M.: “Algebraic Topology - Homotopy and Homology”, Die Grundlehren der mathematischen Wissenschaften 212, Springer (1975)

[22] Whitehead, G.W.: “Elements of homotopy theory”, Graduate Texts in Mathematics 61, Springer (1978)

Address Wolfgang Lück und Thomas Schick Fachbereich für Mathematik und Informatik, Westfälische Wilhelms-Universität Münster, Einsteinstr. 62, 48149 Münster, Germany, email: lueck@math.uni-muenster.de, thomas.schick@math.uni-muenster.de Fax: 0251 83-38370

Version of March 4, 1998