THE GEOMETRIC COMPLEX OF A

MORSE–BOTT–SMALE PAIR AND AN EXTENSION OF A THEOREM BY BISMUT–ZHANG

DAN BURGHELEA AND STEFAN HALLER

Abstract. We define the geometric complex associated to a Morse– Bott–Smale vector field, cf. [AB95], and its associated spectral sequence. We prove an extension of the Bismut–Zhang theorem to Morse–Bott–Smale functions, see Theorem 2 below. The proof is based on the Bismut–Zhang theorem for Morse–Smale functions, see [BZ92].

Contents

1. Introduction 1 2. Analysis of the ODE 6 3. The geometric complex 15 4. Torsion 22 5. Proof of main Theorem 5 32 6. Torsion of smooth bundles 37 Appendix A. Proofs of Theorems 3 and 4 43 Appendix B. Some homological algebra 47 References 51

1. Introduction

Let M be a closed connected manifold, g a Riemannian metric on M, E a flat vector bundle over M and μ a fiber metric on E. This permits to define the formal adjoint of the deRham differential on Ω^{∗}(M; E) and

Date: September 24, 2004. 2000 Mathematics Subject Classification. 57R99, 57J99. Key words and phrases. Morse–Bott theory; Cheeger–Müller theorem; Bismut–

Zhang theorem; Analytic torsion; Combinatorial torsion; Geometric complex;

Part of this work was done while both authors enjoyed the hospitality of the Max Planck Institute for Mathematics. The second author is supported by the Fonds zur Förderung der wissenschaftlichen Forschung (Austrian Science Fund), project

number P14195-MAT.

the Laplacian Δ^{q} acting on Ω^{q}(M; E). In this situation the Ray–Singer torsion [RS71] is defined as:

log T_{an}^{M}(E, g, μ) := \frac{1}{2} ∑_{q}(−)^{q+1}q log det′Δ^{q}

where det′Δ^{q} denotes the zeta regularized determinant of Δ^{q} with zero modes discarded. The main motivation of this paper is the calculation of the Ray–Singer torsion. This will be done with the help of the geometric complex associated to a Morse–Bott–Smale vector field described below, which is an object of independent interest by itself.

Suppose X is a Morse–Bott–Smale vector field on M, i.e. X = − gradg̃(f) where (f, g̃) is a Morse–Bott–Smale pair, cf. Definition 3 in section 2.4. Let Σ denote the critical manifold of X and N^{−} → Σ the negative normal bundle. For every connected component S ⊆ Σ we have a Morse index ind(S) ∈ ℕ_{0} := ℕ ∪ {0} which coincides with the rank of N^{−} over S. Every connected component S ⊆ Σ inherits a Riemannian metric g_{S} from g. Let O_{N^{−}} denote the orientation bundle of N^{−}. The flat vector bundle E_{S} := (E|_{Σ} ⊗ O_{N^{−}})|_{S} over S inherits a fiber metric μ_{S} from the fiber metric μ|_{S} on E|_{S} and the canonic parallel fiber metric on O_{N^{−}}.^{1} So we have Ray–Singer torsions log T_{an}^{S}(E_{S}, g_{S}, μ_{S}) for every connected component S ⊆ Σ.

The Morse–Bott–Smale vector field X gives rise to a geometric complex C^{∗}(X; E) whose underlying graded vector space is

C^{∗}(X; E) := ⨁_{S⊆Σ} Ω^{∗−ind(S)}(S; E_{S})

where the sum is over all connected components S ⊆ Σ. Fiber integration over the unstable manifolds provides a homomorphism of complexes Int : Ω^{∗}(M; E) → C^{∗}(X; E), see section 2 below. The Morse index provides a decreasing finite filtration

C_{p}^{∗}(X; E) := ⨁_{ind(S)≥p} Ω^{∗−ind(S)}(S; E_{S})

on the geometric complex. This gives rise to a spectral sequence (E_{k}C^{∗}(X; E), δ_{k}), see for instance [BT82].

Theorem 1.

1The frame bundle of N^{−} induces a principal ℤ_{2}–covering of Σ. We think of the orientation bundle O_{N^{−}} as the flat line bundle associated to this principal covering with respect to the standard non-trivial ℤ_{2}–action on ℝ. It thus inherits a canonic parallel fiber metric from the standard Euclidean inner product on ℝ.

(i) The integration map induces an isomorphism in cohomology Int : H^{∗}(M; E) = HC^{∗}(X; E).

(ii) The spectral sequence (E_{k}C^{∗}(X; E), δ_{k}) converges to HC^{∗}(X; E). Moreover E_{1}C^{∗}(X; E) = ⨁_{S⊆Σ} H^{∗−ind(S)}(S; E_{S}). Particularly E_{k}C^{∗}(X; E) is finite dimensional for all k ≥ 1 and we have E_{k}C^{∗}(X; E) = E_{∞}C^{∗}(X; E) for sufficiently large k.

The above theorem was first stated in [AB95] but known to experts. Unfortunately, despite its usefulness and simplicity there is no complete (satisfactory) treatment of the geometric complex in the existing literature. Since we need the geometric complex for our main result we will complete the discussion of [AB95] and provide a complete treatment of the geometric complex and its basic properties.

Remark 1. If X is a Morse–Smale vector field the geometric complex is finite dimensional, (E_{1}C^{∗}(X; E), δ_{1}) = (C^{∗}(X; E), δ) is the familiar Morse complex, and δ_{k} = 0 for k ≥ 2. In general (E_{1}C^{∗}(X; E), δ_{1}) has a very simple form, see section 3.4.

Remark 2. If π : M → N is a smooth bundle with M and N closed manifolds, and X_{N} is a Morse–Smale vector field on N then one can construct a Morse–Bott–Smale vector field on X_{M} on M with d_{x}π(X_{M}(x)) = X_{N}(π(x)) for all x ∈ M. The spectral sequence associated with the geometric complex of X_{M} identifies with the Leray–Serre spectral sequence of the bundle π provided by the cell structure on N induced by X_{N}, cf. Propositions 14 and 15 in section 6.1.

The Riemannian metric g and the fiber metric μ induce inductively scalar products on every E_{k}–term of the spectral sequence. Indeed, the restriction of g and μ to Σ provide a scalar product on the deRham complex Ω^{∗}(Σ; E_{Σ}). Its restriction to harmonic forms provides the scalar product on E_{1}C^{∗}(X; E), cf. Theorem 1(ii). Now suppose inductively that E_{k−1}C^{∗}(X; E) is equipped with a scalar product. Then E_{k}C^{∗}(X; E) = H(E_{k−1}C^{∗}(X; E), δ_{k−1}) inherits a scalar product as a subquotient of the finite dimensional Euclidean vector space. Equivalently, one can describe this scalar product via finite dimensional Hodge theory on the complex (E_{k−1}C^{∗}(X; E), δ_{k}).

The scalar products on E_{k}C^{∗}(X; E) permit to define the adjoint of the differential δ_{k} on E_{k}C^{∗}(X; E) and a Laplacian Δ_{k}^{q} acting on the finite dimensional E_{k}C^{q}(X; E), k ≥ 1. Define the combinatorial torsion by:

log T_{comb}^{M}(E, X, g, μ) := ∑_{k≥1} \frac{1}{2} ∑_{q}(−)^{q+1}q log det′Δ_{k}^{q}

The deRham cohomology inherits a scalar product via Hodge decomposition which provides a scalar product on the one dimensional real vector space det H^{∗}(M; E) referred to as the Hodge scalar product.^{2} The canonical identification det HC^{∗}(X; E) = det E_{∞}C^{∗}(X; E) and the scalar product on E_{∞}C^{∗}(X; E) = E_{k}C^{∗}(X; E) for k large, define a scalar product on det HC^{∗}(X; E) referred to as the geometric scalar product. The isomorphism in cohomology induced by integration provides an isomorphism det Int : det H^{∗}(M; E) → det HC^{∗}(X; E). Define^{3}

log T_{met}^{M}(E, X, g, μ) := log vol(det Int).

The fiber metric μ on E provides a closed one form θ ∈ Ω^{1}(M; ℝ) which measures to what extend the induced length one section on det E ⊗ O_{E} is not parallel, see section 4.2. Define

R(θ, X, g) := ∫_{M\Σ} θ ∧ X^{∗}Ψ_{g}

where Ψ_{g} ∈ Ω^{dim M−1}(T M \ M; O_{M}) denotes the global angular form, cf. [BT82], also called Mathai–Quillen form in [BZ92]. The integral on the right hand side need not be convergent but can be regularized, see section 4.1.

The main theorem of this paper is the following extension of a result by Bismut–Zhang [BZ92], Cheeger [C77], [C79] and Müller [M78].

Theorem 2. Let (M, g) be a closed Riemannian manifold, E a flat vector bundle with fiber metric μ over M, and let X be a Morse–Bott– Smale vector field on M, see Definition 3 below. Then

log T_{an}^{M}(E, g, μ) = ∑_{S⊆Σ}(−)^{ind(S)} log T_{an}^{S}(E_{S}, g_{S}, μ_{S})

+ log T_{comb}^{M}(E, X, g, μ) + log T_{met}^{M}(E, X, g, μ) + R(θ, −X, g)

where the sum is over all connected components S ⊆ Σ.

One can regard Σ as a graded manifold where the degree of the component S is ind(S) and write log T_{an}^{Σ} := ∑_{S⊆Σ}(−)^{ind(S)} log T_{an}^{S}. Then one can view the theorem above as a localization theorem stating that

log T_{an}^{M} = log T_{an}^{Σ} + log T_{comb} + log T_{met} + R(θ, −X, g)

2Here we use the notation det V = Λ^{dim(V^{even})}V^{even} ⊗ Λ^{dim(V^{odd})}(V^{odd})^{∗} for a finite dimensional ℤ_{2}–graded vector space. See appendix B for more details on determinant lines.

3Recall that for an isomorphism α : V → W between Euclidean vector spaces one defines vol α := \sqrt{det(α^{∗}α)}.

with the term R(θ, −X, g) computable in terms of differential geometry; the term log T_{comb} computable in terms of finite dimensional linear algebra; and, in case H^{∗}(M; E) = 0, vanishing term log T_{met}.

Theorem 2 can be extended to the case E is a flat bundle of Hilbertian A–modules of finite type where A is a finite von Neuman algebra using the extension of the Bismut–Zhang theorem to this case as formulated in [BFK01]. The proof and the results are exactly the same.

Theorem 2 can be extended to G–manifolds where G is a finite group and the torsions replaced by the G–equivariant torsions which are class functions on G with positive real values. This will use the corresponding extension of the Bismut–Zhang theorem cf. [BZ94]. The particular case G = ℤ_{2} when applied to the double of a manifold with boundary resp. to a bordism will imply an extension of Theorem 2 to compact manifolds with boundary resp. to bordisms and, as application, will provide a very simple derivation of all known glueing formulae for analytic torsion cf. [BFK99]. This is contained in [BH04].

Theorem 2 permits to recover and generalize results of Lück–Schick– Thielmann cf. [LST98] about the torsion of smooth bundles, cf. section 6. In a forthcoming paper we will use the above results and reduce the calculation of G–equivariant torsion for G a compact Lie group to finite dimensional linear algebra.

There is a more conceptual way to formulate Theorem 2 and understand the extend to which the result is a localization theorem for torsion. To explain this consider M a closed manifold, E a flat vector bundle over M, x_{0} ∈ M a base point, X a Morse–Bott–Smale vector field which gives rise to the geometric complex C^{∗}(X; E), and 𝔢 ∈ 𝔈𝔲𝔩_{x_{0}}(M) an Euler structure based at x_{0}, see section 4.1. The analytic torsion τ_{an}^{E,𝔢} can be regarded as an element defined up to multiplication by ±1 in the one dimensional vector space^{4}

Det_{x_{0}}(M; E) := det H^{∗}(M; E) ⊗ (det E_{x_{0}})^{−χ(M)}

where H^{∗}(M; E) denotes the deRham cohomology with coefficients in E and det H^{∗}(M; E) its determinant line in the graded sense, see appendix B. The geometric torsion τ_{geom}^{E,𝔢,X} is an element well defined up to multiplication by ±1 in the one dimensional vector space

Det_{x_{0}}(X; E) := det HC^{∗}(X; E) ⊗ (det E_{x_{0}})^{−χ(M)}

4Here we use the following convention. For a line L, i.e. a one dimensional vector space, we write L^{k} := L ⊗ ⋯ ⊗ L, L^{−k} = L^{∗} ⊗ ⋯ ⊗ L^{∗} and we write V^{0} := ℝ.

Note that there is a canonic isomorphism L^{k_{1}+k_{2}} = L^{k_{1}} ⊗ L^{k_{2}} for all k_{1} and k_{2}.

where HC^{∗}(X; E) denotes the cohomology of the geometric complex, see section 3 for definition. The integration induces an isomorphism

det Int : Det_{x_{0}}(M; E) → Det_{x_{0}}(X; E)

and Theorem 2 is equivalent to the statement that det Int(τ_{an}^{E,𝔢}) = τ_{geom}^{E,𝔢,X}, cf. Theorem 5 below.

The geometric torsion τ_{geom}^{E,𝔢,X} is obtained from the analytic torsions τ_{an}^{E_{S},𝔢_{S}} for a collection of Euler structures 𝔢_{S} ∈ 𝔈𝔲𝔩_{x_{S}}(S), one for every connected component S ⊆ Σ. Precisely, the geometric complex C^{∗}(X; E) provides an isomorphism:

I_{1} : det HC^{∗}(X; E) → ⨂_{S⊆Σ}(det H^{∗}(S; E_{S}))^{(−)^{ind(S)}}

The Euler structure 𝔢 and the Euler structures 𝔢_{S} provide a one chain c ∈ C_{1}(M; ℝ) which one writes as c = c_{0} +∑_{S⊆Σ}(−)^{ind(S)}χ(S)c_{S} where c_{S} is a path from x_{0} to x_{S} and c_{0} ∈ C_{1}(M; ℝ) which then satisfies ∂c_{0} = 0. Parallel transport along the c_{S} and proper scaling determined by c_{0} provides an isomorphism well defined up to sign:

I_{2} : (det E_{x_{0}})^{−χ(M)} → ⨂_{S⊆Σ}((det E_{S})_{x_{S}}^{−χ(S)})^{(−)^{ind(S)}}

The geometric torsion is then defined by:

τ_{geom}^{E,𝔢,X} := (I_{1} ⊗ I_{2})^{−1}(⨂_{S⊆Σ}(τ_{an}^{E_{S},𝔢_{S}})^{(−)^{ind(S)}})

This explains the localization aspect of the result.

The paper contains a number of intermediate results of independent interest such as: the canonic compactification of the unstable sets and the space of trajectories of a Morse–Bott–Smale vector field, see Propositions 2 and 3 and Theorems 3 and 4; the invariant R and the discussion of Euler structures.

2. Analysis of the ODE

2.1. Morse–Bott functions. Let M be a closed manifold, i.e. M is compact and without boundary. Let f : M → ℝ be a smooth function and let Σ := {x ∈ M | df_{x} = 0} denote the critical set of f. Suppose x is a critical point of f and Y, Z ∈ T_{x}M. Then H(Y, Z) := Y · df(Z) = Z · df(Y) for df is closed and vanishes at x. So we obtain a fiber wise symmetric bilinear form H on the vector bundle T M|_{Σ} referred to as the Hessian of f.^{5} If Σ ⊆ M happens to be a closed submanifold the Hessian will provide a smooth fiber wise bilinear form on the smooth vector bundle T M|_{Σ}. Moreover we obviously have T Σ ⊆ ker H. Therefore the Hessian induces a fiber wise bilinear form on the normal bundle N := T M|_{Σ}/T Σ of Σ which we will denote by H too.

Definition 1 (Morse–Bott functions). A function f : M → ℝ is called Morse–Bott if its critical set Σ is a closed submanifold of M and the Hessian considered as fiber wise bilinear form on the normal bundle N of Σ is non-degenerate. In this case Σ will also be referred to as the critical manifold or the rest manifold. The index, i.e. the number of negative eigen values, of the Hessian provides a locally constant ind : Σ → ℕ_{0} called the Morse index of f. Finally, define the critical manifold of index q by Σ_{q} := ⨆_{ind(S)=q} S where the disjoint union is over all connected components S ⊆ Σ of index q.

The following well known result totally clarifies the local behavior of f near its critical submanifold.

Proposition 1 (Morse lemma). Let f be a Morse–Bott function on M, let p_{N} : N → Σ denote the normal bundle of its critical manifold Σ and let H denote the Hessian. Define a smooth function \frac{1}{2}H^{2} : N → ℝ on the total space of N by \frac{1}{2}H^{2}(Y) := \frac{1}{2}H(Y, Y). Then there exists a tubular neighborhood φ : N → M such that φ^{∗}df = d(\frac{1}{2}H^{2}) holds in a neighborhood of the zero section Σ ⊆ N. Particularly φ^{∗}f = \frac{1}{2}H^{2} + p_{N}^{∗}(f|_{Σ}) with f|_{Σ} locally constant. A tubular neighborhood like this is called a Morse chart for f.

2.2. Compatible Riemannian metrics. Suppose f is a Morse–Bott function on a close manifold M. It is easy to see that there are H– orthogonal subbundles N^{+} ⊕ N^{−} = N such that g_{±} := ±H|_{N^{±}} > 0. Certainly H = g_{+} ⊖ g_{−} with respect to this splitting. The bundle N^{+} is called the positive normal bundle and the bundle N^{−} is called the negative normal bundle of f. Note however, that these bundles are well defined only up to isomorphisms of vector bundles. For the rank of N^{−} we have rank(N^{−}) = ind : Σ → ℕ_{0}.

Suppose we have a Riemannian metric g_{Σ} on Σ and linear connections ∇^{±} on N^{±} such that ∇^{±}g_{±} = 0. Define a fiber metric g_{N} := g_{+} ⊕ g_{−} and a linear connection ∇^{N} := ∇^{+} ⊕ ∇^{−} on N. Then ∇^{N}g_{N} = 0 = ∇^{N}H. Define a Riemannian metric g̃ on the total space of N using g_{N}, g_{Σ} and ∇^{N} in the most obvious way. Clearly the restriction of g̃ to Σ coincides with g_{Σ}. On the normal bundle of Σ ⊆ N, which canonically identifies to N → Σ, the Riemannian metric g̃ induces a fiber metric and a linear connection which coincide with g_{N} and ∇^{N}. The splitting of N into positive and negative H–eigen spaces using g_{N} coincides with N^{+} ⊕N^{−} = N. The second fundamental form of Σ ⊆ N vanishes. The endomorphism −(id_{N^{+}} ⊖ id_{N^{−}}) of N can be considered as a vertical vector field on the total space of N which coincides with X := − gradg̃(\frac{1}{2}H^{2}). The unstable set of Σ, i.e. the points in N whose X trajectory departs at Σ, coincides with N^{−} ⊆ N hence is a smooth submanifold of N. Similarly the stable set of Σ coincides with the smooth submanifold N^{+} ⊆ N.

5This bilinear form can be extended (in a non-canonical way) to a smooth bilinear form on T M as follows. Pick any linear torsion free connection ∇ on T M and set H(Y, Z) := (∇df)(Y, Z). This will be symmetric for df is a closed one form and ∇ was supposed to be torsion free. It will coincide with our previous definition over Σ since df vanishes along Σ.

Definition 2 (Compatible Riemannian metrics). Let f be a Morse– Bott function on M. A Riemannian metric g on M is called compatible with f if there exists a Morse chart φ : N → M such that in addition to φ^{∗}df = d(\frac{1}{2}H^{2}) we have φ^{∗}g = g̃ on a neighborhood of Σ ⊆ N. Here g̃ is a Riemannian metric on the total space of N constructed from an H–orthogonal splitting N^{+} ⊕ N^{−} = N for which ±H|_{N^{±}} > 0, a Riemannian metric on Σ and linear connections on N^{±} for which the restrictions of H are parallel as explained in the previous paragraph.

Note that a Riemannian metric g on M induces a Riemannian metric g_{Σ} on Σ, a fiber metric g_{N} on the normal bundle N = (T Σ)^{⊥}, an orthogonal splitting N^{+}⊕N^{−} = N and a smooth map (the exponential map) exp : N → M from the total space N of the normal bundle of Σ to M. If g is compatible then the parallel transport along a path in Σ leaves N and N^{±} invariant and therefore define the connection ∇^{N}, and ∇^{±}. The Riemannian metric g_{Σ}, the fiber metric g_{N} and the connection ∇^{N} induce a Riemannian metric on N. If g is compatible then exp is an isometry when restricted to a small neighborhood of the zero section of the normal bundle. This can also be taken as an equivalent definition of compatible metric with respect to f.

Note that given an arbitrary metric g and an arbitrary neighborhood U of Σ one can produce a Riemannian metric g′ which is compatible for f and agrees with g on Σ and on M \ U.

2.3. Stable and unstable manifold. Let f be a Morse–Bott function on a closed manifold M and let g be a compatible Riemannian metric on M, see Definition 2. Define a vector field X := − grad_{g}(f). For a connected component S ⊆ Σ define its stable resp. unstable set by

W_{S}^{±} := {x ∈ M | lim_{t→±∞} ϕ_{t}(x) ∈ S}

where ϕ_{t} denotes the flow of the vector field X at time t. W_{S}^{±} is a smooth boundary-less submanifold of M and we let i_{±}^{S} : W_{S}^{±} → M denote the inclusions. Moreover we get smooth fiber bundles p_{±}^{S} : W_{S}^{±} → S diffeomorphic to N_{S}^{±} → S. Define W^{±} := ⨆_{S⊆Σ} W_{S}^{±} where the disjoint union is over all connected components S of Σ. The i_{±}^{S} give rise to a smooth immersion i_{±} : W^{±} → M and the p_{±}^{S} give rise to a smooth fiber bundle p_{±} : W^{±} → Σ diffeomorphic to N^{±} → Σ.

Let V_{W^{±}} → W^{±} denote the vertical (vector) bundle of p_{±} : W^{±} → Σ. For its rank we find rank(V_{W^{−}}) = p_{−}^{∗} ind : W^{−} → ℕ_{0}. Here we write p_{−}^{∗} ind for the composition ind ∘p_{−} : W^{−} → Σ → ℕ_{0}. The Lie transport along the normalized \frac{X}{|X|} provides an isomorphism of vector bundles V_{W^{±}} = p_{±}^{∗}N^{±} and thus a canonic isomorphism of orientation bundles:

O_{V_{W^{±}}} = p_{±}^{∗} O_{N^{±}}  (1)

Moreover, if E is a flat vector bundle over M then parallel transport along X provides a canonic isomorphism of flat vector bundles:

i_{±}^{∗}E = p_{±}^{∗}E|_{Σ}  (2)

2.4. The space of trajectories. Let f be a Morse–Bott function on a closed manifold M and let g be a compatible Riemannian metric on M, cf. Definition 2. For x ∈ Σ let i_{−}^{x} : W_{x}^{−} → M denote the restriction of i_{−} : W^{−} → M to the fiber W_{x}^{−} of p_{−} : W^{−} → Σ over x.

Definition 3 (Morse–Bott–Smale vector fields). A pair (f, g) consisting of a Morse–Bott function f and a compatible Riemannian metric g on a closed manifold M is called Morse–Bott–Smale pair if the mappings i_{+} : W^{+} → M and i_{−}^{x} : W_{x}^{−} → M are transversal for all x ∈ Σ.^{6} A vector field X is called Morse–Bott–Smale if there exists a Morse– Bott–Smale pair (g, f) such that X = − grad_{g}(f).

6Given a Morse–Bott function f one can ask if there exists a Riemannian metric g so that the pair (f, g) is a Morse–Bott–Smale pair. Such a function can be called ‘good’ Morse–Bott function. All Morse functions are good Morse–Bott functions but, as expected, not all Morse–Bott functions are good. To decide when a Morse– Bott function is good is an interesting problem. Here are a few examples:

1) If f : N → ℝ is a Morse function and π : M → N is a smooth bundle with compact fibers then f ∘ π is a good Morse–Bott function.

2) If M is a compact smooth G–manifold with G a compact Lie group, any normal G–Morse function f : M → ℝ (i.e. for any critical point the negative isotropy representation is trivial) is a good Morse–Bott function cf. [BH04]. It is known that normal G–Morse functions are C^{0}–dense in the space of all smooth G–invariant functions in C^{0}–topology.

3) If f is a Morse–Bott function so is −f but it is possible that f is good and −f is not.

Suppose X is a Morse–Bott–Smale vector field on M. Let S ≠ S′ be two different connected components of Σ. Then M(S, S′) := W_{S}^{−} ∩W_{S′}^{+} is a smooth submanifold of M. Define M := ⨆_{S≠S′⊆Σ} M(S, S′) where the disjoint union is over all pairs of different connected components S and S′ of Σ. Let i : M → M denote the obvious smooth immersion, p_{+} : M → Σ the obvious smooth mapping and let p_{−} : M → Σ denote the obvious smooth fiber bundle.

Let V_{M} → M denote the vertical bundle of p_{−} : M → Σ. We have a short exact sequence of vector bundles:

0 → V_{M} → V_{W^{−}}|_{M} → T M|_{M}/T W^{+}|_{M} → 0

As in section 2.3, Lie transport along \frac{X}{|X|} provides canonic isomorphisms V_{W^{−}}|_{M} = p_{−}^{∗}N^{−} and T M|_{M}/T W^{+}|_{M} = p_{+}^{∗}N^{−}. So we obtain a short exact sequence of vector bundles:

0 → V_{M} → p_{−}^{∗}N^{−} → p_{+}^{∗}N^{−} → 0

Particularly rank(V_{M}) = p_{−}^{∗} ind −p_{+}^{∗} ind : M → ℕ_{0}. Moreover, this short exact sequence provides us with a canonic isomorphism of orientation bundles:^{7}

(−)^{rank(V_{M})·rank(p_{+}^{∗}N^{−})} : O_{V_{M}} ⊗p_{+}^{∗} O_{N^{−}} = p_{−}^{∗} O_{N^{−}}  (3)

If E is a flat vector bundle over M we get canonic isomorphisms p_{−}^{∗}E|_{Σ} = i^{∗}E = p_{+}^{∗}E|_{Σ} using parallel transport along X.

The flow ϕ_{t} restricts to a smooth action of ℝ on M. This action is free and we have a smooth orbit space T := M/ℝ. The mapping p_{+} : M → Σ factors to a smooth map π_{+} : T → Σ. The fiber bundle p_{−} : M → Σ factors to a smooth fiber bundle π_{−} : T → Σ. Let q : M → T denote the projection and let [X] ⊆ V_{M} denote the line bundle over M generated by the vector field X. We have a short exact sequence of vector bundles:

0 → [X] → V_{M} → q^{∗}V_{T} → 0

THE GEOMETRIC COMPLEX OF A MORSE–BOTT–SMALE PAIR … 11

So the rank of the vertical bundle V_{T} is rank(V_{T}) = π_{−}^{∗} ind −π_{+}^{∗} ind −1 : T → ℕ_{0}. Since the line bundle [X] is oriented via X we obtain an isomorphism of orientation bundles O_{V_{M}} = q^{∗} O_{V_{T}}, that is we use the X last convention here. Using the isomorphism (3) we get an isomorphism q^{∗}(O_{V_{T}} ⊗π_{+}^{∗} O_{N^{−}}) = q^{∗}π_{−}^{∗} O_{N^{−}} which descends to a canonic isomorphism of orientation bundles:

7Throughout the paper we use the following convention. If 0 → E → F → G → 0 is a short exact sequence of vector bundles we use an orientation of G_{x} followed by an orientation of E_{x} to yield the compatible orientation of F_{x}, x in the common base space. We refer to this as the G_{x} first convention. The notation in (3) should indicate that we deviate from this convention and use the V_{M} first convention here.

O_{V_{T}} ⊗π_{+}^{∗} O_{N^{−}} = π_{−}^{∗} O_{N^{−}}  (4)

If E is a flat vector bundle over M then the isomorphism p_{−}^{∗}E|_{Σ} = p_{+}^{∗}E|_{Σ} descends to a canonic isomorphism of flat vector bundles:

π_{−}^{∗}E|_{Σ} = π_{+}^{∗}E|_{Σ}  (5)

For two connected components S and S′ of Σ set T (S, S′) := π_{−}^{−1}(S)∩ π_{+}^{−1}(S′) ⊆ T . Clearly this is the disjoint union over several connected components of T . The rank of its vertical bundle V_{T (S,S′)} is constant and rank(V_{T (S,S′)}) = ind(S) − ind(S′) − 1. Note that this implies T (S, S′) = ∅ unless ind(S) > ind(S′), a consequence of the strong transversality condition. Also note, that we have T (S, S) = ∅ by our very definitions.

2.5. Compactification of T . Let X be a Morse–Bott–Smale vector field on M. For k ≥ 0 define the manifold of unparameterized k–times broken trajectories by:

T̂_{k} := \underbrace{T ×_{Σ} T ×_{Σ} ⋯ ×_{Σ} T}_{k + 1 copies}

Here T ×_{Σ} T := {(x, y) ∈ T × T | π_{+}(x) = π_{−}(y)}, and similarly for more factors. Define a smooth map (π̂_{+})_{k} : T̂_{k} → Σ by composing the projection to the last factor with π_{+} : T → Σ. Define a smooth fiber bundle (π̂_{−})_{k} : T̂_{k} → Σ by composing the projection to the first factor with π_{−} : T → Σ. Note that by definition T̂_{0} = T , (π̂_{+})_{0} = π_{+} and (π̂_{−})_{0} = π_{−}.

Let VT̂_{k} denote the vertical bundle of (π̂_{−})_{k} : T̂_{k} → Σ. Define an isomorphism of orientation bundles

(−)^{p_{1}^{∗} rank(V_{T} )} : O_{VT̂_{1}} ⊗(π̂_{+})_{1}^{∗} O_{N^{−}} = (π̂_{−})_{1}^{∗} O_{N^{−}}  (6)

as follows. Consider the commutative diagram

\xymatrix{T̂_{1} = T ×_{Σ} T \ar[r]^{p_{2}} \ar[d]_{p_{1}} & T \ar[r]^{π_{+}} \ar[d]^{π_{−}} & Σ \\ T \ar[r]^{π_{+}} \ar[d]_{π_{−}} & Σ \\ Σ}

where p_{1} projects onto the first component and p_{2} projects onto the second component of T̂_{1} = T ×_{Σ} T . Notice that (π̂_{+})_{1} = π_{+} ∘ p_{2} and (π̂_{−})_{1} = π_{−} ∘ p_{1}. Clearly we have p_{2}^{∗}V_{T} = V_{p_{1}} and thus p_{2}^{∗} O_{V_{T}} = O_{V_{p_{1}}}.

Moreover the short exact sequence 0 → V_{p_{1}} → VT̂_{1} → p_{1}^{∗}V_{T} → 0 provides O_{V_{p_{1}}} ⊗p_{1}^{∗} O_{V_{T}} = O_{VT̂_{1}}. Finally recall the isomorphism (4). Now define the isomorphism (6) as (−)^{p_{1}^{∗} rank(V_{T} )} times the following composition of isomorphisms:

O_{VT̂_{1}} ⊗(π̂_{+})_{1}^{∗} O_{N^{−}} = O_{V_{p_{1}}} ⊗p_{1}^{∗} O_{V_{T}} ⊗p_{2}^{∗}π_{+}^{∗} O_{N^{−}}
= p_{2}^{∗} O_{V_{T}} ⊗p_{1}^{∗} O_{V_{T}} ⊗p_{2}^{∗}π_{+}^{∗} O_{N^{−}}
= p_{1}^{∗} O_{V_{T}} ⊗p_{2}^{∗}(O_{V_{T}} ⊗π_{+}^{∗} O_{N^{−}})
= p_{1}^{∗} O_{V_{T}} ⊗p_{2}^{∗}π_{−}^{∗} O_{N^{−}}
= p_{1}^{∗}(O_{V_{T}} ⊗π_{+}^{∗} O_{N^{−}})
= p_{1}^{∗}π_{−}^{∗} O_{N^{−}}
= (π̂_{−})_{1}^{∗} O_{N^{−}}

Suppose E is a flat vector bundle over M. In a similar manner, using (5), we obtain a canonic isomorphism of flat vector bundles

(π̂_{−})_{1}^{∗}E|_{Σ} = (π̂_{+})_{1}^{∗}E|_{Σ}  (7)

as the following composition of isomorphisms:

(π̂_{−})_{1}^{∗}E|_{Σ} = p_{1}^{∗}π_{−}^{∗}E|_{Σ} = p_{1}^{∗}π_{+}^{∗}E|_{Σ} = p_{2}^{∗}π_{−}^{∗}E|_{Σ} = p_{2}^{∗}π_{+}^{∗}E|_{Σ} = (π̂_{+})_{1}^{∗}E|_{Σ}

Set T̂ := ⨆_{k≥0} T̂_{k}. Define π̂_{±} : T̂ → Σ by (π̂_{±})_{k} on T̂_{k}. For connected components S, S′ ⊆ Σ introduce a topology on T̂(S, S′) := π̂_{−}^{−1}(S) ∩ π̂_{+}^{−1}(S′) by parameterizing the possibly broken trajectories with the help of f and taking the topology induced from C^{0}([a, b], M), where a = f(S′), b = f(S). Topologize T̂ = ⨆_{S,S′⊆Σ} T̂(S, S′) as the disjoint union.

Proposition 2. The space T̂ is compact.

Proof. Clearly it suffices to show that T̂(S, S′) is compact, for Σ has only finitely many connected components. The compactness of T̂(S, S′) follows from the theorem of Arzela–Ascoli. Indeed, the subset T̂(S, S′) ⊆ C^{0}([a, b], M) is equicontinuous. For t ∈ [a, b] which is non-critical this follows from the fact that these are flow lines of a locally non-vanishing vector field. For critical t one has to use the very explicit form of the singularities of X, cf. [BH01].

To see that T̂(S, S′) ⊆ C^{0}([a, b], M) is closed suppose σ_{k} ∈ T̂(S, S′) converges uniformly to σ ∈ C^{0}([a, b], M). Then σ too is parametrized by f, can only pass through finitely many critical points, departs at S, ends at S′ and away from the critical points σ too is a flow line of \frac{X}{|X|}. Hence σ ∈ T̂(S, S′). □

Recall that a smooth manifold with corners P of dimension n is a paracompact Hausdorff space modeled on {(x_{1}, …, x_{n})|x_{i} ≥ 0} ⊆ ℝ^{n}. The subset of points which in one (and hence every) chart have exactly k coordinates zero is called the k–corner of P and denoted by ∂_{k}P. It inherits the structure of a smooth manifold of dimension n − k. The subset ∂P := ∂_{1}P ∪ ⋯ ∪ ∂_{n}P is closed and inherits the structure of a topological manifold. The pair (P, ∂P) is a topological manifold with boundary.

Theorem 3. Let X be a Morse–Bott–Smale vector field on M. There is a canonic way to equip the space T̂ with the structure of a (compact) smooth manifold with corners such that the following hold:

(i) The inclusion T̂_{k} ⊆ T̂ is a diffeomorphism onto the k–corner. (ii) The mapping π̂_{+} : T̂ → Σ is smooth. (iii) The mapping π̂_{−} : T̂ → Σ is a smooth fiber bundle whose fibers

are (compact) smooth manifolds with corners. (iv) rank(VT̂) = π̂_{−}^{∗} ind −π̂_{+}^{∗} ind −1 : T̂ → ℕ_{0}, where VT̂ denotes

the vertical bundle of π̂_{−} : T̂ → Σ.

(v) If X = − grad_{g}(f) and T̂(S, S′) ≠ ∅ then f(S) > f(S′).

(vi) The isomorphism (4) extends (uniquely) to a smooth isomorphism of orientation bundles O_{VT̂} ⊗π̂_{+}^{∗} O_{N^{−}} = π̂_{−}^{∗} O_{N^{−}}. Via O_{VT̂} |T̂_{1} = O_{VT̂_{1}} its restriction to T̂_{1} identifies with (6).

(vii) If E is a flat vector bundle over M then the isomorphism (5) extends (uniquely) to a smooth isomorphism of flat vector bundles π̂_{−}^{∗}E|_{Σ} = π̂_{+}^{∗}E|_{Σ}. Its restriction to T̂_{1} identifies with (7).

The proof of Theorem 3 is contained in Appendix A.

2.6. Compactification of W^{−}. For k ≥ 0 define a smooth manifold

(Ŵ^{−})_{k} := \underbrace{T ×_{Σ} T ×_{Σ} ⋯ ×_{Σ} T}_{k factors} ×_{Σ}W^{−}

Here T ×_{Σ} W^{−} = {(x, y) ∈ T × W^{−} | π_{+}(x) = p_{−}(y)} and similarly for more factors. Define a smooth mapping î_{k} : (Ŵ^{−})_{k} → M by composing the projection to the last factor with i_{−} : W^{−} → M. Define a smooth fiber bundle (p̂_{−})_{k} : (Ŵ^{−})_{k} → Σ by composing the projection to the first factor with the mapping π_{−} : T → Σ in the case k ≥ 1 and composing with p_{−} : W^{−} → Σ if k = 0. Note that by definition we have (Ŵ^{−})_{0} = W^{−}, î_{0} = i_{−} and (p̂_{−})_{0} = p_{−}.

Let V_{(Ŵ^{−})_{k}} denote the vertical bundle of (p̂_{−})_{k} : (Ŵ^{−})_{k} → Σ. Define an isomorphism of orientation bundles

(−)^{p_{1}^{∗} rank(V_{T} )} : O_{V_{(Ŵ^{−})_{1}}} = (p̂_{−})_{1}^{∗} O_{N^{−}}  (8)

as follows. Consider the commutative diagram

\xymatrix{(Ŵ^{−})_{1} = T ×_{Σ} W^{−} \ar[r]^{p_{2}} \ar[d]_{p_{1}} & W^{−} \ar[r]^{i_{−}} \ar[d]^{p_{−}} & M \\ T \ar[r]^{π_{+}} \ar[d]_{π_{−}} & Σ \\ Σ}  (9)

where p_{1} and p_{2} denote the projection to the first and second factor, respectively. Notice that î_{1} = i_{−} ∘ p_{2} and (p̂_{−})_{1} = π_{−} ∘ p_{1}. Clearly we have p_{2}^{∗}V_{W^{−}} = V_{p_{1}} and thus p_{2}^{∗} O_{V_{W^{−}}} = O_{V_{p_{1}}}. Moreover the short exact sequence

0 → V_{p_{1}} → V_{(Ŵ^{−})_{1}} → p_{1}^{∗}V_{T} → 0

provides O_{V_{p_{1}}} ⊗p_{1}^{∗} O_{V_{T}} = O_{V_{(Ŵ^{−})_{1}}}. Finally recall (1) and (4). Now define the isomorphism (8) as (−)^{p_{1}^{∗} rank(V_{T} )} times the following composition of isomorphisms:

O_{V_{(Ŵ^{−})_{1}}} = O_{V_{p_{1}}} ⊗p_{1}^{∗} O_{V_{T}}
= p_{2}^{∗} O_{V_{W^{−}}} ⊗p_{1}^{∗} O_{V_{T}}
= p_{2}^{∗}p_{−}^{∗} O_{N^{−}} ⊗p_{1}^{∗} O_{V_{T}}
= p_{1}^{∗}(π_{+}^{∗} O_{N^{−}} ⊗ O_{V_{T}})
= p_{1}^{∗}π_{−}^{∗} O_{N^{−}}
= (p̂_{−})_{1}^{∗} O_{N^{−}}

Suppose E is a flat vector bundle over M. In a similar manner, using (2) and (5), define an isomorphism of flat vector bundles

î_{1}^{∗}E = (p̂_{−})_{1}^{∗}E|_{Σ}  (10)

as the following composition of isomorphisms:

î_{1}^{∗}E = p_{2}^{∗}i_{−}^{∗}E = p_{2}^{∗}p_{−}^{∗}E|_{Σ} = p_{1}^{∗}π_{+}^{∗}E|_{Σ} = p_{1}^{∗}π_{−}^{∗}E|_{Σ} = (p̂_{−})_{1}^{∗}E|_{Σ}

Set Ŵ^{−} := ⨆_{k≥0}(Ŵ^{−})_{k}. Define î : Ŵ^{−} → M using î_{k} on (Ŵ^{−})_{k}. Define p̂_{−} : Ŵ^{−} → Σ using (p̂_{−})_{k} on (Ŵ^{−})_{k}. Let S ⊆ Σ be a connected component. We are going to topologize Ŵ_{S}^{−} := p̂_{−}^{−1}(S). Set b := f(S) and a := min{f(x) | x ∈ M}. Every element x ∈ Ŵ_{S}^{−} can be considered as an unparametrized path σ from S to î(x) which we parametrize with the help of f, i.e. σ ∈ C^{0}([a, b], M)

f(σ(t)) = \begin{cases}a + b − t & for a ≤ t ≤ a + b − f(î(x)) and \\ f(î(x)) & for a + b − f(î(x)) ≤ t ≤ b. \end{cases}

We equip Ŵ_{S}^{−} with the topology induced from C^{0}([a, b], M) and topologize Ŵ^{−} = ⨆_{S⊆Σ} Ŵ_{S}^{−} as the disjoint union.

Proposition 3. The space Ŵ^{−} is compact.

The proof of Proposition 3 is exactly the same as the proof of Proposition 2.

Theorem 4. Let X be a Morse–Bott–Smale vector field on M. There is a canonic way to equip the space Ŵ^{−} with the structure of a (compact) smooth manifold with corners such that the following hold:

(i) The inclusion (Ŵ^{−})_{k} ⊆ Ŵ^{−} is a diffeomorphism onto the k– corner of Ŵ^{−}.

(ii) The mapping î: Ŵ^{−} → M is smooth.

(iii) The mapping p̂_{−} : Ŵ^{−} → Σ is a smooth fiber bundle whose

fibers are (compact) smooth manifolds with corners. (iv) rank(VŴ^{−}) = p̂_{−}^{∗} ind : Ŵ^{−} → ℕ_{0}, where VŴ^{−} denotes the ver-

tical bundle of p̂_{−} : Ŵ^{−} → Σ.

(v) If X = − grad_{g}(f) then f̂(Ŵ_{S}^{−}) ≤ f(S) for all connected components S ⊆ Σ, where f̂:= f ∘ î: Ŵ^{−} → ℝ.

(vi) The isomorphism (1) extends (uniquely) to a smooth isomorphism of orientation bundles O_{VŴ^{−}} = p̂_{−}^{∗} O_{N^{−}}. Via O_{VŴ^{−}} |_{(Ŵ^{−})_{1}} = O_{V_{(Ŵ^{−})_{1}}} its restriction to (Ŵ^{−})_{1} identifies with (8).

(vii) If E is a flat vector bundle over M then the isomorphism (2) extends (uniquely) to a smooth isomorphism of flat vector bundles î^{∗}E = p̂_{−}^{∗}E|_{Σ}. Its restriction to (Ŵ^{−})_{1} identifies with (10).

The proof of Theorem 4 is contained in Appendix A.

3. The geometric complex

3.1. Fiber integration. We will heavily use fiber integration. So let us fix a sign convention, which will be the same as in [BT82]. Suppose π : F → B is a smooth fiber bundle. Here F, B and the fibers F_{x} = π^{−1}(x) may have corners and for simplicity we assume the fibers to be compact. However none of these manifolds is assumed to be connected or of constant dimension.^{8} Let V_{F} → F denote the vertical bundle of π. Finally suppose we have a flat vector bundle E → B. Given a differential form α ∈ Ω(F; π^{∗}E ⊗ O_{V_{F}}) we are going to define a form π_{∗}α ∈ Ω(B; E) as follows. Suppose x ∈ B and X_{1}, …, X_{k} ∈ T_{x}B. Then the degree k component of π_{∗}α ∈ Ω(B; E) is defined by:

(π_{∗}α)(X_{1}, …, X_{k}) := ∫_{F_{x}} iX̃_{k} ⋯ iX̃_{1}α

Here X̃_{i} are vector fields on the total space of F defined along the fiber F_{x} which project to X_{i}. The pull back of iX̃_{k} ⋯ iX̃_{1}α to the fiber F_{x} is a form Ω(F_{x}; π^{∗}E_{x} ⊗ O_{F_{x}}) which can be integrated over F_{x} to yield an element of E_{x}. The outcome does not depend on the choice of X̃_{i}. For the fiber bundle π : F → B the fiber integration

π_{∗} : Ω(F; π^{∗}E ⊗ O_{V_{F}}) → Ω(B; E)

has the following properties which follow easily from our definition.

Suppose f : B′ → B is a smooth mapping. Then we have a pull back bundle f^{∗}π : f^{∗}F → B′ and a smooth mapping π^{∗}f : f^{∗}F → F which restricts to a diffeomorphism (f^{∗}F)_{x} → F_{f(x)} for every x ∈ B′.

\xymatrix{f^{∗}F \ar[r]^{π^{∗}f} \ar[d]_{f^{∗}π} & F \ar[d]^{π} \\ B′ \ar[r]^{f} & B}

Hence its tangent mapping induces an isomorphism of vector bundles V_{f^{∗}F} = (π^{∗}f)^{∗}V_{F} over the total space of f^{∗}F. Particularly we get a canonic isomorphism of orientation bundles O_{f^{∗}V_{F}} = (π^{∗}f)^{∗} O_{V_{F}} and thus a canonic isomorphism of flat vector bundles (π^{∗}f)^{∗}(π^{∗}E⊗O_{V_{F}}) = (f^{∗}π)^{∗}f^{∗}E ⊗ O_{V_{f^{∗}F}}. Using this identification we have

(f^{∗}π)_{∗} ∘ (π^{∗}f)^{∗} = f^{∗} ∘ π_{∗} : Ω(F; π^{∗}E ⊗ O_{V_{F}}) → Ω(B′; f^{∗}E)

the naturality of fiber integration.

Suppose π̃ : F̃ → F and π : F → B are two fiber bundles. Then π ∘ π̃ : F̃ → B is a fiber bundle too and we have a canonic short exact sequence of vector bundles 0 → Vπ̃ → V_{π∘π̃} → π̃^{∗}V_{π} → 0 over F̃. This provides us with a canonic isomorphism of orientation bundles O_{V_{π∘π̃}} = O_{Vπ̃} ⊗π̃^{∗} O_{V_{π}} and thus with a canonic isomorphism (π∘π̃)^{∗}E⊗ O_{V_{π∘π̃}} = π̃^{∗}(π^{∗}E ⊗ O_{V_{π}}) ⊗ O_{Vπ̃}. Via this identification we have

π_{∗} ∘ π̃_{∗} = (π ∘ π̃)_{∗} : Ω(F̃; (π ∘ π̃)^{∗}E ⊗ O_{V_{π∘π̃}}) → Ω(B; E)

8Thus even if we fiber integrate a form of homogeneous degree the outcome need not be of homogeneous degree.

the (covariant) functoriality of fiber integration.

Suppose π : F → B is a fiber bundle and suppose E′ and E are two flat vector bundles over B. For α ∈ Ω(F; π^{∗}E ⊗O_{V_{F}}) and β ∈ Ω(B; E′) we have

π_{∗}(π^{∗}β ∧ α) = β ∧ π_{∗}α ∈ Ω(B; E′ ⊗ E)

a push–pull formula. Here we only use the natural isomorphism π^{∗}E′ ⊗ π^{∗}E = π^{∗}(E′ ⊗ E).

Suppose F → B is a fiber bundle and suppose B has no boundary. Then the 1–corner π|_{∂_{1}F} : ∂_{1}F → B is a smooth fiber bundle with possibly non-compact fibers. Using the outward pointing normal first convention we get a canonic isomorphism of orientation bundles O_{V_{F}} |_{∂_{1}F} = O_{V_{∂_{1}F}} and thus a canonic isomorphism of flat vector bundles (π^{∗}E ⊗ O_{V_{F}})|_{∂_{1}F} = (π|_{∂_{1}F})^{∗}E ⊗ O_{V_{∂_{1}F}}. Note that this is the same isomorphisms as the one we get from the short exact sequence

0 → V_{∂_{1}F} → V_{F} |_{∂_{1}F} → V_{F} |_{∂_{1}F} /V_{∂_{1}F} → 0

with the right hand side quotient oriented by the outward pointing normal. Via this identification we find

π_{∗} ∘ d = d ∘ π_{∗} + A ∘ (π|_{∂_{1}F})_{∗} : Ω(F; π^{∗}E ⊗ O_{V_{F}}) → Ω(B; E)

where A := (−)^{k} : Ω^{k}(B; E) → Ω^{k}(B; E) is a grading homomorphism.

3.2. Integration and the differential. Let X be a Morse–Bott– Smale vector field on a closed manifold M, cf. Definition 3. Let E be a flat vector bundle over M. Define a vector space

C(X; E) := Ω(Σ; E|_{Σ} ⊗ O_{N^{−}})

and an integration

Int := (p̂_{−})_{∗}î^{∗} = (p_{−})_{∗}i_{−}^{∗} : Ω(M; E) → C(X; E).

Note that for α ∈ Ω(M; E) we have î^{∗}α ∈ Ω(Ŵ^{−}; î^{∗}E) which via the isomorphisms in Theorem 4(vi) and (vii) can be interpreted as an element of Ω(Ŵ^{−}; p̂_{−}^{∗}E|_{Σ} ⊗ p̂_{−}^{∗} O_{N^{−}} ⊗ O_{VŴ^{−}}) and thus (p̂_{−})_{∗}î^{∗}α is indeed an element of Ω(Σ; E|_{Σ} ⊗ O_{N^{−}}) = C(X; E). Similarly, using the isomorphisms in Theorem 3(vi) and (vii), we define

u := (π̂_{−})_{∗}π̂_{+}^{∗} = (π_{−})_{∗}π_{+}^{∗} : C(X; E) → C(X; E)

and

δ := d + uA : C(X; E) → C(X; E)

where A := (−)^{k} : Ω^{k}(Σ; E|_{Σ} ⊗O_{N^{−}}) → Ω^{k}(Σ; E|_{Σ} ⊗O_{N^{−}}) is a grading homomorphism, cf. section 3.1.

Proposition 4. Int ∘d = δ ∘ Int.

Proof. From section 3.1 we obtain (p̂_{−})_{∗}d = d(p̂_{−})_{∗} + A((p̂_{−})|_{∂_{1}Ŵ^{−}})_{∗} and thus:

Int ∘d = d ∘ Int +A((p̂_{−})|_{∂_{1}Ŵ^{−}})_{∗}î^{∗}

Using the description of the 1–boundary of Ŵ^{−} from Theorem 4 and the obvious relation (π_{−})_{∗}(−)^{rank(V_{T} )}π_{+}^{∗} = AuA we get:

((p̂_{−})|_{∂_{1}Ŵ^{−}})_{∗}î^{∗} = (π_{−} ∘ p_{1})_{∗}(−)^{p_{1}^{∗} rank(V_{T} )}(i_{−} ∘ p_{2})^{∗}
= (π_{−})_{∗}(−)^{rank(V_{T} )}(p_{1})_{∗}p_{2}^{∗}i_{−}^{∗}
= (π_{−})_{∗}(−)^{rank(V_{T} )}π_{+}^{∗}(p_{−})_{∗}i_{−}^{∗}
= AuA Int

Therefore Int d = d Int +A^{2}uA Int = (d + uA) Int = δ Int. □

Since the integration Int : Ω(M; E) → C(X; E) is onto we must have δ^{2} = 0 as a consequence of d^{2} = 0. However, we will give a direct argument below.

Proposition 5. δ^{2} = 0.

Proof. Exactly as in the proof of Proposition 4 we derive ud = du+uAu using the description of the 1–boundary of T̂ from Theorem 3. Then δ^{2} = duA+uAd+uAuA = (du−ud+uAu)A = 0, where we also made use of the obvious dA = −Ad. □

Definition 4 (Geometric complex). The complex (C(X; E), δ) is called the geometric complex with values in the flat vector bundle E. It is a generalization of the deRham complex and the classical Morse complex at the same time. The homomorphism of chain complexes Int : Ω(M; E) → C(X; E) is called the integration homomorphism.

3.3. Grading and filtration. The geometric complex C(X; E) can be graded and filtered in a compatible way. For a connected component S ⊆ Σ let us write E_{S} := E_{Σ}|_{S} = (E|_{Σ} ⊗ O_{N^{−}})|_{S}. Define

C_{p}^{q}(X; E) := ⨁_{ind(S)≥p} Ω^{q−ind(S)}(S; E_{S})

where the direct sum is over all connected components S ⊆ Σ of index larger or equal to p. Let us also introduce the notation C^{q}(X; E) := C_{0}^{q}(X; E), and note that ⨁_{q} C^{q}(X; E) = C(X; E). Finally let us define C_{p}(X; E) = ⨁_{q} C_{p}^{q}(X; E). Clearly C_{p}^{q}(X; E) = C^{q}(X; E) ∩ C_{p}(X; E). Of course the grading and the filtration stem from a bigrading on C(X; E). However, since the bigrading will not be compatible with the differential we will not consider it in the sequel.

Proposition 6. The differential of the geometric complex preserves the grading and the filtrations as indicated below:

δ : C_{p}^{q}(X; E) → C_{p}^{q+1}(X; E)

Proof. This is obviously true for the deRham differential. So it suffices to consider u = (π_{−})_{∗}(π_{+})^{∗}. Let α′ be a form of degree |α′| supported on a connected component S′ ⊆ Σ. Let S ⊆ Σ be another connected component and let α denote the restriction of (π_{−})_{∗}(π_{+})^{∗}α′ to S. That is we integrate (π_{+})^{∗}α′ only over T (S, S′). We may assume T (S, S′) ≠ ∅. Since rank(V_{T (S,S′)}) = ind(S) − ind(S′) − 1, see Theorem 3(iv), the degree of α will be |α′| − ind(S) + ind(S′) + 1. This shows δ : C^{q} → C^{q+1}. Since it is the rank of a vector bundle we must have ind(S) − ind(S′) − 1 ≥ 0 and thus δ : C_{p} → C_{p}. □

A Morse–Bott–Smale vector field on M also provides a filtration of Ω^{∗}(M) as follows. For a connected component S ⊆ Σ let Ŵ_{S}^{−} denote its compactified unstable manifold and let î_{S} : Ŵ_{S}^{−} → M denote the restriction of the smooth map î : Ŵ^{−} → M, cf. Theorem 4. Let Ω_{p}^{q}(M; E) denote the space of forms α ∈ Ω^{q}(M; E) which vanish on some open neighborhood of M(p) := ⋃_{ind(S)<p} î_{S}(Ŵ_{S}^{−}). Clearly Ω_{p+1}^{∗}(M; E) ⊆ Ω_{p}^{∗}(M; E) and this filtration is preserved by the deRham differential. Note that (Ω_{p}^{∗}(M; E), d) computes the relative cohomology H^{∗}(M, M(p); E).

Proposition 7. The integration preserves the grading and the filtration as indicated below.

Int : Ω_{p}^{q}(M; E) → C_{p}^{q}(X; E)

Proof. Let α′ be a form on M of homogeneous degree |α′| and let S ⊆ Σ be a connected component. Let α denote the restriction of (p_{−})_{∗}i^{∗}α′ to S. That is we only integrate over W_{S}^{−}. Since rank(V_{W_{S}^{−}}) = ind(S), see Theorem 4(iv) we must have |α| = |α′| − ind(S). This implies Int : Ω^{q} → C^{q}. The rest is obvious. □

3.4. Cohomology of the geometric complex. Let us consider the filtered graded complex C_{p}^{q}(X; E), see Proposition 6. It gives rise to a spectral sequence (E_{k}C_{p}^{q}(X; E), δ_{k}). Since C_{0}^{∗}(X; E) = C^{∗}(X; E) and C_{dim M+1}^{∗}(X; E) = 0 the spectral sequence will converge to HC^{∗}(X; E). Moreover, for the E_{1}–term we obviously have E_{1}C_{p}^{q}(X; E) = H^{q−p}(Σ_{p}; E_{Σ_{p}}). Here Σ_{p} = ⨆_{ind(S)=p} S denotes the critical manifold of index p and E_{Σ_{p}} = E_{Σ}|_{Σ_{p}} = (E|_{Σ} ⊗ O_{N^{−}})|_{Σ_{p}}. This proves the second part of Theorem 1.

Via this identification the differential on the E_{1}–term has a particularly simple form. All components but

δ_{1} : H^{q−p}(Σ_{p}; E_{Σ_{p}}) → H^{q−p}(Σ_{p+1}; E_{Σ_{p+1}})  (11)

vanish, and (11) is given by δ_{1} = (−)^{q−p}(π̂_{−})_{∗} ∘ (π̂_{+})^{∗}. Here

(π̂_{+})^{∗} : H^{∗}(Σ_{p}; E_{Σ_{p}}) → H^{∗}(T̂(Σ_{p+1}, Σ_{p}); (π̂_{+})^{∗}E)

is given by pull back along π̂_{+} : T̂(Σ_{p+1}, Σ_{p}) → Σ_{p}, and

(π̂_{−})_{∗} : H^{∗}(T̂(Σ_{p+1}, Σ_{p}); (π̂_{+})^{∗}E) → H^{∗}(Σ_{p+1}; E_{Σ_{p+1}})

is the map induced by fiber integration along π̂_{−} : T̂(Σ_{p+1}, Σ_{p}) → Σ_{p+1}. For this note that since the index difference is one, the fibers of the bundle π̂_{−} : T̂(Σ_{p+1}, Σ_{p}) → Σ_{p+1} are zero dimensional, hence there is no boundary, and fiber integration does indeed induce a mapping in cohomology.

Remark 3. The differentials δ_{k} for k ≥ 1 should be considered as generalization of the classical counting of instantons. These are differentials of finite dimensional complexes and can be considered the combinatorics of the dynamic. If X is a Morse–Smale vector field then (E_{1}C^{∗}, δ_{1}) is the familiar Morse complex and δ_{k} = 0 for all k ≥ 2.

Recall the filtration Ω_{p}^{q}(M; E) introduced in section 3.3. Clearly we have Ω_{0}^{∗}(M; E) = Ω^{∗}(M; E) and Ω_{dim M+1}^{∗}(M; E) = 0. So the associated spectral sequence (E_{k}Ω^{∗}(M; E), δ_{k}) will converge to H^{∗}(M; E). Let W_{Σ_{p}}^{−} denote the unstable manifold of the critical manifold of index p. Using the description of the boundary of Ŵ_{Σ_{p}}^{−} in Theorem 4 and the fact that i : W_{Σ_{p}}^{−} → M is an embedding we obtain a homomorphism

Ω_{p}^{∗}(M; E)/Ω_{p+1}^{∗}(M; E) ⟶^{i^{∗}} Ω_{c}^{∗}(W_{Σ_{p}}^{−}; i^{∗}E)  (12)

where the latter denotes forms with compact support. This is a quasi isomorphism, i.e. induces an isomorphism in cohomology, since Ω_{p}^{∗}(M; E) computes H^{∗}(M, M(p); E). It follows from the Thom isomorphism theorem that Int : E_{1}Ω_{p}^{q}(M; E) → E_{1}C_{p}^{q}(X; E), cf. Proposition 7, is an isomorphism. The first part of Theorem 1 now follows from a standard spectral sequence argument.

Remark 4. Note that this argument actually shows that

Int : (E_{k}Ω_{p}^{∗}(M; E), δ_{k}) → (E_{k}C_{p}^{∗}(X; E), δ_{k})

is an isomorphism for all k ≥ 1. That is, the spectral sequences induced by Ω_{p}^{∗}(M; E) and C_{p}^{∗}(X; E) are isomorphic.

Corollary 1. For the Euler characteristics we have:

χ(M) = ∑_{q}(−)^{q}χ(Σ_{q})  (13)

Proof. The Euler characteristics χ(M) equals the Euler characteristics of H^{∗}(M) which equals the Euler characteristics of the geometric complex χ(HC^{∗}) by Theorem 1(i). Since the spectral sequence E_{k}C^{∗} converges to HC^{∗} we get χ(HC^{∗}) = χ(E_{1}C^{∗}) which implies (13) in view of Theorem 1(ii). □

Remark 5. Suppose X is a vector field whose zero set Σ is a closed submanifold but not necessarily non-degenerate. For every connected component S ⊆ Σ one can define an index IND(S) = IND_{X}(S) generalizing the Hopf index, see [D95], and one has χ(M) = ∑_{S⊆Σ} IND(S). If the vector field −X is Morse–Bott then the indices are related by IND_{X}(S) = (−)^{ind_{−X}(S)}χ(S). This also implies Corollary 1 without using the Smale condition. Finally notice that if the zero set of X is degenerate IND(S) may be non-zero even if χ(S) = 0.

The generalized Hopf index can easily be defined as follows. Let U be a connected open neighborhood of S. Then the vector field provides a mapping X : (U, U \ S) → (T U, T U \ U). The Hopf index then is the unique integer which makes the following diagram commutative:

[figure: H^{n}(T U, T U \ U; π_{T U}^{∗} O_{U}); X^{∗}; Thom; H^{n}(U, U \ S; O_{U}); Thom; H^{0}(U) = ℝ; IND(S); ℝ = H^{dim(S)}(S; O_{S})]

Corollary 2 (Morse inequalities). For every q_{0} ∈ ℤ we have

∑_{q≥q_{0}}(−)^{q−q_{0}}b^{q}(M) ≤ ∑_{q≥q_{0}}(−)^{q−q_{0}} ∑_{p} b^{q−p}(Σ_{p})

where b^{q}(M) := dim H^{q}(M; E) and b^{q}(Σ_{p}) = dim H^{q}(Σ_{p}; E_{Σ_{p}}).

Proof. Recall that for a finite dimensional complex C^{∗} and any q_{0} ∈ ℤ we have:

∑_{q≥q_{0}}(−)^{q−q_{0}} dim HC^{q} ≤ ∑_{q≥q_{0}}(−)^{q−q_{0}} dim C^{q}

Applying this to the complex (E_{k}C^{∗}(X; E), δ_{k}) we obtain

∑_{q≥q_{0}}(−)^{q−q_{0}} dim E_{k+1}C^{q} ≤ ∑_{q≥q_{0}}(−)^{q−q_{0}} dim E_{k}C^{q}

which in turn implies:

∑_{q≥q_{0}}(−)^{q−q_{0}} dim E_{∞}C^{q} ≤ ∑_{q≥q_{0}}(−)^{q−q_{0}} dim E_{1}C^{q}

The statement now follows from Theorem 1. □

4. Torsion

4.1. Euler structures. The importance of Euler structures for torsion comes from the fact that they permit to remove the metric ambiguity of analytic and geometric torsion, see below. In the case χ(M) = 0 Euler structures where introduced by Turaev [T90] for exactly this purpose. A detailed discussion of the general case can be found in [BH]. In this paper we will only consider Euler structures over ℝ. Hence the space of Euler structures will be an affine version of H_{1}(M; ℝ), i.e. H_{1}(M; ℝ) acts free and transitively on the set of Euler structures.

In [BH] the set of Euler structures 𝔈𝔲𝔩_{x_{0}}(M) based at x_{0} ∈ M was defined as the set of equivalence classes [X, c], where X is a vector field with non-degenerate zeros and c ∈ C_{1}(M; ℝ) is a singular one chain with ∂c = ∑_{x∈X} IND(x)x − χ(M)x_{0}. Here X denotes the (finite) set of zeros of X and IND(x) = IND_{X}(x) its Hopf index at x. Notice that IND(x) = ±1 for we assume X to be non-degenerate. Every such chain c is called Euler chain or Turaev spider for X based at x_{0}. Since ∑_{x∈X} IND(x) = χ(M) every such vector field admits Euler chains. By definition [X_{1}, c_{1}] = [X_{2}, c_{2}] iff c_{2} − c_{1} = c(X_{1}, X_{2}) modulo boundaries. Here c(X_{1}, X_{2}) is the singular one chain obtained from the zero set of a non-degenerate homotopy from X_{1} to X_{2}. Note that

∂c(X_{1}, X_{2}) = ∑_{x∈X_{2}} IND_{X_{2}}(x)x − ∑_{x∈X_{1}} IND_{X_{1}}(x)x

and that changing the homotopy changes c(X_{1}, X_{2}) by a boundary only. The action of [σ] ∈ H_{1}(M; ℝ) on [X, c] ∈ 𝔈𝔲𝔩_{x_{0}}(M) is defined by [X, c] + [σ] := [X, c + σ]. This action clearly is free and transitive. If x_{1} is another base point and τ a path from x_{0} to x_{1} then

[X, c] ↦ [X, c] − χ(M)τ := [X, c − χ(M)τ]  (14)

defines an H_{1}(M; ℝ)–equivariant (affine) bijection 𝔈𝔲𝔩_{x_{0}}(M) → 𝔈𝔲𝔩_{x_{1}}(M). Clearly this does only depend on the homology class of the path from x_{0} to x_{1}. If χ(M) = 0 it does not depend on the base point at all and the set of Euler structures is defined without reference to a base point, cf. [T90].

We will also use another way to represent Euler structures which is better suited to remove ambiguities from the analytic torsion. For a base point x_{0} ∈ M define 𝔈𝔲𝔩_{x_{0}}^{∗}(M) as the set of equivalence classes [g, α], where g is a Riemannian metric on M and α ∈ Ω^{dim M−1}(M \ {x_{0}}; O_{M}) with dα = E_{g}, where E_{g} ∈ Ω^{dim M}(M; O_{M}) denotes the Euler form of g. By definition [g_{1}, α_{1}] = [g_{2}, α_{2}] iff α_{2}−α_{1} = cs(g_{1}, g_{2}) modulo d(Ω^{dim M−1}(M; O_{M})), where cs(g_{1}, g_{2}) denotes the Chern–Simon invariant. There is a free and transitive action of H^{dim M−1}(M; O_{M}) on 𝔈𝔲𝔩_{x_{0}}^{∗} defined by [g, α] + [β] := [g, α − β]. Recall that Poincaré duality provides an isomorphism H_{1}(M; ℝ) = H^{dim M−1}(M; O_{M}), where [σ] ∈ H_{1}(M; ℝ) corresponds to [β] ∈ H^{dim M−1}(M; O_{M}) iff ∫_{σ} ω = ∫_{M} ω ∧ β for all closed one forms ω ∈ Ω^{1}(M; ℝ). There is a canonic affine isomorphism 𝔈𝔲𝔩_{x_{0}}(M) = 𝔈𝔲𝔩_{x_{0}}^{∗}(M) which can be regarded as affine version of Poincaré duality. To define it we need two preliminary propositions. For details see [BH].

For a closed one form ω ∈ Ω^{1}(M; ℝ), a Riemannian metric g, a form α ∈ Ω^{dim M−1}(M \ {x_{0}}; O_{M}) with dα = E_{g} on M \ {x_{0}} and a smooth function f : M → ℝ such that ω′ := ω − df vanishes in a neighborhood of x_{0} define:

S(ω, g, α; f) := ∫_{M\{x_{0}}} ω′ ∧ α − ∫_{M} fE_{g} + χ(M)f(x_{0})

Proposition 8. The quantity S(ω, g, α; f) does not depend on the choice of f and will be denoted by S(ω, g, α). It should be considered a regularization of the possibly divergent integral ∫_{M\{x_{0}}} ω ∧ α. It has the following properties:

(i) S(ω, g, α) is linear in ω.

(ii) S(dh, g, α) = − ∫_{M} hE_{g} + χ(M)h(x_{0}) for all smooth functions

h : M → ℝ. (iii) S(ω, g, α + β) − S(ω, g, α) = ∫_{M} ω ∧ β for all closed β ∈

Ω^{dim M−1}(M; O_{M}). (iv) If [α_{1}, g_{1}] = [α_{2}, g_{2}] ∈ 𝔈𝔲𝔩_{x_{0}}^{∗}(M) then S(ω, g_{2}, α_{2})−S(ω, g_{1}, α_{1}) =

∫_{M} ω ∧ cs(g_{1}, g_{2}) for all closed one forms ω.

Proof. The independence of f follows from a straight forward calculation using dα = E_{g}, ∫_{M} E_{g} = χ(M) and Stokes’ theorem. All of the properties are obvious, except (iv). For this one use d cs(g_{1}, g_{2}) = E_{g_{2}} − E_{g_{1}} and Stokes’ theorem. For details see [BH, Lemma 2]. □

For a closed one form ω ∈ Ω^{1}(M; ℝ), a vector field X with nondegenerate zero set X, a Riemannian metric g and a smooth function f : M → ℝ such that ω′ := ω − df vanishes in a neighborhood of X define:

R(ω, X, g; f) := ∫_{M\X} ω′ ∧ X^{∗}Ψ_{g} − ∫_{M} fE_{g} + ∑_{x∈X} IND(x)f(x)

Here Ψ_{g} denotes the global angular form, cf. [BT82], also called Mathai– Quillen form in [BZ92]. Recall that Ψ_{g} is define as follows: Let n = dim(M) and let π_{T M} : T M → M be the tangent bundle equipped with the Levi–Civita connection. Let vol_{g} ∈ Ω^{n}(T M; π_{T M}^{∗} O_{M}), the global volume form which vanishes when contracted with horizontal vectors on T M and which assigns to an n-tuple of vertical vectors ‘their volume times their orientation’. Moreover let ξ denote the Euler vector field on T M which assigns to a point v ∈ T M the vertical vector −v ∈ T_{v}(T M). Then

Ψ_{g} := \frac{Γ(n/2)}{(2π)^{n/2}|ξ|^{n}} i_{ξ} vol_{g} ∈ Ω^{n−1}(T M \ M; π_{T M}^{∗} O_{M}).

Proposition 9. The quantity R(ω, X, g; f) does not depend on the choice of f and will be denoted by R(ω, X, g).^{9} It should be considered a regularization of the possibly divergent integral ∫_{M\X} ω ∧ X^{∗}Ψ_{g}. It has the following properties:

(i) R(ω, X, g) is linear in ω.

(ii) R(dh, X, g) = − ∫_{M} hE_{g} + ∑_{x∈X} IND(x)h(x) for all smooth

functions h : M → ℝ. (iii) R(ω, X, g_{2})−R(ω, X, g_{1}) = ∫_{M} ω ∧cs(g_{1}, g_{2}) for two Riemann-

ian metrics g_{1} and g_{2}. (iv) R(ω, X_{2}, g) − R(ω, X_{1}, g) = ∫_{c(X_{1},X_{2})} ω for two vector fields X_{1}

and X_{2} with non-degenerate zeros. Proof. The independence of f follows from straight forward calculation using dΨ_{g} = E_{g}, Stokes’ theorem and lim_{ϵ→0} ∫_{∂B_{ϵ}(x)} X^{∗}Ψ_{g} = − IND(x) with B_{ϵ}(x) denoting the ϵ–ball around x. The first two properties are obvious. For (iii) use Ψ_{g_{2}} − Ψ_{g_{1}} = cs(g_{1}, g_{2}), E_{g_{2}} − E_{g_{1}} = d cs(g_{1}, g_{2}) and Stokes’ theorem. Concerning (iv) it is easy to see that it suffices to consider the case where there is a homotopy 𝕏 from X_{1} to X_{2} such that ω is exact on a neighborhood of the zero set of 𝕏. Let p : I × M → M denote the projection. The homotopy 𝕏 provides a section of p^{∗}T M which we may assume to be transversal to the zero section. Choose f such that p^{∗}ω′ = p^{∗}ω − p^{∗}df vanishes on the zero set of 𝕏. Then

∑_{x∈X_{2}} IND_{X_{2}}(x)f(x) − ∑_{x∈X_{1}} IND_{X_{1}}(x)f(x) = ∫_{c(X_{1},X_{2})} df = ∫_{c(X_{1},X_{2})} ω

9The quantity R(ω, X, g) was considered by Bismut–Zhang [BZ92] for X = − grad_{g}(h) and h a Morse function.

and

∫_{M} ω′ ∧ X_{2}^{∗}Ψ_{g} − ∫_{M} ω′ ∧ X_{1}^{∗}Ψ_{g} = ∫_{I×M} d(p^{∗}ω′ ∧ 𝕏^{∗}p̃^{∗}Ψ_{g})
= − ∫_{I×M} p^{∗}(ω′ ∧ E_{g}) = 0.

Here p̃ : p^{∗}T M → T M denotes the canonic vector bundle homomorphism over p : I × M → M. More details can be found in [BH, Section 3]. □

One can now define the affine isomorphism 𝔈𝔲𝔩_{x_{0}}(M) = 𝔈𝔲𝔩_{x_{0}}^{∗}(M) by saying that [X, c] corresponds to [g, α] iff

R(ω, X, g) − S(ω, α, g) − ∫_{c} ω = 0

for all closed one forms ω ∈ Ω^{1}(M; ℝ). In view of the last two Propositions this is indeed a well defined isomorphism affine over the Poincaré duality H_{1}(M; ℝ) = H^{dim M−1}(M; O_{M}). We will from now on drop the notation 𝔈𝔲𝔩_{x_{0}}^{∗}(M) and feel free to represent an Euler structure 𝔢 ∈ 𝔈𝔲𝔩_{x_{0}}(M) as an equivalence class [X, c] or as an equivalence class [g, α].

For the Morse–Bott case we introduce yet another way to represent Euler structures. This can be considered a localization of the Euler structure at the critical manifold of a Morse–Bott vector field. Fix a Morse–Bott vector field X with critical manifold Σ and define 𝔈𝔲𝔩_{x_{0}}^{X}(M) as the set of equivalence classes [X, c, {𝔢_{S}}], where for every connected component S ⊆ Σ, 𝔢_{S} ∈ 𝔈𝔲𝔩_{x_{S}}(S) with x_{S} ∈ S and c ∈ C_{1}(M; ℝ) is a one chain with ∂c = ∑_{S⊆Σ} IND(S)x_{S} − χ(M)x_{0}. Since we have ∑_{S⊆Σ} IND(S) = χ(M) such c indeed exist. Given two such data (X, c^{1}, {𝔢_{S}^{1}}) and (X, c^{2}, {𝔢_{S}^{2}}) with 𝔢_{S}^{i} ∈ 𝔈𝔲𝔩_{x_{S}^{i}}(S) we choose paths τ_{S} in S from x_{S}^{1} to x_{S}^{2}. Then 𝔢_{S}^{1} − χ(S)τ_{S} ∈ 𝔈𝔲𝔩_{x_{S}^{2}}(S) and hence (𝔢_{S}^{1} − χ(S)τ_{S}) − 𝔢_{S}^{2} ∈ H_{1}(S; ℝ). We define [X, c^{1}, {e_{S}^{1}}] = [X, c^{2}, {e_{S}^{2}}] if for one (and hence every) choice of paths τ_{S} as above we have

∑_{S⊆Σ}(−)^{ind_{−X}(S)}((𝔢_{S}^{1} − χ(S)τ_{S}) − 𝔢_{S}^{2}) = c^{2} − c^{1} − ∑_{S⊆Σ} IND(S)τ_{S}

in H_{1}(M; ℝ). This is indeed independent of the paths {τ_{S}} for we have IND(S) = (−)^{ind_{−X}(S)}χ(S) for every S ⊆ Σ. For [σ] ∈ H_{1}(M; ℝ) the action is defined by [X, c, {𝔢_{S}}] + [σ] := [X, c + σ, {𝔢_{S}}]. Note that for [σ_{S}] ∈ H_{1}(S; ℝ) we have

[X, c, {𝔢_{S} + [σ_{S}]}] = [X, c + ∑_{S⊆Σ}(−)^{ind_{−X}(S)}σ_{S}, {𝔢_{S}}]  (15)

Note how this specializes to the previous definition if X has nondegenerate zeros only, for in this case there is a unique base point and a unique Euler structure on every S = {pt}. Also note how this localization degenerates for the Morse–Bott–Smale vector field X ≡ 0.

Suppose [X, c, {𝔢_{S}}] as above and suppose the Euler structures 𝔢_{S} ∈ 𝔈𝔲𝔩_{x_{S}}(S) are represented as 𝔢_{S} = [Y_{S}, c_{S}]. With the help of a tubular neighborhood of Σ, a connection and a fiber metric on the normal bundle of Σ and a bump function we can construct a globally defined vector field Y which restricts to Y_{S} and is supported in a small neighborhood of Σ. Consider the homotopy t ↦ X + tY. Using the Morse–Bott assumption for X we see that for sufficiently small ϵ > 0 the zeros X^{+} of X^{+} := X + ϵY will all be non-degenerate and coincide with the set of zeros of Y_{S}. For the index of such a zero x we have ind_{−X^{+}}^{M}(x) = ind_{−Y_{S}}^{S}(x)+ind_{−X}^{M}(S). Therefor c^{+} := c+∑_{S⊆Σ}(−)^{ind_{−X}(S)}c_{S} will be an Euler chain for X^{+}, and hence [X^{+}, c^{+}] ∈ 𝔈𝔲𝔩_{x_{0}}(M). This Euler structure does not depend on the choices we made and the construction provides a natural affine isomorphism 𝔈𝔲𝔩_{x_{0}}^{X}(M) = 𝔈𝔲𝔩_{x_{0}}(M). Obviously we have

Lemma 1. Suppose X is a Morse–Bott vector field, 𝔢 ∈ 𝔈𝔲𝔩_{x_{0}}(M) and 𝔢_{S} ∈ 𝔈𝔲𝔩_{x_{S}}(S) for every connected component S ⊆ Σ. Then there exists an Euler chain c ∈ C_{1}(M; ℝ) such that 𝔢 = [X, c, {𝔢_{S}}]. Moreover, every other such Euler chain differs from c by a boundary.

For a Morse–Bott vector field X, a closed one form ω ∈ Ω^{1}(M; ℝ), a Riemannian metric g and a Morse vector field Y on Σ define

R(ω, X, g; Y) := R(ω, X^{+}, g) − ∑_{S⊆Σ}(−)^{ind_{−X}(S)}R(ω, Y |_{S}, g|_{S})

where X^{+} = X + ϵY is a small perturbation of X as above.

Proposition 10. The quantity R(ω, X, g; Y) does not depend on Y nor on the other choices entering the definition of the perturbation X^{+} and will be denoted by R(ω, X, g). It should be considered a regularization of the possibly divergent integral ∫_{M\Σ} ω ∧ X^{∗}Ψ_{g}. It has the following properties:

(i) R(ω, X, g) is linear in ω.

(ii) R(dh, X, g) = − ∫_{M} hE_{g} + ∑_{S⊆Σ}(−)^{ind_{−X}(S)} ∫_{S} hE_{g|_{S}} for all

smooth functions h : M → ℝ. (iii) R(ω, X, g_{2})−R(ω, X, g_{1}) = ∫_{M} ω∧cs(g_{1}, g_{2})−∑_{S⊆Σ}(−)^{ind_{−X}(S)} ∫_{S} ω∧

cs(g_{1}|_{S}, g_{2}|_{S}) for two Riemannian metrics g_{1} and g_{2}.

(iv) If there exists a function f such that ω′ = ω − df vanishes in a neighborhood of Σ then:

R(ω, X, g) = ∫_{M\Σ} ω′ ∧ X^{∗}Ψ_{g} − ∫_{M} fE_{g} + ∑_{S⊆Σ}(−)^{ind_{−X}(S)} ∫_{S} fE_{g|_{S}}

Proof. All this is an immediate consequences of Proposition 9. For the independence of Y note that a homotopy from Y_{1} to Y_{2} as vector fields on Σ provides a homotopy from X_{1}^{+} to X_{2}^{+} via a similar construction. The zero set of this homotopy will coincide with the zero set of the homotopy from Y_{1} to Y_{2} we started with. However, the orientations they inherit differ by (−)^{ind_{−X}(S)}. Now apply Proposition 9(iv) twice.

□

Proposition 11. Suppose 𝔢′ = [g, α] and 𝔢 = [X, c, {𝔢_{S}}] are two Euler structures based at x_{0} and suppose 𝔢_{S} = [g|_{S}, α_{S}]. Then 𝔢 = 𝔢′ iff

0 = R(ω, X, g) − S(ω, α, g) − ∫_{c} ω + ∑_{S⊆Σ}(−)^{ind_{−X}(S)}S(ω, α_{S}, g|_{S})

for all closed one forms ω ∈ Ω^{1}(M; ℝ).

Proof. This is immediate from our definitions. □

4.2. Analytic torsion. Let M be a closed manifold, E a flat vector bundle over M and let 𝔢 ∈ 𝔈𝔲𝔩_{x_{0}}(M) be a an Euler structure based at x_{0} ∈ M. Consider the deRham complex Ω^{∗}(M; E), the determinant line of its cohomology det H^{∗}(M; E) and the one dimensional vector space:

Det_{x_{0}}(M; E) := det H^{∗}(M; E) ⊗ (det E_{x_{0}})^{−χ(M)}  (16)

The analytic torsion will be a non-zero element in Det_{x_{0}}(M; E) defined up to multiplication by ±1, equivalently a scalar product on Det_{x_{0}}(M; E), associated with the Euler structure 𝔢.

To define the analytic torsion choose a Riemannian metric g on M, a fiber metric μ on E and α ∈ Ω^{dim M−1}(M \ {x_{0}}; O_{M}) so that dα = E_{g} on M \ {x_{0}} and so that 𝔢 = [g, α] ∈ 𝔈𝔲𝔩_{x_{0}}(M), cf. section 4.1.

The deRham complex Ω(M; E) inherits a scalar product which permits to define the formal adjoint d^{♯} : Ω(M; E) → Ω(M; E) of the deRham differential d and the Laplacian Δ := d ∘ d^{♯} + d^{♯} ∘ d. Clearly Δ^{q} : Ω^{q}(M; E) → Ω^{q}(M; E) is a symmetric positive semi-definite elliptic second order differential operator. Therefor we have the Ray–Singer torsion, see [RS71]:

log T_{an}^{E,g,μ} := \frac{1}{2} ∑_{q}(−)^{q+1}q log det′Δ^{q} ∈ ℝ

The fiber metric μ on E determines a fiber metric on its determinant bundle det E and a canonic length one section e of det E⊗O_{E}. There is a unique one form θ = θ_{μ}^{E} ∈ Ω^{1}(M; ℝ) such that ∇_{Y}^{det E⊗O_{E}}e = θ(Y)e.^{10} Since the connection on det E ⊗ O_{E} is flat θ is a closed one form. It measures to what extend the fiber metric μ on det E is not parallel. Indeed, one can easily check θ(Y) = −\frac{1}{2} tr_{μ} ∇_{Y}^{E}μ, which could serve as alternative definition, cf. [BH], [BZ92]. The cohomology class Θ_{E} := [θ] ∈ H^{1}(M; ℝ) does not depend on μ and is a complete obstruction to the existence of a fiber metric on E such that the induced fiber metric on det E is parallel.^{11} For later use let us state

Lemma 2.

(i) Suppose c is a closed path starting and ending at x_{0}. Then parallel transport along c provides an automorphism of det E_{x_{0}} which is given by multiplication with ±e^{−⟨Θ_{E},[c]⟩} where [c] ∈ H_{1}(M; ℝ) denotes the homology class represented by c. (ii) Let c be a path from x_{0} to x_{1}, suppose μ is a fiber metric on E and let || · ||_{det E_{x_{i}}}^{μ} denote the induced scalar product on det E_{x_{i}}. Parallel transport along c will map ||·||_{det E_{x_{0}}}^{μ} to e^{∫_{c} θ_{E}^{μ}} ·||·||_{det E_{x_{1}}}^{μ}.

Define a scalar product on H^{∗}(M; E) by restricting the scalar product on Ω^{∗}(M; E) to harmonic forms. Let || · ||_{Hodge}^{E,g,μ} denote the induced scalar product on det H^{∗}(M; E). Let || · ||_{x_{0}}^{μ} denote the scalar product on (det E_{x_{0}})^{χ(M)} induced from μ|_{E_{x_{0}}}. Define a linear isomorphism

τ_{an}^{E,g,α,μ} : (det E_{x_{0}})^{χ(M)} = det H^{∗}(M; E)

so that it maps || · ||_{x_{0}}^{μ} to T_{an}^{E,g,μ}e^{−S(θ,g,α)}|| · ||_{Hodge}^{E,g,μ}. This is well defined up to sign and will be considered as element in Det_{x_{0}}(M; E), cf. (16).

Proposition 12. τ_{an}^{E,g,α,μ} ∈ Det_{x_{0}}(M; E) is independent of μ and does not depend on the representative of the Euler structure 𝔢 = [g, α]. Hence we will denote it by τ_{an}^{E,𝔢}.

Proof. Let g_{1} and g_{2} be two Riemannian metrics on M, let μ_{1} and μ_{2} be two fiber metrics on E and suppose [g_{1}, α_{1}] = 𝔢 = [g_{2}, α_{2}], see section 4.1. Let h denote the unique smooth function so that ||·||_{det E}^{μ_{2}} = e^{h}|| · ||_{det E}^{μ_{1}} where || · ||_{det E}^{μ_{i}} denotes the fiber metric on det E induced by μ_{i}. Then we clearly have log(|| · ||_{x_{0}}^{μ_{2}}/|| · ||_{x_{0}}^{μ_{1}}) = χ(M)h(x_{0}). Moreover, −dh = θ_{2} −θ_{1} where θ_{i} := θ_{E}^{μ_{i}}. With our notation the anomaly formula for the Ray–Singer torsion, see [BZ92, Theorem 0.1] reads:

10The orientation bundle O_{E} just comes in to ensure the existence of a canonic globally defined length one section e. The bundle det E has locally defined length one sections with a sign ambiguity. One could as well use these to first locally define θ. Then one has to observe that the sign ambiguity does not affect θ and that the locally defined one forms actually define a global one form. To circumvent these inconveniences we made use of O_{E}.

11Note that the theta in [BZ92] is −2 times the θ here.

log \frac{T_{an}^{E,g_{2},μ_{2}}|| · ||_{Hodge}^{E,g_{2},μ_{2}}}{T_{an}^{E,g_{1},μ_{1}}|| · ||_{Hodge}^{E,g_{1},μ_{1}}} = ∫_{M} hE_{g_{1}} + ∫_{M} θ_{2} ∧ cs(g_{1}, g_{2})

Using Proposition 8 this implies

log \frac{e^{−S(θ_{2},g_{2},α_{2})}T_{an}^{E,g_{2},μ_{2}}|| · ||_{Hodge}^{E,g_{2},μ_{2}}}{e^{−S(θ_{1},g_{1},α_{1})}T_{an}^{E,g_{1},μ_{1}}|| · ||_{Hodge}^{E,g_{1},μ_{1}}} = χ(M)h(x_{0})

and the proposition follows. □

Definition 5 (Analytic torsion). The element τ_{an}^{E,𝔢} ∈ Det_{x_{0}}(M; E) is well defined up to a sign and will be called the analytic torsion.^{12}

Remark 6. For every σ ∈ H_{1}(M; ℝ) and every 𝔢 ∈ 𝔈𝔲𝔩_{x_{0}}(M) we have τ_{an}^{E,𝔢+σ} = τ_{an}^{E,𝔢}·e^{−⟨Θ_{E},σ⟩}. A path τ from x_{0} to x_{1} provides an isomorphism 𝔈𝔲𝔩_{x_{0}}(M) = 𝔈𝔲𝔩_{x_{1}}(M) denoted by 𝔢 ↦ 𝔢 − χ(M)τ, see (14) in section 4.1. Moreover parallel transport along τ provides an identification Det_{x_{0}}(M; E) = Det_{x_{1}}(M; E). Via the latter we have τ_{an}^{E,𝔢} = τ_{an}^{E,𝔢−χ(M)τ}. This follows from Proposition 8 and Lemma 2(ii).

4.3. Geometric torsion. Let X be Morse–Bott–Smale vector field on a closed manifold M, let E be a flat vector bundle over M and let 𝔢 ∈ 𝔈𝔲𝔩_{x_{0}}(M) be an Euler structure based at x_{0} ∈ M. Consider the geometric complex C^{∗}(M; E), cf. section 3.2, let det HC^{∗}(X; E) denote the determinant line of its cohomology and consider the one dimensional vector space:

Det_{x_{0}}(X; E) = det HC^{∗}(X; E) ⊗ (det E_{x_{0}})^{−χ(M)}

The geometric torsion will be a non-zero element in Det_{x_{0}}(X; E) defined up to multiplication by ±1, equivalently a scalar product on Det_{x_{0}}(X; E), associated to the Euler structure 𝔢.

In order to define the geometric torsion choose an Euler structure 𝔢_{S} ∈ 𝔈𝔲𝔩_{x_{S}}(S) for every connected component S ⊆ Σ and choose c such that 𝔢 = [−X, c, {𝔢_{S}}], see Lemma 1. This gives rise to an isomorphism, well defined up to multiplication by ±1,

(det E_{x_{0}})^{χ(M)} =^{c} ⨂_{S⊆Σ}((det(E_{S})_{x_{S}})^{χ(S)})^{(−)^{ind(S)}}  (17)

12The sign ambiguity of the analytic torsion can be removed using a homology orientation of M, i.e. an orientation of det H^{∗}(M; ℝ), see [T02].

as follows. We choose a path c_{S} from x_{0} to x_{S} and use parallel transport along c_{S} to get an isomorphism det E_{x_{0}} = det E_{x_{S}}. Taking tensor products (and duals) and using Corollary 1 we obtain an isomorphism

(det E_{x_{0}})^{χ(M)} ⟶^{{c_{S}}} ⨂_{S⊆Σ}((det E_{x_{S}})^{χ(S)})^{(−)^{ind(S)}}  (18)

Choose an orientation of (O_{N^{−}})_{x_{S}} to get an identification (O_{N^{−}})_{x_{S}} = ℝ and hence an identification (E_{S})_{x_{S}} = E_{x_{S}}. Set

c_{0} := c − ∑_{S⊆Σ}(−)^{ind(S)}χ(S)c_{S} ∈ C_{1}(M; ℝ)

and note that ∂c_{0} = 0. Now define (17) as e^{−⟨Θ_{E},[c_{0}]⟩} times the isomorphism (18). It follows immediately from Lemma 2(i) that the resulting isomorphism (17) does not depend on the choice of paths {c_{S}} and actually only depends on 𝔢 and {𝔢_{S}} — up to sign.

As explained in section 3.3 the geometric complex is a filtered graded complex C_{p}^{q}(X; E). In view of Theorem 1(ii) the associated spectral sequence converges to HC^{∗}(X; E). This provides us with a natural identification

det HC^{∗} = det GHC^{∗} = det E_{∞}C^{∗} = ⋯ = det E_{2}C^{∗} = det E_{1}C^{∗}

where C^{∗} = C^{∗}(X; E) and GHC^{∗} denotes the associated graded of HC^{∗} which inherits a filtration as usual from the filtration on C^{∗}. For more details on determinant lines see appendix B. Together with the description of the E_{1}–term in Theorem 1(ii) this yields an isomorphism:

det HC^{∗}(X; E) = ⨂_{S⊆Σ}(det H^{∗}(S; E_{S}))^{(−)^{ind(S)}}  (19)

The tensor product of the analytic torsions provides an isomorphism

⨂_{S⊆Σ}((det E_{x_{S}})^{χ(S)})^{(−)^{ind(S)}} = ⨂_{S⊆Σ}(det H^{∗}(S; E_{S}))^{(−)^{ind(S)}}  (20)

Let τ_{geom}^{E,X,c,{𝔢_{S}}} ∈ Det_{x_{0}}(X; E) denote the element corresponding to the composition of (17), (20) and (19). Using Remark 6 and (15) we immediately obtain

Proposition 13. The element τ^{E,X,c,{𝔢_{S}}} ∈ Det_{x_{0}}(X; E) does not depend on the choice of 𝔢_{S}, c and hence will be denoted by τ_{geom}^{E,𝔢,X}.

Definition 6 (Geometric torsion). The element τ_{geom}^{E,𝔢,X} ∈ Det_{x_{0}}(X; E) is well defined up to a sign and called the geometric torsion.^{13} Remark 7. For every σ ∈ H_{1}(M; ℝ) and every 𝔢 ∈ 𝔈𝔲𝔩_{x_{0}}(M) we have τ_{geom}^{E,𝔢+σ,X} = τ_{geom}^{E,𝔢,X} · e^{−⟨Θ_{E},σ⟩}. A path τ from x_{0} to x_{1} provides an isomorphism 𝔈𝔲𝔩_{x_{0}}(M) = 𝔈𝔲𝔩_{x_{1}}(M) denoted by 𝔢 ↦ 𝔢 − χ(M)τ, see (14) in section 4.1. Moreover parallel transport along τ provides an identification Det_{x_{0}}(X; E) = Det_{x_{1}}(X; E). Via the latter we have τ_{geom}^{E,𝔢,X} = τ_{geom}^{E,𝔢−χ(M)τ,X}. Note that this is analogous to the properties of the analytic torsion, cf. Remark 6.

13The sign ambiguity of the geometric torsion can be removed using a homology orientation, that is an orientation of det H^{∗}(M; ℝ), see [T02].

Remark 8. Note how the geometric torsion specializes to the analytic torsion for the Morse–Bott–Smale vector field X ≡ 0.

Also note how the geometric torsion specializes to the combinatorial torsion for a Morse–Smale vector field X. More precisely, if 𝔢 = [−X, c] then the geometric torsion coincides with the composition:

(det E_{x_{0}})^{χ(M)} =^{c} det C^{∗}(X; E) = det HC^{∗}(X; E)

This follows from Proposition 20 in appendix B, which is a very week statement in this special case.

Recall from Theorem 1 that the integration induces an isomorphism H^{∗}(M; E) = HC^{∗}(X; E). So we get an induced isomorphism:

Det_{x_{0}}(M; E) = Det_{x_{0}}(X; E)  (21)

The following is the main result of this paper.

Theorem 5. Let X be a Morse–Bott–Smale vector field on a closed manifold M, let E a flat vector bundle over M and suppose 𝔢 is an Euler structure based at x_{0} ∈ M. Then the isomorphism (21) maps the analytic torsion τ_{an}^{E,𝔢} to the geometric torsion τ_{geom}^{E,𝔢,X}, up to sign.

If X is a Morse–Smale pair then this is due to Bismut–Zhang, see [BZ92], which itself is an extension of a result of Cheeger [C77], [C79] and Müller [M78]. In this formulation it can be found in [BH, Theorem 5]. The proof of Theorem 5 is contained in section 5 below and will make use of the classical Bismut–Zhang theorem, that is we will assume Theorem 5 proved for Morse–Smale vector fields.

4.4. Derivation of Theorem 2 from Theorem 5. We continue to use the notation from section 4.3. So we have an Euler structure 𝔢 ∈ 𝔈𝔲𝔩_{x_{0}}(M) and Euler structures 𝔢_{S} ∈ 𝔈𝔲𝔩_{x_{S}}(S) for every connected component S ⊆ Σ and we have a one chain c such that 𝔢 = [−X, c, {𝔢_{S}}].

Choose a fiber metric μ on E and let μ_{S} denote the induced fiber metric on E_{S}. Choose a Riemannian metric g on M and let g_{S} denote the restriction to S ⊆ Σ. Choose α so that 𝔢 = [g, α] and for every connected component S ⊆ Σ choose α_{S} with 𝔢_{S} = [g_{S}, α_{S}], cf. section 4.1. From Proposition 11 we have:

R(θ, −X, g) − S(θ, α, g) − ∫_{c} θ + ∑_{S⊆Σ}(−)^{ind(S)}S(θ, α_{S}, g_{S}) = 0 (22)

In view of Theorem 5 we have a commutative diagram:

\xymatrix{⨂((det(E_{S})_{x_{S}})^{χ(S)})^{±} \ar[d]_{⨂ τ_{an}^{E_{S} ,𝔢_{S}}} & (det E_{x_{0}})^{χ(M)} \ar[l]_{c} \ar[dr]^{τ_{an}^{E,𝔢}} \ar[d]^{τ_{geom}^{E,𝔢,X}} \\ ⨂(det H^{∗}(S; E_{S}))^{±} & det HC^{∗}(X; E) \ar[l]_{(19)} & det H^{∗}(M; E) \ar[l]_{Int}}

The tensor products are over all connected components S ⊆ Σ and the powers ± are abbreviations for (−)^{ind(S)}.

Note that Int maps || · ||_{Hodge}^{E,g,μ} to (T_{met}^{M}(E, X, g, μ))^{−1}|| · ||_{geom}^{E,X,g,μ} by the very definition of the metric torsion. Applying Proposition 21 in appendix B once for every E_{k}–term we see that (19) maps || · ||_{geom}^{E,X,g,μ} to:

(T_{comb}^{M}(E, X, g, μ))^{−1} · ⨂_{S⊆Σ}(|| · ||_{Hodge}^{E_{S},g_{S},μ_{S}})^{(−)^{ind(S)}}

It follows from Lemma 2(ii) that c maps || · ||_{x_{0}}^{μ} to:

e^{∫_{c} θ} · ⨂(|| · ||_{x_{S}}^{μ_{S}})^{(−)^{ind(S)}}

Therefore the commutativity of the diagram implies:

T_{an}^{M}(E, g, μ)e^{−S(θ,α,g)}(T_{met}^{M}(E, X, g, μ))^{−1}(T_{comb}^{M}(E, X, g, μ))^{−1}

= e^{∫_{c} θ} · ∏_{S⊆Σ}(T_{an}^{M}(E_{S}, g_{S}, μ_{S})e^{−S(θ,α_{S},g_{S})})^{(−)^{ind(S)}}

Together with (22) Theorem 2 follows at once.

Remark 9. The proof actually shows that Theorem 5 is equivalent to Theorem 2. In view of the Bismut–Zhang theorem, see [BZ92, Theorem 0.2] we may assume Theorem 5 proved for Morse–Smale vector fields, see also [BH, Theorem 5].

5. Proof of main Theorem 5

5.1. Homotopies. A Morse–Bott homotopy 𝕏 = {X_{s}}_{s∈ℝ} from a Morse–Bott–Smale vector field X^{−} to another Morse–Bott–Smale vector field X^{+} is a smooth family of vector fields for which there exists ϵ > 0 so that X_{s} = X^{−} when s ≤ −1 + ϵ and X_{s} = X^{+} when s ≥ 1 − ϵ.

To such a homotopy one associate a Morse–Bott vector field Y on the smooth manifold N := M × [−1, 1] defined by

Y (x, s) := X_{s}(x) + 1/2(s^{2} − 1)\frac{∂}{∂s}.

The homotopy is called Morse–Bott–Smale homotopy if Y is a Morse– Bott–Smale vector field. Be aware that it is in general not possible to find a Morse–Bott–Smale homotopy 𝕏 connecting two given Morse– Bott–Smale vector fields X^{−} and X^{+}.

It is obvious from the definition of Y that the discussion in sections 2 and 3 as well as the compacity of T̂ and Ŵ^{−} continue to hold. We will decorate Σ, T , u and Int by X^{±} or Y to indicate the relevant vector field.

We will use several canonic identifications. First, Σ_{Y} identifies with the disjoint union of Σ_{X^{−}} and Σ_{X^{+}}. Note that N_{Y}^{−}|_{Σ_{X^{−}}} = N_{X^{−}}^{−} hence ind_{Y} |_{Σ_{X^{−}}} = ind_{X^{−}} : Σ_{X^{−}} → ℕ_{0} and O_{N_{Y}^{−}} |_{Σ_{X^{−}}} = O_{N_{X^{−}}^{−}}. Moreover N_{X^{+}}^{−} ⊆ N_{Y}^{−}|_{Σ_{X^{+}}} with a canonically oriented one dimensional quotient. Therefore ind_{Y} |_{Σ_{X^{+}}} = 1 + ind_{X^{+}} : Σ_{X^{+}} → ℕ_{0} and O_{N_{Y}^{−}} |_{Σ_{X^{−}}} = O_{N_{X^{+}}^{−}}. For the last one we use the inward pointing normal last convention.

Let p : N → M denote the canonical projection and suppose E is a flat vector bundle over M. From our identifications we obtain

C^{∗}(Y; p^{∗}E) = C^{∗}(X^{−}; E) ⊕ C^{∗−1}(X^{+}; E)

as graded vector spaces and δ_{Y} = ( \begin{matrix}δ_{X^{−}} & 0 \\ v_{𝕏} & δ_{X^{+}} \end{matrix} ). Define a grading homomorphism B = (−)^{k} : C^{k}(X^{+}; E) → C^{k}(X^{+}; E) and let

φ_{𝕏} := B ∘ v_{𝕏} : C^{∗}(X^{−}; E) → C^{∗}(X^{+}; E).

The equation δ_{Y}^{2} = 0 implies v_{𝕏} ∘ δ_{X^{−}} + δ_{X^{+}} ∘ v_{𝕏} = 0 and hence

φ_{𝕏} ∘ δ_{X^{−}} = δ_{X^{+}} ∘ φ_{𝕏}.  (23)

Moreover the integration Int_{Y} : Ω(N; p^{∗}E) → C(Y; p^{∗}E) decomposes as Int_{Y} = ( \begin{matrix}Int_{X^{−}} ∘ inc_{−}^{∗} \\ h_{𝕏} \end{matrix} ) where inc_{±} : M → M × \{±1\} ⊆ N denotes the canonical inclusion. Slightly generalizing Proposition 4 to the boundary case we have:

Int_{Y} ∘d = δ_{Y} ∘ Int_{Y} −B ∘ Int_{X^{+}} ∘ inc_{+}^{∗}  (24)

The extra term stems from the additional 1–boundary component of Y ’s unstable manifold which coincides with the main stratum of the unstable manifold of X^{+}. The sign B stems partially from Stokes’ theorem for fiber integration, see section 3.1, and partially from our identification O_{N_{Y}^{−}} |_{Σ_{X^{+}}} = O_{N_{X^{+}}^{−}} using the inward pointing normal last convention. Define:

H_{𝕏} := B ∘ h_{𝕏} ∘ p^{∗} : Ω^{∗}(M; E) → C^{∗−1}(X^{+}; E)

Equation (24) implies:

h_{𝕏} ∘ d = v_{𝕏} ∘ Int_{X^{−}} ∘ inc_{−}^{∗} +δ_{X^{+}} ∘ h_{𝕏} − B ∘ Int_{X^{+}} ∘ inc_{+}^{∗}

and thus:

H_{𝕏} ∘ d + δ_{X^{+}} ∘ H_{𝕏} = φ_{𝕏} ∘ Int_{X^{−}} − Int_{X^{+}}  (25)

So we get an induced mapping in cohomology φ_{𝕏} : HC^{∗}(X^{−}; E) → HC^{∗}(X^{+}; E) which satisfies φ_{𝕏} ∘ Int_{X^{−}} = Int_{X^{+}} and in view of Theorem 1(i) is an isomorphism which does not depend on the particular homotopy from X^{−} to X^{+}. Notice also, that for a constant homotopy 𝕏 we have φ_{𝕏} = id on chain level.

Also note that φ_{𝕏} : C_{p}^{∗}(X^{−}; E) → C_{p}^{∗}(X^{+}; E) preserves the filtration and hence it induces homomorphism intertwining the spectral sequences: φ_{𝕏} : E_{k}C^{∗}(X^{−}; E) → E_{k}C^{∗}(X^{+}; E).

5.2. A small perturbation. Let X be a Morse–Bott–Smale vector field on a closed manifold M with critical manifold Σ := Σ_{X}. Let Y be a Morse–Smale vector field on Σ. As in section 4.1 use a tubular neighborhood of Σ to extend Y to a globally defined vector field which is supported in a small neighborhood of Σ. Define a homotopy 𝕏 from X to X^{+} := X + Y by 𝕏_{t} := X + tY. By choosing the extension of Y to M carefully we may assume that the following hold true:

(i) X^{+} is Morse–Smale and Σ_{X^{+}} ⊆ Σ.

(ii) We have N_{X^{+}}^{−} = N_{X}^{−} ⊕ N_{Y |_{Σ}}^{−}, ind_{X^{+}}^{M} = ind_{X}^{M} |_{Σ_{X^{+}}} + ind_{Y |_{Σ}}^{Σ} and O_{N_{X^{+}}^{−}} = O_{N_{X}^{−}} ⊗ O_{N_{Y |_{Σ}}^{−}}.

(iii) X^{+} is tangential to Σ. For x and x′ in a connected component

S ⊆ Σ we have T̂_{X^{+}}(x, x′) = T̂_{Y |_{S}}(x, x′). (iv) Suppose S ≠ S′ ⊆ Σ are two different connected components

of Σ, x ∈ Σ_{X^{+}} ∩ S, x′ ∈ Σ_{X^{+}} ∩ S′ and suppose T̂_{X}(S, S′) = ∅. Then T̂_{X^{+}}(x, x′) = ∅.

(v) The homotopy 𝕏 is Morse–Bott–Smale.

(vi) The homotopy 𝕏 is tangential to Σ, i.e. every 𝕏_{t} is tangential to Σ. For every connected component S ⊆ Σ and x ∈ S we have T̂_{𝕏}(x, S) = Ŵ_{Y |_{S}}^{−}(x).

(vii) Suppose S ≠ S′ ⊆ Σ are two different connected components of Σ, x ∈ Σ_{X^{+}} ∩ S and T̂_{X}(S, S′) = ∅. Then T̂_{𝕏}(x, S′) = ∅.

With the help of (i) we can define an interesting filtration on the geometric complex C(X^{+}; E) by:

C(X^{+}; E)_{p} = ⨁_{p′≥p} Ω(Σ_{X^{+}} ∩ Σ_{p′}; E ⊗ O_{N_{X^{+}}^{−}})

Because of (iv) the differential on C(X^{+}; E) preserves this filtration. We thus get a spectral sequence E_{k}C(X^{+}; E) converging to HC(X^{+}; E). In view of (ii) and (iii) we can compute the E_{1}–term as:

E_{1}C^{q}(X^{+}; E)_{p} = HC^{q−p}(Y |_{Σ_{p}}; E_{Σ_{p}})

Here E_{Σ_{p}} = E|_{Σ_{p}} ⊗ O_{N_{X}^{−}} |_{Σ_{p}}. As explained in section 5.1 the Morse– Bott–Smale homotopy, cf. (v), provides a homomorphism of graded complexes which in view of (vii) preserves the filtrations:

φ_{𝕏} : C_{p}^{q}(X; E) → C^{q}(X^{+}; E)_{p}  (26)

From (vi) we see that the induced mapping on E_{1}–terms, cf. Theorem 1(ii), is the isomorphism induced by integration:

[figure: E_{1}C_{p}^{q}(X; E); φ_{𝕏}; E_{1}C^{q}(X^{+}; E)_{p}; H^{q−p}(Σ_{p}; E_{Σ_{p}}); Int; HC^{q−p}(Y |_{Σ_{p}}; E_{Σ_{p}})]

5.3. Proof of Theorem 5. The main ingredient for the proof will be the Bismut–Zhang theorem for Morse–Smale vector fields. We will apply it once for every connected component S ⊆ Σ and once for M. Another major input will be a theorem of Maumary [M69]. This piece of homological algebra describes the behavior of torsion with respect to spectral sequences and generalizes the well known behavior of torsion with respect to short exact sequences. The geometric ingredient is the very special Morse–Bott–Smale homotopy constructed in section 5.2.

We continue to use the notation from section 5.2. Let 𝔢 ∈ 𝔈𝔲𝔩_{x_{0}}(M) be the given Euler structure and choose Euler structures 𝔢_{S} ∈ 𝔈𝔲𝔩_{x_{S}}(S). Choose a one chain c such that 𝔢 = [−X, c, {𝔢_{S}}] and choose one chains c_{S} such that 𝔢_{S} = [−Y_{S}, c_{S}], cf. Lemma 1. Set

c^{+} := c + ∑_{S⊆Σ}(−)^{ind(S)}χ(S)c_{S}.  (27)

This is an Euler chain for X^{+} and in view of section 4.1 we know 𝔢 = [−X^{+}, c^{+}].

[figure: det H^{∗}(M; E); _{Int_{X^{+}}}; _{Int_{X}}; (det E_{x_{0}})^{χ(M)}; τ_{an}^{E,𝔢}; c^{+}; τ_{geom}^{E,𝔢,X^{+}}; det HC^{∗}(X; E); φ_{𝕏}; det HC^{∗}(X^{+}; E); det C^{∗}(X^{+}; E); det E_{∞}C^{∗}(X; E); φ_{𝕏}; det E_{∞}C^{∗}(X^{+}; E); det E_{1}C^{∗}(X; E); φ_{𝕏}; det E_{1}C^{∗}(X^{+}; E); det GC^{∗}(X^{+}; E); ⨂(det H^{∗}(S; E_{S}))^{±}; _{Int_{Y_{S}}}; ⨂(det HC^{∗}(Y_{S}; E_{S}))^{±}; ⨂(det C^{∗}(Y_{S}; E_{S}))^{±}; ⨂((det(E_{S})_{x_{S}})^{χ(S)})^{±}; ⨂(τ_{an}^{E_{S} ,𝔢_{S}} )^{±}; ⨂(c_{S})^{±}; ⨂(τ_{geom}^{E_{S} ,𝔢_{S}} )^{±}; (det E_{x_{0}})^{χ(M)}; c]

Figure 1. A commutative diagram for the proof of Theorem 5. The tensor products are over all connected components S ⊆ Σ, and the powers ± are abbreviations for (−)^{ind(S)}.

Consider the diagram in figure 1. The vertical identifications in the left column stem from the spectral sequence of the filtered complex C_{p}^{q}(X; E). The corresponding vertical identifications in the middle column stem from the spectral sequence of the filtered complex C^{q}(X^{+}; E)_{p}. The isomorphisms labeled c, c^{+} and c_{S} are the ones constructed using parallel transport along these one chains, see section 4.3. We claim that this diagram commutes — up to sign.

Indeed, the upper right part commutes because of the Bismut–Zhang theorem applied to the Morse–Smale vector field X^{+}, but see also Remark 8. The same argument applied to the Morse–Smale vector field Y_{S} on S (and taking tensor products) shows that the lower half circle commutes. The upper left part commutes in view of section 5.1. The squares involving the mapping φ_{𝕏} commute because these mappings are induced from a filtration preserving map φ_{𝕏} : C_{p}^{q}(X; E) → C^{q}(X^{+}; E)_{p} and because of the description of φ_{𝕏} on the E_{1}–term in section 5.2. The big square in the right central part of the diagram commutes in view of a theorem due to Maumary, see Proposition 20 in appendix B. The remaining lower right square obviously commutes.

Note that τ_{geom}^{E,𝔢,X} ∈ Det_{x_{0}}(X; E) can be seen in the diagram as the composition of ⊗τ_{an}^{E_{S},𝔢_{S}} ∘c with the left vertical isomorphisms. In order to prove Theorem 5 we have to show that this isomorphism coincides with the composition Int_{X} ∘τ_{an}^{E,𝔢} at the very top. In view of the commutativity of the diagram it remains to show that the isomorphism obtained by starting at the very bottom and going to the upper right corner along the right edge coincides with the identity on (det E_{x_{0}})^{χ(M)}. But this follows immediately from (27) and the proof is complete.

6. Torsion of smooth bundles

6.1. Bundles. Suppose π : M → N is a smooth bundle with M and N closed manifolds, and suppose (f′, g′) is a Morse–Smale pair on N. Denote by Y := − grad_{g′}(f′) the corresponding Morse–Smale vector field, and let Σ′ denote its critical manifold which is a discrete set. Define a Morse–Bott function f on M by f = f′ ∘ π. Let us write Σ for the critical manifold of f. Clearly Σ = π^{−1}(Σ′), and ind_{Σ} = π^{∗} ind_{Σ′} : Σ → ℕ_{0}. Let us also introduce the notation M_{z} := π^{−1}(z) for the fiber over z ∈ N.

Proposition 14.

(i) There exists a Riemannian metric g on M so that (f, g) is a Morse–Bott–Smale pair and so that X := − grad_{g}(f) satisfies d_{x}π(X(x)) = Y (π(x)) for all x ∈ M.

(ii) One can choose g so that d_{x}π restricted to (T_{x}(M_{π(x)}))^{⊥} is an isometry onto T_{π(x)}N, and so that the fibers M_{z} for z ∈ Σ′ are isometric.

Proof. One chooses a connection in the bundle π (i.e. for any x ∈ M a subspace H_{x} ⊂ T_{x}M depending smoothly on x, so that d_{x}π : H_{x} → T_{π(x)}N is an isomorphism), and a smooth family of Riemannian metrics g_{z}, z ∈ N, on the fibers M_{z}. Clearly T_{x}M decomposes as T_{x}M = H_{x} ⊕ V_{x} with V_{x} = T_{x}(M_{π(x)}). Declare H_{x} and V_{x} orthogonal and define g_{x} so that d_{x}π is an isometry when restricted H_{x} and equal to (g_{z})_{x} on V_{x}.

To insure that the metric g is compatible with f one chooses a sufficiently small contractible neighborhood U ⊂ N of Σ′ and an identification of the bundle π : π^{−1}(U) → U with the trivial bundle pr_{1} : U × F → U. We chose our connection so that its restriction to π^{−1}(U) corresponds to the trivial connection in the bundle pr_{1}, and the family of metrics g_{z}, for z ∈ U, corresponds to a constant family of Riemannian metrics on F. Clearly if g′ is compatible with f′ then, with the above choices, g is compatible with f and satisfies the desired properties. □

Suppose E is a flat vector bundle over M. For z ∈ N let E_{z} := E|_{M_{z}}. Denote by H^{r} the flat bundle over N whose fiber above z is H^{r}(M_{z}; E_{z}). Note that H^{r} is equipped with an obvious flat connection; any path from z_{1} to z_{2} induces an isomorphism from H^{r}(M_{z_{1}}; E_{z_{1}}) to H^{r}(M_{z_{2}}; E_{z_{2}}) which depends only on the homotopy class of the path, and therefore defines a flat connection. Note that g_{z} and μ_{z} induce, via Hodge theory, a scalar product on H^{r}(M_{z}; E_{z}) and therefore a fiber metric μ^{r} on the bundle H^{r}.

Proposition 15. Suppose X and Y are as in Proposition 14. Then:

(i) The spectral sequence of the geometric complex C^{∗}(X; E) is isomorphic to the Leray–Serre spectral sequence of the bundle π : M → N associated with the cell structure given by Y.

(ii) (E_{1}C^{∗}(X; E), δ_{1}) identifies to ⨁_{r} S^{r}(C^{∗}(Y; H^{r}), δ).^{14}

Proof. Observe that the filtration on Ω^{∗}(M; E) induced by X, see section 3.3, induces the Leray–Serre spectral sequence of the bundle. But the spectral sequence of Ω^{∗}(M; E) is isomorphic to the spectral sequence associated to C^{∗}(X; E), see Remark 4 in section 3.4. This proves (i).

Using π^{∗}(N_{z}^{−}) = N_{M_{z}}^{−} for z ∈ Σ′, we get π^{∗}(O_{N_{z}^{−}}) = O_{N_{Mz}^{−}} and hence H^{∗}(M_{z}; E_{z} ⊗O_{N_{Mz}^{−}}) = H^{∗}(M_{z}; E_{z})⊗O_{N_{z}^{−}}. Now (ii) follows easily from the description of the differential δ_{1} on E_{1}C^{∗}(X; E) in section 3.4. □

Suppose that X resp. Y are as in Proposition 14. The isomorphisms between the deRham cohomology and the cohomology of the geometric complex provided by ‘integration theory’ induces the isomorphism

det Int_{E}^{X} : det H^{∗}(M; E) → det H^{∗}(X; E)

and for any r ≥ 0 the isomorphisms

det Int_{H^{r}}^{Y} : det H^{∗}(N; H^{r}) → det H^{∗}(Y; H^{r}).

The last isomorphisms induce the isomorphism

det Int_{E}^{Y} : ⨂_{r}(det H^{∗}(N; H^{r}))^{ϵ(r)} → ⨂_{r}(det H^{∗}(Y; H^{r}))^{ϵ(r)}

14Here S^{r}(C^{∗}, δ^{∗}) denotes the r–suspension of the cochain complex (C^{∗}, δ^{∗}), i.e.

the degree q component of of S^{r}(C^{∗}, δ^{∗}) is exactly (C^{q−r}, δ^{q−r}).

defined by ⨂_{r}(det Int_{H^{r}}^{Y})^{ϵ(r)}. Here V^{ϵ(r)} resp. α^{ϵ(r)} denote V resp. α if r even and the dual V^{∗} resp. (α^{∗})^{−1} if r is odd.

The identification in Proposition 15(ii) induces an isomorphism

E_{2}C^{∗}(X; E) → ⨁_{r} H^{∗−r}(Y; H^{r})

and then an isomorphism:

det(π, X, Y) : det H^{∗}(X; E) → ⨂_{r}(det H^{∗}(Y; H^{r}))^{ϵ(r)}

Proposition 16. The bundle π induces a canonical isomorphism det(π) : det H^{∗}(M; E) → ⨂_{r}(det H^{∗}(B; H^{r}))^{ϵ(r)} so that for X and Y as in Proposition 14 the following diagram is commutative:

\xymatrix{det H^{∗}(M; E) \ar[r]^{det(π)} \ar[d]_{_{det Int_{E}_{X}}} & ⨂_{r}(det H^{∗}(N; H^{r}))^{ϵ(r)} \ar[d]^{_{det Int_{Y}_{E}}} \\ det H^{∗}(X; E) \ar[r]^{det(π,X,Y)} & ⨂_{r}(det H^{∗}(Y; H^{r}))^{ϵ(r)}}

Proof. Choose a pair (X, Y) as in Proposition 14 and define

det(π) := (det Int_{Y}^{E})^{−1} ∘ det(π, X, Y) ∘ Int_{X}^{E} .

We claim that this does not depend on the choice of (X, Y). Since given two such pairs (X_{−}, Y_{−}) and (X_{+}, Y_{+}) one can, by Proposition 14, construct a Morse–Bott–Smale homotopy which satisfies d_{x}π(𝕏(x, t)) = 𝕐(π(x), t) the arguments in section 5.1 imply the independence of the choice of (X, Y). □

Suppose now that g resp. g′ are Riemannian metrics on M resp. N. Recall that g and μ induce a scalar product on det H^{∗}(M; E). Similarly g′ and μ^{r} induce scalar products on det H^{∗}(N; H^{r}). Denote by

log vol_{an}(π) := log vol det(π)

and

log vol_{comb}(π) := log vol det(π, X, Y).

Proposition 16 implies

log vol_{an}(π) − log vol_{comb}(π) = log T_{met}^{M}(E, X, g, μ)

− ∑_{r}(−)^{r} log T_{met}^{N}(H^{r}, Y, g′, μ^{r}) (28)

6.2. Torsion for bundles. As in section 6.1, let π : M → N be a smooth bundle with M and N closed smooth manifolds, E a flat vector bundle over M equipped with a fiber metric μ, and let g resp. g′ be a Riemannian metric on M resp. N. In the previous section we have introduced flat bundles H^{r} with fiber metrics μ^{r}. Denote by θ resp. θ^{r} the closed one forms associated to the fiber metric μ resp. μ^{r}. Let X and Y be vector fields as in Proposition 14.

Consider the spectral sequence associated with the geometric complex C(X; E). For any k ≥ 1 denote

ρ_{k}^{E,X,g,μ} := \frac{1}{2} ∑_{q}(−)^{q+1}q log det′Δ_{k}^{q}

where Δ_{k}^{q} denote the Laplacians in the cochain complexes of Euclidean vector spaces (E_{k}C^{∗}(X; E), δ_{k}) and det′ denotes the product of nonzero eigen values. Note that for k ≥ 1 E_{k}C^{∗}(X; E) is a finite dimensional vector space, and for k large log det′Δ_{k}^{q} = 0. Then the calculation of ρ_{k} for k ≥ 1 is, in principle, a matter of finite dimensional linear algebra.

Theorem 2 applied to X combined with Proposition 15 gives

log T_{an}^{M}(E, g, μ) = ∑_{z∈Σ′}(−)^{ind(z)} log T_{an}^{M_{z}}(E_{z}, g_{z}, μ_{z})

+ ∑_{r}(−)^{r} log T_{comb}^{N}(H^{r}, Y, μ^{r}) + ∑_{k≥2} ρ_{k}^{E,X,g,μ}

+ log T_{met}^{M}(E, X, g, μ) + R(θ, −X, g) (29)

Since Σ′ is zero dimensional, the definition of T_{comb}^{N}(·, Y, ·) does not involve any Riemannian metric on N. For the metric g′ on N the Bismut–Zhang theorem gives

log T_{comb}^{N}(H^{r}, Y, μ^{r}) = log T_{an}^{N}(H^{r}, g′, μ^{r})

− log T_{met}^{N}(H^{r}, Y, g′, μ^{r}) − R(θ^{r}, −Y, g′). (30)

Combining (28), (29) and (30) one obtains the following statement

Theorem 6. Let π : M → N be a smooth fiber bundle with M and N closed manifolds. Suppose g resp. g′ is a Riemannian metric on M resp. N, X and Y are vector fields as in Proposition 14, and suppose E is a flat vector bundle over M equipped with a fiber metric μ. Then:

log T_{an}^{M}(E, g, μ) = ∑_{r}(−)^{r} log T_{an}^{N}(H^{r}, g′, μ^{r})

+ R(θ, −X, g) − ∑_{r}(−)^{r}R(θ^{r}, −Y, g′) + log vol_{an}(π)

− log vol_{comb}(π) + ∑_{k≥2} ρ_{k}^{E,X,g,μ}

+ ∑_{z∈Σ′}(−)^{ind(z)} log T_{an}^{M_{z}}(E_{z}, g_{z}, μ_{z})

Combining with the metric anomaly for torsion,^{15} cf. [BZ92], Theorem 6 implies the main result of [LST98, Theorems 0.2 and 0.4]. As in [LST98], two cases are of particular interest.

Bundles with base a sphere. Suppose that N is homotopy equivalent to S^{n}. Then for k ≥ 1 all d_{k}^{p,r} : E_{k}^{p,r} → E_{k}^{p+k,r−k+1} are zero but d_{n}^{0,r} : E_{n}^{0,r} → E_{n}^{n,r−n+1}. Here we use the notation E_{k}^{p,r} := E_{k}C_{p}^{p+r}(X; E) and d_{k}^{p,r} for the restriction δ_{k} : E_{k}^{p,r} → E_{k}^{p+k,r−k+1} in order to stay with the traditional notation. Then one obtains the following long exact sequence

⋯ → E_{n}^{0,r−1} ⟶^{d_{n}^{0,r−1}} E_{n}^{n,r−n} ⟶^{i^{r−n}} H^{r}(X; E) ⟶^{p^{r}} E_{n}^{0,r} ⟶^{d_{n}^{0,r}} E_{n}^{n,r−n+1} → ⋯

(31)

When H^{r}(X; E) is identified to H^{r}(M; E) via the integration isomorphism we refer to the sequence

⋯ → E_{n}^{0,r−1} ⟶^{d_{n}^{0,r−1}} E_{n}^{n,r−n} ⟶^{i^{r−n}} H^{r}(M; E) ⟶^{p^{r}} E_{n}^{0,r} ⟶^{d_{n}^{0,r}} E_{n}^{n,r−n+1} → ⋯

(32)

as the Wang exact sequence.

The Wang exact sequence can be regarded as an acyclic cochain complex of finite dimensional Euclidean spaces where each term is equipped with the scalar product described above. In particular H(M; E) is equipped with the scalar product induced from μ and g. Denote by T_{W} the torsion of the complex (32). Recall that for an acyclic cochain complex of Euclidean vector spaces (C^{∗}, d^{∗}) the logarithm of the torsion is defined by 1/2 ∑_{q}(−)^{q+1}q log det Δ^{q} = ∑_{q}(−)^{q} log vol(\overline{d^{q}}) where Δ^{q} is the Laplacian acting on C^{q}. Recall that given a linear map α : V → W between two finite dimensional Euclidean spaces one denotes by log vol(\overline{α}) the quantity, cf. appendix B.2,

15which describes how the analytic torsion T_{an} changes when one changes the Riemannian metric g

log vol(\overline{α}) := log vol(α : (ker α)^{⊥} → α(V)).

Proposition 17. One has:

log T_{W} = log T_{met}^{M}(E, X, g, μ) + ρ_{n}^{E,X,g,μ}

Proof. When H^{∗}(X; E) is equipped with a scalar product compatible to the scalar product on E_{∞}^{∗} the torsion of the exact sequence (31) is

ρ_{n}^{E,X,μ,g} = ∑_{r}(−)^{r} log vol(\overline{d_{n}^{0,r}})

simply because log vol(\overline{i^{n−r}}) = 0 = log vol(\overline{p^{r}}).

To see this observe that in the commutative diagram of Euclidean vector spaces below the left side arrow is surjective, the right side arrow injective and the lower horizontal row is short exact. Observe that by construction the scalar product on E_{∞} is induced from the scalar product on E_{n} and the scalar product on H(X; E) is chosen so that it induces the scalar products on E_{∞} in the lower horizontal row sequence.

[figure: E_{n}^{n,r−n}; i^{r−n}; H^{r}(X; E); p^{r}; E_{n}^{n,r−n}; E_{∞}^{n,r−n}; i_{∞}^{n,r−n}; H^{r}(X; E); p_{∞}^{0,r}; E_{∞}^{0,r}.]

Give all these we conclude that

log vol(\overline{i^{r−n}}) = log vol(\overline{i_{∞}^{n,r−n}})
log vol(\overline{p^{r}}) = log vol(\overline{p_{∞}^{0,r}})
log vol(\overline{i_{∞}^{n,r−n}}) =0 = log vol(\overline{p_{∞}^{0,r}})

which imply the statement.

Since in the Wang exact sequence H^{r}(X; E) is replaced by H^{r}(M; E) with the scalar product induced by Hodge theory the torsion T_{W} of the Wang sequence is corrected by T_{met}^{M}(E, X, g, μ). □

Consequently, for n ≥ 2, one obtains from (29):

log T_{an}^{M}(E, μ, g) = ∑_{z∈Σ′}(−)^{ind(z)} log T_{an}^{M_{z}}(E_{z}, g_{z}, μ_{z})

+ ∑_{r}(−)^{r} log T_{comb}^{S^{n}}(H^{r}, Y, μ^{r}) + log T_{W} + R(θ, −X, g).

For n = 1 one obtains from Theorem 2:

log T_{an}^{M}(E, μ, g) = ∑_{z∈Σ′}(−)^{ind(z)} log T_{an}^{M_{z}}(E_{z}, g_{z}, μ_{z})

+ log T_{W} + R(θ, −X, g).

Sphere bundles. Suppose that the fiber M_{z} is homotopy equivalent to S^{n}. Then for k ≥ 1 all d_{k}^{p,r} : E_{k}^{p,r} → E_{k}^{p+k,r−k+1} are zero but d_{n+1}^{p,n} : E_{n+1}^{p,n} → E_{n+1}^{p+n+1,0}. From this we derive the following long exact sequence

⋯ → E_{n+1}^{p−n−1,n} ⟶^{d_{n+1}^{p−n−1,n}} E_{n+1}^{p,0} ⟶^{i^{p}} H^{p}(X; E) ⟶^{p^{p}} E_{n+1}^{p−n,n} ⟶^{d_{n+1}^{p−n,n}} E_{n+1}^{p+1,0} → ⋯

When H^{q}(X; E) is identified to H^{q}(M; E) via ‘integration theory’ the sequence

⋯ → E_{n+1}^{p−n−1,n} ⟶^{d_{n+1}^{p−n−1,n}} E_{n+1}^{p,0} ⟶^{i^{p}} H^{p}(M; E) ⟶^{p^{p}} E_{n+1}^{p−n,n} ⟶^{d_{n+1}^{p−n,n}} E_{n+1}^{p+1,0} → ⋯

(33)

will be referred to as the Gysin long exact sequence.

The Gysin long exact sequence can be regarded as an acyclic cochain complex of finite dimensional Euclidean spaces where each term is regarded with the scalar product described above. Denote by T_{G} the torsion of the complex (33).

Proposition 18. One has:

log T_{G} = log T_{met}^{M}(E, X, g, μ) + ρ_{n+1}^{E,X,g,μ}

The proof is identical to the proof of Proposition 17. Consequently we obtain from (29):

log T_{an}^{M}(E, μ, g) = ∑_{z∈Σ′}(−)^{ind(z)} log T_{an}^{M_{z}}(E_{z}, g_{z}, μ_{z})

+ log T_{comb}^{N}(H^{0}, Y, μ^{0}) + (−)^{n} log T_{comb}^{N}(H^{n}, Y, μ^{n})

+ log T_{G} + R(θ, −X, g)

Note that if E = π^{∗}E′ is the pull back of a flat bundle E′ over N then, as flat bundles over N, we have H^{0} = H^{n} = E′. Also note that for n ≥ 2 every flat bundle is E over M is the pull back of a flat bundle E′ over N.

Appendix A. Proofs of Theorems 3 and 4

The following is a sometimes a simple way to recognize a smooth manifold with corners.

Lemma 3. If P is a smooth manifold with corners, O and S smooth manifolds, p : P → O and s : S → O smooth maps so that p and s are transversal then p^{−1}(s(S)) is a smooth submanifold with corners of P. Here p is called transversal to s if its restriction to each k–boundary ∂_{k}P is transversal to s.

A.1. Some notations. Let ⋯ < c_{i} < c_{i+1} < ⋯ denote the set of all critical values of f. Choose ϵ > 0 sufficiently small, so that c_{i}^{+} < c_{i+1}^{−}, for all i ∈ ℤ, where c_{i}^{±} := c_{i} ± ϵ. Define:

M_{i} := f^{−1}(c_{i})
M_{i}^{±} := f^{−1}(c_{i}^{±})
M(i) := f^{−1}(c_{i−1}, c_{i+1})
Σ_{i} := Σ ∩ M_{i}
W_{Σ_{i}}^{±} := p_{±}^{−1}(Σ_{i}) = ⨆_{S⊆Σ_{i}} W_{S}^{±}
S_{i}^{±} := W_{Σ_{i}}^{±} ∩ M_{i}^{±}
S_{i} := S_{i}^{+} ×_{Σ_{i}} S_{i}^{−}
W^{±}(i) := W_{Σ_{i}}^{±} ∩ M(i)
SW (i) := S_{i}^{+} ×_{Σ_{i}} W^{−}(i)

These definitions are to some extent illustrated in Figure 2. We have: (i) S_{i} ⊆ M_{i}^{+} × M_{i}^{−}. (ii) SW (i) ⊆ S_{i}^{+} × W^{−}(i) ⊆ M_{i}^{+} × M(i). (iii) M_{i}^{±} ⊆ M are smooth submanifolds of codimension 1. (iv) M(i) is an open submanifold of M.

(v) M_{i} is not a manifold, however Ṁ_{i} := M_{i} \ Σ_{i} and Ṁ_{i}^{±} := M_{i}^{±} \ S_{i}^{±} are smooth codimension 1 submanifolds of M.

Let Φ_{t} be the flow associated to the vector field \frac{X}{|X|^{2}} on M \ Σ and consider the diffeomorphisms, see Figure 3,

ψ_{i} : M_{i}^{−} → M_{i−1}^{+}, ψ_{i}(x) := Φ_{c_{i}−ϵ−(c_{i−1}−ϵ)}(x)

and

φ_{i}^{±} : Ṁ_{i}^{±} → Ṁ_{i}, φ_{i}^{±}(x) := Φ_{±ϵ}(x),

as well as the submersion

φ(i) : M(i) \ (W^{+}(i) ∪ W^{−}(i)) → Ṁ_{i}, φ(i)(x) := Φ_{h(x)−c_{i}}(x).

The maps φ_{i}^{±} and φ(i) extend to continuous maps

φ_{i}^{±} : M_{i}^{±} → M_{i} and φ(i) : M(i) → M_{i}.

Figure 2. Notations.

Figure 3. Retractions.

This can be seen from explicit formulae for φ_{i}^{±} and φ(i) in the neighborhood of S_{i}^{±} resp. (W^{+}(i) ∪ W^{−}(i)), cf. [BFK]. Define

P_{i} := {(x, y) ∈ M_{i}^{+} × M_{i}^{−} | φ_{i}^{+}(x) = φ_{i}^{−}(y)},

and denote by p_{i}^{±} : P_{i} → M_{i}^{±} the canonical projections. One can verify the following

Lemma 4. P_{i} ⊆ M_{i}^{+} × M_{i}^{−} is a smooth submanifold with boundary ∂P_{i} diffeomorphic to S_{i} ⊂ M_{i}^{+} × M_{i}^{−}. Precisely we have:

(P1) p_{i}^{±} : P_{i} \ ∂P_{i} → Ṁ_{i}^{±} are diffeomorphisms.

(P2) The restriction of p_{i}^{+} × p_{i}^{−} to ∂P_{i} is a diffeomorphism onto S_{i},

each p_{i}^{±} restricted to ∂P_{i} identifies with the projection onto S_{i}^{±}. Geometrically P_{i} represents the space of all trajectories, possibly broken, from M_{i}^{+} to M_{i}^{−}.

Next we define

Q(i) := {(x, y) ∈ M_{i}^{+} × M(i) | φ_{i}^{+}(x) = φ(i)(y)}.

Equivalently, Q(i) consists of pairs of points (x, y), x ∈ M_{i}^{+}, y ∈ M(i), which lie on the same possibly broken trajectory. Moreover let l_{i} : Q(i) → M_{i}^{+} and r_{i} : Q(i) → M(i) denote the canonical projections. One can verify the following

Lemma 5. Q(i) ⊆ M_{i}^{+} × M(i) is a smooth submanifold with boundary ∂Q(i) diffeomorphic to SW (i) ⊂ M_{i}^{+} × M(i). Precisely we have:

(Q1) l_{i} : Q(i) \ ∂Q(i) → Ṁ_{i}^{+} is a smooth bundle with fiber an open

segment. r_{i} : Q(i)\∂Q(i) → M(i)\W^{−}(i) is a diffeomorphism. (Q2) The restriction of l_{i} × r_{i} to ∂Q(i) is a diffeomorphism onto

SW (i), i.e. l_{i} resp. r_{i} restricted to ∂Q(i) identifies with the projection onto S_{i}^{+} resp. W^{−}(i).

Since P_{i} and Q(i) are smooth manifolds with boundaries

P_{r,r−k} := P_{r} × P_{r−1} × ⋯ × P_{r−k}  (34)

and

P_{r}(r − k) := P_{r} × ⋯ × P_{r−k+1} × Q(r − k)  (35)

are smooth manifolds with corners.

A.2. Proof of Theorem 3. We will equip

T̂(Σ_{r+1}, Σ_{r−k−1}) := π̂_{−}^{−1}(Σ_{r+1}) ∩ π̂_{+}^{−1}(Σ_{r−k−1}) = ⨆_{{S⊆Σ_{r+1} \atop S′⊆Σ_{r−k−1}}} T̂(S, S′)

with the structure of a smooth manifold with corners. If k ≤ −2 the statement is empty. If k = −1 there is nothing to check. So we suppose k ≥ 0. We are going to apply Lemma 3.

Consider the smooth manifold with corners P = P_{r,r−k} from (34) above, the smooth manifold O := ∏_{i=r}^{r−k}(M_{i}^{+} × M_{i}^{−}) and the smooth manifold S := S_{r+1}^{−} × M_{r}^{−} × ⋯ × M_{r−k+1}^{−} × S_{r−k−1}^{+}. In order to define the maps p and s we consider

ω_{i} : M_{i}^{−} → M_{i}^{−} × M_{i+1}^{+}, ω_{i}(x) := (x, ψ_{i}(x))

and

p̃_{i} : P_{i} → M_{i}^{+} × M_{i}^{−}, p̃_{i}(y) := (p_{i}^{+}(y), p_{i}^{−}(y)).

We also denote by α : S_{r+1}^{−} → M_{r}^{+} resp. β : S_{r−k−1}^{+} → M_{r−k}^{−} the restriction of ψ_{r+1} resp. ψ_{r−k}^{−1} to S_{r+1}^{−} resp. S_{r−k−1}^{+}. Finally we set,

s := α × ω_{r} × ⋯ × ω_{r−k+1} × β : S → O

and

p := p̃_{r} × ⋯ × p̃_{r−k} : P → O.

The verification of the transversally of p and s follows easily from (P1), (P2) and the Morse–Smale condition and is done exactly as in [BH01]. It is easy to see that p^{−1}(s(S)) identifies to T̂(Σ_{r+1}, Σ_{r−k−1}) as topological spaces and we leave this verification to the reader. The rest is straight forward.

A.3. Proof of Theorem 4. We are going equip the compact topological space X := Ŵ_{Σ_{r+1}}^{−} with the structure of a smooth manifold with corners. Let us write î:= î_{Σ_{r+1}}^{−} : X → M. For any integer k denote by X(r − k) := î^{−1}(M(r − k)). First we will put the structure of smooth manifold with corners on X(r−k) so that the restriction of î is smooth. Second we check that X(r−k) and X(r−k′) induce on the intersection X(r − k) ∩ X(r − k′) the same smooth structure. These facts imply that X has a canonical structure of smooth manifold with corners and that î is a smooth map.

To accomplish first step we proceed in exactly the same way as in the proof of Theorem 3. Consider the smooth manifold with corners P := P_{r}(r − k) from (35), the smooth manifold O := ∏_{i=r}^{r−k+1}(M_{i}^{+} × M_{i}^{−})×M_{r−k}^{+} and the smooth manifold S := S_{r+1}^{−} ×M_{r}^{−} ×⋯×M_{r−k+1}^{−}. Define

p := p̃_{r} × ⋯ × p̃_{r−k+1} × l_{r−k} : P → O

and

s := α × ω_{r} × ⋯ × ω_{r−k+1} : S → O.

Again we will apply Lemma 3. The verification of the transversality follows from (P1), (P2), (Q1), (Q2) and the Morse–Smale condition, exactly as explained in [BH01]. It is easy to see that p^{−1}(s(S)) identifies to X(r − k). The rest is straight forward.

Appendix B. Some homological algebra

B.1. Finite dimensional complexes. All vector spaces in this section are supposed to be finite dimensional and over a field 𝕂 of characteristics zero. For such a vector space V we define its determinant by det V := Λ^{dim(V)}V. If we have a short exact sequence of vector spaces 0 → U → V → W → 0 we obtain a canonic isomorphism det V = det U ⊗ det W. This is constructed using a section W → V to get an isomorphism V = U ⊕ W. This induces an isomorphism det V = det U ⊗ det W which does not depend on the section.

If V = V^{even} ⊕ V^{odd} is a ℤ_{2}–graded vector space we define its determinant by det V := det V^{even} ⊗ det(V^{odd})^{∗}. Here (V^{odd})^{∗} denotes the dual of V^{odd}. Note that we have a natural identification det((V^{odd})^{∗}) = (det V^{odd})^{∗}. Again, a short exact sequence 0 → U → V → W → 0 of ℤ_{2}–graded vector spaces gives rise to a natural isomorphism det V = det U ⊗ det W by combining the isomorphisms for the even and the odd part.

Suppose C = C^{even} ⊕ C^{odd} is a ℤ_{2}–graded complex with a degree differential of odd degree d : C^{even} → C^{odd} and d : C^{odd} → C^{even}. Then we get three ℤ_{2}–graded vector spaces; Z := ker d, B := img d and HC := Z/B. Moreover we get two short exact sequences of ℤ_{2}–graded vector spaces 0 → B → Z → HC → 0 and 0 → Z → C →^{d} B^{op} → 0. Here B^{op} denotes the ℤ_{2}–graded vector space B equipped with the opposite grading, (B^{op})^{even} := B^{odd} and (B^{op})^{odd} := B^{even}. Note that we have a natural isomorphism det B^{op} = (det B)^{∗}. The two short exact sequences provide natural isomorphisms

det Z = det B ⊗ det HC and det C = det Z ⊗ det B^{op}.

Together with det B ⊗ det B^{op} = 𝕂 they give rise to a natural isomorphism

det C = det HC.  (36)

Suppose we have a short exact sequence of ℤ_{2}–graded complexes 0 → C_{0} → C_{1} → C_{2} → 0. Its long exact sequence in cohomology

\xymatrix{H^{even}C_{0} \ar[r] & H^{even}C_{1} \ar[r] & H^{even}C_{2} \ar[d] \\ H^{odd}C_{2} \ar[u] & H^{odd}C_{1} \ar[l] & H^{odd}C_{0} \ar[l]}

can be used to make the ℤ_{2}–graded vector space HC_{0} ⊕(HC_{1})^{op} ⊕HC_{2} a ℤ_{2}–graded complex with vanishing cohomology. From (36) we thus obtain a natural isomorphism:

det HC_{0} ⊗ det HC_{2} = det HC_{1}  (37)

The following is a reformulation of a well known fact, which for instance can be found in [M66, Theorem 3.2].

Proposition 19. Suppose 0 → C_{0} → C_{1} → C_{2} → 0 is a short exact sequence of ℤ_{2}–graded complexes. Then the following diagram of canonic isomorphisms commutes up to sign:

det C_{0} ⊗ det C_{2}  det C_{1} det HC_{0} ⊗ det HC_{2}  det HC_{1}

Suppose C is a ℤ_{2}–graded complex equipped with a filtration

⋯ ⊇ C_{p} ⊇ C_{p+1} ⊇ ⋯

which is preserved by the differential. We assume C_{p} = 0 for p sufficiently large and C_{p} = C for p sufficiently small. Moreover we assume that the filtration is compatible with the grading, that is C_{p} = C_{p}^{even} ⊕ C_{p}^{odd} where C_{p}^{even} := C_{p} ∩ C^{even} and C_{p}^{odd} := C_{p} ∩ C^{odd}. In this situation we have a spectral sequence (E_{k}C, δ_{k}) converging to HC. Every E_{k}C is a ℤ_{2}–graded complex with differential δ_{k}. For the E_{1}–term we have E_{1}C = HGC as ℤ_{2}–graded vector spaces. Here GC := ⨁_{p} C_{p}/C_{p+1} denotes the associated graded of C considered as ℤ_{2}–graded complex, and HGC its cohomology, a ℤ_{2}–graded vector space. We will not consider the ℤ–grading GC and HGC inherit from the filtration. Applying (36) we obtain a canonic isomorphism det C = det GC = det HGC = det E_{1}C. The cohomology HC inherits the structure of a ℤ_{2}–graded vector space with filtration from C. Let GHC denote the associated graded considered as ℤ_{2}–graded vector space. Again, we will not consider the ℤ–grading GHC inherits. Since the spectral sequence converges we have a canonic isomorphism GHC = E_{∞}C which provides us with a canonic isomorphism det C = det HC = det GHC = det E_{∞}C. Since HE_{k}C = E_{k+1}C we also have a canonic isomorphism det E_{k} = det E_{k+1}C for every k ≥ 1. Also note that E_{k} = E_{∞} for k sufficiently large.

The following can be found in [F92, Theorem 1.3], but see also [M69, Théorèm 4.4] for a more general result.

Proposition 20. Let C be a filtered ℤ_{2}–graded complex. Then the following diagram of canonic isomorphism commutes up to sign:

det HGC  det GC  det C  det HC  det GHC det E_{1}C  det E_{2}C  det E_{3}C  ⋯  det E_{∞}C

B.2. Finite dimensional complexes with scalar product. All vector spaces (complexes) in this section are suppose to be finite dimensional and over ℝ. We will work with ℤ_{2}–graded complexes and all determinant lines are understood in the graded sense. If a ℤ_{2}–graded complex C = C^{even}⊕C^{odd} comes with a scalar product g we will assume g = g^{even} ⊕ g^{odd}, i.e. the even and the odd part are orthogonal. In such a situation B = img d, Z = ker d and HC = Z/B are ℤ_{2}–graded vector spaces with scalar products. We let || · ||_{det C}, || · ||_{det B}, || · ||_{det Z} and ||·||_{det HC} denote the induced norms on det C, det B, det Z and det HC, respectively. Moreover let d^{♯} denote the adjoint of the differential d and Δ = d^{♯}d + dd^{♯} the Laplacian.

For a linear isomorphism between finite dimensional Euclidean spaces α : V → W we set vol α := \sqrt{det(α^{♯}α)}. For an arbitrary linear mapping α : V → W between Euclidean spaces we let \overline{α} : (ker α)^{⊥} → img α denote its restriction and thus have a volume vol(\overline{α}). Moreover, if β : U → U is a linear mapping between finite dimensional vector spaces we write det′β for the product of non-zero eigen values. Clearly vol(\overline{α}) = \sqrt{det′(α^{♯}α)}. For a ℤ_{2}–graded complex C with scalar product define:

T_{C} := vol(\overline{d : C^{even} → 01zC^{odd}}) ; vol(\overline{d : C^{odd} → C^{even}}) = \sqrt{det′(d^{♯}d : C^{even} → C^{even})} ; \sqrt{det′(d^{♯}d : C^{odd} → C^{odd})}

Remark 10. Suppose we have a ℤ–graded complex C = ⨁_{q} C^{q} with scalar product. It gives rise to a ℤ_{2}–graded complex with scalar product in the usual way. Let Δ^{q} denote the Laplacian acting on C^{q}. A two line computation using

det′(Δ^{q}) = det′(d^{♯}d : C^{q} → C^{q}) det′(dd^{♯} : C^{q} → C^{q})

and

det′(dd^{♯} : C^{q} → C^{q}) = det′(d^{♯}d : C^{q−1} → C^{q−1})

shows

log T_{C} = \frac{1}{2} ∑_{q}(−)^{q+1}q log det′Δ^{q}

However, for this to make sense it is essential to have a ℤ–grading at hand.

Proposition 21. Let C be a ℤ_{2}–graded complex with scalar product. Then the canonic isomorphism det C = det HC maps || · ||_{C} to T_{C}|| · ||_{HC}.

Proof. The short exact sequence 0 → B → Z → HC → 0 of ℤ_{2}– graded vector spaces induces an isomorphism det Z = det B ⊗ det HC which clearly maps || · ||_{det Z} to || · ||_{det B} ⊗ || · ||_{det HC}. The short exact sequence 0 → Z → C → B^{op} → 0 of ℤ_{2}–graded vector spaces induces an isomorphism det C = det Z ⊗ det B^{op} which maps || · ||_{det C} to T_{C}|| · ||_{det Z} ⊗||·||_{det B^{op}}. Putting this together we find ||·||_{det C} = T_{C}||·||_{det HC} via det C = det HC. □

Proposition 22. Suppose 0 → C_{0} → C_{1} → C_{2} → 0 is a short exact sequence of ℤ_{2}–graded complexes with compatible scalar products, i.e. C_{0} → C_{1} is an isometry onto its image and C_{2} carries the quotient scalar product. Then the canonic isomorphism

det HC_{1} = det HC_{0} ⊗ det HC_{2}

maps T_{C_{1}}|| · ||_{HC_{1}} to T_{C_{0}}|| · ||_{HC_{0}} ⊗ T_{C_{2}}|| · ||_{HC_{2}}.

Proof. The compatibility condition for the scalar products guarantees that the isomorphism det C_{0} ⊗ det C_{2} = det C_{1} maps || · ||_{det C_{0}} ⊗ || · ||_{det C_{2}} to ||·||_{det C_{1}}. Now the statement follows from Proposition 19 and Proposition 21. □

Let C be a filtered ℤ_{2}–graded complex as considered in section B.1 and suppose C is equipped with a scalar product. Then HC, GC and HGC inherit scalar products in an obvious way.

Proposition 23. Suppose C is a filtered ℤ_{2}–graded complex with scalar product. Then the canonic isomorphism

det HC = det GHC = det E_{∞}C = ⋯ = det E_{1}C = det HGC

maps T_{C}|| · ||_{HC} to T_{GC}|| · ||_{HGC}.

Proof. This follows immediately from Proposition 20, Proposition 21 applied to C and GC as well as the fact that the natural isomorphism det C = det GC maps || · ||_{det C} to || · ||_{det GC}. □

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Dan Burghelea, Dept. of Mathematics, The Ohio State University, 231 West Avenue, Columbus, OH 43210, USA.

  E-mail address: burghele@mps.ohio-state.edu

Stefan Haller, Department of Mathematics, University of Vienna, Nordbergstraße 15, A-1090, Vienna, Austria.

  E-mail address: stefan.haller@univie.ac.at