EULER STRUCTURES, THE VARIETY OF REPRESENTATIONS AND THE MILNOR–TURAEV TORSION

DAN BURGHELEA AND STEFAN HALLER

Abstract. In this paper we extend, and Poincaré dualize, the concept of Euler structures, introduced by Turaev for manifolds with vanishing Euler– Poincaré characteristic, to arbitrary manifolds. We use the Poincaré dual concept, co-Euler structures, to remove all geometric ambiguities from the Ray–Singer torsion by providing a slightly modified object which is a topological invariant. We show that the modified Ray–Singer torsion when regarded as a function on the variety of complex representations of the fundamental group of the manifold is actually the absolute value of a rational function which we call in this paper the Milnor–Turaev torsion. As a consequence one obtains for a closed manifold M and an acyclic unimodular representation of its fundamental group ρ, a computable numerical invariant S(M, ρ) with value in ℝ/πℤ analoguous to the Atiyah–Patodi–Singer eta invariant, cf. section 4.4. Some examples and applications are also discussed, see sections 4.3, 4.5, and 4.6.

Contents

1. Introduction 1 2. A few characteristic forms and the invariant R 5 3. Euler and co-Euler structures 13 4. Torsion and representations 20 Appendix A. Some homological algebra 29 References 30

1. Introduction

This paper was motivated by a question in geometric analysis, whose answer brings to attention a topological invariant which is a rational function on the variety of (some) complex representations of the fundamental group of a closed manifold.

Date: November 26, 2004. 2000 Mathematics Subject Classification. 57R20, 58J52. Key words and phrases. Euler structure; co-Euler structure; combinatorial torsion; analytic

torsion; theorem of Bismut–Zhang; Chern–Simons theory; geometric regularization; mapping torus; rational function.

Part of this work was done while both authors enjoyed the hospitality of the Max Planck Institute for Mathematics in Bonn. A previous version was written while the second author enjoyed the hospitality of the Ohio State University. The second author is supported by the Fonds zur Förderung der wissenschaftlichen Forschung (Austrian Science Fund), project number

P14195-MAT.

Question: Is the Ray–Singer torsion of an odd dimensional manifold,^{1} when regarded as a positive real valued function on the space of complex representations of its fundamental group the absolute value of a holomorphic/meromorphic function, and if so, what is the meaning of its argument?^{2}

Our answer to the first part of the question goes as follows, cf. section 4.6. With the help of a co-Euler structure, as defined in section 3.2, we decompose the Ray– Singer torsion as a product of two positive real valued functions, and verify that the first one is the absolute value of a rational function while the second is the absolute value of a holomorphic function. The rational function is what we call in this paper Milnor–Turaev torsion.^{3} The values of the second function are obtained from an integral on a non-compact manifold, possibly divergent, but always regularizable and referred to as the invariant R defined in section 2.5.

The rational function is a topological invariant of the manifold and carries significant topological information, see for example Theorem 4 in section 4.5.

To grasp the flavor of this invariant let us briefly describe it. For a collection of non-negative integers (k_{0}, k_{1}, …, k_{n}) which satisfy (1): k_{i}−k_{i−1}±⋯+(−1)^{i}k_{0} ≥ 0, i ≤ n − 1 and (2): k_{n} − k_{n−1} ± ⋯ + (−1)^{n}k_{0} = 0. We consider the variety 𝔻̂_{ac}(k_{0}, …, k_{n}) of generically acyclic complexes, C ≡ {0 → ℂ^{k_{0}} → ⋯ → ℂ^{k_{n}} → 0}, and the rational function τ where τ(C) is the torsion of C when acyclic, cf. section 4.2.^{4}

For a finitely presented group Γ we denote by Rep(Γ; V) the complex algebraic set of the complex representations of Γ on the complex vector space V, cf. section 4.1.

For a closed based pointed manifold (M, x_{0}) with Γ = π_{1}(M, x_{0}) denote by Rep^{M}(Γ; V) the algebraic closure of Rep_{0}^{M}(Γ; V), the Zariski open set of representation ρ ∈ Rep(Γ; V) so that H^{∗}(M; ρ) = 0.

By choosing a triangulation τ of M, a collection of paths π_{τ} = {π_{x_{σ}}} from x_{0} to the barycenters x_{σ} of each simplex σ, an ordering o of the set of of simplexes and a base (frame) ϵ of V one obtains a regular map t_{π_{τ} ,o,ϵ} : Rep^{M}(Γ; V) → 𝔻̂_{ac}(k_{0}, …, k_{n}) with k_{i} = dim(V) × ♯(X_{i}) where X_{i} denotes the set of i–simplexes and ♯ denotes cardinality. One can show that while the map t_{π_{τ} ,o,ϵ} depends on (π_{τ} , o, ϵ) the pull-back of τ by t_{π_{τ} ,o,ϵ} depends only on the Euler structure defined by (τ, π_{τ}) (more precisely by (X_{τ} , π_{τ}) cf. section 3.1) and the cohomology orientation in H^{∗}(M; ℝ) induced by o. If one restricts to unimodular unitary representations, as also noticed by W. Müller, not even on the Euler structure.

The absolute value of the Milnor–Turaev torsion is related to Ray–Singer torsion. To describe this relation one needs again the invariant R. The argument is an unexplored invariant which for manifolds of dimension 3, but not in general, might be related to the Atiyah–Patodi–Singer eta invariant, cf. [APS75], at least when considered for a unimodular representation.

1The question can be formulated and positively answered for all dimensions

2A similar question was treated in [Q85]. Instead of the variety of representation Quillen considers the affine space of complex structures in a smooth complex vector bundle over a Riemann surface, and instead of Ray–Singer torsion the square root of the determinant of D^{∗}D where D is the appropriate Cauchy–Riemann operator.

3The sign ambiguity in Milnor’s definition of torsion was ingeniously fixed by Turaev.

4This function is the obvious generalization of det : 𝔻̂_{ac}(r, r) = ℂ^{r^{2}} → ℂ.

The main results are contained in Theorems 1, 3, 4, and Corollary 1. The paper contains a number of additional facts/results of independent interest which expand on the concepts described in the paper.

Let us make a more detailed presentation of the contents of this paper.

Section 2 is largely expository. We collect a number of facts about Riemannian metrics and vector fields with isolated zeros and about the Kamber–Tondeur form, cf. sections 2.1–2.3. In section 2.4 we describe the regularization of some possibly divergent integrals which is essential in the subsequent considerations (the invariant R, the comparison of Euler and co-Euler structures, the modifications of torsions). In section 2.5 we define the invariant R. The invariant R was first introduced by Bismut and Zhang in [BZ92].

We regard the invariant R as providing a pairing between a Riemannian metric and a vector field with isolated zeros with values in one currents rel. boundary. This is in analogy with the Chern–Simons form which provides such pairing for two Riemannian metrics and with the Chern–Simons one chain rel. boundary which provides a pairing between two vector fields with isolated zeros.

Section 3 discusses Euler and co-Euler structures and what they are good for. Turaev [T90] introduced the concept of Euler structures for manifolds with vanishing Euler–Poincaré characteristics χ(M). It was observed in [B99] that Euler structures can be defined for manifolds with arbitrary Euler characteristics at the expense of a base point x_{0} ∈ M. The space of Euler structures 𝔈𝔲𝔩_{x_{0}}(M; ℤ) is an affine version of H_{1}(M; ℤ) in the sense that is H_{1}(M; ℤ) acts free and transitively on 𝔈𝔲𝔩_{x_{0}}(M; ℤ). There is also a real version of Euler structures which is an affine version of H_{1}(M; ℝ) and will be denoted by 𝔈𝔲𝔩_{x_{0}}(M; ℝ). There is a canonic map 𝔈𝔲𝔩_{x_{0}}(M; ℤ) → 𝔈𝔲𝔩_{x_{0}}(M; ℝ) which is affine over the homomorphism H_{1}(M; ℤ) → H_{1}(M; ℝ), see section 3.1.

Given an Euler structure 𝔢 ∈ 𝔈𝔲𝔩_{x_{0}}(M; ℝ), a cohomology orientation 𝔬, i.e. an orientation of H^{∗}(M; ℝ), and a representation ρ ∈ Rep(Γ; V) we use the Milnor– Turaev construction [M66] [T90] to provide an element τ_{comb}^{ρ,𝔢,𝔬} in the determinant line^{5}

Det_{x_{0}}(M; ρ) := det H^{∗}(M; ρ) ⊗ det V^{−χ(M)}  (1)

which will be called Milnor–Turaev torsion, see Definition 1 in section 3.3. For the dependence on 𝔢 and 𝔬 we find

τ_{comb}^{ρ,𝔢+σ} = τ_{comb}^{ρ,𝔢} · [det ∘ρ](σ)^{−1} and τ_{comb}^{ρ,𝔢,−𝔬} = (−1)^{dim V} τ_{comb}^{ρ,𝔢,𝔬}  (2)

for all σ ∈ H_{1}(M; ℤ). Here [det ∘ρ] : H_{1}(M; ℤ) → ℂ^{∗} denotes the homomorphism induced from the homomorphism det ∘ρ : Γ → ℂ^{∗}.

In section 3.2 we introduce the concept of co-Euler structures. The set of co-Euler structures 𝔈𝔲𝔩_{x_{0}}^{∗}(M; ℝ) is an affine version of H^{n−1}(M; O_{M}) and depends on the choice of a base point x_{0} ∈ M if χ(M) ≠ 0. By definition a co-Euler structure is an equivalence class of pairs (g, α), where g is a Riemannian metric on M, and α ∈ Ω^{n−1}(M \ {x_{0}}; O_{M}) with dα = E(g) where E(g) ∈ Ω^{n}(M; O_{M}) denotes the Euler form of g. Two such pairs (g_{1}, α_{1}) and (g_{2}, α_{2}) are equivalent if α_{2} − α_{1} = cs(g_{1}, g_{2}) where cs(g_{1}, g_{2}) ∈ Ω^{n−1}(M; O_{M})/dΩ^{n−2}(M; O_{M}) denotes the Chern–Simons class. The main properties of Euler and co-Euler structures, and the relationship between them are collected in Theorem 1 in section 3.2. Co-Euler structures are nicely suited to remove the metric ambiguities of the analytic torsion, and finally provide a topological invariant the modified Ray–Singer metric/torsion.

5Here we use the following notation. For a finite dimensional vector space V we write det V := Λ^{dim V} V for its determinant line, a one dimensional vector space. For a finite dimensional graded vector space W^{∗} we define the (graded) determinant line by det W^{∗} := det W^{even} ⊗ (det W^{odd})^{∗}.

Moreover, for a line L, i.e. one dimensional vector space, and k ∈ ℕ we write L^{k} := L ⊗ ⋯ ⊗ L and L^{−k} := (L^{k})^{∗}. Then there are natural identifications L^{k+l} = L^{k} ⊗ L^{l} for all k, l ∈ ℤ.

More precisely, let 𝔽_{ρ} denote the flat complex vector bundle over M associated to a representation ρ ∈ Rep(Γ; V). Recall that the Ray–Singer metric is a Hermitian metric ||·||_{RS}^{ρ,g,μ} on the determinant line det H^{∗}(M; 𝔽_{ρ}) = det H^{∗}(M; ρ), but depends on a Riemannian metric g on M, and a Hermitian structure μ on 𝔽_{ρ}, see [BZ92]. Using a co-Euler structure 𝔢^{∗} ∈ 𝔈𝔲𝔩_{x_{0}}^{∗}(M; ℝ) permits to define a modified Ray– Singer metric || · ||_{an}^{ρ,𝔢^{∗}} on the determinant line (1), see Definition 2 in section 3.4. The anomaly formulas for the Ray–Singer torsion imply that this Hermitian metric indeed only depends on the representation ρ and the co-Euler structure 𝔢^{∗}. For the dependence on 𝔢^{∗} we have

|| · ||_{an}^{ρ,𝔢^{∗}+β} = || · ||_{an}^{ρ,𝔢^{∗}} · e^{⟨log |[det ∘ρ]|,P D(β)⟩}

for all β ∈ H^{n−1}(M; O_{M}). Here log |[det ∘ρ]| ∈ H^{1}(M; ℝ) denotes the class corresponding to the homomorphism log |[det ∘ρ]| : H_{1}(M; ℤ) → ℝ, which is paired with the Poincaré dual of β.

There is an isomorphism P : 𝔈𝔲𝔩_{x_{0}}^{∗}(M; ℝ) → 𝔈𝔲𝔩_{x_{0}}(M; ℝ) which is affine over the Poincaré duality isomorphism H^{n−1}(M; O_{M}) → H_{1}(M; ℝ), see section 3.2. This permits to reformulate the Bismut–Zhang theorem [BZ92] as a statement which compares two topological invariants, see Theorem 2 in section 3.4. More precisely, it tells that analysis is able to recover the absolute value of the Milnor–Turaev torsion:

||τ_{comb}^{ρ,𝔢,𝔬}||_{an}^{ρ,𝔢^{∗}} = e^{⟨log |[det ∘ρ]|,P (𝔢^{∗})−𝔢⟩}

Here log |[det ∘ρ]| ∈ H^{1}(M; ℝ) denotes the cohomology class corresponding to the homomorphism log |[det ∘ρ]| : H_{1}(M; ℤ) → ℝ. If H^{∗}(M; ρ) = 0 then Det_{x_{0}}(M; ρ) = ℂ, and τ_{comb}^{ρ,𝔢,𝔬} is a non-zero complex number.

Section 4 is dedicated to the Milnor–Turaev torsion viewed as a rational function. Let Rep_{o}^{M}(Γ; V) ⊆ Rep(Γ; V) denotes the Zariski open subset of representations ρ for which H^{∗}(M; ρ) = 0, and let Rep^{M}(Γ; V) denote its Zariski closure in Rep(Γ; V), cf. section 4.1. Rep_{o}^{M}(Γ; V) and Rep^{M}(Γ; V) only depend on the homotopy type of M, but might be empty, e.g. if χ(M) ≠ 0. The Milnor–Turaev torsion gives rise to a function

T_{comb}^{𝔢,𝔬} : Rep_{o}^{M}(Γ; V) → ℂ^{∗}

which depends on the simple homotopy type of M, 𝔢 and 𝔬. In section 4.2 we observe that this function is a rational function on Rep^{M}(Γ; V) whose zeros and poles are therefore contained in Rep^{M}(Γ; V) \ Rep_{o}^{M}(Γ; V), see Theorem 3.

The function T_{comb}^{𝔢,𝔬} contains relevant topological information, even in the simplest possible examples. In section 4.3 we compute this function for a mapping torus in terms of the Lefschetz zeta function of the glueing diffeomorphism.

If M is obtained by surgery on a framed knot, and dim V = 1, the function T_{comb}^{𝔢,𝔬} coincides with the Alexander polynomial of the knot, see [T02]. We hope that passing to higher dimensional representations T_{comb}^{𝔢,𝔬} captures more subtle knot invariants.

The Bismut–Zhang theorem implies that the absolute value of T_{comb}^{𝔢,𝔬} is a modified Ray–Singer torsion:

|T_{comb}^{𝔢,𝔬}(ρ)| = T_{an}(∇_{ρ}, g, μ) · e^{− ∫_{M} ω(∇_{ρ},μ)∧α}

Here 𝔽_{ρ} = (F_{ρ}, ∇_{ρ}) denotes the flat complex vector bundle associated to the representation ρ ∈ Rep_{o}^{M}(Γ; V), g is a Riemannian metric on M, μ is a Hermitian structure on F_{ρ}, T_{an}(∇_{ρ}, g, μ) denotes the Ray–Singer torsion, ω(∇_{ρ}, μ) is the Kamber–Tondeur form, see section 2.3, and α is such that the pair (g, α) represents the co-Euler structure P^{−1}(𝔢).

We use this fact to conclude (under some additional hypotheses) that the Ray– Singer torsion, when viewed as a function on the analytic set of flat connections, is the absolute value of a meromorphic function. The precise statement is contained in Corollary 1. This partially answers the question formulated at the very beginning of this introduction. This was also known to W. Müller at least in the case when one restricts to unimodular flat connections.

The argument of the complex number T_{comb}^{𝔢,𝔬}(ρ) is an apparently unexplored invariant which is analoguous to the Atiyah–Patodi–Singer eta invariant.

As another application we provide a derivation of Marcsik’s theorem from the Bismut–Zhang theorem and the computation of T_{comb}^{𝔢,𝔬} for mapping tori, see section 4.5.

We would like to thank V. Turaev for pointing out a sign mistake and some lacking references in a previous version of the paper.

2. A few characteristic forms and the invariant R

2.1. Euler, Chern–Simons, and the global angular form. Let M be smooth closed manifold of dimension n. Let π : T M → M denote the tangent bundle, and O_{M} the orientation bundle, a flat real line bundle over M. For a Riemannian metric g denote by

E(g) ∈ Ω^{n}(M; O_{M})

its Euler form, and for two Riemannian metrics g_{1} and g_{2} by

cs(g_{1}, g_{2}) ∈ Ω^{n−1}(M; O_{M})/d(Ω^{n−2}(M; O_{M}))

their Chern–Simons class. The definition of both quantities is implicit in the formulae (6) and (7) below. They have the following properties which follow immediately from (6) and (7) below.

d cs(g_{1}, g_{2}) = E(g_{2}) − E(g_{1})  (3)
cs(g_{2}, g_{1}) = − cs(g_{1}, g_{2})  (4)
cs(g_{1}, g_{3}) = cs(g_{1}, g_{2}) + cs(g_{2}, g_{3})  (5)

Let ξ denote the Euler vector field on T M which assigns to a point x ∈ T M the vertical vector −x ∈ T_{x}T M.

A Riemannian metric g determines the Levi–Civita connection in the bundle π : T M → M. There is a canonic vol(g) ∈ Ω^{n}(T M; π^{∗}O_{M}), which vanishes when contracted with horizontal vectors and which assigns to an n-tuple of vertical vectors their volume times their orientation. The global angular form, see for instance [BT82], is the differential form

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

This form was also considered by Mathai and Quillen [MQ86], and was referred to as the Mathai–Quillen form in [BZ92].

Note that Ψ(g) is the pull back of a form on (T M \ M)/ℝ_{+}. Moreover, we have the equalities

dΨ(g) = π^{∗}E(g).  (6)
Ψ(g_{2}) − Ψ(g_{1}) = π^{∗} cs(g_{1}, g_{2}) mod π^{∗}dΩ^{n−2}(M; O_{M})  (7)

Suppose X is a vector field with isolated zero x, and let B_{ϵ}(x) denote the ball of radius ϵ centered at x, with respect to some chart. Then

lim_{ϵ→0} ∫_{∂(M\B_{ϵ}(x))} X^{∗}Ψ(g) = IND_{X}(x),  (8)

where IND_{X}(x) denotes the Hopf index of X at x. For M = ℝ^{n} equipped with the standard Riemannian metric g_{0} we have

Ψ(g_{0}) = \frac{Γ(n/2)}{(2π)^{n/2}} ∑_{i=1}^{n}(−1)^{i} \frac{ξ_{i}}{(∑ ξ_{i}^{2})^{n/2}} dξ_{1} ∧ ⋯ ∧ d̂ξ_{i} ∧ ⋯ ∧ dξ_{n}.

in standard coordinates x_{1}, …, x_{n}, ξ_{1}, …, ξ_{n} on T M.

The same definition provides a global angular form on every real vector bundle equipped with a linear connection and a parallel Hermitian (fiber) metric.

2.2. Euler and Chern–Simons class for vector fields. Let M be a closed manifold and suppose X is a vector field with isolated set of zeros X ⊆ M. For every x ∈ X we have a Hopf index IND_{X}(x) of X at x. Define a singular^{6} zero chain

E(X) := ∑_{x∈X} IND_{X}(x)x ∈ C_{0}(M; ℤ).

Given two vector fields X_{1} and X_{2} we are going to define a singular one chain, well defined up to boundaries,

cs(X_{1}, X_{2}) ∈ C_{1}(M; ℤ)/∂C_{2}(M; ℤ)  (9)

with the properties:

∂ cs(X_{1}, X_{2}) = E(X_{2}) − E(X_{1})  (10)
cs(X_{2}, X_{1}) = − cs(X_{1}, X_{2})  (11)
cs(X_{1}, X_{3}) = cs(X_{1}, X_{2}) + cs(X_{2}, X_{3})  (12)

This should be compared with the equalities (3)–(5) in section 2.1.

The construction of (9) is accomplished by first reducing to vector fields with non-degenerate zeros with the help of a small perturbation, and then taking the zero set of a generic homotopy connecting X_{1} with X_{2}.

More precisely, suppose we have two vector fields X′_{1} and X′_{2} with non-degenerate zeros. Consider the vector bundle p^{∗}T M → I×M, where I := [1, 2] and p : I×M → M denotes the natural projection. Choose a section 𝕏 of p^{∗}T M which is transversal to the zero section and which restricts to X′_{i} on {i} × M, i = 1, 2. The zero set of 𝕏 is a canonically oriented one dimensional submanifold with boundary. Its fundamental class, when pushed forward via p, gives rise to c(𝕏) ∈ C_{1}(M; ℤ)/∂C_{2}(M; ℤ) satisfying (10).

6Here, and throughout the paper, singular chain stands for smooth singular chain.

Suppose 𝕏_{1} and 𝕏_{2} are two non-degenerate homotopies from X′_{1} to X′_{2}. Then c(𝕏_{1}) = c(𝕏_{2}) ∈ C_{1}(M; ℤ)/∂C_{2}(M; ℤ). Indeed, consider the vector bundle q^{∗}T M → I × I × M, where q : I × I × M → M denotes the natural projection. Choose a section of q^{∗}T M which is transversal to the zero section, restricts to 𝕏_{i} on {i}×I ×M, i = 1, 2, and which restricts to X′_{i} on {s} × {i} × M for all s ∈ I and i = 1, 2. The zero set of such a section then gives rise to σ satisfying c(𝕏_{2}) − c(𝕏_{1}) = ∂σ. So we get a well defined cs(X′_{1}, X′_{2}) := c(𝕏) ∈ C_{1}(M; ℤ)/∂C_{2}(M; ℤ) satisfying (10), provided X′_{1} and X′_{2} have non-degenerate zero set.

Now suppose X is a vector field with isolated zeros. For every zero x ∈ X we choose a contractible open neighborhood B_{x} of x, assuming all B_{x} are disjoint. Set B := ⋃_{x∈X} B_{x}. Choose a vector field with non-degenerate zeros X′ that coincides with X on M \ B. Let X′ denote its zero set. For every x ∈ X we have IND_{X}(x) = ∑_{y∈X′∩B_{x}} IND_{X′}(y). So we can choose a one chain c(X, X′) in B which satisfies ∂c(X, X′) = E(X′) − E(X). Since H_{1}(B; ℤ) vanishes, the one chain c(X, X′) is well defined up to a boundary.

Given two vector fields X_{1} and X_{2} with isolated zeros we choose B, X′_{1}, X′_{2}, c(X_{1}, X′_{1}) and c(X_{2}, X′_{2}) as above and set

cs(X_{1}, X_{2}) := c(X_{1}, X′_{1}) + cs(X′_{1}, X′_{2}) − c(X_{2}, X′_{2}).

Note that for two vector fields which coincide on M \ B and have non-degenerate zeros only, one can construct a non-degenerate homotopy between them which is constant on M \ B. Together with H_{1}(B; ℤ) = 0 this immediately implies that cs(X_{1}, X_{2}) ∈ C_{1}(M; ℤ)/∂C_{2}(M; ℤ) is independent of these choices. Clearly (10)– (12) hold too.

2.3. Kamber–Tondeur one form. Let E be a real or complex vector bundle over M. For a connection ∇ and a Hermitian structure μ on E we define a real valued one form ω(∇, μ) := −\frac{1}{2} tr_{μ}(∇μ) ∈ Ω^{1}(M; ℝ). More explicitly, for a tangent vector X ∈ T M

ω(∇, μ)(X) := −\frac{1}{2} tr_{μ}(∇_{X}μ).

This convention differs from the one in [BZ92] by a factor −\frac{1}{2}. Note that if ∇ is flat then ω(∇, μ) will be a closed one form. This and the following formulas are easily seen by noticing that ω(∇^{det E}, μ^{det E}) = ω(∇, μ) where ∇^{det E} and μ^{det E} denote the induced connection and Hermitian structure on the determinant line det E := Λ^{rank E}E.

Recall that the space of connections on E is an affine space over Ω^{1}(M; E^{∗} ⊗ E). For two connections ∇^{1} and ∇^{2} we find

ω(∇^{2}, μ) − ω(∇^{1}, μ) = ℜ(tr(∇^{2} − ∇^{1}))  (13)

where the trace is with respect to the last two variables.

If μ_{1} and μ_{2} are two Hermitian structures then there exists a unique section A ∈ C^{∞}(E^{∗} ⊗ E) such that μ_{2}(e, f) = μ_{1}(Ae, f). We set V (μ_{1}, μ_{2}) := | det A| which is a positive real valued function on M. Its value at x ∈ M is the volume, with respect to (μ_{2})_{x}, of a parallelepiped obtained from an orthonormal base, with respect to (μ_{1})_{x}. We have:

ω(∇, μ_{2}) − ω(∇, μ_{1}) = −\frac{1}{2}d log V (μ_{1}, μ_{2})  (14)

Suppose ∇ is flat, and let U ⊆ M be a contractible open set. Then we find a parallel Hermitian structure μ̃ over U, and can use (14) to obtain a local potential of the closed one form ω(∇, μ)|_{U} = d(−\frac{1}{2} log V (μ̃, μ)).

Remark 1. Suppose ∇ is flat, and let x_{0} be a base point in M. Parallel transport defines a right action of π_{1}(M, x_{0}) on E_{x_{0}}, the fiber over x_{0}. Composing with the inversion we obtain a representation ρ : π_{1}(M, x_{0}) → GL(E_{x_{0}}). Let [det ∘ρ] : H_{1}(M; ℤ) → ℂ^{∗} denote the homomorphism induced by the homomorphism det ∘ρ : π_{1}(M, x_{0}) → ℂ^{∗}. Let log |[det ∘ρ]| ∈ H^{1}(M; ℝ) denote the cohomology class corresponding to the homomorphism log |[det ∘ρ]| : H_{1}(M; ℤ) → ℝ. This class is called the Kamber–Tondeur class and is represented by the Kamber– Tondeur form ω(∇, μ) for every Hermitian metric μ. Put another way, the cohomology class of ω(∇, μ) captures the absolute value of the holonomy in (det E, ∇^{det E}).

The following observation will be useful in the proof of Corollary 1.

Proposition 1. Let E be a complex vector bundle over M equipped with a Hermitian structure μ. Then there exist mappings ∇ ↦ ω̃(∇, μ) which are affine over the map tr : Ω^{1}(M; E^{∗} ⊗ E) → Ω^{1}(M; ℂ) such that ℜ(ω̃(∇, μ)) = ω(∇, μ), and such that ω̃(∇, μ) is closed whenever ∇ is flat. Particularly, the assignment ∇ ↦ ω̃(∇, μ) is holomorphic.

Proof. Choose a reference connection ∇′ on E, and define

ω̃(∇, μ) := tr(∇ − ∇′) + ω(∇′, μ).

This is certainly affine over the trace, and its real part coincides with ω(∇, μ) in view of (13). If E admits flat connections we can choose ∇′ to be flat. Then ω̃(∇, μ) will be closed provided ∇ was flat. □

2.4. Regularization of an integral. Let M be a closed manifold of dimension n. Suppose α ∈ Ω^{n−1}(M \ X; O_{M}) is a form with isolated set of singularities X ⊆ M, and suppose dα ∈ Ω^{n}(M; O_{M}) is globally smooth. Here O_{M} denotes the orientation bundle of M, a flat real line bundle over M. For x ∈ X define a real number

IND_{α}(x) := lim_{ϵ→0} ∫_{∂(M\B_{ϵ}(x))} α = ∫_{∂(M\B_{ϵ′} (x))} α + ∫_{B_{ϵ′} (x)} dα

where B_{ϵ}(x) denotes the ball of radius ϵ in a chart centered at x. Note that for a Riemannian metric g, and a vector field X with isolated zero x, the form X^{∗}Ψ(g) has an isolated singularity at x, dX^{∗}Ψ(g) = E(g) is globally smooth, see (6), and in view of (8) we have

IND_{X^{∗}Ψ(g)}(x) = IND_{X}(x).  (15)

Let α be as above, and suppose ω ∈ Ω^{1}(M) is a closed real or complex valued one form. Choose a function f such that f vanishes on X and such that ω′ := ω−df vanishes locally around X. Define:

S(ω, α) := S(ω, α; f) := ∫_{M\X} ω′ ∧ α − ∫_{M} fdα

The notation is justified by the following

Lemma 1. The quantity S(ω, α; f) does not depend on the choice of f. It is linear in ω, and α. It satisfies:

(i) S(dh, α) = ∑_{x∈X} h(x) IND_{α}(x) − ∫_{M} hdα for every smooth function h.

(ii) S(ω, β) = ∫_{M} ω ∧ β for all β ∈ Ω^{n−1}(M; O_{M}). Particularly S(ω, dγ) = 0

for all γ ∈ Ω^{n−2}(M; O_{M}).

Proof. Let us first show that S(ω, α; f) does not depend on f. So suppose f_{i} are functions which vanish on X and are such that ω′_{i} := ω −df_{i} vanishes locally around X. Then ω′_{2} − ω′_{1} = −d(f_{2} − f_{1}) and f_{2} − f_{1} vanishes locally around X. Using Stokes’ theorem for forms with compact support on M \ X we get

S(ω, α; f_{2}) − S(ω, α; f_{1}) = − ∫_{M\X} d(f_{2} − f_{1}) ∧ α − ∫_{M}(f_{2} − f_{1})dα = 0.

Now let us turn to (i). Choose a function f so that dh − df vanishes locally around X and so that f vanishes on X. For every x ∈ X choose an embedded ball B_{ϵ}(x) around x such that dh − df vanishes on B_{ϵ}(x). Then h − f is constant equal to h(x) on B_{ϵ}(x), and Stokes’ theorem implies

S(dh, α) = ∫_{M\X} (dh − df) ∧ α − ∫_{M} fdα = ∑_{x∈X} h(x) ∫_{∂(M\B_{ϵ}(x))} α − ∫_{M} hdα

With ϵ → 0 the statement follows. The remaining assertions are obvious. □

2.5. The invariant R. Let M be a closed manifold of dimension n. For a closed one form ω, and two Riemannian metrics g_{1} and g_{2} we set

R(ω, g_{1}, g_{2}) := ∫_{M} ω ∧ cs(g_{1}, g_{2}).  (16)

Note that even though cs(g_{1}, g_{2}), see section 2.1, is only defined up to dΩ^{n−2}(M; O_{M}) the quantity (16) is unambiguously defined according to Stokes’ theorem.

For a closed one form ω, and two vector fields X_{1} and X_{2} with isolated zeros we set

R(ω, X_{1}, X_{2}) := ∫_{cs(X_{1},X_{2})} ω.  (17)

Again, even though cs(X_{1}, X_{2}), see section 2.2, is only defined up to a boundary the quantity (17) is well defined in view of Stokes’ theorem.

For a Riemannian metric g and a vector field X with isolated zeros X ⊆ M we have X^{∗}Ψ(g) ∈ Ω^{n−1}(M \ X; O_{M}) and dX^{∗}Ψ(g) = E(g), see section 2.1, especially (6). So we may define, see section 2.4,

R(ω, X, g) := S(ω, X^{∗}Ψ(g))

for every closed one form ω. The invariant R has first appear in Bismut–Zhang work [BZ92].

Proposition 2. Let ω be a closed one form, h a smooth function, g and g_{i} Riemannian metrics, and let X and X_{i} be vector fields with isolated zeros, on M. Then:

(i) R(ω, g_{1}, g_{2}), R(ω, X_{1}, X_{2}) and R(ω, X, g) are linear in ω.

(ii) R(ω, g_{2}, g_{1}) = −R(ω, g_{1}, g_{2}). (iii) R(ω, X_{2}, X_{1}) = −R(ω, X_{1}, X_{2}). (iv) R(dh, X, g) = h(E(X)) − ∫_{M} hE(g).

(v) R(dh, g_{1}, g_{2}) = ∫_{M} hE(g_{1}) − ∫_{M} hE(g_{2}). (vi) R(dh, X_{1}, X_{2}) = h(E(X_{2})) − h(E(X_{1})). (vii) R(ω, X_{1}, X_{3}) = R(ω, X_{1}, X_{2}) + R(ω, X_{2}, X_{3}). (viii) R(ω, g_{1}, g_{3}) = R(ω, g_{1}, g_{2}) + R(ω, g_{2}, g_{3}).

(ix) R(ω, X, g_{2}) − R(ω, X, g_{1}) = R(ω, g_{1}, g_{2}).
(x) R(ω, X_{2}, g) − R(ω, X_{1}, g) = R(ω, X_{1}, X_{2}).

Here we wrote h(E(X)) = ∑_{x∈X} h(x) IND_{X}(x) where X denotes the zero set of X.

Proof. Statements (i)–(iii) and (v)–(viii) follow immediately from Stokes’ theorem, Lemma 1, equations (3)–(5) and equations (10)–(12).

Assertion (iv) follows from Lemma 1(i), equation (15), and from dX^{∗}Ψ(g) = E(g), see (6).

Statement (ix) follows from Lemma 1(ii) and X^{∗}Ψ(g_{2}) − X^{∗}Ψ(g_{1}) = cs(g_{1}, g_{2}), see (7). Statements (iv) and (ix) can be also found in [BZ92].

Remains to prove (x). In view of (iv) and (vi) we may assume that ω vanishes in a neighborhood of the zero set X_{1}∪X_{2}. It is then clear that we may also assume that X_{1} and X_{2} have non-degenerate zeros only. Choose a non-degenerate homotopy 𝕏 from X_{1} to X_{2}. Perturbing the homotopy slightly, cutting it into several pieces and using (vii) we may assume that the zero set 𝕏^{−1}(0) ⊆ I × M is actually contained in a closed simply connected I × V. Again, we may assume that ω vanishes on V. Then obviously R(ω, X_{1}, X_{2}) = 0. Moreover, in this situation Stokes’ theorem implies

R(ω, X_{2}, g) − R(ω, X_{1}, g) = ∫_{M\V} ω ∧ X_{2}^{∗}Ψ(g) − ∫_{M\V} ω ∧ X_{1}^{∗}Ψ(g)
= ∫_{I×(M\V)} d(p^{∗}ω ∧ 𝕏^{∗}p̃^{∗}Ψ(g))
= − ∫_{I×(M\V)} p^{∗}(ω ∧ E(g)) = 0

where p : I ×M → M denotes the natural projection, and p̃: p^{∗}T M → T M denotes the natural vector bundle homomorphism over p. For the last calculation note that d𝕏^{∗}p̃^{∗}Ψ(g) = p^{∗}E(g) in view of (6), and that ω ∧ E(g) = 0 because of dimensional reasons. □

Remark 2. A similar definition of R(X, g, ω) works for any vector field X with arbitrary singularity set X := {x ∈ M | X(x) = 0} provided ω is exact when restricted to a sufficiently small neighborhood of X. This situation might lead to interesting invariants for holomorphic vector fields on a Kähler manifolds or even more general, for symplectic vector fields on a symplectic manifold equipped with an almost complex structure which tames the symplectic structure.

2.6. Extension of Chern–Simons theory. The material in this section will not be used in the rest of the paper, but puts the observations in Proposition 2 in a more natural perspective.

Let M be a closed manifold of dimension n. We equip Ω^{k}(M; ℝ) with the C^{∞}– topology. The continuous linear functionals on Ω^{k}(M; ℝ) are called k currents. We denote the space of all k currents by D_{k}(M). Let δ : D_{k}(M) → D_{k−1}(M) be given by

(δφ)(α) := φ(dα).

Clearly δ^{2} = 0. By the deRham theorem for currents the chain complex (D_{∗}(M), δ) computes the homology of M with real coefficients.

We have a morphism of chain complexes

C_{∗}(M; ℝ) → D_{∗}(M), σ ↦ σ̂, σ̂(α) := ∫_{σ} α.

Here C_{∗}(M; ℝ) denotes the space of singular chains with real coefficients. Moreover, we have a morphism of chain complexes

Ω^{n−∗}(M; O_{M}) → D_{∗}(M), β ↦ β̂, β̂(α) := (−1)^{\frac{1}{2} |α|(|α|+1)} ∫_{M} α ∧ β.

Here |α| denotes the degree of α.

Note that any zero current represents a degree zero homology class, and H_{0}(M; ℝ) identifies canonically to ⨁_{α} ℝ_{α} where ℝ_{α} is a copy of ℝ corresponding to a connected component of M = ⨆_{α} M_{α}. One denotes by \overline{χ}(M) the vector \{χ(M_{α})\} ∈ ⨁_{α} ℝ_{α}, and by E_{0} the affine subspace of D_{0}(M) consisting of zero currents which represent \overline{χ}(M). Elements in E_{0} are called Euler currents.

Every vector field X with isolated set of zeros X_{τ} gives rise to a zero chain E(X), see section 2.2. Via the first morphism we get a zero current Ê(X). More explicitly, Ê(X)(h) = ∑_{x∈X} IND_{x}(X)h(x) for all functions h ∈ Ω^{0}(M; ℝ). By Hopf’s theorem this is an Euler current.

A Riemannian metric g has an Euler form E(g) ∈ Ω^{n}(M; O_{M}) see section 2.1. Via the second morphism we get a zero current Ê(g) which, by Gauss–Bonnet, is an Euler current. More explicitly, Ê(g)(h) = ∫_{M} hE(g) for all functions h ∈ Ω^{0}(M; ℝ).

Let Z^{k}(M; ℝ) ⊆ Ω^{k}(M; ℝ) denote the space of closed k forms on M equipped with the C^{∞}–topology. The continuous linear functionals on Z^{k}(M; ℝ) are referred to as k currents rel. boundary and identify to D_{k}(M)/δ(D_{k+1}(M)). The two chain morphisms provide mappings

C_{k}(M; ℝ)/∂(C_{k+1}(M; ℝ)) → D_{k}(M)/δ(D_{k+1}(M))  (18)

and

Ω^{n−k}(M; O_{M})/d(Ω^{n−k−1}(M; O_{M})) → D_{k}(M)/δ(D_{k+1}(M))  (19)

For two vector fields, X_{1} and X_{2}, with isolated zeros we have constructed cs(X_{1}, X_{2}) ∈ C_{1}(M; ℤ)/∂(C_{2}(M; ℤ)), cf. section 2.2, which gives rise to cs(X_{1}, X_{2}) ∈ C_{1}(M; ℝ)/∂(C_{2}(M; ℝ)) and via (18) we get a one current rel. boundary which we will denote by ĉs(X_{1}, X_{2}). More precisely, ĉs(X_{1}, X_{2})(ω) = ∫_{cs(X_{1},X_{2})} ω = R(ω, X_{1}, X_{2}) for all closed one forms ω ∈ Z^{1}(M; ℝ).

For two Riemannian metrics, g_{1} and g_{2}, we have the Chern–Simons form cs(g_{1}, g_{2}) ∈ Ω^{n−1}(M; O_{M})/d(Ω^{n−2}(M; O_{M})), see section 2.1. Via (19) we get a one current rel. boundary which we will denote by ĉs(g_{1}, g_{2}). More precisely, ĉs(g_{1}, g_{2})(ω) = − ∫_{M} ω ∧ cs(g_{1}, g_{2}) = −R(ω, g_{1}, g_{2}) for all closed one forms ω ∈ Z^{1}(M; ℝ).

Suppose X is a vector field with isolated zeros and g is a Riemannian metric. We define a one current rel. boundary by ĉs(X, g)(ω) := −R(ω, X, g) for all closed one forms ω ∈ Z^{1}(M; ℝ), see section 2.5. Moreover set ĉs(g, X) := −ĉs(X, g).

Proposition 2 can now be reformulated as

Proposition 3. Let any of the symbols x, y, z denote either a Riemannian metric or a vector field with isolated zeros. Then one has:

(i) δĉs(x, y) = Ê(y) − Ê(x).

(ii) ĉs(y, x) = −ĉs(x, y)
(iii) ĉs(x, z) = ĉs(x, y) + ĉs(y, z)

2.7. Smooth triangulations. Smooth triangulations provide a remarkable source of vector fields with isolated zeros. To any smooth triangulation τ of a smooth manifold M one can associate a smooth vector field X_{τ} called Euler vector field, with the following properties:

P1: The zeros of X_{τ} are all non-degenerate and are exactly the barycenters x_{σ}

of the simplexes σ of τ. P2: The flow of −X_{τ} is defined for all t ∈ ℝ.^{7} P3: For each zero x_{σ} the unstable set with respect to −X_{τ} is exactly the open simplex σ, consequently the zeros are hyperbolic. The Morse index of −X_{τ} at x_{σ} equals dim(σ) and the (Hopf) index of X_{τ} at x_{σ} equals (−1)^{dim(σ)}. P4: The piecewise differentiable function f_{τ} : M → ℝ defined by f_{τ} (x_{σ}) = dim(σ), extended to M by linearity on the barycentric subdivision of τ, is a Lyapunov function for −X_{τ}, i.e. strictly decreasing on non-constant trajectories of −X_{τ}.

The space of vector fields satisfying P1–P4 is a contractible space. The convex combination provides a contraction.

To construct such a vector field we begin with a standard simplex Δ_{n} of vectors (t_{0}, …, t_{n}) ∈ ℝ^{n+1} satisfying 0 ≤ t_{i} ≤ 1 and ∑ t_{i} = 1.

(i) Let E_{n} denote the Euler vector field of the corresponding affine space (∑ t_{i} = 1) centered at the barycenter O (of coordinates (1/(n+1), …, 1/(n+ 1)) and restricted to Δ_{n}.

(ii) Let e : Δ_{n} → [0, 1] denote the function which is 1 on the barycenter O and

zero on all vertices. (iii) Let r : Δ_{n} \ {O} → ∂Δ_{n} denote the radial retraction to the boundary. Set X′_{n} := e · E_{n}, which is a vector field on Δ_{n}. By induction we will construct a canonical vector field X_{n} on Δ_{n} which at any

point x ∈ Δ_{n} is tangent to the open face the point belongs to and vanishes only at the barycenter of each face. We proceed as follows:

Suppose we have constructed such canonical vector fields on all Δ(k), k ≤ n − 1. Using the canonical vector fields X_{n−1} we define the vector field X_{n} on the boundary ∂Δ_{n} and extend it to the vector field X′′_{n} by taking at each point x ∈ Δ_{n} the vector parallel to X_{n}(r(x)) multiplied by the function (1 − e) and at O the vector zero. Clearly such vector field vanishes on the radii \overline{OP} (P the barycenter of any face). We finally put

X_{n} := X′_{n} + X′′_{n}.

The vector field X_{n} is continuous and piecewise differential (actually Lipschitz) and has a well defined continuous flow.

Putting together the vector fields X_{n} on all simplexes (cells) we provide a piecewise differential (and Lipschitz) vector field X on any simplicial (cellular) complex or polyhedron and in particular on any smoothly triangulated manifold. The vector field X has a flow and f_{τ} is a Lyapunov function for −X. The vector field X is not necessary smooth but by a small (Lipschitz) perturbation we can approximate it by a smooth vector field which satisfies P1–P4. Any of the resulting vector fields is referred to as the Euler vector field of a smooth triangulation τ and denoted by X_{τ}. It was pointed out to us that the vector field X_{τ} has first appeared in [S99].

7This is always the case if M is closed.

Note that the above construction extends to smooth cell structures. Recall that a smooth cell structure on a smooth manifold M is a CW-complex structure provided by cells whose characteristic map χ : D^{k} → M are smooth embeddings (one to one and of maximal rank) from the compact manifolds with corners (homeomorphic to the unit disc in the Euclidean space). One denotes by M(r) the r–skeleton and for any cell one calls barycenter, the image by the characteristic map of the center of D^{k}. By an entirely similar procedure one can construct piecewise differential vector fields X on M which are tangent to M(r) and have the barycenters as the only zeros. Moreover one can construct piecewise differential functions which are Lyapunov for X and when composed with the characteristic maps are Morse functions on the interior of any cell with only one critical point, the center of the cell. One can show that the space of all these vector fields with the Lipschitz topology is contractible.

3. Euler and co-Euler structures

3.1. Euler structures. Euler structures have been introduced by Turaev [T90] for manifolds with vanishing Euler Poincaré characteristic. In the presentation below we remove the vanishing hypothesis at the expense of a base point.

Let (M, x_{0}) be a base pointed closed connected manifold of dimension n. Let X be a vector field and let X denote its zero set. Suppose the zeros of X are isolated and recall the singular zero chain E(X) = ∑_{x∈X} IND_{X}(x)x from section 2.2. An Euler chain for X is a singular one chain c ∈ C_{1}(M; ℤ) so that

∂c = E(X) − χ(M)x_{0}.

Since ∑_{x∈X} IND_{X}(x) = χ(M) every vector field with isolated zeros admits Euler chains.

Consider pairs (X, c) where X is a vector field with isolated zeros, and c is an Euler chain for X. We call two such pairs (X_{1}, c_{1}) and (X_{2}, c_{2}) equivalent if

c_{2} = c_{1} + cs(X_{1}, X_{2}) ∈ C_{1}(M; ℤ)/∂(C_{2}(M; ℤ)).

For the definition of cs(X_{1}, X_{2}) see section 2.2. We will write 𝔈𝔲𝔩_{x_{0}}(M; ℤ) for the set of equivalence classes as above and [X, c] ∈ 𝔈𝔲𝔩_{x_{0}}(M; ℤ) for the element represented by the pair (X, c). Elements of 𝔈𝔲𝔩_{x_{0}}(M; ℤ) are called Euler structures of M based at x_{0}. There is an obvious H_{1}(M; ℤ) action on 𝔈𝔲𝔩_{x_{0}}(M; ℤ) defined by

[X, c] + [σ] := [X, c + σ],

for [σ] ∈ H_{1}(M; ℤ) and [X, c] ∈ 𝔈𝔲𝔩_{x_{0}}(M; ℤ). Obviously this action is free and transitive. In this sense 𝔈𝔲𝔩_{x_{0}}(M; ℤ) is an affine version of H_{1}(M; ℤ).

Considering Euler chains with real coefficients one obtains, in exactly the same way, an affine version of H_{1}(M; ℝ) which we will denote by 𝔈𝔲𝔩_{x_{0}}(M; ℝ). Moreover, there is a natural map 𝔈𝔲𝔩_{x_{0}}(M; ℤ) → 𝔈𝔲𝔩_{x_{0}}(M; ℝ) which is affine over the homomorphism H_{1}(M; ℤ) → H_{1}(M; ℝ).

Remark 3. There is another way of understanding the H_{1}(M; ℤ) action on 𝔈𝔲𝔩_{x_{0}}(M; ℤ). Suppose n > 2 and represent [σ] ∈ H_{1}(M; ℤ) by a simple closed curve σ. Choose a tubular neighborhood N of σ(S^{1}) considered as vector bundle N → S^{1}. Choose a fiber metric and a linear connection on N. Choose a representative of [X, c] ∈ 𝔈𝔲𝔩_{x_{0}}(M; ℤ) such that X|_{N} = \frac{∂}{∂θ}, the horizontal lift of the canonic vector field on S^{1}. Choose a function λ : [0, ∞) → [−1, 1], which satisfies λ(r) = −1 for r ≤ \frac{1}{3} and λ(r) = 1 for r ≥ \frac{2}{3}. Finally choose a function μ : [0, ∞) → ℝ satisfying μ(r) = r for r ≤ \frac{1}{3}, μ(r) = 0 for r ≥ \frac{2}{3} and μ(r) > 0 for all r ∈ (\frac{1}{3}, \frac{2}{3}). Now construct a new smooth vector field X̃ on M by setting

X̃ := \begin{cases}X & on M \ N \\ λ(r)\frac{∂}{∂θ} + μ(r)\frac{∂}{∂r} & on N, \end{cases}

where r : N → [0, ∞) denotes the radius function determined by the fiber metric on N and −r \frac{∂}{∂r} is the Euler vector field of N. This construction is known as Reeb surgery, see e.g. [N03]. If the zeros of X are all non-degenerate the homotopy X_{t} := (1 − t)X + tX̃ is a non-degenerate homotopy from X_{0} = X to X_{1} = X̃ from which one easily deduces that [X̃, c] = [X, c] + [σ]. Particularly all the choices that entered the Reeb surgery do not affect the out-coming Euler structure [X̃, c].

Let us consider a change of base point. Let x_{0}, x_{1} ∈ M and choose a path σ from x_{0} to x_{1}. Define

𝔈𝔲𝔩_{x_{0}}(M; ℤ) → 𝔈𝔲𝔩_{x_{1}}(M; ℤ), [X, c] ↦ [X, c − χ(M)σ].  (20)

This is an H_{1}(M; ℤ) equivariant bijection but depends on the homology class of σ. However, if χ(M) = 0, then 𝔈𝔲𝔩_{x_{0}}(M; ℤ) does not depend on the base point, cf. [T90].

Let 𝔛(M, x_{0}) denote the space of vector fields which vanish at x_{0} and are nonzero elsewhere. We equip this space with the C^{∞}–topology. Let π_{0}(𝔛(M, x_{0})) denote the set of homotopy classes of such vector fields. If X ∈ 𝔛(M, x_{0}) we will write [X] for the corresponding class in π_{0}(𝔛(M, x_{0})). The following proposition is due to Turaev in the case χ(M) = 0.

Proposition 4. Suppose n > 2. Then there exists a natural bijection

π_{0}(𝔛(M, x_{0})) = 𝔈𝔲𝔩_{x_{0}}(M; ℤ),  [X] ↦ [X, 0].  (21)

Proof. Clearly (21) is well defined. Let us prove that it is onto. So let [X, c] represent an Euler structure. Choose an embedded disk D ⊆ M centered at x_{0} which contains all zeros of X and the Euler chain c. For this we may have to change c, but without changing the Euler structure [X, c]. Choose a vector field X′ which equals X on M \ D and vanishes just at x_{0}. Since H_{1}(D; ℤ) = 0 we clearly have [X′, 0] = [X, c] ∈ 𝔈𝔲𝔩_{x_{0}}(M; ℤ) and thus (21) is onto.

Let us prove injectivity of (21). Let X_{1}, X_{2} ∈ 𝔛(M, x_{0}) and suppose cs(X_{1}, X_{2}) = 0 ∈ H_{1}(M; ℤ). Let D ⊆ M denote an embedded open disk centered at x_{0}. Consider the vector bundle p^{∗}T M → I × M and consider the two vector fields as a nowhere vanishing section of p^{∗}T M defined over the set ∂I ×Ṁ, where Ṁ := M \D. We would like to extend it to a nowhere vanishing section over I × Ṁ. The first obstruction we meet is an element in

H^{n}(I × Ṁ, ∂I × Ṁ; {π_{n−1}}) = H_{1}(I × Ṁ, I × ∂D; ℤ)
= H_{1}(M, D̄; ℤ)
= H_{1}(M; ℤ)

which corresponds to cs(X_{1}, X_{2}) = 0. Here {π_{n−1}} denotes the system of local coefficients determined by the sphere bundle of p^{∗}T M with π_{n−1} = π_{n−1}(S^{n−1}). Since this obstruction vanishes by hypothesis the next obstruction is defined and is an element in:

H^{n+1}(I × Ṁ, ∂I × Ṁ; {π_{n}}) = H_{0}(I × Ṁ, I × ∂D; π_{n}(S^{n−1}))
= H_{0}(M, D̄; π_{n}(S^{n−1}))
= 0

So obstruction theory, see e.g. [W78], tells us that we find a nowhere vanishing section of p^{∗}T M defined over I ×Ṁ, which restricts to X_{i} on {i}×Ṁ, i = 1, 2. Such a section can easily be extended to a globally defined section of p^{∗}T M → I × M, which restricts to X_{i} on {i} × M, i = 1, 2, and whose zero set is precisely I × {x_{0}}.^{8} Such a section can be considered as homotopy from X_{1} to X_{2} showing [X_{1}] = [X_{2}]. Hence (21) is injective. □

Remark 4. If n > 2, Reeb surgery defines an H_{1}(M; ℤ) action on π_{0}(𝔛(M, x_{0})) which via (21) corresponds to the H_{1}(M; ℤ) action on 𝔈𝔲𝔩_{x_{0}}(M; ℤ), cf. Remark 3.

Let 𝔛_{0}(M) denote the space of nowhere vanishing vector fields on M equipped with the C^{∞}–topology. Let π_{0}(𝔛_{0}(M)) denote the set of its connected components.

Proposition 5. Suppose χ(M) = 0 and n > 2. Then we have a surjection:

π_{0}(𝔛_{0}(M)) → 𝔈𝔲𝔩_{x_{0}}(M; ℤ),  [X] ↦ [X, 0].  (22)

Proof. The assignment (22) is certainly well defined. Let us prove surjectivity. Let [X, c] be an Euler structure. Choose an embedded disk D ⊆ M which contains all zeros of X and its Euler chain c, cf. the proof of Proposition 4. Since χ(M) = 0 the degree of X : ∂D → T D \ D vanishes. Modifying X only on D we get a nowhere vanishing X′ which coincides with X on M \ D. Certainly X′ has an Euler chain c′ which is also contained in D and satisfies [X, c] = [X′, c′]. Since X′ has no zeros we get ∂c′ = 0 and since H_{1}(D; ℤ) = 0 we arrive at [X, c] = [X′, c′] = [X′, 0] which proves that (22) is onto. □

3.2. Co-Euler structures. We will now describe another approach to Euler structures which is in some sense Poincaré dual to the one in section 3.1. We still consider a closed connected base pointed manifold (M, x_{0}) of dimension n. Consider pairs (g, α) where g is a Riemannian metric on M and α ∈ Ω^{n−1}(M \ x_{0}; O_{M}) with dα = E(g) where E(g) ∈ Ω^{n}(M; O_{M}) denotes the Euler class of g, see section 2.1. We call two pairs (g_{1}, α_{1}) and (g_{2}, α_{2}) equivalent if

cs(g_{1}, g_{2}) = α_{2} − α_{1} ∈ Ω^{n−1}(M \ x_{0}; O_{M})/dΩ^{n−2}(M \ x_{0}; O_{M}).

We will write 𝔈𝔲𝔩_{x_{0}}^{∗}(M; ℝ) for the set of equivalence classes and [g, α] for the equivalence class represented by the pair (g, α). Elements of 𝔈𝔲𝔩_{x_{0}}^{∗}(M; ℝ) are called co-Euler structures based at x_{0}. There is a natural H^{n−1}(M; O_{M}) action on 𝔈𝔲𝔩_{x_{0}}^{∗}(M; ℝ) given by

[g, α] + [β] := [g, α − β]

for [β] ∈ H^{n−1}(M; O_{M}). Since H^{n−1}(M; O_{M}) = H^{n−1}(M \ x_{0}; O_{M}) this action is obviously free and transitive. In this sense 𝔈𝔲𝔩_{x_{0}}^{∗}(M; ℝ) is an affine version of H^{n−1}(M; O_{M}).

8For example trivialize the bundle p^{∗}T M over I × D̄ and extend the section on ∂(I × D̄) to I × D̄ by scaling radially centered at ( \frac{1}{2} , x_{0}) ∈ I × D̄.

Suppose [X, c] ∈ 𝔈𝔲𝔩_{x_{0}}(M; ℝ) and [g, α] ∈ 𝔈𝔲𝔩_{x_{0}}^{∗}(M; ℝ). In view of Lemma 1 and Proposition 2, the quantity

R(ω, X, g) − S(ω, α) − ∫_{c} ω  (23)

does only depend on the Euler structure [X, c] ∈ 𝔈𝔲𝔩_{x_{0}}(M; ℝ), the co-Euler structure [g, α] ∈ 𝔈𝔲𝔩_{x_{0}}^{∗}(M; ℝ), and the cohomology class [ω] ∈ H^{1}(M; ℝ). Thus (23) defines a coupling

𝕋 : 𝔈𝔲𝔩_{x_{0}}(M; ℝ) × 𝔈𝔲𝔩_{x_{0}}^{∗}(M; ℝ) → H_{1}(M; ℝ).

From the very definition we have

⟨[ω], 𝕋([X, c], [g, α])⟩ = ∫_{M} ω ∧ (X^{∗}Ψ(g) − α) − ∫_{c} ω,  (24)

where ω is any representative of [ω] which vanishes locally around the zeros of X and vanishes locally around the base point x_{0}. Moreover, we have

𝕋(𝔢 + σ, 𝔢^{∗} + β) = 𝕋(𝔢, 𝔢^{∗}) − σ + PD(β)  (25)

for all 𝔢 ∈ 𝔈𝔲𝔩_{x_{0}}(M; ℝ), 𝔢^{∗} ∈ 𝔈𝔲𝔩_{x_{0}}^{∗}(M; ℝ), σ ∈ H_{1}(M; ℝ) and β ∈ H^{n−1}(M; O_{M}). Here PD is the Poincaré duality isomorphism PD : H^{n−1}(M; O_{M}) → H_{1}(M; ℝ).

We have the following affine version of Poincaré duality:

Proposition 6. There is a natural isomorphism

P : 𝔈𝔲𝔩_{x_{0}}^{∗}(M; ℝ) → 𝔈𝔲𝔩_{x_{0}}(M; ℝ)

affine over the Poincaré duality isomorphism PD : H^{n−1}(M; O_{M}) → H_{1}(M; ℝ). More precisely, for β ∈ H^{n−1}(M; O_{M}) and 𝔢^{∗} ∈ 𝔈𝔲𝔩_{x_{0}}^{∗}(M; ℝ) we have P (𝔢^{∗} + β) = P (𝔢^{∗}) + PD(β). Moreover, 𝕋(𝔢, 𝔢^{∗}) = P (𝔢^{∗}) − 𝔢.

Proof. In view of (25) we may define P by 𝕋(P (𝔢^{∗}), 𝔢^{∗}) = 0. The equivariance property and the last equation follow at once. □

Let us collect the observations from the last two sections in the following

Theorem 1. Let (M, x_{0}) be a closed connected base pointed manifold.

(i) Let π_{0}(𝔛(M, x_{0})) denote the set of connected components of the space of vector fields which just vanish in x_{0}, equipped with the C^{∞}–topology. If dim M > 2 then we have a bijection:

π_{0}(𝔛(M, x_{0})) → 𝔈𝔲𝔩_{x_{0}}(M; ℤ), [X] ↦ [X, 0].

(ii) Let π_{0}(𝔛_{0}(M)) denote the set of connected components of the space of nowhere vanishing vector fields, equipped with the C^{∞}–topology. If χ(M) = 0 and dim M > 2 we have a surjection:

π_{0}(𝔛_{0}(M)) → 𝔈𝔲𝔩_{x_{0}}(M; ℤ), [X] ↦ [X, 0].

(iii) There exists an isomorphism

P : 𝔈𝔲𝔩_{x_{0}}^{∗}(M; ℝ) → 𝔈𝔲𝔩_{x_{0}}(M; ℝ)

affine over the Poincaré duality isomorphism PD : H^{n−1}(M; O_{M}) → H_{1}(M; ℝ), i.e. P (𝔢^{∗} + β) = P (𝔢^{∗}) + PD(β), for all β ∈ H^{n−1}(M; O_{M}).

(iv) The assignment 𝕋(𝔢, 𝔢^{∗}) := P (𝔢^{∗}) − 𝔢
𝕋 : 𝔈𝔲𝔩_{x_{0}}(M; ℝ) × 𝔈𝔲𝔩_{x_{0}}^{∗}(M) → H_{1}(M; ℝ)

is a corrected version of the invariant R. More precisely, suppose 𝔢 = [X, c], 𝔢^{∗} = [g, α]. Then for every [ω] ∈ H^{1}(M; ℝ) we have

⟨[ω], 𝕋(𝔢, 𝔢^{∗})⟩ = ∫_{M} ω ∧ (X^{∗}Ψ(g) − α) − ∫_{c} ω

where ω ∈ Ω^{1}(M) is any representative of [ω] which vanishes locally around x_{0} and vanishes locally around the zeros of X.

3.3. Milnor–Turaev torsion. Let M be a closed connected manifold of dimension n. Fix a base point x_{0} ∈ M and let Γ = π_{1}(M, x_{0}) denote the fundamental group. Let 𝕂 be a field and let V be a finite dimensional vector vector space over 𝕂. Let ρ : Γ → GL(V) be a representation. Consider the graded 𝕂–vector space H^{∗}(M; ρ), its determinant line det H^{∗}(M; ρ), and the one dimensional 𝕂–vector space^{9}

Det_{x_{0}}(M; ρ) := det H^{∗}(M; ρ) ⊗ (det V)^{−χ(M)}.  (26)

Remark 5. If χ(M) = 0 then Det_{x_{0}}(M; ρ) = det H^{∗}(M; ρ). If H^{∗}(M; ρ) = 0 then Det_{x_{0}}(M; ρ) is canonically isomorphic to 𝕂.^{10}

Suppose we have a smooth triangulation τ of M. It gives rise to a cochain complex of 𝕂–vector spaces C_{τ}^{∗}(M; ρ) which computes the cohomology H^{∗}(M; ρ). By standard linear algebra we get a canonical isomorphism of 𝕂–vector spaces:

φ_{τ}^{ρ} : det C_{τ}^{∗}(M; ρ) → det H(C_{τ}^{∗}(M; ρ)) = det H^{∗}(M; ρ)  (27)

This isomorphism is described in details in [FT00, section 2, formula (2.2)] where a sign correction (−1)^{N(C)} of previously used definitions is added, cf. the appendix of this paper.

Recall the Euler vector field X_{τ} of τ from section 2.7. Its zero set X_{τ} coincides with the set of barycenters of τ. For a cell σ of τ we let x_{σ} ∈ X_{τ} denote its barycenter. We have IND_{X_{τ}} (x_{σ}) = (−1)^{dim σ} for every cell σ of τ. Suppose we have given an Euler structure 𝔢 ∈ 𝔈𝔲𝔩_{x_{0}}(M; ℤ). Choose a collection of paths π_{x}, from x_{0} to x ∈ X_{τ}, so that that with c := ∑_{x∈X_{τ}} IND_{X_{τ}} (x)π_{x} we have 𝔢 = [X_{τ} , c].^{11}

Let v be a non-zero element in det V.^{12} Using parallel transport along π_{x} we get a non-zero element in det C_{τ}^{∗}(M; ρ) with a sign ambiguity. If the set of barycenters X_{τ} was ordered we would get a well defined non-zero element in det C_{τ}^{∗}(M; ρ).

Suppose 𝔬 is a cohomology orientation of M, i.e. an orientation of det H^{∗}(M; ℝ). We say an ordering of the zeros is compatible with 𝔬 if the non-zero element in det C_{τ}^{∗}(M; ℝ) provided by this ordered base is compatible with the orientation 𝔬 via the canonic isomorphism

φ_{τ}^{ℝ} : det C_{τ}^{∗}(M; ℝ) → det H(C_{τ}^{∗}(M; ℝ)) = det H^{∗}(M; ℝ).

So given a collection of paths π_{x} with 𝔢 = [X_{τ} , c], an ordering of X_{τ} compatible with 𝔬, and a non-zero element v ∈ det V one obtains (using Milnor’s construction modified by Turaev) a non-zero element in det C_{τ}^{∗}(M; ρ) which corresponds to a non-zero element in det H^{∗}(M; ρ) via (27). We thus get a mapping

9For the definition of (graded) determinant lines see the footnote in the introduction.

10Note that the convention that the determinant line of a zero dimensional vector space is canonically isomorphic to the field 𝕂 is consistent with the calculus with determinant lines.

11Such a representative for the Euler structure is called spray or Turaev spider.

12Note that a frame (basis) in V determines such an element. Note that if instead of v we pick up a frame ϵ of V, we will obtain a base in C_{τ}^{∗}(M; ρ).

det V \ 0 → det H^{∗}(M; ρ) \ 0.  (28)

This mapping is obviously homogeneous of degree χ(M). A straight forward calculation shows that it does not depend on the choice of π_{x} or the ordering on X_{τ}. As a matter of fact this mapping does not depend on τ either, only on the Euler structure 𝔢 and the cohomology orientation 𝔬.^{13} This is a non-trivial fact, and its proof is contained in [M66] and [T86] for the acyclic case, and implicit in the existing literature cf. [FT00], but see also Corollary 2 in the appendix.

Definition 1 (Milnor–Turaev torsion). The element τ_{comb}^{ρ,𝔢,𝔬} in Det_{x_{0}}(M; ρ), see (26), corresponding to the homogeneous mapping (28) is referred to as the Milnor–Turaev torsion.

We obviously have, but see Remark 1 in section 2.3 for the sign,

τ_{comb}^{ρ,𝔢+σ,𝔬} = τ_{comb}^{ρ,𝔢,𝔬} · [det ∘ρ](σ)^{−1}

for all σ ∈ H_{1}(M; ℤ). Here [det ∘ρ] : H_{1}(M; ℤ) → 𝕂^{∗} denotes the homomorphism induced from the homomorphism det ∘ρ : Γ → 𝕂^{∗}. In particular, if ρ is an unimodular representation then τ_{comb}^{ρ,𝔢,𝔬} is independent of 𝔢. Moreover

τ_{comb}^{ρ,𝔢,−𝔬} = (−1)^{dim V} τ_{comb}^{ρ,𝔢,𝔬}.  (29)

Remark 6. Note that when H^{∗}(M; ρ) = 0 then, in view of the canonical isomorphism Det_{x_{0}}(M; ρ) = 𝕂, see Remark 5, the Milnor–Turaev torsion is an element in 𝕂^{∗}. We will denote the inverse of this number by T_{comb}^{𝔢,𝔬}(ρ) := (τ_{comb}^{ρ,𝔢,𝔬})^{−1}, and later consider it as a function on the space of acyclic representations. The sign is added in order to stay with traditional notation.

3.4. Modified Ray–Singer metric. Let us briefly recall the definition of the Ray–Singer torsion. Suppose 𝔽 = (F, ∇) is a real or complex vector bundle F equipped with a flat connection ∇. Let g be a Riemannian metric on M and let μ be a Hermitian metric on F. The Ray–Singer torsion T_{an}(∇, g, μ) is the positive real number given by:

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

Here Δ^{k} denotes the Laplacian in degree k of the elliptic complex (Ω^{∗}(M; F), d_{∇}) when equipped with the scalar product induced from the Riemannian metric g and the Hermitian metric μ, and det′Δ^{k} denotes its zeta regularized determinant, see [RS71].

Recall that H^{∗}(M; 𝔽) canonically identifies with the harmonic forms and thus inherits a Hermitian metric from g and μ. Let us write || · ||_{Hodge}^{𝔽,g,μ} for the induced Hermitian metric on det H^{∗}(M; 𝔽). The Ray–Singer metric, see [BZ92], is the Hermitian metric

|| · ||_{RS}^{𝔽,g,μ} := T_{an}(∇, g, μ) · || · ||_{Hodge}^{𝔽,g,μ}

on det H^{∗}(M; 𝔽).

Suppose 𝕂 is ℝ or ℂ. Let V be a finite dimensional vector space over 𝕂. Every representation ρ : Γ → GL(V) gives rise to an associated flat vector bundle 𝔽_{ρ} = (F_{ρ}, ∇_{ρ}) over M. Moreover the fiber over x_{0} is canonically identified with V. Note that holonomy at x_{0} provides a right Γ–action on the fiber over x_{0} which, when composed with the inversion in GL(V), reproduces ρ.

13In fact it depends on 𝔬 only when dim V is odd, cf. (29) below.

Choose a Riemannian metric on g on M, and a Hermitian metric on F_{ρ}. This induces a Hermitian metric || · ||_{x_{0}}^{μ} on (det V)^{−χ(M)}. Moreover, we have the Ray– Singer metric || · ||_{RS}^{𝔽_{ρ},g,μ} on det H^{∗}(M; 𝔽_{ρ}) = det H^{∗}(M; ρ). Suppose 𝔢^{∗} ∈ 𝔈𝔲𝔩_{x_{0}}(M; ℝ) is a co-Euler structure. Choose α so that 𝔢^{∗} = [g, α], see section 3.2. Recall the Kamber–Tondeur form ω(∇_{ρ}, μ) ∈ Ω^{1}(M; ℝ) from section 2.3. Consider the Hermitian metric

|| · ||_{an}^{ρ,𝔢^{∗}} := || · ||_{RS}^{𝔽_{ρ},g,μ} ⊗ || · ||_{x_{0}}^{μ} · e^{−S(ω(∇_{ρ},μ),α)}  (30)

on Det_{x_{0}}(M; ρ), see (26) and Lemma 1. The known anomaly formulas for the Ray–Singer metric, see [BZ92, Theorem 0.1], imply that this is indeed independent of μ and only depends on the co-Euler structure 𝔢^{∗}. For β ∈ H^{n−1}(M; O_{M}) we obviously have

|| · ||_{an}^{ρ,𝔢^{∗}+β} = || · ||_{an}^{ρ,𝔢^{∗}} · e^{⟨log |[det ∘ρ]|,PD(β)⟩}

where log |[det ∘ρ]| ∈ H^{1}(M; ℝ) is the Kamber–Tondeur class corresponding to the homomorphism log |[det ∘ρ]| : H_{1}(M; ℝ) → ℝ, see Remark 1 in section 2.3.

Definition 2 (Modified Ray–Singer metric). The Hermitian metric || · ||_{an}^{ρ,𝔢^{∗}} on Det_{x_{0}}(M; ρ), see (26) and (30), will be referred to as the modified Ray–Singer metric.

The Bismut–Zhang theorem [BZ92, Theorem 0.2] can now be reformulated as a statement comparing two topological invariants, one defined with the help of analysis, and one defined with the help of a triangulation.

Theorem 2 (Bismut–Zhang). Let (M, x_{0}) be a closed connected manifold with base point, and let ρ : π_{1}(M, x_{0}) → GL(V) be a finite dimensional real or complex representation. Suppose 𝔢 ∈ 𝔈𝔲𝔩_{x_{0}}(M; ℤ) is an Euler structure, and 𝔢^{∗} ∈ 𝔈𝔲𝔩_{x_{0}}^{∗}(M; ℝ) a co-Euler structure. Then one has:

||τ_{comb}^{ρ,𝔢,𝔬}||_{an}^{ρ,𝔢^{∗}} = e^{⟨log |[det ∘ρ]|,𝕋(𝔢,𝔢^{∗})⟩}

Particularly, if 𝕋(𝔢, 𝔢^{∗}) = 0 then ||τ_{comb}^{ρ,𝔢,𝔬}||_{an}^{ρ,𝔢^{∗}} = 1.

For an alternative proof of the (original) Bismut–Zhang theorem see also [BFK01].

Remark 7. If H^{∗}(M; ρ) = 0 then Det_{x_{0}}(M; ρ) = 𝕂, see Remark 5. In this case the modified Ray–Singer metric gives rise to a positive real number T_{an}^{𝔢^{∗}}(ρ) := ||1||_{an}^{ρ,𝔢^{∗}} > 0. From the very definition we have

T_{an}^{𝔢^{∗}}(ρ) = T_{an}(∇_{ρ}, g, μ) · e^{−S(ω(∇_{ρ},μ),α)}

where ∇_{ρ} denotes the flat connection on the associated bundle F_{ρ}, g is a Riemannian metric on M, μ is a Hermitian structure on F_{ρ}, and α is such that [g, α] = 𝔢^{∗}. Recall the number T_{comb}^{𝔢,𝔬}(ρ) ∈ 𝕂^{∗} from Remark 6. Provided 𝕋(𝔢, 𝔢^{∗}) = 0, the Bismut–Zhang theorem implies |T_{comb}^{𝔢,𝔬}(ρ)| = T_{an}^{𝔢^{∗}}(ρ).

Remark 8. Note that given a vector bundle F over M, equipped with a flat connection ∇, a Hermitian metric μ, a Riemannian metric g on M, and a co-Euler structure 𝔢^{∗} = [g, α] ∈ 𝔈𝔲𝔩_{x_{0}}^{∗}(M; ℝ), we can consider the quantity:

T_{an}(∇, g, μ, 𝔢^{∗}) := T_{an}(∇, g, μ) · e^{−S(ω(∇,μ),α)}

Then T_{an}(∇, g, μ, 𝔢^{∗}) is independent of μ. If H^{∗}(M; ρ_{∇}) = 0, then T_{an}(∇, g, μ, 𝔢^{∗}) coincides with T_{an}^{𝔢^{∗}}(ρ_{∇}), is gauge invariant in ∇, and independent of g.

4. Torsion and representations

Warning: Following standard conventions in algebraic geometry cf. [H77] we reserve the name affine algebraic variety for an irreducible variety in the affine space, and the term algebraic set for any finite union of varieties. By a rational function on an algebraic set we simply mean a rational function on each irreducible component.

4.1. Complex representations. Let Γ be a finitely presented group with generators g_{1}, …, g_{r} and relations R_{i}(g_{1}, g_{2}, …, g_{r}) = e, i = 1, …, p, and V be a complex vector space of dimension N. Let Rep(Γ; V) be the set of linear representations of Γ on V, i.e. group homomorphisms ρ : Γ → GL_{ℂ}(V). By identifying V to ℂ^{N} this set is in a natural way an algebraic set inside the space ℂ^{rN^{2}+1} given by pN^{2} + 1 equations. Precisely if A_{1}, …, A_{r}, z represent the coordinates in ℂ^{rN^{2}+1} with A := (a^{ij}), a^{ij} ∈ ℂ, so A ∈ ℂ^{N^{2}} and z ∈ ℂ, then the equations defining Rep(Γ; V) are

z · det(A_{1}) · det(A_{2}) ⋯ det(A_{r}) = 1
R_{i}(A_{1}, …, A_{r}) = id,  i = 1, …, p

with each of the equalities R_{i} representing N^{2} polynomial equations.

Suppose Γ = π_{1}(M, x_{0}), M a closed manifold. Denote by Rep_{0}^{M}(Γ; V) the set of representations ρ ∈ with H^{∗}(M; ρ) = 0 and notice that they form a Zariski open set in Rep(Γ, V) and denote the closure of this set by Rep^{M}(Γ; V). This is again an algebraic set which depends only on the homotopy type of M, can be empty (for example if χ(M) ≠ 0) but in some interesting cases can be the full space Rep(Γ; V), cf. section 4.3.

Definition 3 (V –acyclicity). Given a complex vector space V we say that the closed manifold M is V –acyclic if there exist representations ρ ∈ Rep(Γ; V) with H^{∗}(M; ρ) = 0. Equivalently Rep^{M}(Γ; V) ≠ ∅.

Recall that every representation ρ ∈ Rep(Γ; V) induces a canonical vector bundle F_{ρ} equipped with a canonical flat connection ∇_{ρ}. They are obtained from the trivial bundle M̃ × V → M̃ and the trivial connection by passing to the Γ quotient spaces. Here M̃ is the canonical universal covering provided by the base point x_{0}. The Γ–action is the diagonal action of deck transformations on M̃ and the action ρ on V. The fiber of F_{ρ} over x_{0} identifies canonically with V. The holonomy representation determines a right Γ–action on the fiber of F_{ρ} over x_{0}, i.e. an anti homomorphism Γ → GL(V). When composed with the inversion in GL(V) we get back the representation ρ. The pair (F_{ρ}, ∇_{ρ}) will be denoted by 𝔽_{ρ}.

If ρ_{0} is a representation in the connected component Rep_{α}(Γ; V) one can identify Rep_{α}(Γ; V) to the connected component of ∇_{ρ_{0}} in the complex analytic space of flat connections of the bundle F_{ρ_{0}} modulo the group of bundle isomorphisms of F_{ρ_{0}} which restrict to the identity on the fiber above x_{0}.

4.2. The space of cochain complexes and Turaev functions. Let (k_{0}, k_{1}, …, k_{n}) be a string of non-negative integers which satisfy the following requirements:

k_{0} − k_{1} + k_{2} ∓ ⋯ + (−1)^{n}k_{n} = 0  (31)
k_{i} − k_{i−1} + k_{i−2} ∓ ⋯ + (−1)^{i}k_{0} ≥ 0 for any i ≤ n − 1.  (32)

Denote by 𝔻(k_{0}, …, k_{n}) the collection of cochain complexes of the form

C = (C^{∗}, d^{∗}) : 0 → C^{0} ⟶^{d^{0}} C^{1} ⟶^{d^{1}} ⋯ ⟶^{d^{n−2}} C^{n−1} ⟶^{d^{n−1}} C^{n} → 0

with C^{i} := ℂ^{k_{i}}, and by 𝔻_{ac}(k_{0}, …, k_{n}) ⊆ 𝔻(k_{0}, …, k_{n}) the subset of acyclic complexes. The cochain complex C is determined by the collection {d^{i}} of linear maps d^{i} : ℂ^{k_{i}} → ℂ^{k_{i+1}}. If regarded as the subset of {d^{i}} ∈ 𝕃(k_{0}, …, k_{n}) = ⨁_{i=0}^{n−1} L(ℂ^{k_{i}}, ℂ^{k_{i+1}})^{14} which satisfy the quadratic equations

d^{i+1} · d^{i} = 0,

the set 𝔻(k_{0}, …, k_{n}) is an affine algebraic set given by degree two homogeneous polynomials and 𝔻_{ac}(k_{0}, …, k_{n}) is a Zariski open set.

The map

π_{0} : 𝔻_{ac}(k_{0}, …, k_{n}) → Emb(C^{0}, C^{1})

which associates to C ∈ 𝔻(k_{0}, …, k_{n}) the linear map d^{0}, is a bundle whose fiber is isomorphic to 𝔻_{ac}(k_{1} − k_{0}, k_{2}, …, k_{n}). Note also that Emb(C^{0}, C^{1}) is the Stiefel manifold of k_{0}–frames in C^{1}, and is the total space of a GL(k_{0}) principal bundle over the Grassmann manifold Gr_{k_{0}}(k_{1}).

As a consequence we have

Proposition 7.

(i) 𝔻_{ac}(k_{0}, …, k_{n}) is a connected smooth quasi algebraic set of dimension

k_{0} · k_{1} + (k_{1} − k_{0}) · k_{2} + ⋯ + (k_{n−1} − k_{n−2} + ⋯ ± k_{0}) · k_{n}.

(ii) The closure 𝔻̂_{ac}(k_{0}, …, k_{n}) is an irreducible algebraic set, hence an affine algebraic variety

For any cochain complex in C ∈ 𝔻_{ac}(k_{0}, …, k_{n}) denote by

B^{i} := img(d^{i−1}) ⊆ C^{i} = ℂ^{k_{i}}

and consider the short exact sequence

0 → B^{i} ⟶^{inc} C^{i} ⟶^{d^{i}} B^{i+1} → 0.

Choose a bases b_{i} for each B_{i}, and choose lifts \overline{b}_{i+1} of b_{i+1} in C^{i} using d^{i}, i.e. d^{i}(\overline{b}_{i+1}) = b_{i+1}. Clearly \{b_{i}, \overline{b}_{i+1}\} is a base of C^{i}.

Consider the base \{b_{i}, \overline{b}_{i+1}\} as a collection of vectors in C^{i} = ℂ^{k_{i}} and write them as a columns of a matrix [b_{i}, \overline{b}_{i+1}]. Define the torsion of the acyclic complex C, by

τ(C) := (−1)^{N} ∏_{i=0}^{n} det[b_{i}, \overline{b}_{i+1}]^{(−1)^{i}}

where (−1)^{N} is Turaev’s sign, see [FT00]. The result is independent of the choice of the bases b_{i} and of the lifts \overline{b}_{i} cf. [M66] [FT00], and leads to the function

τ : 𝔻_{ac}(k_{0}, …, k_{n}) → ℂ \ 0.

14We denote by L(V, W) the space of linear maps from V to W.

Turaev provided a simple formula for this function, cf. [T01], which permits to recognize τ as the restriction of a rational function on 𝔻̂_{ac}(k_{0}, k_{1}, …, k_{n}).

Following Turaev [T01], one considers symbols α ≡ (α_{0}, α_{1}, …, α_{n}) with α_{i} subsets the set of integers {1, 2, …, k_{i}} so that:

α_{n} = ∅
♯(α_{i}) = k_{i+1} − ♯(α_{i+1}).

Such α is called a τ–chain. There are, of course, only finitely many such τ–chains for a given collection (k_{0}, …, k_{n}) as above.

For any cochain complex C ∈ 𝔻(k_{0}, …, k_{n}), and in fact for any element d ≡ {d^{i}} ∈ 𝕃, a τ–chain defines the collection of of square matrices A_{i}(d^{i}) representing the ♯(α_{i}) × (k_{i+1} − ♯(α_{i+1}))–minor of the matrix d^{i} consisting of the columns whose indices are in α_{i} and the rows not in α_{i+1}.

For example if k_{i} = 2, k_{i+1} = 4, α_{i} = {2}, α_{i+1} = {1, 3, 4}, and d^{i} is the matrix whose entries are

(\begin{matrix}d_{1,1}^{i} & d_{1,2}^{i} \\ d_{2,1}^{i} & d_{2,2}^{i} \\ d_{3,1}^{i} & d_{3,2}^{i} \\ d_{4,1}^{i} & d_{4,2}^{i} \end{matrix})

with A_{i}(d^{i}) the 1 × 1 matrix (d_{2,2}^{i}).

One considers the expression

F_{α}(d) = ϵ(α) ∏_{i=0}^{n}(det A_{i}(d^{i}))^{(−1)^{i+1}}

where ϵ(α) = ±1 and is chosen according to the strategy explained in [T01, Remark 2.4, page 9]. Here d ≡ {d^{i}} ∈ 𝕃. Clearly F_{α} is a rational function on 𝕃.

One says that the cochain complex is α–non-degenerate if all quantities det A_{i}(d^{i}) are non-zero. Turaev has shown that:

(i) For an α–non-degenerate cochain complex C ∈ 𝔻_{ac}(k_{0}, …, k_{n}), τ(C) = F_{α}(d) where d = {d_{i}} is collection of the differentials in the complex C, cf. [T01, Theorem 2.2].

(ii) If U_{α} denotes the subsets of 𝔻_{ac}(k_{0}, …, k_{n}) consisting α–non-degenerate

complexes then U_{α} is open, and ⋃_{α} U_{α} = 𝔻_{ac}(k_{0}, …, k_{n}). (iii) For any τ–chain α there exists a cochain complex C ∈ 𝔻_{ac}(k_{0}, …, k_{n})

which is α–non-degenerate. In view of the connectivity of 𝔻_{ac}(k_{0}, …, k_{n}) and (i)–(iii) above, F_{α}, are rational

functions on 𝕃 which agree on the connected set 𝔻_{ac}(k_{0}, …, k_{n}). Therefore they define a rational function τ on the irreducible variety 𝔻̂_{ac}(k_{0}, …, k_{n}) whose zero and poles are in Σ, the algebraic subset of non-acyclic cochain complexes.

Consider now a smooth triangulation τ of M whose set of simplexes of dimension q is denoted by X_{q} and the collection of integers k_{i} = ♯(X_{i})·dim V where V is a fixed complex vector space. Let 𝔢 ∈ 𝔈𝔲𝔩_{x_{0}}(M; ℤ) be an Euler structure, 𝔬 a cohomology orientation, and suppose that M is V –acyclic. Then the integers (k_{0}, …, k_{n}) satisfy (31) and (32). As in section 3.3, choose a collection of paths π_{𝔢} ≡ {π_{x}} from x_{0} to the barycenters x, so that 𝔢 = [X_{τ} , c] where c is the Euler chain defined by {π_{x}}. Choose also an ordering o of the barycenters (zeros of X_{τ}) compatible with 𝔬, and a framing ϵ of V.

Consider the chain complex (C_{τ}^{∗}(M; ρ), d_{τ} (ρ)) associated with the triangulation τ which computes the cohomology H^{∗}(M; ρ). Using π_{x}, o and ϵ one can identify C_{τ}^{q}(M; ρ) with ℂ^{k_{q}}. We obtain in this way a map

t_{π_{𝔢},o,ϵ} : Rep(Γ; V) → 𝔻(k_{0}, …, k_{n})

which sends Rep^{M}(Γ; V)\Σ(M) to 𝔻_{ac}(k_{0}, …, k_{n}). A look at the explicit definition of d_{τ} (ρ) implies that t_{π_{𝔢},o,ϵ} is actually a regular map between two algebraic sets. Change of ϵ, π_{𝔢} and o changes the map t_{π_{𝔢},o,ϵ}. Since we are in the case H^{∗}(M; ρ) = 0 we easily recognize that the composition τ · t_{π_{𝔢},o,ϵ}, when restricted to Rep^{M}(Γ; V) \ Σ(M), is exactly the Milnor–Turaev torsion T_{comb}^{𝔢,𝔬}, see Remark 6 in section 3.3, hence independent of all these choices π_{𝔢}, o, ϵ.

We summarize these observations in the following

Theorem 3. Let M be a closed manifold, x_{0} a base point, Γ := π_{1}(M, x_{0}) the fundamental group, 𝔢 ∈ 𝔈𝔲𝔩_{x_{0}}(M; ℤ) an Euler structure, 𝔬 a cohomology orientation, and let V be a finite dimensional complex vector space. Consider the Milnor–Turaev torsion T_{comb}^{𝔢,𝔬} : Rep^{M}(Γ; V) \ Σ(M) → ℂ \ 0.

(i) T_{comb}^{𝔢,𝔬} is a rational function on the algebraic set Rep^{M}(Γ; V) whose poles and zeros are contained in Σ(M).

(ii) The absolute value of T_{comb}^{𝔢,𝔬} is the modified Ray–Singer torsion T_{an}^{𝔢^{∗}}, where 𝔢^{∗} = P^{−1}(𝔢) ∈ 𝔈𝔲𝔩_{x_{0}}^{∗}(M; ℝ), see Remark 7. More precisely, for ρ ∈ Rep^{M}(Γ; V) \ Σ(M) one has

|T_{comb}^{𝔢,𝔬}(ρ)| = T_{an}^{𝔢^{∗}}(ρ) = T_{an}(∇_{ρ}, g, μ, 𝔢^{∗}) = T_{an}(∇_{ρ}, g, μ) · e^{−S(ω(∇_{ρ},μ),α)}

where (F_{ρ}, ∇_{ρ}) denotes the flat complex vector bundle associated to the representation ρ, g is a Riemannian metric on M, μ is a Hermitian structure on F_{ρ}, ω(∇_{ρ}, μ) the Kamber–Tondeur form, and α is such that the pair (g, α) represents the co-Euler structure P^{−1}(𝔢). (iii) For σ ∈ H_{1}(M; ℤ), and ρ ∈ Rep^{M}(Γ; V) \ Σ(M) we have

T_{comb}^{𝔢+σ,𝔬}(ρ) = T_{comb}^{𝔢,𝔬}(ρ) · [det ∘ρ](σ) and T_{comb}^{𝔢,−𝔬}(ρ) = (−1)^{dim V} · T_{comb}^{𝔢,𝔬}(ρ)

where [det ∘ρ] : H_{1}(M; ℤ) → ℂ \ 0 is the homomorphism induced from det ∘ρ : Γ → ℂ \ 0.

4.3. Milnor–Turaev torsion for mapping tori. Let N be a closed connected manifold and φ : N → N a diffeomorphism. Define the mapping torus M = N_{φ} by glueing the boundaries of N × I with the help of φ, more precisely identifying (x, 1) with (φ(x), 0). The manifold M comes with a projection p : M → S^{1}, and an inclusion N → M, x ↦ (x, 0). Set Γ := π_{1}(S^{1}, 0). Let 𝕂 be a field of characteristics zero, and suppose V is a finite dimensional vector space over 𝕂. For ρ ∈ Rep(Γ; V) = GL(V) define

P_{φ}^{k}(ρ) := det(φ^{∗} ∘ ρ_{∗} − id : H^{k}(N; V) → H^{k}(N; V))

and the Lefschetz zeta function:

ζ_{φ}(ρ) := \frac{∏_{k even} P_{φ}^{k}(ρ)}{∏_{k odd} P_{φ}^{k}(ρ)}

This is a rational function ζ_{φ} : GL(V) → 𝕂.

Choose a base point x_{0} ∈ M with p(x_{0}) = 0 ∈ S^{1}. Every representation ρ ∈ Rep(Γ; V) gives rise to a representation p^{∗}ρ of π_{1}(M, x_{0}) on V. It is not hard to see that H^{∗}(M; p^{∗}ρ) = 0 iff P_{φ}^{k}(ρ) ≠ 0 for all k. Indeed, we have a long exact sequence, the Wang sequence:

⋯ → H^{∗}(M; p^{∗}ρ) → H^{∗}(N; V) ⟶^{φ^{∗}∘ρ_{∗}−id} H^{∗}(N; V) → H^{∗+1}(M; p^{∗}ρ) → ⋯ (33)

Recall that this sequence is derived from the long exact sequence of the pair (M, N), and the isomorphism H^{∗}(M, N; p^{∗}ρ) = H^{∗}(N × I, N × ∂I; V) = H^{∗−1}(N; V). The latter is the Thom isomorphism with the standard orientation on I, and the first comes from a trivialization of the coefficient system on N ×I from N ×{0} upwards.

Let ω := p^{∗}dt ∈ Ω^{1}(M). The space of vector fields X on M which satisfy ω(X) < 0 is contractible and hence defines an Euler structure 𝔢 ∈ 𝔈𝔲𝔩(M; ℤ), cf. Theorem 1(ii). Note that we do not have to specify a base point since the Euler characteristics of M vanishes.

Recall that a finite dimensional long exact sequence of 𝕂 vector spaces

⋯ → A^{n} → B^{n} → C^{n} → A^{n+1} → ⋯

induces a canonical isomorphism of graded determinant lines^{15}

u : det(A^{∗}) ⊗ det(C^{∗}) → det(B^{∗}), a ⊗ b → u(a ⊗ b).

If in addition B^{∗} = C^{∗} we obtain a canonical isomorphism det(A^{∗}) → 𝕂, a → tr(u(a ⊗ ·)). Applying this to the Wang exact sequence for the trivial one dimensional real representation

⋯ → H^{∗}(M; ℝ) → H^{∗}(N; ℝ) → H^{∗}(N; ℝ) → H^{∗+1}(M; ℝ) → ⋯

we obtain a canonic isomorphism det H^{∗}(M; ℝ) = ℝ. Thus M has a canonic cohomology orientation 𝔬, the one corresponding to the standard orientation of ℝ via this isomorphism.

The rest of this section is dedicated to the proof of

Proposition 8. With this notations, if H^{∗}(M; p^{∗}ρ) = 0, then T_{comb}^{𝔢,𝔬}(p^{∗}ρ) = (−1)^{dim(V)·z_{φ}}ζ_{φ}(ρ) where

z_{φ} = ∑_{q} dim H^{q}(N; ℝ) · dim ker(φ^{∗} − id : H^{q}(N; ℝ) → H^{q}(N; ℝ))

To get a hold on the Euler structure 𝔢 we start with the following

Lemma 2. Suppose τ̃ is a triangulation of M such that N is a subcomplex. For every simplex σ ∈ τ̃ let π_{σ} be a path in N × [0, 1) considered as path in M from the base point x_{0} ∈ N to the barycenter x_{σ}. Consider the Euler structure 𝔢̃ = [Xτ̃, cτ̃] where cτ̃ := ∑_{σ∈τ̃}(−1)^{dim σ}π_{σ}. Then [det ∘p^{∗}ρ](𝔢̃ − 𝔢) = 1. Particularly T_{comb}^{𝔢,𝔬}(p^{∗}ρ) = T_{comb}^{𝔢̃,𝔬}(p^{∗}ρ).

Proof. Consider N_{ϵ} ⊆ M, x ↦ (1 − ϵ, x), where ϵ > 0 is sufficiently small such that ω(X_{τ}) < 0 on N_{ϵ} and such that no zeros of X_{τ} are contained in (1 − ϵ, 1) × N. This is possible since N is a sub-complex, hence the Euler vector field X_{τ} is tangential to N and points away from N in a small neighborhood of N. We thus find a homotopy 𝕏 from X to X_{τ} which has the property that ω(𝕏_{t}) < 0 on N_{ϵ} for all t ∈ I. We conclude that cs(X, X_{τ}) has a representative contained in M \ N_{ϵ}. Since 𝔢̃ − 𝔢 ∈ H_{1}(M; ℤ) is represented by c_{τ} − cs(X, X_{τ}) we conclude p_{∗}(𝔢̃ − 𝔢) = 0 ∈ H_{1}(S^{1}; ℤ), and hence [det ∘p^{∗}ρ](𝔢̃− 𝔢) = 1. The last assertion follows from Theorem 3(iii). □

15For more details consult the appendix.

Note that the statement remains the same if one replaces triangulation by cell structure.

Lemma 3. Let φ_{i} : N → N be diffeomorphisms, and suppose H^{∗}(N_{φ_{i}}; p^{∗}ρ) = 0, i = a, b. Then T_{comb}^{N_{φ_{b}} ,𝔢,𝔬}(p^{∗}ρ) · T_{comb}^{N_{φa} ,𝔢,𝔬}(p^{∗}ρ)^{−1} = (−1)^{dim(V)·(z_{φ_{b}} −z_{φa})}ζ_{φ_{b}}(ρ) · ζ_{φ_{a}}(ρ)^{−1}.

Proof. We may assume, by changing φ_{i} up to isotopy, that there are smooth triangulations τ_{0} and τ_{1} of N such that both φ_{a} and φ_{b} are cellular, i.e. they send the r–skeleton of τ_{0} to the r–skeleton of τ_{1}. Let τ be a smooth triangulation on N × [0, 1] such that N × {0} and N × {1} are subcomplexes and the triangulation induced by τ on N × {0} resp. N × {1} is exactly τ_{0} resp. τ_{1}. Let τ̃_{a} and τ̃_{b} denote the cell structures on N_{φ_{a}} and N_{φ_{b}} induced by τ. Note that τ̃_{a} and τ̃_{b} are smooth cell structures but not smooth triangulations. The characteristic maps are smooth embeddings of compact affine simplexes which satisfy the axioms of a CW–complex but not of a triangulation. Also recall that the torsion T_{comb}^{N_{φ_{i}} ,𝔢,𝔬}(p^{∗}ρ) can be computed using any cell structure [M66].

Recall that we denote by X, X_{0} and X_{1} the set of simplexes (cells) of τ, τ_{0} and τ_{1}. The cells of both τ̃_{a} and τ̃_{b} are indexed by X \ X_{0}. Hence the graded vector spaces C∗ ; τ̃_{a}(N_{φ_{a}}; p^{∗}ρ) and C∗ ; τ̃_{b}(N_{φ_{b}}; p^{∗}ρ) are tautologically the same. Moreover the relative

cochain complexes C∗ ; τ̃_{a}(N_{φ_{a}}, N; p^{∗}ρ), C∗ ; τ̃_{b}(N_{φ_{b}}, N; p^{∗}ρ) and C_{τ}^{∗}(N × I, N × ∂I; p^{∗}ρ)

are tautologically the same; their cells are indexed by X \ {X_{0} ∪ X_{1}} and the differentials coincide.

The two short exact sequences of cochain complexes

C_{τ}^{∗}(N × I, N × ∂I; p^{∗}ρ) = C∗ ; τ̃_{i}(N_{φ_{i}}, N; p^{∗}ρ) → C∗ ; τ̃_{i}(N_{φ_{i}}; p^{∗}ρ) → C_{τ_{1}}^{∗}(N; V)

i = a, b, give rise to a commutative diagram:

\xymatrix{det C∗ ; τ̃a (N_{φa}; p^{∗}ρ) \ar[d]_{φτ̃a} & det C_{τ}^{∗}(N × I, N × ∂I; p^{∗}ρ) ⊗ det C_{τ_{1}}^{∗} (N; V ) \ar[d]^{φ⊗φτ_{1}} \ar[l]_{(−1)^{ya} ψa} \ar[r]^{(−1)^{y_{b}} ψ_{b}} & det C∗ ; τ̃_{b} (N_{φ_{b}}; p^{∗}ρ) \ar[d]^{φτ̃_{b}} \\ ℝ = det H^{∗}(N_{φa}; p^{∗}ρ) & det H^{∗}(N × I, N × ∂I; p^{∗}ρ) ⊗ det H^{∗}(N; V ) \ar[l]_{φ_{H^{∗}_{a}}} \ar[r]^{φ_{H^{∗}_{b}}} & det H^{∗}(N_{φ_{b}}; p^{∗}ρ) = ℝ}

Here each square is a fusion diagram as in Lemma 5 in the appendix with the middle complex C_{1}^{∗} acyclic. Note that (−1)^{y_{a}} = (−1)^{y_{b}} in view of Lemma 5. It follows from (33) that φ_{H_{b}^{∗}} ∘ φ_{H_{a}^{∗}}^{−1} = ζ_{φ_{a}}(ρ) ∘ ζ_{φ_{b}}(ρ)^{−1}.

Consider now the diagram

\xymatrix{det C∗ ; τ̃_{b}(N_{φ_{b}}; p^{∗}ρ) & det C_{τ}^{∗}(N × I, N × ∂I; p^{∗}ρ) ⊗ det C_{τ_{1}}^{∗}(N; V) \ar[l]_{(−1)^{y_{b}} ψ_{b}} \ar[d]^{(−1)^{ya} ψ_{a}} \\ ℝ = det V^{χ} \ar[u]^{β_{φ_{b}}} \ar[r]^{β_{φa}} & det C∗ ; τ̃_{a}(N_{φ_{a}}; p^{∗}ρ)}  (34)

where β_{φ_{a}} and β_{φ_{b}} are induced from the homogeneous maps of degree χ = χ(N_{φ_{a}}) = χ(N_{φ_{b}}) = 0 which associates to a frame of V the equivalence class of bases induced from the frame, a collection of paths representing the canonical Euler structure of N_{φ_{a}} resp. N_{φ_{b}} and the ordering of simplexes of τ representing the canonical cohomology orientation of N_{φ_{a}} resp. N_{φ_{b}}. Observe that the diagram is commutative up to sign because the canonical Euler structures on N_{φ_{i}} can be realized by the same Euler chain in N × I cf. Lemma 2. We conclude T_{comb}^{N_{φ_{b}} ,𝔢,𝔬}(p^{∗}ρ) · T_{comb}^{N_{φa} ,𝔢,𝔬}(p^{∗}ρ)^{−1} = ±ζ_{φ_{b}}(ρ) · ζ_{φ_{a}}(ρ)^{−1}.

To decide about the sign we have to compare the orderings of X \ X_{0} induced from the canonical cohomology orientation on N_{φ_{a}} and on N_{φ_{b}}. To do this we have to consider the fusion diagram for the trivial one dimensional real representation:

\xymatrix{det C∗ ; τ̃_{a}(N_{φ_{a}}; ℝ) \ar[d]_{φℝ ; τ̃a} & det C_{τ}^{∗}(N × I, N × ∂I; ℝ) ⊗ det C_{τ_{1}}^{∗}(N; ℝ) \ar[d]^{φ^{ℝ}⊗φ_{τ_{1}}^{ℝ}} \ar[l]_{(−1)^{y^{ℝ}_{a}} ψ_{a}^{ℝ}} \ar[r]^{(−1)^{y^{ℝ}_{b}} ψ_{b}^{ℝ}} & det C∗ ; τ̃_{b}(N_{φ_{b}}; ℝ) \ar[d]^{φℝ ; τ̃_{b}} \\ det H^{∗}(N_{φ_{a}}; ℝ) & det H^{∗}(N × I, N × ∂I; ℝ) ⊗ det H^{∗}(N; ℝ) \ar[l]_{φ_{H^{∗}_{a}}^{ℝ}} \ar[r]^{φ_{H^{∗}_{b}}^{ℝ}} & det H^{∗}(N_{φ_{b}}; ℝ)}

Using Lemma 5 one can show that y_{b}^{ℝ} − y_{a}^{ℝ} = z_{b} − z_{a} mod 2. Together with y_{b} − y_{a} = 0 mod 2 this implies that the digram (34) commutes up to the sign (−1)^{dim(V)·(z_{b}−z_{a})}. This completes the proof. □

Lemma 4. We have T_{comb}^{N_{id},𝔢,𝔬}(p^{∗}ρ) = (−1)^{dim(V)·z_{id}}ζ_{id}(ρ) whenever H^{∗}(N_{id}; p^{∗}ρ) = 0.

Proof. Equip the interval I with the three cell structure, let τ_{1} be a smooth triangulation of N, and let τ denote the product cell structure on N × I. Let τ̃ denote the induced (product) cell structure on N_{id} = N × S^{1}. Then the complex C_{τ}^{∗}(N × I, N × ∂I; p^{∗}ρ) is the suspension of C_{τ_{1}}^{∗}(N; p^{∗}ρ). Using Lemma 5 and Lemma 2 it is easy to see that T_{comb}^{N_{id},𝔢,𝔬}(p^{∗}ρ) = ±ζ_{id}(ρ).

As in the proof of Lemma 3 consider the fusion diagram of the short exact sequence C_{τ}^{∗}(N × I, N × ∂I; p^{∗}ρ) → C∗ ; τ̃(N_{id}; p^{∗}ρ) → C_{τ_{1}}^{∗}(N; V) for an acyclic ρ and for the trivial one dimensional real representation. Let (−1)^{y^{ρ}} and (−1)^{y^{ℝ}} denote their signs. Using Lemma 5 one can show dim(V) · y^{ℝ} − y^{ρ} = dim(V) · z_{id} mod 2. This completes the proof. □

Proposition 8 follows from the previous two lemmas, and the simple observation that H^{∗}(N_{φ}; p^{∗}ρ) = 0 implies H^{∗}(N_{id}; p^{∗}ρ) = 0. For the last statement consider cohomology in degree zero where φ^{∗} = id^{∗} for all φ.

4.4. An invariant with values in ℝ/πℤ. If M is a closed manifold with Γ = π_{1}(M, x_{0}) and ρ an acyclic (i.e. H^{∗}(M, ρ) = 0) unimodular representation on a finite dimensional complex Hilbert space then T_{comb}^{𝔢,𝔬} is by Theorem 3(iii) independent on 𝔢 and its argument mod π independent on 𝔬. Denote by

S(M; ρ) := arg(T_{comb}^{𝔢,𝔬}) ∈ ℝ/πℤ

This is a topological invariant computable with the help of a triangulation. Proposition 8 gives a simple formula for the case M is a mapping torus and ρ is the pullback by p of a unimodular representation of ℤ and implicitly establishes its non-triviality.

In a forthcoming paper an analytic construction of this invariant will be given; such construction justifies the name of eta invariant of the DeRham complex.

It is interesting to compare this invariant with the one defined using the spectral flow in [APS75] for an odd dimensional manifold and a unitary representation. More will be said in the forthcoming paper.

4.5. Marcsik’s theorem. In this section we will use the Bismut–Zhang theorem and Proposition 8 to give a proof of a theorem due to J. Marcsik. We continue to use the notations from section 4.3.

Let g be a Riemannian metric on M. For z ∈ ℂ denote by d_{ω}(z)α := dα+zω ∧α the Witten deformed differential on Ω^{∗}(M; ℂ) and by δ_{ω}(z) its adjoint. Denote by

Δ_{ω}(z) := d_{ω}(z)δ_{ω}(z) + δ_{ω}(z)d_{ω}(z)

the Laplacian, and by Δ_{ω}^{k}(z) the Laplacian on k forms. Introduce the Ray–Singer torsion:

T_{an}(ω, g)(z) := exp(\frac{1}{2} ∑_{k}(−1)^{k+1}k log det′Δ_{ω}^{k}(z))

Here det′Δ denotes the regularized determinant of the non-negative self-adjoint elliptic operator Δ obtained by ignoring the zero modes.

Identifying ℂ^{∗} = GL(ℂ) we have P_{φ}^{k}(w) and ζ_{φ}(w) defined for all w ∈ ℂ^{∗}, see section 4.3.

Theorem 4 (Marcsik, [M98]). Let N be a closed manifold, φ : N → N a diffeomorphism, M the mapping torus of φ, p : M → S^{1} the canonic projection, and ω the pull back of the canonic volume form on S^{1}. Let g be a Riemannian metric on M, and suppose X is a vector field on M with ω(X) < 0. Then

log T_{an}(ω, g)(z) = log |ζ_{φ}(e^{z})| + ℜ(z) ∫_{M} ω ∧ X^{∗}Ψ(g)

for all z ∈ ℂ which satisfy P_{φ}^{k}(e^{z}) ≠ 0 for all k.^{16}

Proof. Consider the trivial line bundle M × ℂ over M. Let ∇^{0} denote the trivial flat connection, and let μ denote the standard Hermitian structure on M × ℂ. For z ∈ ℂ define a connection ∇^{z} := ∇ + zω on M × ℂ. This connection is flat for ω is closed. Clearly d^{∇^{z}} = d_{ω}(z), and hence

T_{an}(ω, g)(z) = T_{an}(∇^{z}, g, μ).

Define a co-Euler structure 𝔢^{∗} := [g, X^{∗}Ψ(g)] ∈ 𝔈𝔲𝔩^{∗}(M; ℝ). Recall the Euler structure 𝔢 = [X, 0] ∈ 𝔈𝔲𝔩(M; ℤ). From Theorem 1(iv) we see that 𝕋(𝔢, 𝔢^{∗}) = 0. Assume P_{φ}^{k}(e^{z}) ≠ 0 for all k. Then the Bismut–Zhang theorem implies, see Remark 7:

|T_{comb}^{𝔢,𝔬}(e^{z})| = T_{an}(∇^{z}, g, μ) · e^{−S(ω(∇^{z},μ),X^{∗}Ψ(g))}

Note that ∇^{z}μ = −2ℜ(zω)μ, hence ω(∇^{z}, μ) = ℜ(z)ω, and thus

S(ω(∇^{z}, μ), X^{∗}Ψ(g)) = ℜ(z) ∫_{M} ω ∧ X^{∗}Ψ(g).

From Proposition 8 we get T_{comb}^{𝔢,𝔬}(e^{z}) = ±ζ_{φ}(e^{z}). Putting everything together and taking the logarithm completes the proof. □

16This statement is slightly more precise than the one formulated in [M98].

4.6. An application. Suppose that p : F → M is a smooth complex vector bundle which admits flat connections equipped with a fiber metric μ over a closed manifold M. Let g be a Riemannian metric on M. Denote by C(F) the space of all connections on F. This is an affine Fréchet space over ℂ when equipped with the C^{∞}–topology, and consider FC(F) the subset of all flat connections. This is a closed complex analytic subspace of C(F). Let Σ(F) denote the subset of flat connections ∇ which are not a local minimum for the function d : FC(F) → ℤ defined by d(∇) := ∑_{i=0}^{n} dim H^{i}(M; ρ_{∇}). Here ρ_{∇} denotes the holonomy representation of ∇. The set Σ(F) is a closed analytic subspace of FC(F).

Denote by FC^{M}(F) the flat connections whose holonomy representations are in Rep^{M}(Γ; F_{x_{0}}) and by G_{x_{0}}(F) the gauge group which fixes the fiber over the base point x_{0}. Observe that FC^{M}(F) is a closed analytic subspace of FC(F) and that G_{x_{0}}(F) acts freely and holomorphically on FC(F) and FC^{M}(F). The quotient space in both cases is a complex analytic space which identifies to a union of connected components of Rep^{M}(Γ; F_{x_{0}}), resp. of irreducible components of Rep^{M}(Γ; F_{x_{0}}).

The Ray–Singer torsion T_{an}(∇, g, μ) defines a positive real valued function T_{an}^{g,μ} : FC(F) → ℝ, and if the dimension of M is odd it also defines a positive real valued function T_{an}^{g} : Rep(Γ; V) → ℝ by the formula

T_{an}^{g}(ρ) := T_{an}(∇_{ρ}, g, μ)

where μ is any fiber metric in F_{ρ}. Indeed, in this case in view of the Hermitian anomaly [BZ92, Theorem 0.1] and because the Euler form E(g) is identically zero, T_{an}(∇_{ρ}, g, μ) is independent on μ. Both functions have jumps on Σ(F) resp. Σ(M). One can ask however if they are the absolute values of meromorphic functions with poles and zeros located on Σ(F) resp. Σ(M). It is not hard to verify that the polyharmonicity property is satisfied away from the jumps, so these functions are at least locally and away from the jumps absolute values of holomorphic maps. The following is a straightforward consequence of Theorem 3 and Proposition 1.

Corollary 1.

(i) There exists a meromorphic function on FC^{M}(F) whose restriction to FC^{M}(F) \ Σ(F) has T_{an}^{g,μ} as the absolute value the function.

(ii) If M has odd dimension then there exists a meromorphic function on Rep^{M}(Γ; V) whose restriction to Rep^{M}(Γ; V) \ Σ(M) has T_{an}^{g} as the absolute value function.

Proof. To prove the result we fix a co-Euler structure 𝔢^{∗} = [g, α] ∈ 𝔈𝔲𝔩_{x_{0}}^{∗}(M; ℝ), and write

T_{an}^{g,μ}(∇) = T_{an}(∇, g, μ) = T_{an}(∇, g, μ, 𝔢^{∗}) · e^{S(ω(∇,μ),α)}

resp.

T_{an}^{g}(ρ) = T_{an}(∇_{ρ}, g, μ) = T_{an}(∇_{ρ}, g, μ, 𝔢^{∗}) · e^{S(ω(∇,μ),α)}

if M is of odd dimension. Theorem 3 shows that the first factor is the absolute value of a function which, in the first case factors via the gauge group by, in the second case is a rational function, hence is meromorphic. The second factor is holomorphic in ∇ and in second case factors via gauge group by a holomorphic function in view of Proposition 1. Indeed in this second case the second factor is gauge invariant given the fact that E(g) is identically zero. □

Appendix A. Some homological algebra

Let 𝕂 be a field of characteristics zero. Suppose C^{∗} is a finite dimensional ℤ– graded complex over 𝕂, with differential of degree one, d^{q} : C^{q} → C^{q+1}. We will always assume C^{q} = 0 for q < 0. Let us write Z^{q} = ker d^{q} ⊆ C^{q} for the cocycles, B^{q} := img d^{q−1} ⊆ C^{q} for the coboundaries, and H^{q} := Z^{q}/B^{q} for the cohomology.

Let 𝔠^{q} be a base of C^{q}, let 𝔥^{q} be a base of H^{q}, and let 𝔟^{q} be a base of B^{q}. Recall the short exact sequences

0 → Z^{q} → C^{q} ⟶^{d} B^{q+1} → 0 and 0 → B^{q} → Z^{q} → H^{q} → 0.

Choose lifts 𝔥̃^{q} in Z^{q} of 𝔥^{q}. Then 𝔟^{q}𝔥̃^{q} is a base of Z^{q}. Choose lifts 𝔟̃^{q+1} in C^{q} of 𝔟^{q+1}. Then (𝔟^{q}𝔥̃^{q})𝔟̃^{q+1} is a base of C^{q}. For two bases 𝔞_{1} and 𝔞_{2} of a common vector space we write [𝔞_{1}/𝔞_{2}] for the determinant of the matrix expressing 𝔞_{1} in terms of 𝔞_{2}. The quantity ∏_{q}[𝔠^{q}/(𝔟^{q}𝔥̃^{q})𝔟̃^{q+1}]^{(−1)^{q}} does not depend on the choice of 𝔟^{q} and does not depend on the choice of lifts 𝔟̃^{q} and 𝔥̃^{q}, see [M66].

Recall that the determinant line of a finite dimensional vector space V is det V := Λ^{dim V} V. Moreover, we will write L^{1} := L and L^{−1} := Hom(L; 𝕂) for a line, i.e. one dimensional vector space, L. Consider the graded determinant line of C^{∗}

det C^{∗} := ⨂_{q}(det C^{q})^{(−1)^{q}}

and let 𝔠 denote the base of det C^{∗} induced from the bases 𝔠^{q}. Similarly, let 𝔥 denote the base of det H^{∗} induced from the bases 𝔥^{q}. Consider the isomorphism

φ_{C^{∗}} : det C^{∗} → det H^{∗}, 𝔠 ↦ (−1)^{N} ∏_{q}[𝔠^{q}/(𝔟^{q}𝔥̃^{q})𝔟̃^{q+1}]^{(−1)^{q}}𝔥

where N := ∑_{q≥0} α^{q}β^{q} with α^{q} := ∑_{j≥q} dim C^{j} and β^{q} := ∑_{j≥q} dim H^{j}, see [FT00] and [T01]. The isomorphism φ_{C^{∗}} : det C^{∗} → det H^{∗} does not depend on the choice of bases 𝔠^{q} or 𝔥^{q}. The sign (−1)^{N} is referred to as Turaev’s sign refinement. Note that (−1)^{N} = 1 if C^{∗} is acyclic.

Suppose

0 → C_{0}^{∗} → C_{1}^{∗} → C_{2}^{∗} → 0  (35)

is a short exact sequence of finite dimensional ℤ–graded cochain complexes over 𝕂. Then we have isomorphisms φ_{C_{i}^{∗}} : det C_{i}^{∗} → det H_{i}^{∗}, i = 0, 1, 2. Let H^{∗} denote the corresponding long exact sequence

0 → H_{0}^{0} → H_{1}^{0} → H_{2}^{0} ⟶^{δ^{0}} H_{0}^{1} → ⋯ ⟶^{δ^{q−1}} H_{0}^{q} → H_{1}^{q} → H_{2}^{q} ⟶^{δ^{q}} H_{0}^{q+1} → ⋯ (36)

considered as (acyclic) ℤ–graded cochain complex, the grading so that H^{3q+i} = H_{i}^{q}. As an acyclic complex it provides an isomorphism φ_{H^{∗}} : det H^{∗} → 𝕂 which we regard as an isomorphism φ_{H^{∗}} : det H_{0}^{∗} ⊗ det H_{2}^{∗} → det H_{1}^{∗}.

The short exact sequence (35) provides isomorphisms det C_{0}^{q} ⊗ det C_{2}^{q} = det C_{1}^{q} for all q, and then an isomorphism ψ : det C_{0}^{∗} ⊗ det C_{2}^{∗} → det C_{1}^{∗}. For the proof of the following lemma look carefully at the proof of Theorem 3.2 in [M66].

Lemma 5. The following diagram commutes:

\xymatrix{det C_{0}^{∗} ⊗ det C_{2}^{∗} \ar[r]^{(−1)^{y}ψ} \ar[d]_{φ_{C^{∗}_{0}} ⊗φ_{C^{∗}_{2}}} & det C_{1}^{∗} \ar[d]^{φ_{C^{∗}_{1}}} \\ det H_{0}^{∗} ⊗ det H_{2}^{∗} \ar[r]^{φ_{H^{∗}}} & det H_{1}^{∗}}

Here

y = N(C_{0}^{∗}) + N(C_{1}^{∗}) + N(C_{2}^{∗}) + ∑_{q} f_{1}^{q} · b_{2}^{q} + b_{2}^{q} · b_{0}^{q+1} + f_{2}^{q} · b_{0}^{q+1}

with b_{i}^{q} := dim B^{q}(C_{i}^{∗}), and f_{i}^{q} := dim B^{3q+i}(H^{∗}).

Following Farber–Turaev [FT00] we call the above diagram fusion diagram, and (−1)^{y}ψ the fusion isomorphism. Note however, that the fusion homomorphism in general depends on the differentials of the complexes. If the short exact sequence of complexes splits is does not, see [FT00].

Lemma 6. In the situation of Lemma 5 suppose in addition that C_{2}^{∗} is acyclic and two dimensional, concentrated in degree k and k + 1. Then the sign (−1)^{y} does not depend on the differentials on C_{0}^{∗} and C_{1}^{∗}.

Proof. Indeed we have

N(C_{0}^{∗}) + N(C_{1}^{∗}) = β^{k+1}(C_{0}^{∗}) mod 2
N(C_{2}^{∗}) = 0
f_{1}^{q} = dim H_{1}^{q} = dim H_{0}^{q}
f_{2}^{q} = 0
b_{2}^{q} = 0 for q ≠ k + 1
b_{2}^{k+1} = 1

Using the general fact α^{q}(C_{0}^{∗}) + β^{q}(C_{0}^{∗}) + b_{0}^{q} = 0 mod 2 this implies

y = β^{k+1}(C_{0}^{∗}) + dim H_{0}^{k+1} + b_{0}^{k+2} = β^{k+2}(C_{0}^{∗}) + b_{0}^{k+2} = α^{k+2}(C_{0}^{∗}) mod 2

Hence (−)^{y} does not depend on the differentials. □

Since elementary expansions give rise to short exact sequences as in Lemma 6, we have

Corollary 2. The Milnor–Turaev torsion as defined in section 3.3, Definition 1 does indeed not depend on the triangulation.

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Dan Burghelea, Dept. of Mathematics, The Ohio State University, 231 West 18th 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