LAPLACE TRANSFORM, TOPOLOGY AND (OF A CLOSED ONE …LAPLACE TRANSFORM, TOPOLOGY AND SPECTRAL...

LAPLACE TRANSFORM, TOPOLOGY ANDSPECTRAL GEOMETRY

(OF A CLOSED ONE FORM).

DAN BURGHELEA AND STEFAN HALLER

Abstract. We consider a system (M, g, ω,X) where (M, g) is aclosed Riemannian manifold, ω a closed one form whose all zerosare nondegenerate and X an (−ω, g) gradient like vector field. Fort ∈ R+ large enough, the Witten complex (Ω∗, d∗t = d∗ + tω∧)associated with (ω, g) decomposes canonically as a direct sum oftwo one parameter family of cochain complexes (Ωsm(t), d∗t ) and(Ωla(t), d∗t ).

We show that under a mild hypotheses, generically satisfied, theinverse Laplace transform of (Ωsm(t), d∗(t)) determines completelyNovikov’s counting of the instantons of the vector field X (orga-nized as a Dirichlet series, see Theorem 3). The inverse Laplacetransform of the analytic torsion of (Ωla(t), d∗(t)) properly cor-rected, determines the counting function of closed trajectories ofX, see Theorem 4.

Contents

1. Introduction 21.1. The Witten–Laplacian 21.2. Novikov complex and integration 31.3. A few relevant smooth real valued functions 61.4. A geometric invariant associated with (ω, g) 71.5. Topology and dynamics 81.6. Dirichlet series and their Laplace transform 102. Topology of the space of trajectories and stable/unstable

sets 123. Proof of Theorems 2 and 3 and few additional remarks

about Int∗X,ω,O 184. Homotopy between gradient like vector fields 22

Date: October 21, 2003.2000 Mathematics Subject Classification. PPPPP, SSSSS.Key words and phrases. Morse–Novikov theory, Dirichlet series.The second author is supported by the ‘Fonds zur Forderung der wis-

senschaftlichen Forschung’ (Austrian Science Fund), project number P14195-MAT.1

2 DAN BURGHELEA AND STEFAN HALLER

5. The regularization R(X,ω, g) 286. Proof of Theorem 4 336.1. A summary of Hutchings-Pajintov results 336.2. The geometry of closed one form 376.3. Recallection of results from [1] and [3] 406.4. Proof of Theorem 4 407. The invariant ρ 42Appendix A. 46Appendix B. 47Appendix C. 48References 48

1. Introduction

NOTE: This manuscript is ”WORK IN PROGRESS”. Many signshave to be corrected and better proofs are yet to be provided .

1.1. The Witten–Laplacian. Let (M,ω, g) be a system consisting ofa closed manifold M , a Riemannian metric g and a closed one form ω.We suppose that ω is a Morse form. This means that locally ω = dh, hsmooth function with all critical points nondegenerate. A critical pointor a zero of ω is a critical point of h and since nondegenerate, has anindex, the index of the Hessian d2

xh, denoted by ind(x).Denote by X be the set of critical points of ω and by Xq be the subset

of critical points of index q.For t ∈ [0,∞) consider the complex (Ω∗(M), d∗ω(t)) with differential

dqω(t) : Ωq(M)→ Ωq+1(M) given by

dqω(t)(α) := dα + tω ∧ α.Clearly dqω(0) = dq is the usual exterior differential. Using the Rie-mannian metric g one constructs the formal adjoint of dqω(t), (dqω(t))] :Ωq+1(M)→ Ωq(M), and defines the Witten–Laplacian ∆q

ω(t) : Ωq(M)→Ωq(M) (associated to the closed 1-form ω) by:

∆qω(t) := (dqω(t))] · dqt + dq−1

ω (t) · (dq−1ω (t))].

Thus, ∆qω(t) is a second order differential operator, with ∆q

ω(0) =∆q, the Laplace–Beltrami operator. The operators ∆q

ω(t) are elliptic,selfadjoint and nonnegative, hence their spectra Spect ∆q

ω(t) lie in theinterval [0,∞).

It is not hard to see that

∆qt = ∆q + t(L+ L]) + t2||ω||2 Id

LAPLACE TRANSFORM, TOPOLOGY AND SPECTRAL GEOMETRY 3

where L denotes the Lie derivative along the vector field − gradg ω, L]

the formal adjoint of L and ||ω||2 is the fiberwise norm of ω.The following result extends a result due to E. Witten (cf. [20]) in

the case that ω is exact and its proof was sketched in [6].

Theorem 1. Let ω be a Morse form. Then there exist constantsC1, C2, C3, T > 0 so that for t > T we have:

(i) Spect ∆qω(t) ∩ [C1e

−C2t, C3t] = ∅.(ii) ]

(Spect ∆q

ω(t) ∩ [0, C1e−C2t]

)= ](Xq).

(iii) 1 ∈ (C1e−C2t, C3t).

Here ] denotes cardinality of the set in parenthesis. This Theoremcan be complemented with the following proposition proven also byMityagin and Novikov (cf. [15]) whose proof is included in Appendix B.

Proposition 1. For all but finitely many t, dim(ker ∆ω(t)) is constantin t.

Denote by Ω∗sm(M)(t) the R–linear span of the eigenforms whichcorrespond to eigenvalues smaller than 1 and referred to bellow as thesmall eigenvalues. Denote by Ω∗la(M)(t) the orthogonal complement ofΩ∗sm(M)(t) which, by elliptic theory, is a closed subspace of Ω∗(M) withrespect to C∞ topology (in fact, with any Sobolev) topology. The spaceΩ∗la(M)(t) is the closure of the span of the eigenforms which correspondto eigenvalues larger than one.

The immediate consequence of Theorem 1 is that for t > T we have:

(Ω∗(M), dω(t)) =(Ω∗sm(M)(t), dω(t)

)⊕(Ω∗la(M)(t), dω(t)

)(1)

With respect to this decomposition the Witten–Laplacian is diagonal-ized

∆qω(t) = ∆q

ω,sm(t)⊕∆qω,la(t). (2)

and by Theorem 1(ii), we have

dim(Ωq(M)sm(t)) = ](Xq)for any t > T .

1.2. Novikov complex and integration. Consider X a vector fieldwhich is (−ω, g) gradient like, i.e. X := − gradg ω in some neighbor-hood of X and ω(X)(x) < 0 for any x ∈ M \ X . Here X is the set ofthe rest points of X, equivalently of the zeros of X, equivalently of thecritical points of ω. The (−ω, g) gradient like vector fields are actuallygradients of −ω with respect to some metric g′ which agrees to g in aneighborhood of X , but it is convenient to regard them in this way. InAppendix A we prove the following


Proposition 2. If X is (−ω, g) gradient like then one can modify theRiemannian metric g into g′ with g′ equal to g in a small neighborhoodof X so that X = − gradg′ ω.

Let Φt be the flow of X. Then for any x ∈ Xq consider

W±x :=

y ∈M

∣∣ limt→±∞

Φt(y) = x,

the stable/unstable set of X associated with the critical point x. Ob-serve that the set W±

x has a unique structure of smooth manifold sothat the inclusion i±x : W±

x → M is a smooth one to one immersion.The smooth manifold W+

x resp. W−x is diffeomorphic to Rn−q resp. Rq

with q = ind(x).Denote by gx the pullback of g onW−

x and by hx : W−x → (−∞, 0] the

unique smooth function which satisfies hx(x) = 0 and dhx = (i−x )∗(ω).Introduce ρ(ω,X, g) ∈ [0,∞] defined by

ρ(ω,X, g) := inft ∈ [0,∞]

∣∣∣ ∀x ∈ X :

∫W−x

ethx dvolgx <∞, (3)

and observe that it is independent of g and independent of the form ωup to its cohomology class, see section 7. So ρ defined by (3) providesan invariant ρ([ω], X) ∈ [0,∞], cf. Section 7.1

In view of this observation we make the following

Definition 1. A vector field X is −ξ gradient like, ξ ∈ H1(M,R) ifξ contains a closed one form ω so that X is (−ω, g) gradient like forsome Riemannian metric g, or equivalently if X = − gradg′ ω for someRiemannian metric g′ with ω a Morse form.

For such vector fields the invariant ρ(ξ,X) can be defined.

Definition 2 (Hypothesis MS). The −ξ gradient like vector field Xsatisfies MS (Morse–Smale) property if for any x, y ∈ X the maps i−xand i+y are transversal.

It is well known that generically this is always the case and we have

Proposition 3. If X is (−ω, g) gradient like then one can find aMorse–Smale vector field X ′ arbitrary closed to X in the C1–topologysuch that:

(i) X ′ is (−ω, g) gradient like,(ii) X ′ equal to X in the complement of a given compact neighbor-

hood of X ,(iii) X ′ equal to X in some neighborhood of X.

1The invariant ρ([ω], X) can be defined for any vector field X whose all zeros arehyperbolic and nondegenerate and not only for (−ω, g) gradient like vector fields.


Note that:N1: X ′ also agrees with X in a smaller neighborhood of X .N1: If the Morse Smale condition is satisfied when checked in an

open set U and K is a compact subset of U the vector field X ′ withthe properties mentioned above can be chosen to be equal to X on K.

N3: Proposition 3 and the previous statements remain true if onerequire the Morse Smale condition be satisfied only for a specified col-lection of pairs of rest points.

The proof is similar with the proof of Proposition 1 in [6] and is leftto the reader.

We conjecture, cf. section 7 below, that for a −ξ gradient like Morse–Smale X we always have ρ(ξ,X) <∞.

A homotopy X from the vector fieldX1 to a vector fieldX2 consists ofa smooth one parameter family of vector fields Xt so that Xt = X1 fort in a neighborhood of (−∞,−1] and Xt = X2 for t in a neighborhoodof [1,∞). To the homotopy X one can associate the vector field Y onM × R given by Y (x, t) = X(x, t) + 1/2(t2 − 1) ∂

∂twhich is −p∗(ξ)

gradient like, cf section 4, and we define

ρ(ξ,X) := ρ(p∗(ξ), Y ). (4)

If X is a −ξ gradient like vector field we also consider

ρ(ξ,X) := sup(ρ(ξ,X), infXρ(ξ,X))

where X is a homotopy from X to a −ξ gradient like vector field ofthe form − gradg f where f : M → R is a Morse function and g aRiemannian metric.

We expect that if X is a −ξ gradient like vector field which is MSthen ρ(ξ,X) = ρ(ξ,X).

Choose O = Ox, x ∈ X a collection of orientations of the unstablemanifolds of the critical points, with Ox orientation of W−

x .When ρ([ω], X) <∞ and for t > ρ([ω], X) denote by

IntqX,ω,O(t) : Ωq(M)→ Maps(Xq,R)

the linear map defined by:

IntqX,ω,O(t)(a)(x) :=

∫W−x

ethx(i−x )∗a, a ∈ Ωq(M). (5)

The integral is absolutely convergent in view of (3).

Theorem 2. Suppose X is a (−ω, g) gradient like vector field whichsatisfies MS and suppose O is a set of orientations as above. Assumeρ([ω], X) <∞.


Then there exists T ′ > ρ([ω], X),2 so that for t ≥ T ′ and any q, thelinear map IntqX,ω,O(t) defined by (5), when restricted to Ωq(M)sm(t),is an isomorphism and an O(1/t) isometry with respect to the scalarproduct induced by g on Ω∗(M) and the ”obvious scalar product” onMaps(Xq,R).3 In particular Ωq

sm(M)(t) has a canonical base EOx (t), x ∈X with EOx (t) = (IntqX,ω,O(t))−1(Ex), where Ex ∈ Maps(Xq,R) is thecharacteristic function of x ∈ Xq.

As a consequence we have

dq−1ω (EOy (t)) =:

∑x∈Xq

IO(x, y)(t)EOx (t), (6)

where IO(x, y) : [T ′,∞) → R are smooth (actually analytic cf. Theo-rem 3 below) functions.

Note. These results can be routinely extended to the case of compactbordisms (M,∂M−, ∂M+), cf section 2.

1.3. A few relevant smooth real valued functions. In additionto the functions IO(x, y)(t) defined for t ≥ T ′, cf. (6), we consider alsothe function

logV(t) = logV(ω, g,X)(t) :=∑q

(−1)q log VolEx(t)|x ∈ Xq (7)

Observe that the change in the orientations O does not change theright side of (7), so O does not appear in the notation V(t).4

The cochain complex (Ω∗la(M)(t), dω(t)) is acyclic and by Theorem 1(ii)of finite codimension in the elliptic complex (Ω∗(M), dω(t)). Thereforewe can define the function

log Tan,la(t) = log Tan,la(ω, g, t) := 1/2∑q

(−1)q+1q log det ∆qω,la(t),

where det ∆qω,la(t) is the zeta-regularized product of all eigenvalues of

∆qω,la(t) larger than one 5. This quantity will be referred to as the ”large

torsion”. More about torsion will be said in section 6.

2and larger than T provided by Theorem 13the scalar product which makes orthonormal the base provided by the charac-

teristic functions of the critical points4It might be noticed that V(t) is the torsion of the ‘mapping cone’ of Int∗X,ω,O :

Ω∗sm(M)→ Maps(X ,R), which is an acyclic complex of finite dimensional Euclideanspaces.

5which by the ellipticity are all eigenvalues of ∆qω(t) but finitely many


1.4. A geometric invariant associated with (ω, g). Recall thatMathai–Quillen [12] (cf. also [1]) have introduced a form Ψ(g) ∈ Ωn−1(TM\M ;OM) for any (M, g) an n–dimensional Riemannian manifold. Forany closed one form ω on M we consider the n–differential form ω ∧X∗(Ψ(g)) ∈ Ωn(M\X ;OM). Here X is regarded as a map X : M\X →TM \M and M is identified with the image of the zero section of thetangent bundle.

The integral ∫M\X

ω ∧X∗(Ψ(g)).

The above integral is in general divergent, however it does have a reg-ularization defined by the formula

R(X,ω, g) := (−1)n+1

(∫M

ω0 ∧X∗(Ψ(g)) +

∫M

fE(g)−∑x∈X

(−1)ind(x)f(x)

),

(8)where

(i) f is a smooth function whose differential df is equal to ω in asmall neighborhood of X and therefore ω0 := ω − df vanishesin a small neighborhood of X .

(ii) E(g) ∈ Ωn(M) is the Euler form associated with g.

It will be shown in section 5 cf below that the definition is indepen-dent of the choice of f , cf also [7]. Finally we introduce the function

log Tan(t) := log Tan(X,ω, g, t), defined by the formula:

log Tan(X,ω, g, t) := log Tan,la(ω, g, t)− logV(t)− tR(X,ω, g) (9)

and referred to as as the corrected large torsion.The main results of this paper, Theorems 3 and 4, state that if

ρ([ω], X) < ∞ the smooth functions IO(x, y)(t) resp. log Tan(ω, t) arerestrictions of holomorphic functions on a half plane z ∈ C | <z > R,where R a positive real number (larger than ρ([ω], X)) which haveinverse Laplace transforms. Their inverse Laplace transforms are thecounting functions for the instantons6 from x ∈ Xq to y ∈ Xq−1 resp.for the closed trajectories of X.

To formulate precisely these results we need a few additional defini-tions, like Dirichlet series, and counting functions for instantons andclosed trajectories.

6A trajectory γ(t) from x ∈ X to y ∈ X is called instanton if it is isolated inthe space of trajectories from x to y. Under the Hypothesis MS the existence of aninstanton from x to y implies that indx = ind y + 1.


1.5. Topology and dynamics. The closed one form ω induces thehomomorphism [ω] : H1(M,Z) → R and then the injective group ho-momorphism

[ω] : Γ := H1(M,Z)/ ker[ω]→ R.

For any two points x, y ∈ M denote by Px,y the space of continuouspaths from x to y and by ω : Px,y → R the map defined by

ω(α) :=

∫[0,1]

α∗ω, α ∈ Px,y.

We say that α ∈ Px,y is equivalent to β ∈ Px,y, iff the closed pathβ−1 ?α7 represents an element in ker([ω]) or equivalently iff

∫[0,1]

α∗ω =∫[0,1]

β∗ω. We denote by Px,y the set of equivalence classes of elements

in Px,y and observe that ω induces the injective map

[ω] : Px,y → R.

Clearly Px,y depends only on the cohomology class of ω.8 One can seethat Γ acts freely and transitively (both from the left and from the

right) on Px,y. The action ? is defined by juxtaposing at x resp. y aclosed curve representing an element γ ∈ Γ to a path representing theelement α ∈ Px,y. We have

[ω](γ ? α) = [ω](γ) + [ω](α)

[ω](α ? γ) = [ω](α) + [ω](γ)

Recall that a trajectory9 γ from x ∈ X to y ∈ X is called instantonif it is isolated in the space of trajectories from x to y.

Proposition 4. Suppose X satisfies the Hypothesis MS. If instantonsfrom x ∈ Xq to y ∈ Xr exist then q − r = 1 and the set of instantons

from x to y in each class α ∈ Px,y is finite.

Once a collection of orientations O is given any instanton γ fromx ∈ Xq to y ∈ Xq−1 has a sign ε(γ) = ±1 defined as follows: Theorientations Ox and Oy induce an orientation on γ. Take ε(γ) = +1

7Here ? denotes the juxtaposition of paths. Precisely if α, β : [0, 1] → M ,β(0) = α(1), then β ∗α : [0, 1]→M is given by α(2t) for 0 ≤ t ≤ 1/2 and β(1− 2t)for 1/2 ≤ t ≤ 1.

8An equivalence relation in Px,y which lead to Px,y can be defined for any cov-ering π : M → M not necessary principal cf. section 2. In the above situation thecohomology class of the closed one form ω defines a principal Γ–covering.

9A smooth map α : (−∞,∞)→M is a trajectory from x to y if it is of the formα(t) = Φt(γ(0)) with limt→±∞ γ(t) = x/y.


if this orientation is compatible with the orientation ‘from x to y’ andε(γ) = −1 otherwise. One defines the integer

IO(x, y)(α) :=

∑γ∈α

ε(γ). (10)

We call the function IO(x, y) : Px,y → Z the counting function of theinstantons from x to y. Note that by changing the orientations O thefunction IO(x, y) might change but only up to a sign.

Definition 3. A closed trajectory is a pair θ := (θ, T ), where T is apositive real number, and θ : R → M is a trajectory θ(t) := Ψt(x)so that θ(t + T ) = θ(t). (Then θ(t) = Ψt(x) defines a smooth mapθ : S1 = R/TZ → M .) The closed trajectory (Ψt(x), T ) is nonde-generate iff Dx(ΨT ) : TxM → TxM is invertible with the eigenvalue1 of multiplicity one, equivalently iff the map ΨT : M → M × Mdefined by ΨT (x) := (x,ΨT (x)) is transversal to the diagonal map∆ : M →M ×M defined by ∆(x) = (x, x) at x ∈M .

Definition 4 (Hypothesis NCT). The vector field X satisfies NCT(nondegenerate closed trajectories) property if all closed trajectories ofX are nondegenerate.

Proposition 5. If X satisfies both MS and NCT properties then forany γ ∈ Γ = H1(M : Z)/ ker[ω], the set of closed trajectories represent-ing the class γ, is finite 10.

Any nondegenerate closed trajectory θ has a period p(θ) ∈ N and a

sign ε(θ) := ±1 defined as follows:

(i) p(θ) is the largest positive integer p such that θ : S1 → Mfactors through a self map of S1 of degree p.

(ii) ε(θ) := sign det(Dx(ΨT )).

We write

ZX(γ) :=∑θ∈γ

(−1)ε(θ)

p(θ)∈ Q.

The function ZX : Γ → Q will be called the ”counting function forclosed trajectories”.

Theorems 3 and 4 below will show how to use spectral geometry todetermine the counting functions IO(x, y) : Px,y → Z ⊂ C and ZX :Γ → Q ⊂ C. To explain the results we will need few considerationsabout Dirichlet series and their Laplace transform.

10This fact is implied by Proposition 6 2)below


1.6. Dirichlet series and their Laplace transform. A Dirichletseries f is given by a pair of finite or infinite sequences(

λ1 < λ2 < · · · < λk < λk+1 · · ·a1 a2 · · · ak ak+1 · · ·

)with the first sequence, a sequence of real numbers, the second se-quence, a sequence of nonzero complex numbers, with the propertythat λk →∞ if the sequences are infinite. The associated series

L(f)(z) :=∑i

e−zλiai

has an abscissa of convergence ρ(f) ≤ ∞, characterized by the follow-ing properties:

(i) If <z > ρ(f) then f(z) is convergent and defines a holomorphicfunction

(ii) If <z < ρ(f) then f(z) is divergent. cf. [18] and [19].

A Dirichlet series can be regarded as a complex valued measure withsupport on the discrete set

λ1 < λ2 < · · · < λk < λk+1 < · · ·,

where the ‘measure’ of λi equal to ai, in which case the above series isthe Laplace transform of this measure cf. [19]. The following Proposi-tion is a reformulation of results which lead to the Novikov theory andto the work of Hutchings–Lee and Pajintov etc.

Proposition 6.

(i) (Novikov) If X is MS then for any x ∈ Xq and y ∈ Xq−1 thecollection of pairs of numbers(

[ω](α), IO(x, y)(α)) ∣∣∣ [ω](α) 6= 0, α ∈ Px,y

defines a Dirichlet series. The sequence of λ’s consists of thenumbers [ω](α) when IO(x, y)(α) is nonzero, and the sequencea’s consists of the numbers IO(x, y)(α) ∈ Z.

(ii) (D. Fried, M. Hutchings) If X is MS and NCT then the col-lection of pairs of numbers(

[ω](γ),ZX(γ)) ∣∣∣ γ ∈ Γ, [ω](γ) 6= 0

defines a Dirichlet series. The sequence of λ’s consists of thereal numbers [ω](γ) when ZX(γ) is nonzero and the sequenceof a’s consists of the numbers ZX(γ) ∈ Q.

We continue to denote these Dirichlet series by IO(x, y) and ZX .The main results of this paper are Theorems 3 and 4 below.


Theorem 3. Suppose ρ([ω], X) < ∞. Then if X is an (−ω, g) gra-dient like vector field which satisfies MS then there exists a positivereal number R > ρ([ω], X), so that for any x ∈ Xq, y ∈ Xq−1 thefunction IO(x, y)(t) is the restriction of a holomorphic function onz ∈ C | <z > R which has inverse Laplace transform. The in-verse Laplace transform is exactly the Dirichlet series IO(x, y) definedabove.

NOTE: Theorem 3 will be proven here under the hypothesis thata slightly larger invariant tildeρ([ω], X) < ∞ We hope to show in animproved version of this paper that tildeρ([ω], X) < ∞ = ρ([ω], X) <∞.

Theorem 4. Suppose ρ([ω], X) < ∞. If X is (−ω, g) gradient likeand satisfies MS and NCT, then there exists a positive real numberR > ρ([ω], X) so that the function log Tan(t), cf. (9), is the restriction ofa holomorphic function on z ∈ C | <z > R which has inverse Laplacetransform. The inverse Laplace transform is exactly the Dirichlet seriesZX defined above. If H∗(M, t[ω]) is trivial for t large then the sameconclusion holds under the (apparently)weaker hypothesis, ρ(ξ,X) <∞.

Remark 1.

(i) Both collections of numbers IO(x, y) and ZX can be definedunder weaker hypothesis than MS and NCT, however with-out these hypotheses we were unable to check that they areDirichlet series.

(ii) The Dirichlet series ZX depends only on X and ξ = [ω], whileIO(x, y) depends only on X and ξ up to multiplication with

a constant (with a real number r for the sequence of λ’s andwith ε = ±1 for the sequence of a’s.

Corollary 1 (J. Marcsik cf. [11]). Suppose X is a −ξ gradient likevector field, ξ ∈ H1(M,R), ω a closed one form representing ξ and ga Riemannian metric. Suppose all closed trajectories of X are nonde-generate and denote by

log Tan(t) := 1/2∑

(−1)q+1q log Det(∆qω(t)).

Then

log Tan(t) = (−1)n+1t

∫M

ω ∧X∗(Ψ(g))

is the Laplace transform of the Dirichlet series ZX which counts the setof closed trajectories of X with the help of ξ.


Strictly speaking J. Marsick proved the above result (see [11]) in thecase X = − gradg ω. The same arguments could also yield the resultin the generality stated above.

Remark 2. In case that M is the mapping torus of a diffeomorphismφ : N → N , M = Nφ whose periodic points are all nondegenerate,the Laplace transform of the Dirichlet series ZX is the Lefschetz zetafunction Lef(Z) of φ, with the variable Z replaced by e−z.

All Theorems 3,4 and Corollary 1 can be routinely extended to thecase of a compact bordism as considered in section 2.

The proof of Theorem 1 as stated is contained in [6] and so is theproof of Theorem 2 but in a slightly different formulation and (in appar-ently) less generality. For this reason and for the sake of completenesswe will review and complete the arguments (with proper references to[6] when necessary) in section 3. The proof of Theorem 2 is based ona topological result, Theorem 5, which was also proven in [6]. A sig-nificant short cut in the proof and a slightly more general formulationmakes appropriate to have the proof reconsidered. This will be done insection 2. Section 3 contains the proof of the Theorems 2 and 3. Sec-tion 4 contains a few results of independent interest. More preciselywe discuss the morphism from the cochain complex C(X1, ω,O1)(t) toC(X2, ω,O2)(t) induced by a homotopy between the gradient like vec-tor fields X1 and X2 when X i are two −ξ gradient like vector fields andω realizes ξ. Section 5 treats the regularization (8) and is also of inde-pendent interest. These results will be used in the proof of Theorem 4which is presented in section 6. Section 7 contains a discussion of theinvariant ρ([ω], X). The proof of Theorem 4 as presented in this paperrelies on some previous work of Hutchings–Lee, Pajintov [10], [9], [17]and the work of Bismut–Zhang and Burghelea–Friedlander–Kappeler[1].

2. Topology of the space of trajectories and

stable/unstable sets

Let X be an (−ω, g) gradient like vector field on a closed manifoldM . Let π : M →M be a covering satisfying the following property (P):

M is connected and ω := π∗ω is exact. (P)

Denote by X the vector field X := π∗X, and by h a smooth functionso that dh = ω. We write X = π−1(X ) and Xq = π−1(Xq). Clearly

Cr(h) = π−1(Cr(ω)) are the zeros of X.Given x ∈ X let i+x : W+

x → M and i−x : W−x → M , be the one

to one immersions whose images define the stable and the unstable


sets of x with respect to the vector field X. The maps i±x are actually

smooth embeddings because X is gradient like for the function h, andthe manifold topology on W±

x coincide with the topology induced from

M . Clearly for any x with π(x) = x one can canonically identify W±x to

W±x and then we have π · i±x = i±x .We will suppose that X is MS, i.e. for any x, y ∈ X the maps i−x and

i+y are transversal, equivalently, for any x, y ∈ X the maps i−x and i+y are

transversal. This implies thatM(x, y) := W−x ∩W+

y is a submanifold of

M of dimension ind(x)−ind(y). The manifoldM(x, y) is equipped withthe action µ : R×M(x, y)→M(x, y), defined by the flow generated byX. If x 6= y the action µ is free and we denote the quotientM(x, y)/Rby T (x, y). The quotient T (x, y) is a smooth manifold of dimensionind(x) − ind(y) − 1, possibly empty, which, in view of the fact thatX ·h = ω(X) < 0 is diffeomorphic to the submanifold h−1(c)∩M(x, y),where c is any regular value of h with h(x) > c > h(y).

Note that if ind(x) ≤ ind(y), and x 6= y, in view the transversalityrequired by the Hypothesis MS, the manifold M(x, y) is empty.

An unparametrized broken trajectory from x ∈ X to y ∈ X , is anelement of the set

B(x, y) :=⋃

k ≥ 0, y0, . . . , yk+1 ∈ Xy0 = x, yk+1 = y

ind(yi) > ind(yi+1)

T (y0, y1)× · · · × T (yk, yk+1).

For any x ∈ X introduce the completed unstable set W−x defined by

W−x := W−

x ∪⋃

k ≥ 1, y0, . . . , yk ∈ Xy0 = x

ind(yi) > ind(yi+1)

T (y0, y1)×· · ·×T (yk−1, yk)×W−yk.

To study W−x we introduce the set B(x;λ) of unparametrized broken

trajectory from x ∈ X to the level λ ∈ R.

B(x;λ) :=⋃

k ≥ 0, y0, . . . , yk ∈ Xy0 = x

ind(yi) > ind(yi+1)

T (y0, y1)×· · ·×T (yk−1, yk)×(W−yk∩h−1(λ)).

Clearly, if λ > h(x) then B(x;λ) = ∅.


Since any broken trajectory of X intersects each level of h in atmost one point one can view the set B(x, y) resp. B(x;λ) as a subsetof C0

([h(y), h(x)], M

)resp. C0

([λ, h(x)], M

). One parametrizes the

points of a broken trajectory by the value of the function h on thesepoints. This leads to the following characterization (and implicitly to acanonical parameterization) of an unparameterized broken trajectoriy.

Remark 3. Let x, y ∈ X and set a := h(y), b := h(x). The parame-terization above defines a one to one correspondence between B(x, y)and the set of continuous mappings γ : [a, b] → M , which satisfy thefollowing two properties:

(i) h(γ(s)) = a+ b− s, γ(a) = x and γ(b) = y.(ii) There exists a finite collection of real numbers a = s0 < s1 <· · · < sr−1 < sr = b, so that γ(si) ∈ X and γ restricted to(si, si+1) has derivative at any point in the interval (si, si+1),and the derivative satisfies

dγ

ds(s) =

X

−ω(X)

(γ(s)

).

Similarly the elements of B(x;λ) correspond to continuous mappingsγ : [λ, b]→ M , which satisfies (i) and (ii) with ‘a replaced by λ’.11

We have the following

Proposition 7. For any x, y ∈ X and λ ∈ R, the spaces

• B(x, y) with the topology induced from C0([h(y), h(x)], M), and

• B(x;λ) with the topology induced from C0([λ, h(x)], M

)are compact.

Let i−x : W−x → M denote the map whose restriction to T (y0, y1) ×

· · · × T (yk−1, yk) × W−yk

is the composition of the projection on W−yk

with i−yk and let hx := hx i−x : W−x → R, where hx = h− h(x).

Recall that an n–dimensional manifold with corners P , is a para-compact Hausdorff space equipped with a maximal smooth atlas withcharts ϕ : U → ϕ(U) ⊆ Rn+, where Rn+ = (x1, x2, . . . , xn) | xi ≥ 0.The collection of points of P which correspond (by some chart andthen by any other chart) to points in Rn with exactly k coordinatesequal to zero is a well defined subset of P and it will be denoted byPk. It has a structure of a smooth (n− k)–dimensional manifold. Theunion ∂P = P1 ∪ P2 ∪ · · · ∪ Pn is a closed subset which is a topologicalmanifold and (P, ∂P ) is a topological manifold with boundary ∂P . The

11The condition γ(b) = y has to be ignored.


following theorem was proven in [6].12 In this section we will shortenthis proof.

Theorem 5. Let X be an (−ω, g) a gradient like vector field which sat-isfies MS and let π : M →M be a covering which satisfies property (P).Then:

(i) For any two rest points x, y ∈ X the smooth manifold T (x, y)has B(x, y) as a canonical compactification. Moreover B(x, y)has the structure of a compact smooth manifold with corners,and

B(x, y)k =⋃

8>>><

>>>:

y0, . . . , yk+1 ∈ Xy0 = x, yk+1 = y

ind(yi) > ind(yi+1)

9>>>=

>>>;

T (y0, y1)× · · · × T (yk, yk+1),

in particular B(x, y)0 = T (x, y).(ii) For any rest point x ∈ X , the smooth maps i−x : W−

x → Mand hx : W−

x → R− have canonical completions to the smooth

maps i−x : W−x → M and hx : W−

x → R− so that W−x has a

structure of a smooth manifold with corners, and

(W−x )k =

⋃y0, . . . , yk ∈ X

y0 = xind(yi) > ind(yi+1)

T (y0, y1)× · · · × T (yk−1, yk)×W−yk,

in particular (W−x )0 = W−

x . Moreover i−x is a smooth proper

map whose image is a closed set in M .(iii) The maps hx are proper.

Proof. First we verify Theorem 5 in the case that ω has degree ofrationality one in which case all critical values of h form a discretesubset of R. This is done in all details in [6] section 4 paragraphs 4.1–4.3.

Next, we observe that as long as part (i) and (ii) of Theorem 5are concerned the result depends on X and not on the form ω. Wealso observe that if Theorem 5(i) and (ii) are true for one coveringπ : M → M which satisfies (P) then they are true for the universalcovering which clearly satisfies (P) and therefore for any other coveringwhich satisfies (P).

12In [6] this result was proven for X = −gradgω. By Proposition 2 if X is (−ω, g)gradient like one can change g in g′ so that X = − gradg′ ω.


It is therefore sufficient to show that if X is an (−ω, g) gradient like,then it is also (−ω′, g) gradient like for some ω′, a closed one form ofdegree of rationality one. Indeed, choose U a neighborhood of X sothat ω|U = − gradg ω|U and N a compact neighborhood of M \U . As Xis (−ω, g) gradient like, it remains (−ω′, g) gradient like for any ω′ withω′ equal to ω on U and sufficiently C0–close to ω on N . In view of thefact that closed one forms of degree of rationality one which agree withω on a compact contractible neighborhood of X are dense in the spaceof all such closed one form with respect to any Cr–topology, r ≥ 0,one can choose ω′ of degree of rationality one so that X is an (−ω′, g)gradient like.

Note that once (i) and (ii) are verified, the map hx is clearly smooth,

(being equal to h · i−x ) and the inverse image of each closed interval[a, b] ⊂ R is compact hence (iii) is true.

It will be convenient to formulate Theorem 5 without any referenceto the covering π : M →M or to lifts x of rest points x.

Let ξ be a one dimensional cohomology class, ξ ∈ H1(M,R), so thatπ∗ξ = 0. As in section 1 denote by Px,y the set of continuous paths

from x to y and by Px,y the equivalence classes of paths in Px,y withrespect to the following equivalence relation.

Definition 5. Two paths α, β ∈ Px,y are equivalent if for some (and

then for any) lift x of x the lifts α and β of α and β originating fromx end up in the same point y.

The reader might note that the present situation is slightly moregeneral than the one considered in Introduction which correspond tothe case the covering π is the Γ–principal covering with Γ inducedfrom ξ as described in section 1. In this case the two definitions of Px,ycoincide.

Note that any two lifts x, y ∈ M determine an element α ∈ Px,yand the set of trajectories from x to y identifies to the set T (x, y, α) oftrajectories of X from x to y in the class α.

So Theorem 5 can be reformulated in the following way:

Theorem 6 (Reformulation of Theorem 5). Let M be a smooth man-ifold and X a smooth vector field which is −ξ gradient like (cf. Defini-tion 1, section 1) for ξ ∈ H1(M,R). Then:

(i) For any two rest points x, y ∈ X and α ∈ Px,y the set of tra-jectories from x to y in the class α have the structure of a


smooth manifold of dimension ind(x) − ind(y) − 1 which ad-mits a canonical compactification to a compact smooth mani-fold with corners, B(x, y, α). The k–corner of B(x, y, α) is

B(x, y, α)k =⋃

8>>>>><

>>>>>:

y0, . . . , yk+1 ∈ Xαi ∈ Pyi,yi+1

α0 ? · · · ? αk = αy0 = x, yk+1 = y

9>>>>>=

>>>>>;

T (y0, y1, α0)×· · ·×T (yk, yk+1, αk),

in particular B(x, y, α)0 = T (x, y, α).(ii) For any rest point x ∈ X , the smooth maps i−x : W−

x → Mand hx : W−

x → R− have canonical completions to the smooth

maps i−x : W−x → M and hx : W−

x → R− so that W−x has a

structure of a smooth manifold with corners, where

(W−x )k =

⋃8>>><

>>>:

y0, . . . , yk ∈ Xy0 = x

αi ∈ Pyi,yi+1

9>>>=

>>>;

T (y0, y1, α0)×· · ·×T (yk−1, yk, αk−1)×W−yk

and the restriction of i−x to T (y0, y1, α0)×· · ·×T (yk−1, yk, αk−1)×W−yk

is the projection on W−yk

followed by i−yk : W−yk→M .

(iii) The maps hx are proper.

The above results can be easily extended to the case of compactmanifolds with boundary. Given a compact smooth manifold M withboundary ∂M we will consider only ‘admissible metrics’ i.e. Riemann-ian metrics g which are ‘product like’ near the boundary, in which casedenote by g0 the induced metric on the boundary. This means thatthere exists a collar neighborhood ϕ : ∂M × [0, ε) → M with ϕ equalto the identity when restricted to ∂M × 0 and ϕ∗(g) = g0 + ds2.

Definition 6. The vector field X is called (−ω, g) gradient like, whereω is a closed one form and g an admissible metric on M if the followingconditions hold:

(i) X is (−ω, g) gradient like.(ii) The restriction X∂M of X to ∂M is a tangent vector field to

∂M and is a (−ω∂M , g0) gradient like, where ω∂M denotes thepullback of ω on ∂M .

(iii) If we set

X ′′− :=x ∈ X ∩ ∂M

∣∣ ind∂M(x) = ind(x)

X ′′+ :=x ∈ X ∩ ∂M

∣∣ ind∂M(x) = ind(x)− 1


then X ′′− resp. X ′′+ lie in different components ∂M− resp. ∂M+

of M .

This definition implies that X = X ′ tX ′′, where X ′ is the set of restpoints inside M and X ′′ of the rest points on ∂M which is the same asthe set of rest points of X∂M , and M is a bordism between two closedmanifolds, ∂M− and ∂M+. For x ∈ X ′′ denote by i−x : W−

x → M theunstable manifold with respect to X and by j−x : W−

∂M,x → ∂M theunstable manifold with respect to X∂M .

Remark 4.

(i) If x ∈ X ′′− then the unstable manifold of x with respect to Xand X∂M are the same. More precisely i−x : W−

x →M identifiesto j−x : W−

∂M,x → ∂M followed by the inclusion of ∂M ⊂M .(ii) If x ∈ X ′′+ then:

(a) (W−x ,W

−∂M,x) is a smooth manifold with boundary diffeo-

morphic to (Rk+,Rk−1) with k = ind(x),

(b) i−x : W−x →M is transversal to ∂M and i−x : (i−x )−1(∂M)→

∂M can be identified to j−x : W−∂M,x → ∂M .

Theorems 5 and 6 remain true as stated with the following specifi-cations. For x ∈ X ′′+ we have:

(W−x )0 = W−

x \W−∂M,x

and

(W−x )k = (W−

∂M,x)k−1∪

∪⋃

8>>><

>>>:

y0, . . . , yk ∈ Xy0 = x

αi ∈ Pyi,yi+1

9>>>=

>>>;

T (y0, y1, α0)× · · · × T (yk−1, yk, αk−1)× P−yk

where P−y := W−y \W−

∂M,y for y ∈ X ′′+, and P−y := W−y for y ∈ X ′ tX ′′−.

3. Proof of Theorems 2 and 3 and few additional

remarks about Int∗X,ω,O

We begin with an (−ω, g) gradient like vector field X which is MS.We choose a set of orientations O as in introduction (section 1) andsuppose that ρ([ω], X) < ∞. For t > ρ([ω], X) we consider the linearmaps

IntqX,ω,O(t) : Ωq(M)→ Maps(Xq,R)


defined in Section 1 by formula (5).Recall that in Section 1 we have defined the instanton counting func-

tion (or the Novikov incidence) IO(x, y) : Px,y → Z, cf. (10).The following proposition is a reformulation of a basic observation

made by S.P. Novikov [16] in order to define his celebrated complex.

Proposition 8.

(i) For any x ∈ Xq, y ∈ Xq−1 and every real number R the setα ∈ Px,y

∣∣ IOq (x, y)(α) 6= 0, [ω](α) ≥ R

is finite.(ii) For any x ∈ Xq, z ∈ Xq−2 and γ ∈ Px,z one has∑

8>>><

>>>:

y ∈ Xq−1

α ∈ Px,y, β ∈ Py,zα ? β = γ

9>>>=

>>>;

IO(x, y)(α) · IO(y, z)(β) = 0. (11)

Formula (11) implicitly states that the left side of the equality containsonly finitely many nonzero terms.

Proposition 8 above is equivalent to Theorem 2 parts 1 and 2 in [6].A proof can be also found in [6] and is originally due to Novikov.

The following proposition will be the main tool in the proof of The-orem 2.

Proposition 9. Suppose t ∈ R, s ∈ C and <(s), t > ρ([ω], X). Then:

(i) For every a ∈ Ωq(M) and every x ∈ Xq the integral

IntqX,ω,O(s)(a)(x) :=

∫W−x

eshx(i−x )∗a

converges absolutely,13 In particular

IntqX,ω,O(t)(a)(x) :=

∫W−x

ethx(i−x )∗a

defines a linear map IntqX,ω,O(t) : Ωq(M)→ Maps(Xq,R).(ii) The map IntqX,ω,O(t) : Ωq(M)→ Maps(Xq,R) is surjective.

13Recall that for an oriented n–dimensional manifold N and a ∈ Ωn(N) one has|a| := |a′|Vol ∈ Ωn(M), where Vol ∈ Ωn(N) is any volume form and a′ ∈ C∞(N,R)is the unique function satisfying a = a′ ·Vol. The integral

∫Na is called absolutely

convergent, if∫N|a| converges.


(iii) We have

Intq+1X,ω,O(t)(dω(t)a)(x) =

∑α∈Px,y ,y∈Xq

etω(α)IO(x, y)(α) IntqX,ω,O(t)(a)(y)

(12)where the right side of (12) is a potentially infinite sum whichis convergent.

The proof of Proposition 9 is given in details in [6] section 5, (cfProposition 4) and uses in an essential way Theorem 5 and Stokes’theorem. Particular care is necessary in view of the fact that W−

x arenot compact. The integration has to be performed on a non compactmanifold and Stokes’ theorem applied to non-compact manifolds withcorners.

The proof of Theorem 2 boils down to the verification of the followingclaims:

Claim 1. For any t > supρ([ω], X), T ′, the possibly infinite sum∑α∈Px,y I

Oq (x, y)(α)et[ω](α) is convergent and the formula

δX,ω,O(t)(Ey) =∑

x∈Xq+1,α∈Px,y

(IOq (x, y)(α)etω(α))Ex (13)

defines a linear map δX,ω,O(t) : Cq(X,O) → Cq+1(X,O) which makes(C∗(X,O), δ∗X,ω,O(t)) a smooth (actually analytic) family of cochaincomplexes of finite dimensional Euclidean spaces. Recall that Ex, x ∈X denotes the characteristic function of x and E ′xs provide thecanonical base of C∗(X,O) which, implicitly equips each componentCq(X,O) with a scalar product, the unique scalar product which makesthis base orthonormal.

Claim 2. The linear maps IntqX,ω,O, (t) are surjective and define a mor-phism of cochain complexes.

Claim 3. There exists T larger than ρ(ω,X) so that for t > T thelinear map IqX,ω,O(t) when restricted to Ωq(M)sm is an isomorphismand actually an O(1/t)–isometry.

Everything but the ‘O(1/t)–isometry’ statement in Claim 3 is astraightforward consequence of Theorem 1 and Proposition 9 above.To check this part of Claim 3 we have to go back to the proof of The-orems 3 and 4 in [6], section 6. We observe that if t is large enoughthe restriction of IntqX,ω,O(t) to the subspace H1(t) ⊂ Ωq(M) defined in[6], section 4 page 172 (cf. Proof of Theorem 3) is surjective and thenby Lemma 7 in [6] so is the restriction of IntqX,ω,O(t) to Ωq(M)sm(t).


This because H1(t) and Ωq(M)sm(t) by Lemma 7 in [6] section 6 are,for t large enough, as close as we want. Since the spaces Ωq(M)sm(t)and Cq(X,O) have the same finite dimension, by the ‘surjectivity’ inClaim 2, IntqX,ω,O(t) is an isomorphism and actually as shown in [6]

section 4 page 172 an O(1/t)–isometry. We take as the base EOx (t) thedifferential forms EOx (t) = IntqX,ω,O, (t)

−1(Ex). This finishes the proofof Theorem 2. Theorem 3 is a consequence of Theorem 2 and Claim 3.

We will finish this section with the following remarks: Theorem 3implies that for t > R the finite dimensional vector spaces

Cq(X,O) := Maps(Xq,R)

and the linear maps

δqX,ω(t) : Maps(Xq,R)→ Maps(Xq+1,R)

defined by

δX,ω(t)(Ex) :=∑α∈Py,x

IO(y, x)(α)etω(α)Ey

give rise to a cochain complex of finite dimensional vector spaces C(X,ω,O)(t) :=Cq(X,O), δX,ω(t).

In fact one can say a little more. If we denote by ρ(ξ,X)+supx,y ρx,ywhere ρx,y denote the abscissa of convergence of the Dirichlet series as-sociated to the counting functions IO(x, y), then for any complex num-ber z ∈ C (Cq(X,O) ⊗ C, δX,ω(z) with δX,ω(z) defined by ((13)) witht replaced by z is a holomorphic family of finite dimensional cochaincomplexes. Moreover ρ(ξ,X) < T ′ with T ′ as in Theorem 2.

Note also that one can replace ω by any ω′ closed one form inthe same cohomology class ξ = [ω] and obtain the cochain complex(Cq(X,O), δX,ω′(t) or (Cq(X,O)⊗C, δX,ω′(z). Given the fact that ω′ =ω+ df the collections of t for which δX,ω′(t)and δX,ω(t) are well defined(i.e convergent) are the same. Also Int∗Xω′,O(t) : (Ω∗(M), dω′(t)) →C(X,ω′,O)(t) defines a morphism of cochain complexes which inducesan isomorphism in cohomology.

Let X be a −ξ gradient like vector field. If ω1 and ω2 are two closedone forms in the same cohomology class ξ hence ω1−ω2 = df , f : M →R, a smooth function, define the morphisms of cochain complexes

m∗f (t) : (Ω∗(M), d∗ω1(t))→ (Ω∗(M), d∗ω2

(t))

and

s∗f (t) : C(X,O, ω1)(t)→ C(X,O, ω2)(t)

by the formulae

mqf (t)(a) := etfa, a ∈ Ωq(M)


and bysqf (t)(Ex) := etfEx, Ex ∈ Maps(Xq,R)

where Ex denotes the characteristic function of x ∈ Xq.The diagram

(Ω∗(M), dω1(t)m∗f (t)−−−→ (Ω∗(M), dω2(t)

Int∗X,O,ω1,(t)

y yInt∗X,O,ω2(t)

C∗(X,O, ω1)(t)s∗f (t)−−−→ C∗(X,O, ω2)(t)

is clearly well defined and commutative for t > ρ([ω], X): Indeedbecause h′x−hx = (f−f(x))·i−x is bounded, (h′x associated to ω′ and hxassociated to ω,)

∫Rq e

th′x(i−x )∗a is absolutely convergent iff∫Rq e

thx(i−x )∗ais.

4. Homotopy between gradient like vector fields

Let ξ be a one dimensional cohomology class, ξ ∈ H1(M,R), andπ : M →M be a covering so that π∗ξ = 0.

Definition 7.

(i) A smooth family of vector fields X := Xs, s ∈ [−1, 1], will becalled a homotopy from the vector field X1 to the vector fieldX2 if there exists ε > 0 so that Xs = X1 for s < −1 + ε andXs = X2 for s > 1− ε.

To a homotopy X := Xs one associates the vector field Y on thecompact manifold with boundary N := M × [−1, 1], defined by

Y (x, s) := X(x, s) + 1/2(s2 − 1)∂

∂s. (14)

With this notation we have the following

Proposition 10. If X is a homotopy between two −ξ gradient likevector fields then the vector field Y is −p∗ξ gradient like cf. Definition 6,where p : M × [−1, 1]→M is the canonical projection.

Proof. Since X i, i = 1, 2, are both −ξ gradient like we can chooseclosed 1–forms ωi representing ξ, and Riemannian metrics gi on M ,such that X i is (−ωi, gi) gradient like, i = 1, 2. Choose an admissibleRiemannian metric g on N inducing gi on the boundaries. This ispossible. For example one can choose

g := (1− λ)p∗g1 + λp∗g2 + ds2,


where λ : [−1, 1]→ R is a non-negative smooth function satisfying:

λ(s) =0for s ≤ −1 + ε

1for s ≥ 1− ε (15)

Next choose a closed 1–form ω on N which restricts to p∗ω1 onM × [−1,−1 + ε] and which restricts to p∗ω2 on M × [1 − ε, 1]. Thisis possible since ω1 and ω2 define the same cohomology class ξ. Moreexplicitly this can be achieved by choosing a function h on M withω2 − ω1 = dh and setting ω := p∗ω1 + d(λp∗h). Now choose a functionh : [−1, 1]→ R, such that:

• u(s) ≥ −12(s2 − 1) for all s ∈ [−1, 1].

• u(s) = −12(s2 − 1) for all s ≤ −1 + ε

2and for all s ≥ 1− ε

2.

• u(s) ≥−ω(Y )((x,s)

12

(s2−1)

for all s ∈ [−1 + ε, 1− ε] and x ∈M .

This is clearly possible since−ω(Y )((x,s)

12

(s2−1)

is strictly negative for

s = −1 + ε and s = 1− ε.Then ω := ω + u(s)ds represents the cohomology class p∗ξ and it is

straight forward to check that Y is (−ω, g) gradient like.

Let X1 and X2 be two −ξ gradient like vector fields which are MSand let X be a homotopy. Let Y be the vector field defined in (14).With the notations from section 2 (in the case of a compact manifoldwith boundary) we have

Y = Y ′′ = Y ′′− t Y ′′+with

Y ′′− = X 1 × −1 and Y ′′+ = X 2 × 1.

Definition 8. The homotopy X is called Morse–Smale if the vectorfield Y is MS, i.e. X1 and X2 are MS and for any y ∈ Y ′′− and z ∈ Y ′′+the maps i−y and i+z are transversal.

Proposition 11. Let X1 and X2 be two −ξ gradient like vector fieldsand X a homotopy from X1 to X2. Then there exists a Morse–Smalehomotopy X′ from X1 to X2, arbitrarily close to X in the C0–topology.

The proof follows from Proposition 3 (actually from Note after Propositin3). Indeed, first we modify the vector field Y into Y ′ by a small changein the C1− topology, and only in the neighborhood of M × 0, in or-der to have the Morse Smale condition satisfied for any x ∈ X”+ andx ∈ X”−. This can be done using Proposition 3. Unfortunately Y ′

might not be of the form required in order to be if obtained froma homotopy of vector fields. However, the I component of Y ′ is C1


closed from the I component of Y. Therefore, by multiplication witha function which is C1 close to 1 and equal to 1 on the complementof a small compact neighborhood of the locus where Y and Y ′ are notthe same, one obtains a vector field Y ” whose I component is exactly1/2(s2 − 1)∂/∂s. The M component of Y ” defines the desired homo-topy . Note that by mutiplying a vector field with a smooth positivefunction the stable and unstable sets do not change, and neither theirtransversality.

In view of Theorem 6 for compact manifolds with boundary we havethe following

Remark 5. For any y = (x,−1) ∈ Y ′′− the 1–corner of W−y is given by

∂1(W−y ) = V0 t V1 t V2 (16)

where

V0 ' W−x (17)

V1 '⋃

8<

:v ∈ Y ′′−, α ∈ Py,v

ind(v) = ind(y)− 1

9=

;

T (y, v, α)× (W−v \ ∂M) (18)

V2 '⋃

8<

:u ∈ Y ′′+, α ∈ Py,u

ind(u) = ind(y)− 1

9=

;

T (y, u, α)×W−u (19)

It is understood that W−x represents the unstable manifold if x ∈ X 2

in M = M × 1.In view of (14) we introduce the invariant ρ(ξ,X) ∈ [0,∞] for any

homotopy X by defining

ρ(ω,X, g) := inft ∈ [0,∞]

∣∣∣∀y ∈ Y :

∫W−y

etFy dvolgy <∞

(20)

where Fy : W−y → R+ is the unique smooth function with dFy =

(i−y )∗p∗ω and Fy(y) = 0. As previously observed in section 1 (cf. sec-tion 7) this is independent of g and independent of the form ω, up toits cohomology class. Hence we can put ρ(ξ,X) := ρ(ω,X, g). Notethat ρ(ξ,X i) ≤ ρ(ξ,X) = ρ(p∗ξ, Y ).

Suppose that X i, i = 1, 2, and X as above are Morse–Smale. Foreach X i choose the orientations Oi of the corresponding rest points.Observe that the set Px′,x identifies to P(x′,1),(x′,−1). The orientationsO1 and O2 define the orientations O for the unstable manifolds of therest points of Y . For x1 ∈ X 1, x2 ∈ X 2 and α ∈ Px1,x2 define theincidences

IO1,O2

(x2, x1)(α) := IO((x2, 1), (x1,−1)

)(α). (21)


Suppose in addition that supρ(ξ,X), ρ(ξ,X1), ρ(ξ,X2) = ρ(ξ, Y ) <∞. For any t > ρ(ξ,X) and ω a closed one form representing ξ definethe linear maps

uqω(t) := uqX1,X2,O1,O2,ω(t) : Cq(X1,O1)→ Cq(X2,O2)

and the linear maps

hqω(t) := hqω,X1,X2,O1,O2(t) : Ωq(M)→ Maps(Xq−1,R)

by

uqω(t)(Ex1) :=∑

8<

:x2 ∈ X 2

q

α ∈ Px1,x2

9=

;

IO1,O2

(x2, x1)(α)etω(α)Ex2 , x1 ∈ X 1q

(22)and

(hqω(t)(a))(x2) =

∫W−y

etFy(i−y )∗p∗a, x2 ∈ X 2q−1 and y = (x2, 1). (23)

The right side of (22) is a convergent infinite sum since is a sub sumof the right hand side of (13) when applied to the vector field Y .

Proposition 12. Suppose X1, X2 are two −ξ gradient like vector fieldswhich are Morse–Smale and X is a (−ξ gradient like) Morse–Smalehomotopy. Suppose that ρ(ξ,X), ρ(ξ,X1), ρ(ξ,X2) < ρ < ∞. Supposeω is a closed one form which realizes ξ and t > ρ. Then:

(i) The collection of linear maps uqω(t) define a morphism ofcochain complexes:

u∗ω(t) := u∗X1,X2,O1,O2,ω(t) : C(X1,O1, ω)(t)→ C(X2,O2, ω)(t)

(ii) The collection of linear maps hqω(t) define an algebraic homo-topy between Int∗X2,O2,ω(t) and u∗X1,X2,O1,O2,ω(t) Int∗X2,O2,ω(t).Precisely, we have:

h∗+1ω (t) d∗ω(t) + δ∗−1

ω (t) h∗ω(t) = u∗ω(t) Int∗X1,O1,ω(t)− Int∗X2,O2,ω(t)


Proof. Statement (i) follows from the following equality:∑8>>><

>>>:

z′ ∈ X 2q

α ∈ Pz′,x, β ∈ Pz,z′β ? α = γ

9>>>=

>>>;

IO1,O2

(z′, x)(α)IO2

(z, z′)(β)+

+∑

8>>><

>>>:

x′ ∈ X 1q+1

α ∈ Px′,x, β ∈ Pz,x′β ? α = γ

9>>>=

>>>;

IO1,O2

(z, x′)(β)IO1

(x′, x)(α) = 0

for any x ∈ X 1q , z ∈ X 2

q+1 and γ ∈ Py,x which is a reinterpretationof equation (11) when applied to the vector field Y , the rest points

(x,−1) and (z, 1) and γ ∈ Pz,x = P(z,1),(x,−1). To verify statement (ii)we first observe that:

(a) If y, u ∈ Y the restriction of Fy to T (y, u)(α)×W−u , α ∈ Py,u,

when this set is viewed as a subset of W−y is given by

Fu · PrW−u + ω(α).

(b) If y = (x,−1), x ∈ X 1 via the identification of W−x to W−

y , wehave Fy = hx.

In view of the uniform convergence of all integrals which appear inthe formulae below, guaranteed by the hypothesis t > ρ, the Stokestheorem for manifolds with corners gives for any a ∈ Ωq(M) and y ∈(Y ′′+)q ∫

W−y

d(etFyc) =

∫V0

etFyc+

∫V1

etFyc+

∫V2

etFyc, (24)

where c := (i−y )∗p∗a ∈ Ωq(W−y ).

In view of the Remark 5 we have∫V0

etFyc = IntqX2,O2,ω(t)(a), (25)

∫V2

etFyc =∑

8<

:u ∈ Y ′′+, α ∈ Py,u

ind(u) = ind(y)− 1

9=

;

IO(y, u)(α)etω(α)

∫W−u

eFu (i−u )∗p∗a

= −(δq−1ω (t) Hq

ω(t))(a) (26)


and∫V1

eFyc = −∑

8<

:v ∈ Y ′′−, α ∈ Py,v

ind(v) = ind(y)− 1

9=

;

IO(y, v)(α)eω(α)

∫W−v

eFv(i−v )∗p∗a

= −uω(t) IntX1,O1,ω(t)(a). (27)

Since

(hq+1ω (t)dω(t)(a))(y) =

∫W−y

etFy(i−y )∗p∗(db+ tω ∧ b)

=

∫W−y

d(etFy(i−y )∗p∗b) (28)

the statement follows combining the equalities (24)–(28).

The following proposition will be important in the proof of Theo-rem 4.

Proposition 13.

(i) Given X a −ξ gradient like vector field, there exists a −ξ gra-dient like vector field X ′, which is MS, a −ξ gradient like ho-motopy X and a Morse function h : M → R so that X ′ is(−dh, g) gradient like for some metric g. If X was MS onecan chose X to be MS.

It is very likely that if ρ(ξ,X) < ρ one can choose X so that ρ(ξ,X) <ρ.

Proof of (i). Let ω be a Morse closed one form and g a Riemannianmetric so that X = − gradg(ω). This is possible by Proposition 2.Suppose first that there exists a Morse function h : M → R so that:the set of critical points of index q of the function h and the set Xqare the same, and in a small neighborhood of X ω and dh are equal.(This of course is not always possible.) The vector field X ′ will be ofthe form − gradg(ω + Tdh) with T chosen a big enough positive realnumber to make sure that for any x ∈M \ X one has

|| gradg(ω + Tdh)(x)|| > 0. (29)

To explain the choice of T note first that we can choose an openneighborhood U of X so that ω|U = dh|U and outside of an openneighborhood V , whose closure V is contained in U , || gradg h|| > C >

0. Denote by C ′ := supx∈M\V || gradg(ω)(x)||. For any T so that T > C′

C

equation (29) holds. If − gradg h is MS so will be (1/T )X + gradg hfor T large enough, given the ‘openness’ of the MS condition. Finally


choose T so that both requirements hold. To define the homotopyconsider the function λ(s) defined by (15). Define the smooth familyof closed one forms

ωs := ω + λ(s)Tdh, (30)

and choose as homotopy the family of vector fields

Xs = gradg(ωs). (31)

If the above homotopy is not MS one can correct it by using Proposi-tion 11.

So it remains to show that starting with a the vector field − gradg(ω)we can modify it up to homotopy into one which satisfies our assump-tion. This is straightforward and can be done inserting pairs of criticalpoints of index q and q+1 The modification is the same as the insertionof such critical points for a Morse function as described for example inMilnor [13].

Proof of (??). NOT YET DONE!

5. The regularization R(X,ω, g)

Recall from section 1 that for a Riemannian metric g we have con-sidered the Mathai–Quillen form Ψ(g) ∈ Ωn−1(TM \M). The Mathai–Quillen form (see [12]) is actually associated to a a pair ∇ = (∇, µ)consisting of a connection and a parallel Euclidean structure in a vec-tor bundle π : E → M of rank k and is a degree (k − 1) differ-ential form with values in the pull back of the orientation bundle,Ψ(∇) ∈ Ωk−1(E \ 0M ; π∗o(E)). Here 0M denotes the zero section inthe bundle E whose image identifies to M and o(E) the orientationbundle of E. The form Ψ(g) ∈ Ωn−1(TM \M ; π∗o(M)) = Ψ(∇g) is

associated to E = TM and the Levi–Civita pair ∇g induced by the Rie-mannian metric g. Here we write o(M) := o(TM) for the orientationbundle of TM . The Mathai–Quillen form has the following properties:

(i) For the Euler form E(∇) ∈ Ωk(M ; o(E)) associated to ∇ wehave

dΨ(∇) = π∗(E(∇)).

(ii) For two ∇1 and ∇2 we have

Ψ(∇2)−Ψ(∇1) = π∗(cCS(∇1, ∇2)) modulo exact forms,

where cCS(∇1, ∇2) ∈ Ωk−1(M ; o(E))/dΩk−2(M ; o(E)) is theChern–Simon invariant.


(iii) For every x ∈ M the form −Ψ(∇) restricts to the standardgenerator of Hn−1(Ex \0;π∗o(E)), where Ex = π−1(x) denotesthe fiber over x ∈M . Note that it is closed by (i).

(iv) Suppose E = TM , ∇g is the Levi–Civita pair, and supposethat on the open set U we have coordinates x1, . . . , xn in whichthe Riemannian metric g|U is given by gij = δij. Then, withrespect to the induced coordinates x1, . . . , xn, ξ1, . . . , ξn on TU ,the form Ψ(g) is given by

Ψ(g) =Γ(n/2)

2πn/2

∑i

(−1)iξi(∑

j(ξj)2)n/2dξ1 ∧ · · · ∧ dξi ∧ · · · ∧ dξn,

cf. [12].

Let X be a vector field on M , i.e. a section X : M → TM of thetangent bundle π : TM → M . We suppose that it has only isolatedzeros, i.e. X := X−1(0M) is a discrete subset of M . The vector fielddefines an integer valued map IND : X → Z, where IND(x) denotesthe Hopf index of the vector field X at x. This integer IND(x) is thedegree of the map (U,U \x)→ (TxM,TxM \0) obtained by composingX : U → TU with the projection p : TU → TxM induced by a localtrivialization of the tangent bundle near x ∈ M . Here U denotes asmall disc around x in M .

Choose coordinates around x so that we can speak of the disc withradius ε > 0 around x. It is well known that we have

IND(x) = − limε→0

∫∂Uε

X∗(Ψ(g)) (32)

where ∂Uε is oriented as the boundary of Uε. Indeed, by (ii) we mayassume that g is flat on Uε, thus E(g) = 0 and Ψ(g) is closed on Uεby (i). Using (iii) we see that −Ψ(g) gives the standard generator ofHn−1(E|U \ 0U ; π∗o(E)) and thus certainly IND(x) = −

∫∂εU

X∗(Ψ(g)).

The vector field X has its rest points (zeros) nondegenerate andin particular isolated, if the map X is transversal to the zero section0M , in which case X is an oriented zero dimensional manifold, whoseorientation is specified by IND(x). In this case is we have

IND(x) = sign det(∇X : TxM → TxM) ∈ ±1,

where the Hessian ∇X : TxM → TxM is independent of the affineconnection ∇. Particularly if there exist coordinates x1, . . . , xn so that

X = −k∑i=1

xi∂/∂xi +n∑

i=k+1

xi∂/∂xi. (33)


we get IND(x) = (−1)k.Let X1 and X2 be two vector fields and X := Xs, s ∈ [−1, 1]

a homotopy. The homotopy is called nondegenerate if the map X :[−1, 1] ×M → TM defined by X(s, x) := Xs(x) is transversal to thezero section 0M . In this case necessarily X1 and X2 are vector fieldswith nondegenerate zeros and so are all but finitely many Xs. Moreoverall Xs have isolated zeros with indexes in 0, 1,−1 and the set X :=X−1(0M) is an oriented one dimensional smooth submanifold of [−1, 1]×

M .We have also the function IND : X → −1, 0, 1 defined by IND(s, x) :=

IND(x). It is not hard to see that in this case X has an unique ori-entation so that at each regular point the map p1 : X → [−1, 1], therestriction of the projection p1 : [−1, 1]×M → [−1, 1] has the Jacobianpositive/negative according to the value ±1 of the function IND.

Given a closed one form ω on M denote by

I(X, ω) :=

∫Xp∗2ω,

where p2 : X → M denotes the restriction of the projection [−1, 1] ×M →M .

Remark 6. Note that we have

∂X =∑y∈X 2

IND(y)y −∑x∈X 1

IND(x)x.

If X′ is a second homotopy joining X1 with X2 then X ′ − X is theboundary of a smooth 2–cycle. Indeed, if we choose a homotopy ofhomotopies joining X with X′ which is transversal to the zero section,then its zero set will do the job. Particularly I(X, ω) does not dependon the homotopy X — only on X1, X2 and ω.

Remark 7. If there exists a simply connected open set V ⊂M so thatXs ⊂ V for all s ∈ [−1, 1] then one can calculate I(X, ω) as follows:Choose a smooth function f : V → R so that ω|V = df . Then

I(X, ω) =∑y∈X 2

IND(y)f(y)−∑x∈X 1

IND(x)f(x).

The proof of the equality is a straightforward application of Stokes’theorem.

With these considerations we will describe now the ‘regularization’referred to in Section 1.3, cf. (8). First note that for a nonvanishing


vector field X, closed one form ω and Riemannian metric g the quantity

R(X,ω, g) := (−1)n+1

∫M

ω ∧X∗(Ψ(g)) (34)

satisfies:

(i) If ω′ = ω + df then

R(X,ω′, g)−R(X,ω, g) = (−1)n∫M

fE(g)

(ii) If g1 and g2 are two Riemannian metrics then

R(X,ω, g2)−R(X,ω, g1) = (−1)n+1

∫M

ω ∧ cCS(g1, g2)

where cCS(g1, g2) = cCS(∇g1 , ∇g2).

This follows from properties (i) and (ii) of the Mathai–Quillen form.If X has zeros, then the form ω∧X∗(Ψ(g)) is well defined on M \X

but the integral∫M\X ω∧X

∗(Ψ(g)) might be divergent unless ω is zero

on a neighborhood of X .We will define below a ‘regularization’ of the integral

∫M\X ω∧X

∗(Ψ(g))

which in case X = ∅ is equal to the integral (34). For this purposewe choose a smooth function f : M → R so that the closed 1–formω′ := ω − df vanishes on a neighborhood of X , and put

R(X,ω, g; f) := (−1)n+1

(∫M\X

ω′ ∧X∗(Ψ(g))−∫M

fE(g) +∑x∈X

IND(x)f(x)

)(35)

Proposition 14. The quantity R(X,ω, g; f) is independent of f .

Therefore R(X,ω, g; f) can be denoted by R(X,ω, g) which will becalled the regularization of

∫M\X ω ∧X

∗(Ψ(g)).

Proof. Suppose f 1 and f 2 are two functions such that ωi := ω − df i

vanishes in a neighborhood U of X , i = 1, 2. For every x ∈ X wechoose a chart and let Dx(ε) denote the ε-disc around x. Put D(ε) :=⋃x∈X Dx(ε).


For ε small enough D(ε) ⊂ U and f 2− f 1 is constant on each Dx(ε).From (35), Stokes’ theorem and (32) we conclude that

(−1)n+1(R(X,ω, g; f 2)−R(X,ω, g; f 1)

)=

= −∫M\X

d((f 2 − f 1) ∧X∗(Ψ(g)

)+∑x∈X

IND(x)(f 2(x)− f 1(x)

)= − lim

ε→0

∫M\D(ε)

d((f 2 − f 1) ∧X∗(Ψ(g)

)+∑x∈X

IND(x)(f 2(x)− f 1(x)

)=

∑x∈X

(f 2(x)− f 1(x)

)limε→0

∫∂Dx(ε)

X∗(Ψ(g)) +∑x∈X

IND(x)(f 2(x)− f 1(x)

)= 0

and thus R(X,ω, g; f 1) = R(X,ω, g; f 2).

Proposition 15. Suppose that X is a non-degenerate homotopy fromthe vector field X1 to X2 and ω is a closed one form. Then

R(X2, ω, g)−R(X1, ω, g) = (−1)n+1I(X, ω). (36)

Proof. First observe that if Y 1 and Y 2 are two nonzero smooth vec-tor fields on a smooth manifold M and ϕ : TM → TM is a bundleisomorphism so that ϕ X2 = X1. Then

(Y 1)∗(Ψ(g))− (Y 2)∗(Ψ(g)) = (Y 2)∗(cCS(ϕ∗(∇g), ∇g)) = 0

Suppose that our homotopy Xs has the additional property:Additional Property: There exists K ⊂ M a compact contractible

set (smooth embedded disc) so that Xs ⊂ K for any s and there existsϕ : T (M \K)→ T (M \K) a bundle isomorphism so that X1 = ϕ ·X2

By choosing ω′ vanishing on an open neighborhood of K and inthe same cohomology class as ω and by taking f : M → R so thatω = ω′ + df one checks that

R(X1, ω, g)−R(X2, ω, g) = (−1)n+1(∑x∈X 1

IND(x)f(x)−∑y∈X 2

IND(y)f(y)),

which by Remark 7 implies the result. The statement follows then infull generality by observing that one can find an ε > 0 so that the familyXs, t − ε ≤ s ≤ 1 + ε (corrected into a homotopy as defined above)from Xt−ε to Xt−ε satisfies the additional property and by applyingRemark 7 above.

Alternative proof of Proposition 15. We may assume that there exists asimply connected V ⊆M with Xs ⊆ V for all s ∈ [−1, 1]. Indeed, sinceboth sides of (36) do not depend on the homotopy X, cf. Remark 6, wemay first slightly change the homotopy and assume that no component


of X lies in a single s ×M . Then we find −1 = t0, . . . , tk = 1 sothat for every 0 ≤ i < k we find a simply connected Vi ⊆M such thatXs ⊆ Vi for all s ∈ [ti, ti+1].

Assuming a V as above we choose a function f so that ω′ := ω −df vanishes on a neighborhood of every Xs, i.e. p∗2ω

′ vanishes on aneighborhood of X , where p2 : [−1, 1]×M →M denotes the canonicalprojection. Moreover let p2 : [−1, 1]× TM → TM denote the canonicprojection and note that p∗2ω

′ ∧X∗p∗2Ψ(g) is a globally defined form on[−1, 1]× TM . Using Stokes’ theorem and Remark 7 we then get

(−1)n+1(R(X2, ω, g)−R(X1, ω, g)

)=

=

∫[−1,1]×M

d(p∗2ω

′ ∧ X∗p∗2(Ψ(g)))

+ I(X, ω)

=

∫[−1,1]×M

p∗2(ω′ ∧ E(g)) + I(X, ω)

= I(X, ω)

where we used dX∗p∗2(Ψ(g)) = p∗2(E(g)) for the second equality.

6. Proof of Theorem 4

The proof of Theorem 4 we will present here combines on the resultsof Hutchings, Pajintov and other with the results from [3]. Becauseof this, additional notations and preliminaries, some of independentinterest are necessary. They will be collected in three preliminarysubsections. These subsections will be followed by the fourth whereTheorem 4 is proven.

6.1. A summary of Hutchings-Pajintov results. We begin by re-calling the results of Hutchings as formulated in his paper [9] in aslightly more general setting.

Let M be a compact smooth manifold, x ∈ M a base point andξ ∈ H1(M,R) a cohomology class in degree one.

Recall from section 1.4 that ξ defines the free abelian group Γ andinduces the injective homomorphism ξ : Γ→ R. Denote by π : M →Mthe principal Γ-covering canonically associated with x and Γ. To ξ weassociate:

(i)the Novikov ring Λξ with coefficients in R or C which is actuallya field,

(ii)for any ρ ∈ [0,∞) the subring Λξ,ρ ⊂ Λξ,(iii) the multiplicative groups of invertible elements Λ·ξ ⊂ Λξ and

Λ·ξ,ρ ⊂ Λξ,ρ.


The Novikov ring Λξ consists of functions f : Γ → R which satisfythe property that for any real number R ∈ R the cardinality of the setγ ∈ Γ|f(γ) 6= 0, ξ(γ) ≤ R is finite. The multiplication in Λξ is givenby convolution, cf [6].

Note Z[Γ] ⊂ Λξ,ρ ⊂ Λξ and denote by

H∗sing(M ; Λξ) := H∗sing(M ;Z)⊗Z[Γ] Λξ which is a finite dimensional vec-tor space over the field Λξ. Denote by detH∗sing(M ; Λξ) the one di-

mensional vector space over Λξ defined by ⊗(Λi(H iX(M ; Λξ))

ε(i) whereV ε(i) = V if i is even and V ε(i) is the dual of V if i is odd.

We have also shown both in Section 1 and in more details in [6] howto interpret the elements of Λξ as Dirichlet series. In this context Λξ,r

is the subring of Λξ consisting of those elements whose correspondingDirichlet series has the abscissa of convergence smaller than ρ. One canverify that Λ+

ξ,ρ = Λ+ξ ∩ Λξ,ρ.

For t > ρ we denote by evt : Λξ,ρ → R the ring homomorphism whichassociates to each f ∈ Λξ,ρ interpreted as a Dirichlet series f, the valueof the Laplace transform L(f) at t. For any Λξ,ρ− module M we writeM ⊗evt R for M ⊗Λξ,ρ R where R is viewed as an Λξ,ρ module via thehomomorphism evt.

Let X be a (−ξ) gradient like vector field which satisfies MS. Bydefinition X is a vector field which is (−ω, g) gradient like in the senseon Definition 6 section 2 for some ω closed one form representing ξ andfor some admissible metric g. Let X be the pullback of X on M.

We denote by E(M,x) the set of Euler structures based at x, cf [5]or [7] for a definition 14. Choose O a collection of orientations for theunstable manifolds of the rest points of X and therefore of the restpoints of X. Choose an Euler structure e ∈ E(M,x) and an elementoH ∈ detH∗sing(M : Λξ) \ 0.

Denote by (NCqX , ∂

qO) the Novikov cochain complex of free Λξ mod-

ules (actually vector spaces since Λξ is a field) as defined in [6] and byH∗X(M ; Λξ) its cohomology. This cohomology is canonically isomorphicto H∗sing(M ; Λξ). To describe this canonical isomorphism note that for

any two (−ξ) gradient like vector fields X1 and X2 which are MS thereexists MS homotopies X from X1 to X2. As shown in Section 6 eachsuch homotopy induces a morphism from the Novikov complex asso-ciated to X1 to the Novikov complex associated to X2 which induce

14The set of Euler structures with respect to x is the set of connected componentsof vector fields which vanish only at p


isomorphism between their cohomology. This isomorphism is indepen-dent of the homotopy X and will be denoted by [u∗(X1, X2)

u∗(X1, X2) : H∗X1(M : Λξ)→ H∗X2(M : Λξ).

For any three (−ξ) gradient like vector fields X i, i = 1, 2, 3 whichsatisfy MS these canonical isomorphisms satisfy

u∗(X3, X2) · u∗(X2, X1) = u∗(X3, X1).

Let (f, g) be a pair consisting of a Morse function f : M → R

and a Riemannian metric g with −gradgf satisfying MS. By takingC a very large positive constant, one can insure that the vector fieldX2 = X1 − C(gradgf) is (−ξ) gradient like vector field which satisfiesMS and that its Novikov complex identifies to the the geometric cochaincomplex associated to (f , g), the pull back of (f, g) to M, tensored overZ[Γ] by Λξ. This cochain complex is obviously a quotient of C∗sing(M)tensored (over Z[Γ]) by Λξ which calculates H∗sing(M ; Λξ) and we havean obvious isomorphism from H∗sing(M ; Λξ) to H∗X2(M : Λξ). Recall

that the geometric (or Morse) cochain complex associated to (f , g) isa Z cochain complex on which Γ acts so is actually a cochain complexof free Z[Γ] modules.

It is rather straightforward (and standard) to check that the com-position of the two isomorphisms mentioned above is independent ofC and of the choice (f, g) providing a canonical identification betweenthe cohomology of the Novikov cochain complex and the singular co-homology H∗sing(M ; Λξ).

An element oH ∈ detH∗sing(M ; Λξ) \ 0 induces a class of (ordered)bases of the graded vector space H∗X(M ; Λξ). Choose such base hi.

The Euler structure e will induce a class of (unordered) bases 15 forthe Novikov complex (NC∗X , ∂

∗O). The torsion of the Novikov complex

equipped with the (unordered) base induced from e and a base in thecohomology induced from oH provides (cf Milnor in [14]) an element inΛ+ξ /±1 which depends only on e and oH and will be denoted by

τ((NC∗X , ∂∗O), e, oH) ∈ Λ+

ξ /±1.Suppose now that X satisfies also NCT. As noticed in [9] ZX ∈ Λξ

and then eZX ∈ Λ+ξ .

Consider then I(X, e, oH) := ±eZX · τ(NC∗X , c, h) ∈ Λ+ξ /±1.

The main result of Hutchings can be formulated as follows

15A base provide a map s : X → X with the property π · s = id. By choosinga path αy in M from the base point x of M to s(y), y ∈ X and by consideringπ(αy) one obtains an Euler chain as defined in [7] which, together with X, definesan Euler structure. It is shown in [7] that any Euler structure appears in this way


Theorem 7. (Hutchings). The invariant I(X, e, oH) is independent ofX, in particular if X1 and X2 are two (−ξ) gradient like vector fieldswhich are MS and NCT then

I(X1, e, oH) · I(X2, e, oH)−1 = ±eZX1 · e−ZX2

The proof of this theorem is given in [9]. The author considers onlythe acyclic case. however this hypothesis is used only indirectly, toinsure that Milnor definition of torsion, cf [14],works.

Once a base in the cohomology of the Novikov complex associatedto X is provided the arguments in [9] can be repeated word by word.The base is chosen with the help of the canonical isomorphism betweenthe cohomology of the Novikov complex and the singular cohomologyH∗sing(M ; Λξ).

Suppose X1 = − gradg f where f : M → R is a Morse function, andX1 is (−ξ) gradient like and satisfies MS. X1 has no closed trajectorieshence is NCT by default and ZX1 = 0 The above result then implies

Corollary 2.

I(X2, e, oH) · I(X1, e, oH)−1 = ±eZX1

Let X be a homotopy from the vector field X1 to the vector fieldX2 which is MS. The incidences (21) induce from X provide a mor-phism Nu∗X : (NCq

X1 , ∂q) → (NCq

X2 , ∂q) which induces an isomor-

phism in cohomology. In particular the mapping cone of this mor-phism, C(Nu∗X), is an acyclic complex of Λξ vector spaces. By choos-ing a base induced from the Euler structure e in each of the Novikovcomplexes (NCq

Xi , ∂qOi) i = 1, 2, one obtains a base for the mapping

cone complex C(Nu∗X). We write τ(X,O1,O2) ∈ Λ+ξ /±1 for the

torsion defined as in [14] and note that by changing the orientationsO1 or O2 this torsion does not change 16 Consequently we can writeτ(X1, X2, e) := τ(X,O1,O2, e)

Proposition 16. If X2 and X1 are two −ξ gradient like vector fields ean Euler structure as above and oH a nonzero element in detH∗(M,Λxi)and X an homotopy from X1 to X2 which is MS then

τ(X1, X2, e) := I(X2, e, oH) · I(X1, e, oH)−1 = ±eZX2 · e−ZX1 .

Hence τ(X1, X2, e) is independent of e and then it can be denoted byτ(X1, X2)

The result follows easily from the following lemma.

16change only up to multiplication with ±1


Lemma 1. Suppose u : (C∗1 , ∂∗1)→ (C∗2 , ∂

∗2) is an isomorphism of based

cochain complexes which induces an isomorphism H∗u in cohomology.Suppose c1 and c2 are bases of C∗1 and C∗2 and h1 and h2are bases inthe cohomology of these complexes and in addition that∏

(detH iu)(−1)i = 1.

Here (detH iu) is calculated with respect to the bases h1 and h2.Then:

τ(u∗, c∗1, c∗2) = τ(C∗2 , c

∗2, h∗2)− τ(C∗1 , c

∗1, h∗1).

Suppose X1 = − gradg f where f : M → R is a Morse function, andX1, X2 are −ξ gradient like and MS. Suppose X is a homotopy fromX1 to X2 which is MS

Corollary 3.τ(X1, X2) = ±eZX1

It is not hard to see that Hutchings theorem is equivalent to thiscorollary. In this form the result was also established by Pajintov [17]by a different type of arguments.

Suppose now that that we are in the hypotheses of 3 and in additionρ(ξ,X) < ∞, and ρ(ξ,X i) < ∞, i = 1, Then all complexes NC∗X1

NC∗X2 and the morphism Nu∗ are complexes of free modules over Λξ,ρ

for some ρ big enough and, then in view of the previous corollaryτ(X, e) = ±eZX1 ∈ Λ+

ξ,ρ, and then ZX1 ∈ Λξ,ρ. This because an element

in in Λξ whose exponential is in Λ+ξ,ρ is actually in Λξ,ρ. In particular

ZX1 has Laplace transform which can be evaluated at any t > ρ.

6.2. The geometry of closed one form. Suppose M is a connectedsmooth manifold and p ∈ M a base point chosen once for all. Thehomomorphism [ω] : H1(M ;Z) → R induces the one dimensional rep-resentation ρ = ρ[ω] : π1(M, p) → GL1(R) defined by the composition

π1(M, p) → H1(M ;Z)[ω]−→ R

exp−−→ R+ = GL1(R). The representationρ provides a flat rank one vector bundle ξρ : Eρ → M with the fiberabove p identified with R. This bundle is the quotient of trivial flat

bundle M ×R→ M by the group π = π1(M, p) which acts diagonally

on M ×R. Here M denotes the universal covering constructed canoni-cally with respect to p (from the set of homotopy classes of continuous

paths originating from p). The group π acts freely on M with quotientspace M. The action of π on R is given by the representation ρ. We

also denote by p the canonical basepoint of M.


There is a bijective correspondence between the closed one forms ωin the cohomology class represented by ρ and the Hermitian structuresµ in the vector bundle ξρ which agree with the standard Hermitianstructure on the fiber above p (identified to R)Given ω in the cohomology class [ω] ∈ H1(M ;R), one constructs a

Hermitian structure µω on the trivial bundle M × R → M which isπ−invariant; therefore, by passing to quotients one obtains a Hermitianstructure µω in ξρ. The Hermitian structure µω is defined as follows:

(i) Observe that the pull back ω of ω on M is exact and therefore

equal dh where h : M → R is the unique function with h(p) = 0and dh = ω.

(ii) Define µ(x) by specifying the length of the vector 1 ∈ R. Weput ||1x||µ(x) := eh(x).

Given a Hermitian structure µ one construct a closed one form ωµ asfollows:

(i) Choose a covering with base pointed contractible open sets(Uk, xk), xk ∈ Uk ⊂M.

(ii) Extend by parallel transport the scalar product µxk from (ξρ)−1(xk)

to a Hermitian structure µk. on the bundle (ξρ)−1(Uk). Com-

pare µ|Uk and µk; the result is the function αk(x) := ||1||µ(x)/||1||µk(x).(iii) Verify that the degree one forms ωµ|Uk := d log(αk) agree on

the intersection of the open sets Uk and provide a globallydefined closed one form ωµ.

(iv) Verify that the form ωµ is independent of the choices made.

To simplify the writing below we denote (by a slight abuse of notation):

(i) ρ(t) := ρtω(ii) µ(t) := µtω

and when an additional closed one form ω′ is under consider-ation we also write

(iii) ρ′(t) := ρtω′ and µ′(t) := µtω′ .

Remark 8. The cochain complex (Ω∗(M), dω(t)) equipped with thescalar product induced from g is isometric to the cochain complex(Ω∗(M,ρ(t)) equipped with the scalar product induced from g andµ(t) = µ as defined in [3].

In particular we have

log Tan(ω, t) = log Tan(M,ρ(t), µ(t)) (37)


where log Tan(M,ρ, g, µ) is the analytic torsion considered in [3] andassociated with the Riemannian manifold (M, g) the representation ρand the hermitian structure µ in the flat bundle induced from ρ.

Remark 9. Let (ω, g) be a pair consisting of a closed one Morse formω and a Riemannian metric g and X an −[ω] gradient like vector fieldwhich is also (−dh, g′) gradient like i.e. a generalized triangulation inthe sense of [3]. Suppose X is MS . Then

Int∗X,ω,O(t) : (Ω∗(M), dω(t))→ C∗(X,ω, t)

defined in (5) where (Ω∗(M) is equipped with the scalar product in-duced from g and Maps(Xq,R) is equipped with the ”obvious scalarproduct” is isometrically conjugate to

Int∗ : (Ω∗(M,ρ(t)), dρ(t))→ (C∗(ρ(t)), δρ(t))

defined in [3] where (Ω∗(M,ρ(t)) is equipped with the scalar productinduced by (g, µ(t)) and C∗(ρ(t)) is equipped with the scalar productinduced from µ(t) cf [3].

Here C∗(ρ(t), δρ(t)) denotes the cochain complex induced by the cellstructure of M provided by the unstable sets of X and the representa-tion ρ(t). In particular we have

log Ttop(X,ω, t) = log Tcomb(M,ρ(t), τ, µ(t)) (38)

where log Tcomb(M,ρ, τ, µ) is the combinatorial torsion as defined in [3].

Suppose ω′ = ω + dh. By the previous considerations the formsω and ω′ induce the Hermitian structures µω and µω′ . We apply theHermitian anomaly formula of [1] cf.also [3] and (37) and obtain

log Tan(ω, t) = log Tan(ω′, t)− t∫M

hEg + t∑

x∈Cr(ω′)

(−1)indexxh(x)

+ log Vol(Hv(t))∗

(39)where Eg denotes the Euler form given by the Riemannian metric gcf [3] section 5 page 63, and (Hv(t))∗ is the isomorphism induced incohomology by the isomorphism v(t)

v(t) : (Ω∗(M), dω′(t))→ (Ω∗(M)dω(t)),

defined byv(t)(a); = etha, a ∈ (Ω∗(M).


6.3. Recallection of results from [1] and [3]. Suppose that (M, g)is a closed Riemannian manifold X a gradient like vector field whoseunstable sets provides a generalized triangulation τ, ρ a representa-tion of π1(M) and µ a hermitian structure in the flat vector bundleassociated with ρ. Consider Int∗ : (Ω∗(M), dρ) → (C∗(τ, ρ), δρ) andequip each of these complexes with a scalar product, the first complexwith the scalar product induced from the Riemannian metric g andthe Hermitian structure µ and the second with the scalar product in-duced from the generalized triangulation τ and the hermitian structureµ. Then the cohomology are in a natural ways finite dimensional vec-tor spaces equipped with Euclidean structure. Denote by H∗Int theisomorphism induced in cohomology and write

log V H(ρ, µ, g, τ, ) =∑

(−1)q log V ol(HqInt) (40)

Let ω(µ) be the closed one form induced by µ as described in [1] and[3]. We have the following result due to Bismut Zhang (cf [3])

Theorem 8. With the hypothesis above we have

log Tan(M, g, ρ, µ) = log Tcomb(M, τ, ρ, µ)−log V H(ρ, µ, g, τ)+R(ω(µ), g,X)(41)

6.4. Proof of Theorem 4. We begin with a X = X1 an (−ω, g)gradient like vector field which we supposed to be MS and NCT. In viewof Proposition 13 we construct the vector field X2 and the homotopyX from X1 to X2 so that

1) both X2 and X are MS,2) there exists a generalized triangulation τ = (h, g) so that X2 is

(−dh, g) gradient like,3)ρ(X, [ω]) <∞.Choose orientations O1 resp. O2 for the unstable manifolds of X1

resp. X2. By Proposition 12, for t big enough, we have the following ho-motopy commutative diagram of finite dimensional cochain complexeswhose arrows induce isomorphisms in cohomology.

(Ω∗(M)sm, dω(t)Id−−−→ (Ω∗(M)sm, dω(t)

Int∗X1,O,ω1

(t)

y Int∗X2,O,ω2

(t)

yC(X1,O1, ω)(t)

u∗C(X,O1,O2,ω(t)

−−−−−−−−−→ C(X2,O2, ω)(t)

For simplicity of the writing we will use the following abbreviations:


I1(t) := Int∗X1,O,ω(t),

I2(t) := Int∗X2,O2,ω(t) and

u(t) := u∗X,O1,O2,ω(t)

Recall from [3] that for the homotopy equivalence θ∗ : (C∗1 , d∗1) →

(C∗2 , d∗2) of finite dimensional cochain complexes of Euclidean vector

spaces one defines log τ(θ∗) as the log torsion of the mapping conecomplex which is an acyclic complex of finite dimensional Euclideanspaces. It is therefore straightforward, cf [3] to check that

log τ(θ∗) = log τ((C∗1 , d∗1))− log τ((C∗2 , d

∗2)) + log V olH∗(θ∗) (42)

where log V olH∗(θ∗) =∑

i(−1)i log V olH i(θ∗)In case that the homotopy equivalence θ∗ is an isomorphism then

log τ(θ∗) =∑i

(−1)i log V ol(θi)) (43)

In view of this we have

log τ(I1(t)) = logV(t) (44)

and

log τ(I2(t)) = − log τ((X2,O2, ω)(t) + log τan(M, tω, g)

+ log τan,la(M, tω, g) + log V ol(H∗(I2(t))

which by Theorem 8 section 6.3 we obtain

log τ(I2(t)) = + log τan,la(M, tω, g) + tR(X2, ω, g) (45)

By the homotopy commutativity of the above diagram we have

log τ(I1(t)) + log τ(u∗(t)) = log τ(I2(t)) (46)

By Proposition 15 combined with the observations that X2 has noclosed trajectories we have

log τ(u∗) = evt(ZX1) + (−1)nI(X, tω) (47)

Combining (44) - (47) we obtain

− logV(t)+evt(ZX1)+(−1)nI(X, tω) = + log τan,la(M, tω, g)+tR(X2, ω, g)(48)

which in view of Proposition 15 provides the verification of Theorem4.


7. The invariant ρ

Throughout this section M will be a closed manifold and X a vectorfield with Morse singularities. For a critical point x ∈ X , i.e. a zeroof X, we let i−x : W−

x → M denote the inclusion of x’s unstable setinto M . If M is equipped with a Riemannian metric we get an inducedRiemannian metric gx := (i−x )∗g on W−

x thus a volume density µ(gx)on W−

x and hence the spaces Lp(W−x ), p ≥ 1. Though the Lp–norm

depends on the metric g the space Lp(W−x ) and its topology does not.

Indeed for another Riemannian metric g′ on M we find a constant

C > 0 so that 1/C ≤ g′(X,Y )g(X,Y )

≤ C for all tangent vectors X and Y

which implies 1/C ′ ≤ µ(g′x)µ(gx)

≤ C ′ for some constant C ′ > 0.

Given a closed 1–form ω on M we let hωx denote the unique smoothfunction on W−

x which satisfies dhωx = (i−x )∗ω and hωx(x) = 0. Weare interested in the space of 1–forms for which eh

ωx ∈ L1(W−

x ). Thiscondition actually only depends on ω’s cohomology class. Indeed wehave hω+df

x = hωx + (i−x )∗f − f(x) and so |hω+dfx − hωx | ≤ C ′′ and e−C

′′ ≤eh

ω+dfx /eh

ωx ≤ eC

′′for some constant C ′′ > 0. So we define

Rx(X) :=

[ω] ∈ H1(M)∣∣ ehωx ∈ L1(W−

x )

and setR(X) :=

⋂x∈Cr(X)

Rx(X).

Note that we have

ρ(ξ,X) = inft ∈ R | tξ ∈ R(X) ∈ R ∪ ±∞.Let us also define

ρx(ξ,X) = inft ∈ R | tξ ∈ Rx(X) ∈ R ∪ ±∞and notice that ρ(ξ,X) = maxx∈X ρx(ξ,X).

Lemma 2. The sets Rx(X) and R(X) are convex. Particularly

ρ(λξ1 + (1− λ)ξ2, X

)≤ maxρ(ξ1, X), ρ(ξ2, X)

for all 0 ≤ λ ≤ 1.

Proof. Indeed let [ω0], [ω1] ∈ H1(M), λ ∈ [0, 1] and set ωλ := λω1+(1−λ)ω0. Then hωλx = λhω1

x +(1−λ)hω0x . For λ ∈ (0, 1) we set p := 1/λ > 1

and q := 1/(1− λ). Then 1/p+ 1/q = 1 and by Holder’s inequality

||ehωλx ||1 = ||eλh

ω1x e(1−λ)h

ω0x ||1 ≤ ||eλh

ω1x ||p||e(1−λ)h

ω0x ||q = ||eh

ω1x ||λ1 ||eh

ω0x ||1−λ1

So, if [ω0] and [ω1] ∈ Rx(X) then [ωλ] ∈ Rx(X), and thus Rx(X) isconvex. As an intersection of convex sets R(X) is convex too.


Next we introduce

Bx(X) :=

[ω] ∈ H1(M)∣∣ ehωx ∈ L∞(W−

x )

and set B(X) :=⋂x∈Cr(X) Bx(X). Most obviously we have:

Lemma 3. The sets Bx(X) and B(X) are convex cones. Moreover wehave Rx(X) +Bx(X) ⊆ Rx(X) and R(X) +B(X) ⊆ R(X).

Next define

G(X) :=ξ ∈ H1(M)

∣∣ X is ξ gradient like

Recall that ξ ∈ G(X) if there exists a representative ω, [ω] = ξ, whichsatisfies

(i) ω(X) < 0 on M \ X , where X denotes the zeros of X, and(ii) there exists a Riemannian metric g on M so that g(X, ·) = ω

on a neighborhood of X .

Lemma 4. The set G(X) ⊆ H1(M) is open and contained in B(X).Moreover G(X) is a convex cone.

Proof. The subset G(X) ⊆ H1(M) is open, for we can change thecohomology class [ω] by adding a form whose support is disjoint from Xand hence not affecting condition (ii). If the form we add is sufficientlysmall the condition (i) will still be satisfied.

We have G(X) ⊆ B(X) for (i) and (ii) tell that the function hωxattains its maximum at x and thus is bounded from above.

Next note that both conditions (i) and (ii) are convex and homoge-neous conditions on ω. Thus G(X) is a convex cone.

Lemma 5. Suppose G(X) ∩ R(X) 6= ∅. Then every ray of G(X), i.e.a half line starting at the origin which is contained in G(X), intersectsR(X).

Proof. Pick ξ ∈ G(X)∩R(X). Since ξ ∈ R(X), Lemma 4 and Lemma 3imply:

ξ +G(X) ⊆ R(X) +G(X) ⊆ R(X) +B(X) ⊆ R(X) (49)

On the other hand G(X) is open and ξ ∈ G(X), so every ray in G(X)has to intersect ξ + G(X). In view of (49) it has to intersect R(X)too.

Corollary 4. Suppose X is ξ0 gradient like and ρ(ξ0, X) < ∞. Thenρ(ξ,X) <∞ whenever X is ξ gradient like.

Suppose x ∈ Xq. Let B ⊆ W−x denote a small ball centered at

x. The submanifold i−x(W−x \ B

)⊆ M gives rise to a submanifold


Gr(W−x \ B) ⊆ Grq(TM) in the Grassmannified tangent bundle, the

space of q–dimensional subspaces in TM . For a critical point y ∈ Xwe define

Kx(y) := Grq(TyW−y ) ∩Gr(W−

x \B)

where Grq(TyW−y ) ⊆ Grq(TyM) ⊆ Grq(TM). Note that Kx(y) does

not depend on the choice of B.

Remark 10.

(i) Even though we removed a neighborhood of x from the unsta-ble manifold W−

x the set Kx(x) need not be empty. Howeverif we did not remove B the set Kx(x) would never be vacuousfor trivial reasons.

(ii) If q = ind(x) > ind(y) we always haveKx(y) = ∅, for Grq(TyW−y ) =

∅ since q > dim(TyW−y ) = ind(y).

(iii) If dim(M) = n = q = ind(x) we always have Kx(y) = ∅ for ally ∈ X .

(iv) If q = ind(x) = 1 and y ∈ X is in the closure of W−x we will

have Kx(y) 6= ∅.

Proposition 17. Let X be ξ gradient like and x ∈ X . SupposeKx(y) = ∅ for all y ∈ X . Then ρx(ξ,X) <∞.

We start with a little

Lemma 6. Let (V, g) be an Euclidean vector space and V = V + ⊕ V −an orthogonal decomposition. For κ ≥ 0 consider the endomorphismAκ := κ id⊕− id ∈ End(V ) and the function

δAκ : Grq(V )→ R, δAκ(W ) := trg|W (p⊥W Aκ iW ),

where iW : W → V denotes the inclusion and p⊥W : V → W theorthogonal projection. Suppose we have a compact subset K ⊆ Grq(V )for which Grq(V

+) ∩K = ∅. Then there exists κ > 0 and ε > 0 withδAκ ≤ −ε on K.

Proof. Consider the case κ = 0. Let W ∈ Grq(V ) and choose a g|W–orthonormal base ei = (e+

i , e−i ) ∈ V + ⊕ V −, 1 ≤ i ≤, q of W . Then

δA0(W ) =

q∑i=1

g(ei, A0ei) = −q∑i=1

g(e−i , e−i ).

So we see that δA0 ≤ 0 and δA0(W ) = 0 iff W ∈ Grq(V+). Thus

δA0|K < 0. Since δAκ depends continuously on κ and sinceK is compactwe certainly find κ > 0 and ε > 0 so that δAκ|K ≤ −ε.


Proof of Proposition 17. Let S ⊆ W−x denote a small sphere centered

at x. Let X := (i−x )∗X denote the restriction of X to W−x and let Φt

denote the flow of X at time t. Then

ϕ : S × [0,∞)→ W−x , ϕ(x, t) = ϕt(x) = Φt(x)

parameterizes W−x with a small neighborhood of x removed.

Let κ > 0. For every y ∈ X choose a chart uy : Uy → Rn centered

at y so that

X|Uy = κ∑

i≤ind(y)

uiy∂

∂uiy−

∑i>ind(y)

uiy∂

∂uiy.

Let g be a Riemannian metric on M which restricts to∑

i duiy ⊗ duiy

on Uy and set gx := (i−x )∗g. Then

∇X|Uy = κ∑

i≤ind(y)

duiy ⊗∂

∂uiy−

∑i>ind(y)

duiy ⊗∂

∂uiy.

In view of our assumption Kx(y) = ∅ for all y ∈ X Lemma 6 permitsus to choose κ > 0 and ε > 0 so that after possibly shrinking Uy wehave

divgx(X) = trgx(∇X) ≤ −ε < 0 on ϕ(S × [0,∞))∩ (i−x )−1(⋃y∈X

Uy

).

(50)Next choose a closed 1–form ω so that [ω] = ξ and ω(X) < 0 on M \X .Choose τ > 0 so that

τω(X) + ind(x)||∇X||g ≤ −ε < 0 on M \⋃y∈X

Uy. (51)

Using τX · hωx ≤ 0 and

divgx(X) = trgx(∇X) ≤ ind(x)||∇X||gx ≤ ind(x)||∇X||g(50) and (51) yield

τX · hωx + divgx(X) ≤ −ε < 0 on ϕ(S × [0,∞)). (52)

Choose an orientation of W−x and let µ denote the volume form on W−

x

induced by gx. Consider the function

ψ : [0,∞)→ R, ψ(t) :=

∫ϕ(S×[0,t])

eτhωxµ ≥ 0.

For its first derivative we find

ψ′(t) =

∫ϕt(S)

eτhωx iXµ > 0


and for the second derivative, using (52),

ψ′′(t) =

∫ϕt(S)

(τX·hωx+divgx(X)

)eτh

ωx iXµ ≤ −ε

∫ϕt(S)

eτhωx iXµ = −εψ′(t).

So (ln ψ′)′(t) ≤ −ε hence ψ′(t) ≤ ψ′(0)e−εt and integrating again wefind

ψ(t) ≤ ψ(0) + ψ′(0)(1− e−εt)/ε ≤ ψ′(0)/ε.

So we have eτhωx ∈ L1

(ϕ(S × [0,∞)

)and hence eτh

ωx ∈ L1(W−

x ) too.We conclude ρx(ξ,X) ≤ τ <∞.

Corollary 5. Let X be ξ gradient like and suppose Kx(y) = ∅ for allx, y ∈ X . Then ρ(ξ,X) <∞.

Remark 11.

(i) In view of Remark 10(iii) Proposition 17 implies ρx(ξ,X) <∞whenever X is ξ gradient like and ind(x) = dim(M). Howeverthere is a much easier argument for this special case. Indeed, inthis case W−

x is an open subset of M and therefore its volumehas to be finite. Since X is ξ gradient like we immediately evenget ρx(ξ,X) ≤ 0.

(ii) In view of Remark 10(iv) most of the time we can not applyProposition 17 to the case ind(x) = 1. However, in this casewe certainly have ρx(ξ,X) ≤ 0.

(iii) Throughout the whole section we did not make use of a Morse–Smale condition.

Proposition 18. Suppose X is ξ0 gradient like and x ∈ X . LetBr(x) ⊆ W−

x denote the ball of radius r in the unstable manifold W−x

with respect to the induced metric gx. Then the following are equivalent

(i) There exists a constant C > 0, such that Vol(Br(x)) ≤ eCr.(ii) ρx(ξ0, X) <∞.

(iii) ρx(ξ,X) <∞ whenever X is ξ gradient like.

Proof. TODO.

Appendix A

Proposition 19. Let (M,X, ω, g) with M a smooth manifold, X avector field, ω a closed one form and g a Riemannian metric. SupposeU ⊂M is an open set and

(i) the vector fields X and − gradg ω agree on U and(ii) ω(X)(x) > 0 for all x in a neighborhood of M \ U .

Then there exists a Riemannian metric g′ so that:


(i) X = − gradg′ ω(ii) g and g′ agree on U .

Proof. Let N be an open neighborhood of M \U so that ω(X) < 0 andtherefore Xx 6= 0, x ∈ N . For x ∈ N the tangent space Tx decomposesas the direct sum Tx = Vx⊕[Xx] where [Xx] denotes the one dimensionalvector space generated by Xx and Vx = ker(ω(x) : Tx → R). Clearly onU the function −ω(X) is the square of the length of Xx with respect tothe metric g and Xx is orthogonal to Vx and on N it is strictly negative.Define a new Riemannian metric g′ on M as follows: For x ∈ U thescalar product in Tx is the same as the one defined by g. For x ∈ Nthe scalar product on Tx agrees to the one defined by g but make Vxand [Xx] perpendicular and the length of Xx equal to

√−ω(X)(x). It

is clear that the new metric is well defined and smooth.

We can apply the above result to a vector field X which is (−ω, g)gradient like and obtain

Corollary 6. Suppose X is a (−ω, g) gradient like vector field. Thereexists a Riemannian metric g′ so that

(i) the metrics g′ and g agree on a neighborhood of X and(ii) X = − gradg′ ω.

Appendix B

Lemma 7. Suppose a1, . . . , an ∈ R, ai 6= aj for i 6= j. Supposeλ1, . . . , λn ∈ R, λi 6= 0. Consider the function

f : R→ R, f(t) :=n∑i=1

λietai . (53)

Then there are only finitely many t ∈ R for which f(t) = 0.

Proof. W.l.o.g. a1 > ai for i 6= 1. There certainly exists T+ ≥ 0, suchthat

1

n− 1|λ1e

ta1| > |λietai|, for all t ≥ T+ and for all i 6= 1.

Using |x+ y| ≥ |x| − |y| we derive

|f(t)| =∣∣∣ n∑i=1

λietai

∣∣∣ ≥ |λ1eta1|−

n∑i=2

|λietai| > |λ1eta1|−

n∑i=2

1

n− 1|λ1e

ta1| = 0,

for all t ≥ T . Similarly (using the smallest ai instead of the largest)one finds T− ≥ 0, such that |f(t)| > 0 for all t ≤ −T−. SettingT := maxT+, T− we have f(t) 6= 0 for all |t| ≥ T and since the set ofzeros has to be discrete there can only be a finite number of them.


Corollary 7. Let ω be a closed one form and set βi(t) := dimH i(Ω∗(M), dω(t)),where t ∈ R and i = 0, 1, . . . , dim(M). There there are finitely manyti ∈ R, t0 < t1 < · · · < tN and positive integers βi so that βi(t) = βi,for t 6= t1, . . . , tN and so that βi(tk) ≥ βi for all k.

Proof. Take any generalized (in particular smooth) triangulation τ andconsider X an Euler vector field i.e. a vector field with isolated restpoints and whose unstable sets identify to the open cells (simplexes) ofthe triangulation. For any t ∈ R

Int∗X,O,ω(t) : (Ω∗(M), dω(t))→ C(X,O, ω)(t)

is well defined and provides an isomorphism in cohomology by deR-ham’s theorem.

Note that the differential δqX,O,ω(t) with respect to the canonical baseof Cq(X,O) is given by a matrix whose entries are functions in t of theform (53), see (13). Since this t−family, t ∈ R, of cochain complexescan be complexified to a z–family, z ∈ C, the points z where βi(z)changes form a proper complex analytic subspace of C hence of di-mension 0. Indeed, the condition for the change of the dimension ofcohomology of H i(C(X,O, ω)(z)) can be express as vanishing of finitelymany minors of the matrices representing δiX,O,ω. They are a collectionof functions of the form (53), hence a proper complex analytic set ofdimension zero. Therefore βi(z) is an integer βi for all z but a discretesubset K of C and βi(z) ≥ βi, z ∈ K. In particular this is the case fort ∈ R. If one restricts our attention to the points t ∈ R where βi(t)changes they are points of a real analytic set given by finitely manyand then also by one real valued function of type (53) hence by thelemma above a finite set.

Appendix C

References

[1] J.–M. Bismut and W. Zhang, An extension of a theorem by Cheeger and Muller,Asterisque. 205, Societe Mathematique de France, 1992.

[2] D. Burghelea, L. Friedlander, T. Kappeler and P. McDonald, Analytic andReidemeister torsion for representations in finite type Hilbert modules, Geom.Funct. Anal. 6(1996), 751–859.

[3] D. Burghelea, L. Friedlander and T. Kappeler, Relative Torsion, Commun.Contemp. Math. 3(2001), 15–85.

[4] D. Burghelea, L. Friedlander and T. Kappeler, Elementary Morse Theory,Lectures Notes in preparation, 2003.

[5] D. Burghelea, Removing Metric Anomalies from Ray–Singer Torsion, Lett.Math. Phys. 47(1999), 149–158.

[6] D. Burghelea and S. Haller, On the topology and analysis of closed one form.I (Novikov theory revisited), Monogr. Enseign. Math. 38(2001), 133–175.


[7] D. Burghelea and S. Haller, A Riemannian invariant, Euler structures andsome topological applications, Preprint OSU. (2003), 37 pp.

[8] D. Burghelea and S. Haller, On the topology and analysis of closed one form(the case of manifolds with boundary), Manuscript, 2003, 133–175.

[9] M. Hutchings, Reidemeister torsion in generalized Morse Theory, Forum Math.14(2002), 209–244.

[10] M. Hutchings, Y.–J. Lee, Circle–valued Morse Theory, Reidemeister torsionand Seiberg–Witten invariants of 3 manifolds, Topology 38(1999), 861–888.

[11] J. Marsick, ..., Thesis OSU 8(1998).[12] V. Mathai and D. Quillen, Superconnections, Thom Classes, and Equivariant

differential forms, Topology 25(1986), 85–110.[13] J. Milnor, Lectures on the h–cobordism, Princeton University Press, Princeton,

N.J., 1965.[14] J. Milnor, Whitehead trosion, Bull. Amer. Math. Soc. 72(1966), 358–426.[15] S.P. Novikov, ..., 8(2002), xxx–yyy.[16] S.P. Novikov, paper Quasiperiodic structures in topology, in Topological meth-

ods in modern mathematics, Proceedings of the symposium in honor of JohnMilnor’s sixtieth birthday held at the State University of New York, StonyBrook, New York, 1991, eds L. R. Goldberg and A. V. Phillips, Publish orPerish, Houston, TX, 1993, 223–233.

[17] A. Pajintov, ..., ???, xx, yyyy.[18] J.P. Serre, A course in arithmetic, Graduate Texts in Mathematics, No. 7,

Springer Verlag, New York–Heidelberg, 1973.[19] D.V. Wildder, The Laplace Transform, Princeton Mathematical series, v. 6,

Princeton University Press, Princeton, N.J., 1941.[20] E. Witten, Supersymmetry and Morse theory J. Differential Geom. 17(1982),

661–692.

Dan Burghelea, Dept. of Mathematics, The Ohio State University,

231 West Avenue, Columbus, OH 43210, USA.

E-mail address: [email protected]

Stefan Haller, Institute of Mathematics, University of Vienna,

Strudlhofgasse 4, A-1090, Vienna, Austria.

E-mail address: [email protected]

LAPLACE TRANSFORM, TOPOLOGY AND (OF A CLOSED ONE …LAPLACE TRANSFORM, TOPOLOGY AND SPECTRAL...

Documents

Transcript of LAPLACE TRANSFORM, TOPOLOGY AND (OF A CLOSED ONE …LAPLACE TRANSFORM, TOPOLOGY AND SPECTRAL...