ON THE TOPOLOGY AND ANALYSIS OF A CLOSED ONE FORM. I (NOVIKOV S THEORY REVISITED) D. Burghelea and S. Haller

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1 ON THE TOPOLOGY AND ANALYSIS OF A CLOSED ONE FORM. I (NOVIKOV S THEORY REVISITED) D. Burghelea and S. Haller Abstract. We consider systems (M, ω, g) with M a closed smooth manifold, ω a real valued closed one form and g a Riemannian metric, so that (ω, g) is a Morse-Smale pair, Definition 2. We introduce a numerical invariant ρ(ω, g) [0, ] and using the theory of Dirichlet series and extensions of Witten-Helffer-Sjöstrand results improve Morse-Novikov theory for (M, ω, g) provided ρ(ω, g) <. For example we show that the Novikov complex of (M, ω, g) is an extension of a complex of free modules over a ring Λ [ω],ρ (subring of the Novikov field Λ [ω]) of holomorphic functions in {s C Re(s) > ρ} and the Novikov incidence numbers I q(x, y, ˆα), cf. Introduction, can be entirely recovered from the spectral geometry of (M, ω, g). Theorem 2 part (3) and Theorem 4 are the main new results of this paper with Theorem 1 a key ingredient in their proof. Contents 0. Introduction 1. The results 1.1 Novikov rings, Dirichlet series 1.2 Morse-Smale condition and the invariant ρ 1.3 Compactification 1.4 Novikov complexes 1.5 Witten-Helffer-Sjöstrand theory 2. The proof of Proposition 2 3. The proof of Proposition 3 4. The proof of Theorem Some notations 4.2 The proof for degree of rationality The transversality of p and s 4.4 The general case 5. The proof of Theorem 2 6. Sketch of the proof of Theorems 3 and 4 Supported in part by NSF. The second author is supported by the Fonds zur Förderung der wissenschaftlichen Forschung project number P14195-MAT. 1 Typeset by AMS-TEX

2 2 D. BURGHELEA AND S. HALLER 0. Introduction Let (M, ω, g) be a system consisting of a closed connected smooth n-dimensional manifold M, a closed one form ω and a Riemannian metric g. The form ω induces the homomorphism [ω] : H 1 (M; Z) R. Denote by Γ := H 1 (M; Z)/ ker([ω]) and write r(ω) := rank Γ. For any two points x, y M denote by P(x, y) the set of Γ-equivalence classes of smooth paths α : [0, 1] M with α(0) = x and α(1) = y, where we say that α is Γ-equivalent to β iff [ω](αβ 1 ) = 0. Here αβ 1 represents the cycle obtained by going along α and returning along β. The equivalence class of α will be denoted by ˆα. The juxtaposition of paths α and β with α(1) = β(0) defines an action Γ P(x, y) P(x, y) which is free and transitive, and the obvious map P(x, y) P(y, z) P(x, z). The form ω also associates the function [ω] : P(x, y) R defined by [ω](ˆα) := α ω R. Suppose ω is a Morse form. Then each critical point x Cr(ω) := Zeros(ω) is non-degenerated and has an index, ind(x) {0, 1,..., dim(m)}. The unstable set Wx of the vector field X = grad g ω, at the critical point x, is the image in M by the one to one immersion i x : R ind(x) M defined in an obvious way using the trajectories departing from x. For any x Cr(ω) choose an orientation O x on R ind(x). Denote by o := {O x x Cr(ω)} the collection of all these orientations. Suppose that (ω, g) satisfies the Morse-Smale condition, cf. Definition 2 in section 1.2. For any x Cr q (ω), y Cr q 1 (ω) and ˆα P(x, y), S. P. Novikov has associated the integer number I q (x, y, ˆα), cf. section 1.4 for definition, and has noticed the following properties: (1) For any real number R the set {ˆα P(x, y) Iq (x, y, ˆα) 0, [ω](ˆα) R } is finite. (2) For any x Cr q (ω), z Cr q 2 (ω) and ˆγ P(x, z) the sum y Cr q 1(ω) ˆα P(x,y), ˆβ P(y,z) ˆα ˆβ=ˆγ I q (x, y, ˆα)I q 1 (y, z, ˆβ) = 0, which means that in the sum above, the left side contains only finitely many nonzero terms whose sum is zero. As a consequence Novikov has observed that the collections of numbers I q (x, y, ˆα) can be algebraically organized to provide a cochain complex (NC, ) of free modules over the Novikov ring Λ [ω], which is actually a field, see section 1.4. The elements of Λ [ω] can be viewed as formal power series in r(ω) variables with complex coefficients. We introduce a numerical invariant ρ(ω, g) [0, ], see Definition 3 in section 1.2, conjecturally always smaller than, and the first purpose of this paper

3 ON THE TOPOLOGY AND ANALYSIS OF A CLOSED ONE FORM. I 3 is to show that if ρ(ω, g) < (cf. Theorem 2(3)) (3) For any x Cr q (ω) and y Cr q 1 (ω) the sum I q,s (x, y) := I q (x, y, ˆα)e s[ω](ˆα) ˆα P(x,y) is a Dirichlet series, cf. [Se73], which defines a holomorphic function in the half plane {s C Re(s) > ρ}. The main ingredient in the proof is Theorem 1. It was brought to our attention that if r(ω) = 1 this result follows also from the work of Farber cf. [Fa85]. As consequence, we answer positively, in the case ρ <, a question raised by S. P. Novikov, cf. Section 1.4, but more important, we observe that the collections of numbers I q (x, y, ˆα) can be algebraically organized to provide a cochain complex (C, ) of free modules over a smaller ring Λ [ω],ρ, Λ [ω],ρ Λ [ω], cf. section 1.1. The ring Λ [ω],ρ is a ring of holomorphic functions in {s C Re(s) > ρ} and for any complex number s with Re(s) > ρ, the evaluation at s provides a surjective ring homomorphism ev s : Λ [ω],ρ C. This permits to regard (C, ) as a holomorphic family (C, s ) of cochain complexes of finite dimensional C-vector spaces. The component C q is the C-vector space generated by the critical points of ω of index q, hence has a canonical base. The complex (C, s ) is the tensor product of (C, ) with C regarded as a Λ [ω],ρ -module via ev s : Λ [ω],ρ C. With respect to the canonical base of C q, the boundary map s q can be written as a matrix whose entries s(x, q y), x Cr q+1 (ω), y Cr q (ω) are functions s I q+1,s (x, y) which we show are defined by the Dirichlet series obtained from the numbers I q+1 (x, y, ˆα). In particular the numbers I q (x, y, ˆα) are entirely determined by the restriction of the function s(x, q y) to (a, ), for a > ρ. Since (C, s ) is a holomorphic family of complexes it is completely determined by s = t (T, ), T a real number large enough. The second purpose of this paper is to construct (using analysis=spectral geometry) a smooth one parameter family of cochain complexes of finite dimensional R-vector spaces ( ) Ω t,sm(m), d t (Ω (M), d t := d + tω ) and a smooth family of isomorphisms from ( ) ( ) Ω t,sm(m), d t to Maps(Cr (ω), R), t (whose complexification is (C, t ) for t real and large enough. Then ( ) Ω t,sm(m), d t will carry all information provided by the Novikov complex, (NC, ). Precisely, given a system as above (M, ω, g), with ω a Morse form, Theorem 3 claims that there exists a positive real number T so that for t T the de Rham complex ( Ω (M), d t := d + tω ) decomposes canonically as a direct orthogonal sum of two complexes ( ) ( ) Ω t,sm(m), d t and Ω t,la (M), d t. The first complex has the component Ω q t,sm(m), a finite dimensional vector space of dimension equal to the cardinality of Cr q (ω). In the case of an exact form this result is due to E. Witten 1. If (ω, g) satisfies the Morse-Smale conditions, ρ(ω, g) < and one gives the orientations o, by Theorem 4 we show that the integration theory provides an isomorphism between ( ) ( ) Ω t,sm(m), d t and Maps(Cr (ω), R), t for any t T, where 1 It was first noticed in [Pa87] that Witten s agruments hold for a closed one (Morse) form and can be used in Novikov theory

4 4 D. BURGHELEA AND S. HALLER T is some positive real number larger than T and ρ discussed above. Moreover, for t T we construct a base E t,x Ω t,sm(m), x Cr(ω). With respect to this base d q t is a matrix whose entries are exactly the functions t I q+1,t (x, y). This results reformulates and extends results of Helffer and Sjöstrand for ω exact 2, cf. [HeSj85]. Consequently the family ( Ω t,sm(m), d t ) can be viewed as an analytic substitute of the Novikov complex. When the base E t,x is available, which is the case if (ω, g) is Morse-Smale, ρ(ω, g) < and the orientations o are provided, this complex permits the derivation of the numbers I q (x, y, ˆα). All these results are immediate corollaries of Theorems 1 4 stated in section 1 and of Proposition 4 in section 5, which are of independent interest and have other pleasant applications, which will be worked out in the second part of this work. Theorem 2 part (3) and Theorem 4 are the main new results of this paper. Theorems 1 and 3, important ingredients in the proof of Theorems 2 and 4, are not far from what is known. We have included their proof from lack of satisfactory treatment in the existing literature. 1. The results 1.1 Novikov rings, Dirichlet series. Let (M n, ω) be a pair consisting of a closed connected smooth n-dimensional manifold M and a closed real valued 1-form ω Z 1 (M) := {ω Ω 1 (M) dω = 0}. The form ω induces the homomorphism [ω] : H 1 (M; Z) R whose image is a finitely generated free Abelian group of rank r. Denote by Γ := H 1 (M; Z)/ ker([ω]). The integer r = rank(γ) is called degree of rationality of the form ω. We identify Γ to Z r by choosing a base e 1,..., e r Γ with [ω](e i ) = κ i R positive real numbers Q-linearly independent. Let M π M be the regular Γ covering associated with H 1 (M; Z) Γ, i.e. M is a connected covering, such that for one (and hence all) m M img ( π 1 ( M, m) π π 1 (M, m) ) = ker ( π 1 (M, m) [ω] R ), where m = π( m). The group Γ acts freely on M with quotient space M. The pull back of ω on M is exact, i.e. π ω = dh, with h : M R a smooth function. This function is unique up to an additive constant. Given m M there exists a unique function h m, so that π ω = dh m and h m ( m) = 0. In particular ω induces a function H : M M R defined by H( x, ỹ) = h m ( x) h m (ỹ), which is independent of m. When there is no risk of confusion we write h for any of the function h m. Note that (1.1) h(γ x) = h( x) + [ω](γ), 2 in the case ω = dh we have ρ = 0 and (C, t ) is a rescaling of the Morse complex associated to (M, h, g)

5 ON THE TOPOLOGY AND ANALYSIS OF A CLOSED ONE FORM. I 5 for all γ Γ. S. P. Novikov, see [N93], has introduced the ring Λ [ω], consisting of functions f : Γ C with the property that for any R R the set { γ Γ f(γ) 0, [ω](γ) R } is finite. The product in this ring is given by convolution, i.e. (f g)(γ) = γ Γ f( γ)g( γ 1 γ). Because of the Novikov condition above this sum is actually finite and f g is in Λ [ω]. Since [ω] : Γ R is injective, or equivalently the numbers κ i are Q-linearly independent, Λ [ω] is actually a field, cf. [HoSa95]. Each f Λ [ω] defines a Dirichlet series ˆf(s) := γ Γ f(γ)e s[ω](γ) = n i Z f(n 1,..., n r )e s(κ1n1+ +κrnr), where the set of numbers κ 1 n κ r n r with f(n 1,..., n r ) 0 is a strictly increasing sequence of real numbers λ 1 < λ 2 < which is either finite or is tending to +. Recall that if (λ n ) n N is an increasing sequence of real numbers tending to + a Dirichlet series with exponents λ n is a series of the form ˆf := a n e sλn, a n C, s C. If the series converges for s 0, it defines a holomorphic function on the open half plane {s C Re(s) > Re(s 0 )} so there exists ρ( ˆf) R { }, referred to as the abscissa of convergence of ˆf, making the series a holomorphic function on {s C Re(s) > ρ}. Note that f g = ˆf ĝ and ρ( f g) sup{ρ( ˆf), ρ(ĝ)}. Let Λ [ω],ρ be the the subring of Λ [ω] consisting of elements f Λ [ω] whose corresponding Dirichlet series is convergent for any s C with Re(s) > ρ. Any such s gives rise to an evaluation homomorphism (1.2) ev s : Λ [ω],ρ C, f ˆf(s), and let ι : Λ [ω],ρ F be the obvious ring homomorphism obtained by restricting the holomorphic function defined by the element in Λ [ω],ρ to the interval (ρ, ), and where F denotes the ring of germs at + of C-valued analytic functions f : (a, ) C, where a R. Clearly ι is injective and the general theory of Dirichlet series (or almost periodic functions) permits to recover the coefficients f(n 1,..., n r ) from the germ ι(f), cf. [Se73]. 1.2 Morse-Smale condition and the invariant ρ. Recall that for x Cr(ω) := Zeros(ω) the Hessian of ω at x is (1.3) H x ω : T x M T x M R, (H x ω)(x, Y ) := ( X ω)(y ),

6 6 D. BURGHELEA AND S. HALLER where is any linear connection on M. The Hessian H x ω does not depend on the connection and is symmetric since ω is closed. The closed 1-form ω is called Morse form if H x ω is non-degenerate for every x Cr(ω). The index of ω at x Cr(ω) is the index of H x ω. By the Morse lemma, for any x Cr(ω) there exists an open neighborhood U x of x, positive real numbers c x, ɛ x and a diffeomorphism θ x : (U x, x) (D n (ɛ x ), 0), where D n (r) denotes the open disc of radius r in R n centered at 0, so that (a) (θ 1 x ) ω = d ( c x (x x 2 k) + c x (x 2 k x 2 n) ), where k = ind(x). In what follows we consider systems (M, ω, g) where M is a closed manifold, ω a closed 1-form as above and g is a Riemannian metric. Let grad g ω be the unique vector field which corresponds to ω by the bijective correspondence between vector fields and closed 1-forms provided by the Riemannian metric g and set X := grad g ω. For each x M denote by γ x (t) the trajectory of X with γ x (0) = x. For x Cr(ω) denote by W x ± the sets W ± x = { y lim t ± γ y(t) = x }. They will be referred to as the stable resp. unstable sets of the critical point x. Definition 1 (Morse pairs). The pair (ω, g) is called a Morse pair if for any x Cr(ω) there exists ɛ x, c x and θ x, so that (a) and the following condition (b) are satisfied. (b) (θx 1 ) g = dx 1 dx dx n dx n In view of the theorem of existence, uniqueness and smooth dependence on the initial conditions for the solutions of ordinary differential equations, the fact that (ω, g) is a Morse pair implies that W x resp. W + x is the image by a smooth one to one immersion i x : R k M resp. i + x : R n k M, where k = ind(x). Denote by h x : R k R the unique smooth map which satisfies (i x ) ω = dh x, h x (0) = 0 and by g x := (i x ) g the pull back of the Riemannian metric g by the immersion i x, which is a Riemannian metric on R k. Definition 2 (Morse-Smale condition). The pair (ω, g) is called Morse-Smale if it is a Morse pair and in addition for any x, y Cr(ω), i x and i + y are transversal. Note that if (ω, g) is a Morse resp. Morse-Smale pair then so is (tω, g) for any 0 t R. Denote by G be the set of smooth Riemannian metrics on M. Let U M be open and g G. Denote by G g,u the set G g,u := { g G x M \ U : g (x) = g(x) }. The following almost obvious result establishes the existence of Morse forms and Morse pairs.

7 ON THE TOPOLOGY AND ANALYSIS OF A CLOSED ONE FORM. I 7 Proposition 1. Suppose M is a closed manifold. Then the following holds: (1) The set of Morse forms is open and dense subset of Z 1 (M) equipped with the C 1 -topology. (2) Let ω be a Morse form, g a Riemannian metric and let U be a neighborhood of Cr(ω). Then the set of metrics g G g,u, so that (ω, g ) is a Morse pair is dense in G g,u with respect to the C 0 -topology. The following proposition establishes the existence of Morse-Smale pairs. Its proof can be derived from Kupka-Smale s theorem, cf. [Pe67]. In section 2 we will give an alternative proof on the lines of [Sch93]. Proposition 2. Let (ω, g) be a Morse pair, ε > 0 small and set U := z Cr(ω) B(z, ε) \ B(z, ε 2 ). Then there exists a Banach manifold G G g,u of smooth Riemannian metrics, which is dense in G g,u with respect to the L 2 -topology, and a residual subset G G, such that for any g G the pair (ω, g ) is Morse-Smale. Definition 3 (The invariant ρ). For a Morse-Smale pair (ω, g) we denote ρ(ω, g) := inf { a R + x Cr(ω) : e ah x R ind(x) vol g x < }. It is conceivable that there are no such positive real numbers a, in which case we put ρ(ω, g) =. We believe that always ρ(ω, g) <. There are plenty of examples where ρ(ω, g) = The invariant ρ(ω, g) will be discussed in a forthcoming paper. 1.3 Compactification. Let (ω, g) be a Morse-Smale pair. Denote the set of critical points of h by Cr(h) := Cr(dh) = π 1 (Cr(ω)). Recall that for x Cr(h) one has the stable and unstable manifolds W ± x of the negative gradient flow of h. One can also consider the immersions i + x : Rn k M and i x : Rk M, where k = ind( x), which in this case are embeddings. The submanifolds W ± x are exactly their images. Note that with the notation h x introduced in section 1.2, one has h x = h x i x, for any x with π( x) = x. 3 If dim(m) 2 and (ω, g) is a Morse-Smale pair then ρ(ω, g) = 0. Indeed, for ind(x) = 1 this follows from Lemma 3 in section 3, below. Moreover for ind(x) = n = dim(m) one has for all a 0 x volg x vol(m) <, R n e ah since i x : R n M is a one to one immersion. 4 If (M i, ω i, g i ), i = 1, 2 are Morse-Smale pairs then M 1 M 2, π 1 ω 1 + π 2 ω 2, π 1 g 1 + π 2 g 2 is a Morse-Smale pair and ρ π 1 ω 1 + π 2 ω 2, π 1 g 1 + π 2 g 2 sup ρ(ω 1, g 1 ), ρ(ω 2, g 2 ), where π i : M 1 M 2 M i denotes the canonical projection.

8 8 D. BURGHELEA AND S. HALLER Observation 1. Cr(h) M is a discrete subset. Γ acts freely on Cr(h) with quotient set Cr(ω). Observation 2. π : h 1 (c) M is injective, for all c R. In particular any critical level of h contains only finitely many critical points. Observation 3. If x Cr(h), then π : W ± x W ± π(x) M is an injective immersion. The Morse-Smale condition from Definition 2 is equivalent to the transversality of W x and W + ỹ for any two x, ỹ Cr(h). Observation 4. The Morse-Smale condition implies that M( x, ỹ) := W x W + ỹ is a submanifold of M of dimension ind(x) ind(y). The manifold M( x, ỹ) is equipped with the action µ : R M( x, ỹ) M( x, ỹ), defined by µ(t, z) = γ z (t). If x ỹ the action µ is free and we denote the quotient M( x, ỹ)/r by T ( x, ỹ). The quotient T ( x, ỹ) is a smooth manifold of dimension ind(x) ind(y) 1, possibly empty, diffeomorphic to the submanifold h 1 (c) M( x, ỹ), where c is any regular value of h with h( x) > c > h(ỹ). Note that if ind( x) ind(ỹ), and x ỹ, in view of the transversality requested by the Morse-Smale condition, M( x, ỹ) =. If x = ỹ, then W x W + x = x. The elements of T ( x, ỹ) will be referred to as the unparameterized trajectories from x to ỹ. Definition 4 (Broken trajectories). An unparameterized broken trajectory from x Cr(h) to ỹ Cr(h) is an element of B( x, ỹ) := k 0, ỹ 0,...,ỹ k+1 Cr(h) ỹ 0= x, ỹ k+1 =ỹ ind(ỹ i)>ind(ỹ i+1) T (ỹ 0, ỹ 1 ) T (ỹ k, ỹ k+1 ). An unparameterized broken trajectory from x Cr(h) to the level λ R is an element of B( x; λ) := k 0, ỹ 0,...,ỹ k Cr(h) ỹ 0= x ind(ỹ i)>ind(ỹ i+1) T (ỹ 0, ỹ 1 ) T (ỹ k 1, ỹ k ) (W ỹ k h 1 (λ)). Clearly, if λ > h( x) then B( x; λ) =. There is an obvious way to regard B( x, ỹ) resp. B( x; λ) as a subset of C 0( [h(ỹ), h( x)], M ) resp. C 0( [λ, h( x)], M ), by parameterizing a broken trajectory by the value of h. This leads to the following characterization and implicitly to a canonical parameterization of an unparameterized broken trajectory. Observation 5. Let x, ỹ Cr(h) and set a := h(ỹ), b := h( x). The parameterization above defines a one to one correspondence between B( x, ỹ) and the set of continuous mappings γ : [a, b] M, which satisfy the following two properties: (1) h(γ(s)) = a + b s, γ(a) = x and γ(b) = ỹ.

9 ON THE TOPOLOGY AND ANALYSIS OF A CLOSED ONE FORM. I 9 (2) There exists a finite collection of real numbers a = s 0 < s 1 < < s r 1 < s r = b, so that γ(s i ) Cr(h) and γ restricted to (s i, s i+1 ) has derivative at any point in the interval (s i, s i+1 ), and the derivative satisfies (1.4) γ (s) = grad g h grad g h 2 ( γ(s) ). Similarly the elements of B( x; λ) correspond to continuous mappings γ : [λ, b] M, which satisfies (1) and (2), with a replaced by λ. In section 3 we will verify the following Proposition 3. Let (ω, g) be a Morse-Smale pair, x, ỹ Cr(h) and λ R. Then: (1) B( x, ỹ) is compact, with the topology induced from C 0( [h(ỹ), h( x)], M ). (2) B( x; λ) is compact, with the topology induced from C 0( [λ, h( x)], M ). For ỹ 0,..., ỹ k Cr(h) with ind(ỹ i ) > ind(ỹ i+1 ), consider the smooth map iỹ0,...,ỹ k : T (ỹ 0, ỹ 1 ) T (ỹ k 1, ỹ k ) W ỹ k M, M denotes the inclu- defined by iỹ0,...,ỹ k (γ 1,..., γ k, ỹ) := iỹk (ỹ), where i x : W x sion. Definition 5 (Completed unstable manifold). For x Cr(h) define Ŵ x := T (ỹ 0, ỹ 1 ) T (ỹ k 1, ỹ k ) W ỹ k. k 0, ỹ 0,...,ỹ k Cr(h) ỹ 0= x ind(ỹ i)>ind(ỹ i+1) Moreover, let î x : Ŵ x M denote the mapping, whose restriction to T (ỹ 0, ỹ 1 ) T (ỹ k 1, ỹ k ) W ỹ k is given by iỹ0,...,ỹ k and let ĥ x := h x î x : Ŵ x R. To formulate the next result we need an additional concept, smooth manifold with corners. Recall that an n-dimensional manifold P with corners is a paracompact Hausdorff space equipped with a maximal smooth atlas with charts ϕ : U ϕ(u) R n +, where R n + = {(x 1, x 2,..., x n ) x i 0}. The collection of points of P which correspond (by some and then by any chart) to points in R n with exactly k coordinates equal to zero is a well defined subset of P and it will be denoted by P k. It has a structure of a smooth (n k)-dimensional manifold. P = P 1 P 2 P n is a closed subset which is a topological manifold, and (P, P ) is a topological manifold with boundary P. Theorem 1. Let (ω, g) be a Morse-Smale pair. (1) For any two critical points x, ỹ Cr(h) the smooth manifold T ( x, ỹ) has B( x, ỹ) as a canonical compactification. Moreover B( x, ỹ) has the structure of a compact smooth manifold with corners, and B( x, ỹ) k = T (ỹ 0, ỹ 1 ) T (ỹ k, ỹ k+1 ), ỹ 0,...,ỹ k+1 Cr(h) ỹ 0= x, ỹ k+1 =ỹ ind(ỹ i)>ind(ỹ i+1)

10 10 D. BURGHELEA AND S. HALLER especially B( x, ỹ) 0 = T ( x, ỹ). (2) For any critical point x Cr(h), Ŵ x has a canonical structure of a smooth manifold with corners, and (Ŵ x ) k = T (ỹ 0, ỹ 1 ) T (ỹ k 1, ỹ k ) W ỹ k, ỹ 0,...,ỹ k Cr(h) ỹ 0= x ind(ỹ i)>ind(ỹ i+1) especially (Ŵ x ) 0 = W x. Moreover î x and ĥ x are smooth and proper maps, and î x is a closed map. The proof of this theorem will be given in section 4. Propositions 1 3 are known in literature and Theorem 1, can be also found in [L95]. Our proof of Theorem 1 is however different from the one sketched in [L95] and we hope more conceptual. It also has the virtue that it extends essentially word by word to Bott-Smale pairs. 1.4 Novikov complexes. Let (ω, g) be a Morse-Smale pair. For any x Cr(ω) choose an orientation O x in Wx and denote the collection of these orientations by o. Via π these orientations induce orientations O x on W x. Denote by X q := Cr q (h) = { x Cr(h) ind(x) = q}. Theorem 1 implies the existence of the map I q : X q X q 1 Z defined as follows: If T ( x, ỹ) = put I q ( x, ỹ) = 0. If T ( x, ỹ), then for any γ T ( x, ỹ) the set γ W ỹ appears as an open set of the boundary Ŵ x and the orientation O x induces an orientation on it. If this is the same as the orientation Oỹ, we set ε(γ) = +1, otherwise we set ε(γ) = 1. Now define I q ( x, ỹ) by I q ( x, ỹ) := ε(γ), γ T ( x,ỹ) which is a finite sum by Proposition 3(1). The following result establishes the main properties of the numbers I q ( x, ỹ). Theorem 2. Suppose (ω, g) is a Morse-Smale pair. Then: (1) I q (γ x, γỹ) = I q ( x, ỹ), for all γ Γ. (2) For all x X q and ỹ X q 2 the sum below contains only finitely many nonzero terms and one has (1.5) I q ( x, z)i q 1 ( z, ỹ) = 0. z X q 1 (3) For any x X q, ỹ X q 1 and any s C with Re(s) > ρ(ω, g) the sum (1.6) I q (γ x, ỹ)e s[ω](γ) is convergent. γ Γ If ρ(ω, g) < Theorem 2(3) provides a positive answer to the following conjecture formulated by S. P. Novikov, cf. [N93], [A90] and [Pa98].

11 ON THE TOPOLOGY AND ANALYSIS OF A CLOSED ONE FORM. I 11 Conjecture (Novikov). For any Morse-Smale pair (ω, g) and any two critical points x X q and ỹ X q 1 the integers I q (γ x, ỹ), γ Γ = Z r have at most exponential growth. More precisely, there exist constants C x,ỹ, M x,ỹ R, such that for all γ Γ. I q ( γ x, ỹ) C x,ỹ e M x,ỹ[ω](γ), In [Pa98] A. V. Pazhitnov has verified this conjecture for a generic subset of the set of Riemannian metrics g, for which (ω, g) is Morse-Smale. Remark. Note, that the numbers I q ( x, ỹ) can be defined without any reference to the covering π : M M. Indeed, let x M and choose x M, such that π( x) = x. Then there exists a natural one to one correspondence between P(x, y) and π 1 (y), given by lifting a path α from x to y, to a path starting at x and looking at the endpoint, which does only depend on ˆα. In view of Theorem 2(1), I q (x, y, ˆα) := I q ( x, ỹ) is well defined, i.e. independent of the choice of x, and the formulas (1.5) and (1.6) become (2) and (3) in the introduction. Definition 6 (Novikov condition). We say a map f : X q C has property (N) if for any x X q and any R R the set {γ Γ f(γ x) 0, [ω](γ) R} is finite, and we say it has property (N ρ ) if for any x X q and s C with Re(s) > ρ the series γ f(γ x)e s[ω](γ) is convergent. Let NC q denote the C-vector space of functions X q C, which satisfy (N). For λ Λ [ω] and f NC q we set (λ f)( x) := γ Γ λ(γ)f(γ 1 x). In this way NC q becomes a free Λ [ω] -module of finite rank equal to the cardinality Cr q (ω). Moreover let C q denote the subspace of functions, which satisfy (N) and (N ρ ). The formula above also makes C q a free Λ [ω],ρ-module of the same rank as NC q. Note, that every section σ : Cr q (ω) X q, i.e. π σ = id, defines a base for both NC q and C q, namely {δ σ(x) x Cr q (ω)}, where δ x : Cr(h) C is the Kronecker function, δ x (ỹ) = δ x,ỹ. For ỹ X q we define q (δỹ) Maps(X q+1, C) by ( q (δỹ))( x) := I q+1 ( x, ỹ). Theorem 2(3) shows that q (δỹ) satisfies (N ρ ) and Corollary 1 in section 3 shows, that it also satisfies (N), i.e. q (δỹ) C q+1 NC q+1. From Theorem 2(1) one gets q (δ γỹ ) = δ γ q (δỹ). This equivariance property and the fact that NC q and C q are free modules shows, that q extends uniquely to a Λ [ω] resp. Λ [ω],ρ -linear map q : NC q NC q+1 resp. q : C q C q+1, both given by the formula q (f)( x) = ỹ X q I q+1 ( x, ỹ)f(ỹ).

12 12 D. BURGHELEA AND S. HALLER Theorem 2(2) immediately shows q+1 q = 0, by checking it on the elements δỹ. So we have two cochain complexes, the Novikov complex (NC, ) and (C, ) and a natural isomorphism Λ [ω] Λ [ω],ρ (C, ) = (NC, ), λ f λ f. Define C q := Maps(Cr q (ω), C) and note that this is a finite dimensional C-vector space. Suppose ρ := ρ(ω, g) < and let s C with Re(s) > ρ. For x Cr q (ω) and y Cr q 1 (ω) choose ỹ X q 1, such that π(ỹ) = y and define I q,s (x, y) := I q ( x, ỹ)e sh( x,ỹ) C, x π 1 (x) which converges by Theorem 2(3) and does not depend on the choice of ỹ. The map H( x, ỹ) := h( x) h(ỹ) was introduced in section 1.1. Moreover it follows from Theorem 2(2), that one has I q,s (x, z)i q 1,s (z, y) = 0, z Cr q 1(ω) for all x Cr q (ω) and y Cr q 2 (ω). So for every s C with Re(s) > ρ we get another cochain complex (C, s ), where s(f)(x) q := I q+1,s (x, y)f(y). y Cr q(ω) Next define an evaluation map ev h s : C C, by ev h s (f)(x) := f( x)e sh( x). x π 1 (x) This depends on the choice of h, but if one changes h it changes only by a nonzero multiplicative constant in C. One easily checks ev h s q = q s ev h s, i.e. ev h s : (C, ) (C, s ) is a chain mapping. Moreover one has ev h s (λ f) = ev s (λ) ev h s (f), where ev s : Λ [ω],ρ C is the evaluation map from (1.2). Therefore C Λ [ω],ρ (C, ) = (C, s ), z f z ev h s (f) is an isomorphism of cochain complexes over C. Here the Λ [ω],ρ-module structure on C is the one given by ev s : Λ [ω],ρ C. Finally let Ω (M; C) := Ω (M) C denote the C-valued differential forms on M and consider d q s : Ω q (M; C) Ω q+1 (M; C), d q s(α) := dα + sω α.

13 ON THE TOPOLOGY AND ANALYSIS OF A CLOSED ONE FORM. I 13 Since ω is closed one has d q+1 s d q s = 0. So ( Ω (M; C), ds) is a cochain complex and for Re(s) > ρ one has a chain mapping Int s : ( Ω (M; C), d s) (C, s ), Int s (α)(x) := e sh x π α, where x Cr(h), such that π( x) = x, and h x is the unique h, such that h x ( x) = 0. The integral converges because of Proposition 4(1) and is obviously independent of the choice of x. Proposition 4(2) shows that Int s intertwines the differentials. Proposition 4 is stated in section 5 below. Theorem 4 in the next section implies, that Int s induces an isomorphism in cohomology. The cochain complex (C, s ) can be regarded as a holomorphic family of cochain complexes of finite dimensional C-vector spaces which, in view of the fact that the cohomology of (C, s ) does not change dimension for large s, is a holomorphic bundle of cochain complexes over {s C s > R}, R a real number large enough. 1.5 Witten-Helffer-Sjöstrand theory. Let M be a closed manifold and ω a closed 1-form. For t R consider the complex ( Ω (M), d t ) with differential W x d q t : Ω q (M) Ω q+1 (M), d q t (α) := dα + tω α. Clearly d q 0 = dq. Recall, that on an oriented n-dimensional Riemannian manifold (M, g) one has the Hodge-star operator : Ω q (M) Ω n q (M). It is a zero order operator and satisfies = ( 1) q(n q) id : Ω q (M) Ω q (M). One defines the fiberwise scalar product, : Ω q (M) Ω q (M) Ω 0 (M), α 1, α 2 := 1 (α 1 α 2 ) and the formal adjoint of d q t, (d q t ) : Ω q+1 (M) Ω q (M), (d q t ) (α) = ( 1) nq+1 d n q 1 t ( α) = (d q ) (α) + i gradg ωα. The fiberwise scalar products, and the operators (d q t ) are independent of the orientation of M. They can even be defined (first locally and then, being differential operators, globally) for an arbitrary Riemannian manifold, not necessarily orientable. Moreover one has the scalar product Ω q (M) Ω q (M) R, α 1, α 2 := α 1 α 2 = α 1, α 2 vol. The operators (d q t ) are formal adjoints of d q t, more precisely d q t (α 1 ), α 2 = α1, (d q t ) (α 2 ). Next we introduce the Witten Laplacian for the closed 1-form ω, q t : Ω q (M) Ω q (M), M M q t (α) := (d q t ) ( d q t (α) ) + d q 1 t ( (d q 1 t ) (α) ). This is a second order differential operator, and q 0 = q, the Laplace-Beltramy operator. The operators q t are elliptic, selfadjoint and positive, hence their spectra spect( q t ) lie on [0, ). Finally one has ker( q t ) = { α Ω q (M) d q t (α) = 0, (d q 1 t ) (α) = 0 }. The following result extends a result due to E. Witten in the case that ω is exact.

14 14 D. BURGHELEA AND S. HALLER Theorem 3. Suppose that (ω, g) is a Morse pair. There exist constants C 1, C 2, C 3 and T 0 depending on (ω, g), so that for any t T 0 (1) spect( q t ) (C 1 e C2t, C 3 t) =, and (2) the number of the eigenvalues of q t in the interval [0, C 1 e C2t ] counted with their multiplicity is equal to the number of zeros of ω of index q. The above theorem states the existence of a gap in the spectrum of q t, namely the open interval (C 1 e C2t, C 3 t), which widens to (0, ) when t. Clearly C 1, C 2, C 3 and T 0 determine a constant T T 0, so that for t T, 1 (C 1 e C2t, C 3 t) and therefore and spect( q t ) [0, C 1 e C2t ] = spect( q t ) [0, 1] spect( q t ) [C 3 t, ) = spect( q t ) [1, ). For t T we denote by Ω q t,sm(m) the finite dimensional subspace, generated by the eigenforms of q t corresponding to the eigenvalues of q t smaller than 1. The elliptic theory implies that these eigenvectors, a priori elements in the L 2 -completion of Ω q (M), are actually in Ω q (M). Note that d q t : Ω q t,sm(m) Ω q+1 t,sm(m), so that ( Ω t,sm(m), d t ) is a finite dimensional cochain subcomplex of ( Ω (M), d t ). Clearly ( the orthogonal ) complement is also a closed subcomplex, we will denote by Ω t,la (M), d t. One has the following orthogonal decomposition ( Ω (M), d ) ( t = Ω t,sm (M), d ) ( t Ω t,la (M), d ) t, and (Ω t,la (M), d t ) is acyclic. Let (ω, g) be a Morse-Smale pair. Recall that for each critical point x Cr(ω) we have δ x Maps(Cr(ω), R) which takes the value 1 on x and 0 on all other critical points. Clearly {δ x x Cr(ω)} is a base of the vector space Maps(Cr(ω), R). We equip Maps(Cr(ω), R) with the unique scalar product which makes this base orthonormal. The next result is an extension of Helffer-Sjöstrand theorem as formulated in [BFKM96], but for closed one forms instead of functions. Theorem 4. Suppose (ω, g) is a Morse-Smale pair with ρ(ω, g) < and o are orientations as above. Then there exists T 0, depending on (ω, g) so that for t T Int t : ( Ω t,sm(m), d ) ( t Maps(Cr (ω), R), t ) is an isomorphism of cochain complexes. Moreover, there exists a family of isometries R q t : Maps(Cr q (ω), R) Ω q t,sm(m) of finite dimensional vector spaces so that Int t R q t = id +O(1/t). The proof of Theorems 3 and 4 is similar to the one given in [BFKM96] or [BFK] for Witten and Helffer-Sjöstrand theorems. However for the readers convenience we sketch the arguments in section 6.

15 ON THE TOPOLOGY AND ANALYSIS OF A CLOSED ONE FORM. I 15 For t T consider E t,x := (Int t ) 1 (δ x ) Ω t,sm(m). Clearly these forms provide a base for Ω t,sm(m), and the functions t I q,t (x, y) are the unique functions which satisfy the formula d q t (E t,y ) = I q+1,t (x, y)e t,x. x Cr q+1(ω) Consequently the numbers I q ( x, ỹ) or equivalently I q (x, y, ˆα) can be recovered from the family (Ω t,sm(m), d t ) and the base {E t,x x Cr(ω)} by using the theory of Dirichlet series. 2. The proof of Proposition 2 We will begin with few notations. Let R := R {± }, equipped with the structure of a manifold with boundary via the diffeomorphism R [ 1, 1], t t(1 + t 2 ) 1 2. Choose a tubular neighborhood V T M of the zero section, such that (exp, p) : T M V M M becomes a diffeomorphism onto its image. Here exp : T M M is defined with respect to a Riemannian metric g 0 fixed once and for all. Departing from the notation used in section 1 and in order to remain as close as possible to the reference [Sch93] we will write γ for an element of C x,y( R, M) := {γ C ( R, M) γ( ) = x, γ( ) = y}. For γ C x,y( R, M) we have a well defined Sobolev space H 1,2 (γ T M), cf. [Sch93] page 24 and a Sobolev embedding H 1,2 (γ T M) C 0 (γ T M). Here γ T M denotes the pull back of T M M by γ : R M. So H 1,2 (γ V ) := { σ H 1,2 (γ T M) t R : σ(t) V } is an open neighborhood of 0 H 1,2 (γ T M). We set ϕ γ : H 1,2 (γ V ) C 0 x,y( R, M), ϕ γ (σ) := exp σ, define and P 1,2 x,y := E 2 x,y := γ C x,y ( R,M) σ P 1,2 x,y img(ϕ γ ) {σ} L 2 (σ T M) and denote by π : E 2 x,y P 1,2 x,y the obvious projection. For γ C x,y( R, M) let ψ γ : H 1,2 (γ V ) L 2 (γ T M) E 2 x,y, ψ γ (σ, ξ) := ( ϕ γ (σ), Pt 1 ξ ),

16 16 D. BURGHELEA AND S. HALLER where Pt 1 denotes the time 1 parallel transport along the geodesics s ϕ γ (sσ(t)) with respect to the metric g 0. The following facts are not hard to verify, cf. Proposition 2.7 and Proposition 2.9 in [Sch93]. Fact 1. The maps ϕ γ resp. ψ γ, γ Cx,y( R, M), define an atlas which provides a structure of smooth Hilbert manifold on Px,y 1,2 resp. Ex,y 2 as well as a structure of a smooth Hilbert vector bundle for π : Ex,y 2 Px,y 1,2. These structures are, up to an isomorphism, independent on the metric g 0. The inclusions Cx,y( R, M) Px,y 1,2 Cx,y( R, 0 M) are continuous maps, have dense images and are homotopy equivalences. Fact 2. Let ω Z 1 (M) and x, y Cr(ω). The map F : P 1,2 x,y E 2 x,y, defined by c ( c, t c + (grad g ω) c ) is a smooth section. If x and y are non-degenerate, then the zeros of F are precisely the smooth mappings γ : R M, satisfying γ (t) = (grad g ω)(γ(t)), lim γ(t) = x and lim γ(t) = y. t t We will write M(x, y) := F 1 (0), which identifies with W x W + y. If x, y are non-degenerated, then pr 2 ψ 1 γ F ϕ γ : H 1,2 (γ V ) L 2 (γ T M) is a Fredholm mapping of index ind(x) ind(y). Fact 3. For γ M(x, y) the differential of F can be calculated using the charts ϕ γ and ψ γ. Precisely the differential of ψγ 1 F ϕ γ at 0 is the linear map T 0 (ψ 1 γ F ϕ γ ) : H 1,2 (γ T M) H 1,2 (γ T M) L 2 (γ T M) T 0 (ψ 1 γ F ϕ γ )(ξ) = ( ξ, t ξ + ξ grad g ω ), where denotes the connection on γ T M induced from the Levi-Civita connection on M provided by the Riemannian metric g 0, and t the induced connection on γ T M. Suppose ω is a Morse form, x Cr(ω) and let U x := B(x, ε) \ B(x, ε 2 ). We set S Ux := { A C (End sym (T M)) supp(a id) Ux }, where End sym (T M) denotes the endomorphisms of T M, which are symmetric with respect to g. For a sequence (λ k ) k N of real numbers and A C (End sym (T M)) we set A := λ k sup } {{ } A, k 0 k-times where denotes the covariant differentiation induced from g 0, and define S x := {A S Ux A < }. One can choose (λ k ) k N, such that (S x, ) becomes

17 ON THE TOPOLOGY AND ANALYSIS OF A CLOSED ONE FORM. I 17 an affine Banach space, which is dense in S Ux with respect to the L 2 -topology, cf. [Sch93] and Lemma 5.1 in [F88]. Finally let S + U x := { A S Ux A is pos. def. w. r. to g } and S + x := S x S + U x. Note that S + x is an open neighborhood of id in S x. Consider the smooth mapping Φ : S + x P 1,2 x,y E 2 x,y, Φ(A, c) = ( c, t c + (A grad g ω) c ). Notice that g A (, ) := g(a 1, ) is a Riemannian metric, and Φ A ( ) := Φ(A, ) = F ga, since grad ga ω = A grad g ω. So for any y Cr(ω) and any γ C x,y( R, M) pr 2 ψ 1 γ Φ A ϕ γ : H 1,2 (γ V ) L 2 (γ T M) is a Fredholm mapping of index ind(x) ind(y). If Φ(A, γ) = 0 the differential at (A, 0) of the vertical part of Φ in the chart given by ϕ γ and ψ γ is (2.1) D : S x H 1,2 (γ T M) L 2 (γ T M) ( D(B, ξ) := T (A,0) pr2 ψγ 1 Φ (id, ϕ γ ) ) (B, ξ) = t ξ + ξ (A grad g ω) + (B grad g ω) γ. Lemma 1. Let ω be a Morse form and x, y Cr(ω). Then Φ : S + x P 1,2 x,y E 2 x,y intersects the zero section transversally. Proof. Suppose Φ(A, γ) = 0. We have to show, that D : S x H 1,2 (γ T M) L 2 (γ T M) is onto. Since pr 2 ψγ 1 Φ A ϕ γ is a Fredholm mapping, we see that img(d) L 2 (γ T M) is a closed subspace of finite codimension. Suppose there exists 0 η L 2 (γ T M), such that (2.2) D(B, ξ), η = 0, (B, ξ) Sx H 1,2 (γ T M), where, denotes the inner product of L 2 (γ T M). From (2.1) one gets t ξ + ξ (A grad g ω), η = 0, ξ H 1,2 (γ T M). The adjoint of ξ t ξ + ξ (A grad g ω) is of the form η t η + Kη, for some K C (End(γ T M)). So η(t) 0 for all t R by the uniqueness result for ODEs. Choose t 0 R, such that γ(t 0 ) U x. From (2.2) and (2.1) we also get (B gradg ω) γ, η = 0, B S x, and hence g ( (B 0 grad g ω)(γ(t 0 )), η(t 0 ) ) = 0, B 0 End sym (T γ(t0)m). But since (grad g ω)(γ(t 0 )) 0, we must have η(t 0 ) = 0, a contradiction. verifies the surjectivity of D. This

18 18 D. BURGHELEA AND S. HALLER Lemma 2. Let x, y Cr(ω) and suppose F : Px,y 1,2 Ex,y 2 is transversal to the zero section. Then i x : R ind(x) Wx M and i + y : R n ind(y) W y + M are transversal. Proof. Suppose conversely, that they are not transversal at some point. Since every point in the intersection Wx W y + lies on a trajectory γ M(x, y), we may assume that this point is γ(0). Choose η 0 in the orthogonal complement of T γ(0) Wx + T γ(0) W y + and let η be the unique vector field along γ satisfying t η + Kη = 0 and η(0) = η 0, where K is the pointwise adjoint of ξ ξ grad g ω. If we can show, that η L 2 (γ T M), then we would get a contradiction, because t ξ + ξ grad g ω, η = ξ, t η + Kη = 0, but ξ t ξ + ξ grad g ω is onto, since F is transversal to the zero section. For ξ 0 T γ(0) Wx let ξ be the unique vector field along γ satisfying t ξ + ξ grad g ω = 0 and ξ(0) = ξ 0. From t ξ + ξ grad g ω = [ξ, grad g ω] γ it follows, that ξ(t) T γ(t) W x for all t. So t g(ξ, η) = g( t ξ, η) + g(ξ, t η) = g( t ξ + ξ grad g ω, η) = 0, and hence η(t) is orthogonal to T γ(t) Wx for all t. A similar argument shows, that η(t) is orthogonal to T γ(t) W y + for all t, too. Now consider the function α(t) := 1 2g(η(t), η(t)) > 0. Then α (t) = g(kη(t), η(t)) = g ( η(t), η(t) grad g ω ) = ( η(t) ω)(η(t)). Since the Hessian of ω at y, see (1.3), is negativ definite on the orthogonal complement of stable manifold we find a constant k > 0, such that α (t) = ( η(t) ω)(η(t)) 1 2kg(η(t), η(t)) = kα(t), for large t. So t ln α(t) k, hence ln α(t) ln(α(0)) kt and finally α(t) α(0)e kt, for large t. So we see that η(t) converges exponentially to 0 as t. A similar argument shows the exponential convergence for t. This shows η L 2 (γ T M), and the proof is complete. We are now in the position to give the Proof of Proposition 2. Recall first that a residual set in a complete metric space is a countable intersection of open and dense sets. By Baire category theorem it is a dense subset. Clearly a finite intersection of residual sets is residual.

19 ON THE TOPOLOGY AND ANALYSIS OF A CLOSED ONE FORM. I 19 Next note, that every Riemannian metric on M is of the form g A for a unique positive definite A C (End sym (T M)). We set G x := {g A A S + x } and G := x Cr(ω) G x x Cr(ω) G g,ux = G g,u. Since S + x is a Banach manifold which is dense in S + U x with respect to the L 2 -topology, the same is true fro G G g,u. From Lemma 1 and Sard s theorem for Fredholm maps between Banach manifolds, cf. Proposition 2.24 in [Sch93] it follows, that for every y Cr(ω) there exists a residual subset S x,y S + x, such that for any A S x,y the section F ga : P 1,2 x,y E 2 x,y intersects the zero section transversally. So G x := { g A A y Cr(ω) S x,y } is a residual subset of G x, and Lemma 2 implies, that for any g G x and any y Cr(ω) the mappings i x and i + y are transversal. So G := x Cr(ω) G x x Cr(ω) G x = G satisfies the statement of Proposition The proof of Proposition 3 Lemma 3. Let (ω, g) be a Morse-Smale pair. Then there exists a constant C > 0, such that (1) For all x, ỹ Cr(h) for which T ( x, ỹ) one has d( x, ỹ) C(h( x) h(ỹ)). (2) For all x Cr(h) and all z W x one has d( x, z) max{c(h( x) h( z)), 1}. Here d denotes the distance in M given by the Riemannian metric. 5 Proof. For r > 0 denote Ũ r := z Cr(ω) π 1( B(z, r) ), where B(z, r) denotes the open ball of radius r. Now choose 1 2 Ũ ε is a disjoint union of balls. Choose C, such that ε > 0, such that (3.1) 4 (grad g h)(z) C, for all z M \ Ũ ε 2. 5 Actually the proof shows, that there exists a small ball B( x, ε), such that for z W x \B( x, ε) one has d W ( x, z) C h( x) h( z), where d x W denotes the distance given by the induced x Riemannian metric on W x. The only extra argument is needed in (3.4), where one has to use the fact that for every ỹ Cr(h) every trajectory in B(ỹ, ε) has length at most 2ε.

20 20 D. BURGHELEA AND S. HALLER Let x, ỹ Cr(h) and γ T ( x, ỹ), parameterized by the value of h, cf. Observation 5. So γ : [a, b] M, where a = h(ỹ) and b = h( x). Suppose we have [s, t] [a, b]. If γ([s, t]) M \ Ũ ε then in view of (1.4) and 2 (3.1) we get (3.2) d ( γ(s), γ(t) ) t s γ (σ) dσ C (t s). 4 If γ([s, t]) Ũ ε and γ([s, t]) Ũε then there exists s, t [s, t], such that 2 γ([s, t ]) M \ Ũ ε, γ(s ) Ũ ε and γ(t ) Ũε. So (3.2) yields 2 2 (3.3) ε 2 = d( Ũ ε, Ũε) ( 2 d γ(s ), γ(t ) ) C 4 t s C t s. 4 This implies that there exist a = s 0 < t 0 < s 1 < t 1 < < s k < t k = b, such that γ([t i, s i+1 ]) M \Ũ ε, γ((s 2 i, t i )) Ũε, γ([s i, t i ]) Ũ ε and γ([s 2 i, t i ]) Ũε. So (3.2) and (3.3) imply (3.4) d ( γ(t i ), γ(s i+1 ) ) C(s i+1 t i ), d ( γ(s i ), γ(t i ) ) 2ε C(t i s i ). Adding all these estimates together gives d( x, ỹ) C(t k s 0 ) = C(b a) = C ( h( x) h(ỹ) ). This proves part (1). To see part (2) notice, that if z does not lie in the component of Ũε containing x, the argument above works and one gets d( x, z) C(h( x) h( z)). If both lie in the same component, one certainly has d( x, z) 2ε 1. Corollary 1. Let (ω, g) be a Morse-Smale pair. Then the following holds: (1) For all x, ỹ Cr(h) and R R the set {γ Γ T (γ x, ỹ), [ω](γ) R} is finite. (2) Given x, ỹ Cr(h), there exist only finitely many ỹ 1,..., ỹ k 1 Cr(h), such that T ( x, ỹ 1 ) T (ỹ k 1, ỹ). Proof. By Lemma 3(1) and (1.1) the set {γ x T (γ x, ỹ), [ω](γ) R} M is bounded, and since it is discrete too, it must be finite, for M is a complete Riemannian manifold. Statement (1) follows immediately. Part (2) follows from a similar argument, ind(ỹ i ) > ind(ỹ i+1 ) and h( x) h(ỹ i ) h(ỹ). Proof of Proposition 3. We will only prove part (1), the proof of (2) is similar. First we will show that B( x, ỹ) is closed. For notational simplicity set a := h(ỹ) and b := h( x). Suppose γ n B( x, ỹ) converge uniformly to γ C 0( [a, b], M ). Clearly the conditions in Observation 5(1) hold for γ, too. Since γ ([a, b]) is compact and Cr(h) is discrete and because of h(γ (s)) = a + b s, there are only finitely many s i [a, b], with γ (s i ) Cr(h). If γ (s) / Cr(h) then, for large n, the same holds for γ n (s) and (1.4) follows. So B( x, ỹ) is closed.

21 ON THE TOPOLOGY AND ANALYSIS OF A CLOSED ONE FORM. I 21 Lemma 3(2) implies, that {γ([a, b]) γ B( x, ỹ)} M is bounded and since M is a complete Riemannian manifold, its closure is compact. In view of the theorem of Arzela-Ascoli it remains to show that B( x, ỹ) is equicontinuous. So let ε > 0 small and let C ε denote the constant C we have constructed in the proof of Lemma 3, which actually depended on ε. Set δ := ε C ε and suppose s 0, s 1 [a, b], with s 1 s 0 δ. We have to show (3.5) d ( γ(s 0 ), γ(s 1 ) ) 2ε, for all γ B( x, ỹ). If γ([s 0, s 1 ]) M \ Ũ ε this follows from (3.2). If γ([s 2 0, s 1 ]) Ũ ε, we must 2 have γ([s 0, s 1 ]) Ũε, for otherwise we get a contradiction to (3.3). But since the diameter of each component of Ũε is 2ε, (3.5) follows in this case, too. 4. The proof of Theorem 1 For didactical reasons the proof will be given first in the particular case that the set of all critical values, h(cr(h)) is a discrete subset of R, i.e. ω has degree of rationality 1. Then we will show, that the same arguments properly modified hold in the general case as well. For the case where ω is exact the proof below is similar to the one in [BFK]. 4.1 Some notations. Let > c i > c i 1 >, i Z denote the set of all critical values of h. Choose ɛ i > 0 small enough, so that c i ɛ i > c i 1 + ɛ i 1, for all i Z. Denote, see Figure 1, Cr(i) := Cr(h) h 1 (c i ), M i :=h 1 (c i ), M ± i :=h 1 (c i ± ɛ i ) and M(i) :=h 1 (c i 1, c i+1 ). In view of Observation 2, Cr(i) is always a finite set, even when ω has degree of rationality greater than 1. To keep the notation simpler we will denote the critical points of of h by x, y,... instead of x, ỹ,.... There is no danger of confusion since the critical points of ω will not appear in this section. For any x Cr(i) denote, see Figure 1, S x ± :=W x ± M ± i, S x :=S x + Sx, W x ± (i) :=W x ± M(i) and SW x (i) :=S x + Wx (i).

22 22 D. BURGHELEA AND S. HALLER It will be convenient to write S ± i := S x ±, S i := W ± (i) := SW (i) := x Cr(i) x Cr(i) x Cr(i) x Cr(i) S x, W x ± (i) and SW x (i). Observation 6. We have: Figure 1 (1) S i M + i M i. (2) SW (i) S + i W (i) M + i M(i). (3) M ± i are smooth manifolds of dimension n 1, where n = dim( M). (4) M(i) is a smooth manifold of dimension n, actually an open set in M. (5) M i is not a manifold, however Ṁ i := M i \ Cr(i) and Ṁ ± i := M ± i \ S ± i are are smooth manifolds of dimension n, actually submanifolds of M. Let Φ t be the flow associated to the vector field grad g h/ grad g h 2 on M \ Cr(h) and consider the diffeomorphisms, see Figure 2, ψ i : M i M + i 1, ψ i(x) := Φ ci c i 1 ɛ i ɛ i 1 (x) and ϕ ± i : Ṁ ± i Ṁi, ϕ ± i (x) := Φ ±ɛ i (x),

23 ON THE TOPOLOGY AND ANALYSIS OF A CLOSED ONE FORM. I 23 as well as the submersion ϕ(i) : M(i) \ ( W + (i) W (i) ) Ṁi, ϕ(i)(x) := Φ h(x) ci (x). Observation 7. ϕ ± i and ϕ(i) extend to continuous maps ϕ ± i : M ± i M i and ϕ(i) : M(i) M i. Define P i := { (x, y) M + i Figure 2 M i ϕ + i (x) = ϕ i (y)}, and denote by p ± i : P i M ± i the canonical projections. One can verify the following Observation 8. P i is a smooth (n 1)-dimensional manifold with boundary (smooth submanifold of M + i M i ), whose boundary P i is diffeomorphic to S i M + i M i. Precisely we have (P1) p ± i : P i \ P i Ṁ ± i are diffeomorphisms, and (P2) the restriction of p + i p i to P i is a diffeomorphism onto S i, each p ± i restricted to P i identifies with the projection onto S ± i. Next we define Q(i) := { (x, y) M + i M(i) ϕ + i (x) = ϕ(i)(y)}, or equivalently, Q(i) consists of pairs of points (x, y), x M + i, y M(i), which lie on the same (possibly broken) trajectory. Moreover let l i : Q(i) M + i and r i : Q(i) M(i) denote the canonical projections. One can verify the following

24 24 D. BURGHELEA AND S. HALLER Observation 9. Q(i) is a smooth n-dimensional manifold with boundary (smooth submanifold of M + i M(i)), whose boundary Q(i) is diffeomorphic to SW (i) M(i). Precisely we have M + i and (Q1) l i : Q(i) \ Q(i) Ṁ + i is a smooth bundle with fiber an open segment, and r i : Q(i) \ Q(i) M(i) \ W (i) is a diffeomorphism. (Q2) The restriction of l i r i to Q(i) is a diffeomorphism onto SW (i), i.e. l i resp. r i restricted to Q(i) identifies with the projection onto S + i resp. W (i). Since P i and Q(i) are smooth manifolds with boundaries are smooth manifolds with corners. P r,r k := P r P r 1 P r k P r (r k) := P r P r k+1 Q(r k) 4.2 The proof for degree of rationality 1. The proof of Theorem 1 will be based on the following recognition method for a smooth manifold with corners. Observation 10. If P is a smooth manifold with corners, O, S smooth manifolds, p : P O and s : S O smooth maps so that p and s are transversal (p is transversal to s if its restriction to each k-boundary P k is transversal to s), then p 1 (s(s)) is a smooth submanifold with corners of P. Proof of Theorem 1 (1). We want to verify that B(x, y) is a smooth manifold with corners. Let x Cr(r + 1) and y Cr(r k 1), k 2. If k = 2 the statement is empty, if k = 1 there is nothing to check, so we suppose k 0. We consider P = P r,r k as defined above, O := r k i=r (M + i M i ) and S := Sx Mr M r k+1 S+ y. In order to define the maps p and s we consider ω i : M i M i M + i+1, ω i(x) := (x, ψ i (x)) and p i : P i M + i M i, p i(y) := ( p + i (y), p i (y)). We also denote by α : S x M + r resp. β : S + y M k r the restriction of ψ r+1 resp. ψ 1 r k to S x resp. S + y. Finally we set, see Diagram 1, and s := α ω r ω r k+1 β : S O p := p r p r k : P O. The verification of the transversality of p and s follows easily from (P1), (P2) and the Morse-Smale condition, as we will explain in section 4.3 below. It is easy to see that p 1 (s(s)) identifies to B(x, y) as topological spaces and we leave this verification to the reader. The compactness of B(x, y) is stated in Proposition 3.

25 ON THE TOPOLOGY AND ANALYSIS OF A CLOSED ONE FORM. I 25 Proof of Theorem 1 (2). Consider the set X := Ŵ x, the map î x : X = Ŵ x M and ĥ : X R, see Definition 5. For any positive integer k, denote by X(k) := î 1 x (M(k)). First we will topologize X(k) and put on it a structure of smooth manifold with corners, so that the restriction of î x and of ĥx to X(k) are smooth maps. Second we check that X(k) and X(k ) induce on the intersection X(k) X(k ) the same topology and the same smooth structure. These facts imply that X has a canonical structure of smooth manifold with corners and that î x is a smooth map. The properness of ĥx follows from the compactness of ĥ 1 (s), which is in fact the space B(x; s) whose compactness is stated in Proposition 3. To accomplish first step we proceed in exactly the same way as in the proof of part (1). Suppose x Cr(r 1). Consider P := P r (r k), O := r k i=r (M + i M i ) and S := Sx M k M r k+1. Define, cf. Diagram 2, and p := p r p r k+1 l r k : P O s := α ω r ω r k+1 : S O. The verification of the transversality follows from (P1), (P2), (Q1), (Q2) and the Morse-Smale condition, as will be explained in section 4.3, below. It is easy to see and left to the reader, that p 1 (s(s)) identifies to X(r k). The second step is more or less straightforward, so it will be left again to the reader. 4.3 The transversality of p and s. Consider the diagrams: Sr+1 Mr M r 1 M r k+1 α id ψ r id ψ r k+1 β M r + Mr M r 1 + Mr 1 M + r k M r k p + r p r p + r 1 p r 1 p + r k p r k P r 1 P r k P r Diagram 1 S + r k 1 Sr+1 Mr M r 1 M r k+1 α id ψ r id ψ r k+1 M r + Mr M r 1 + Mr 1 M + r k p + l p r r p + r k r 1 pr 1 Q(r k) P r P r 1 Diagram 2