INCIDENCE COEFFICIENTS IN THE NOVIKOV COMPLEX FOR MORSE FORMS: RATIONALITY AND EXPONENTIAL GROWTH PROPERTIES.

Let f : M → S 1 be a Morse map, v a transverse f -gradient. Theconstruction of the Novikov complex associates to these data a free chain complexC ∗ (f, v) over the ring Z[t]][t −1 ], generated by the critical points of f and computingthe completed homology module of the corresponding infinite cyclic covering of M .Novikov’s Exponential Growth Conjecture says that the boundary operators in thiscomplex are power series of non-zero convergence raduis.In [12] the author announced the proof of the Novikov conjecture for the case ofC 0 -generic gradients together with several generalizations. The proofs of the firstpart of this work were published in [13]. The present article contains the proofs ofthe second part.There is a refined version of the Novikov complex, defined over a suitable com-pletion of the group ring of the fundamental group. We prove that for a C 0 -genericf -gradient the corresponding incidence coefficients belong to the image in the Novikovring of a (non commutative) localization of the fundamental group ring.The Novikov construction generalizes also to the case of Morse 1-forms. In thiscase the corresponding incidence coefiicients belong to a certain completion of thering of integral Laurent polynomials of several variables. We prove that for a givenMorse form ω and a C 0 -generic ω-gradient these incidence coefficients are rationalfunctions.The incidence coefficients in the Novikov complex are obtained by counting thealgebraic number of the trajectories of the gradient, joining the zeros of the Morseform. There is V.I.Arnold’s version of the exponential growth conjecture, whichconcerns the total number of trajectories. We confirm this stronger form of theconjecture for any given Morse form and a C 0 -dense set of its gradients.We give an example of explicit computation of the Novikov complex.

series of non-zero convergence raduis. In 1995 the author proved that these boundary operators are rational functions of t for the case of C 0 -generic gradient v. That implies the Novikov Exponential Growth Conjecture for C 0 -generic gradients. This result and its generalizations were announced in [14], and the proofs of first part of these results were published in [15]. The proofs of the second part were available as an e-print [13]. Since then the results of this e-print were used in several papers, see e.g. [5]. Recently this domain attracted again the interest of researchers, see the article [6], where a family of examples of Novikov complex with infinite power series as incidence coefficients was constructed.
The present paper is a revised version of the eprint [13]. Several references were updated, and references to more recent articles were added. Thus this article completes the publication of the results announced in [14].
The paper falls naturally into three parts. The first part (Section 2) contains a recollection of the material of the author's paper [15] in a compressed and rearranged form, suitable for our present purposes. The second part (Section 3, 4) is the hard core of the paper. We prove here that the rationality property of the boundary operators is C 0 -generic 1 . In Section 3 we deal with non-abelian Novikov rings and the rationality property is expressed in terms of a certain non-abelian localization of the group ring of the fundamental group of the manifold. In Section 4 we consider the case of arbitrary Morse forms and free abelian coverings.
In the third part we study the exponential growth properties. We begin by an example of a Novikov complex where the incidence coefficients grow exponentially (Section 5). Then we apply the results of the previous sections to show that the exponential growth property of the incidence coefficients of the gradient of a Morse form is C 0 -generic.
In the rest of the Introduction we give the statement of our main results preceded by short background subsections 1.1 and 1.2.

Morse-Novikov theory for circle-valued Morse functions.
The classical Morse-Thom-Smale construction associates to a Morse function g : M Ñ R on a closed manifold a free chain complex C˚(g) where the number m(C p (g)) of free generators of C p (g) equals the number of the critical points of g of index p for each p. The boundary operator in this complex is defined in a geometric way, counting the trajectories of a gradient of g, joining critical points of g (see [8,12,[17][18][19]).
In the early 80s S. P. Novikov generalized this construction to the case of maps f : M Ñ S 1 (see [9]). The corresponding analogue of Morse complex is a free chain complex C˚(f ) over Z [[t]] [t´1]. Its number of free generators equals the number of critical points of f of index p, and the homology of C˚(f ) equals to the completed homology of the cyclic covering.
Fix some k. The boundary operator B : C k (f ) Ñ C k´1 (f ) is represented by a matrix, which entries are in the ring of Laurent power series. That is B ij = ř 8 n=´N a n t n , where a n P Z, and N is some natural number. Since the beginning S. P. Novikov conjectured that the power series B ij have some nice analytic properties. In particular he conjectured that: ‚ generically the coefficients a n of B ij = 8 ř n=´N a n t n grow at most exponentially with n.
In [15] we have proved that for a C 0 generic f -gradient the incidence coefficients above are actually rational functions. To recall the statement of the Main Theorem of [15] let M be a closed connected manifold and f : M Ñ S 1 a Morse map, non-homotopic to zero. Denote the set of critical points of f by S(f ). The set of f -gradients of the class C 8 , satisfying the transversality assumption (see §1 for terminology), will be denoted by Gt(f ). By Kupka-Smale theorem it is residual in the set of all the C 8 gradients. Choose v P Gt(f ). Denote by M P Ý Ñ M the connected infinite cyclic covering for which f˝P is homotopic to zero. Choose a lift F : M Ñ R of f˝P and let t be the generator of the structure group of P such that F (xt) ă F (x). The t-invariant lift of v to M will be denoted by the same letter v. For every critical point x of f choose a liftx of x to M . Choose orientations of stable manifolds of critical points. Then for every x, y P S(f ) with ind x = ind y + 1 and every k P Z the incidence coefficient n k (x, y; v) is defined (as the algebraic number of (´v)-trajectories joiningx toȳt k ).
Theorem. ([15, p.971]) In the set Gt(f ) there is a subset Gt 0 (f ) with the following properties. (1) Gt 0 (f ) is open and dense in Gt(f ) with respect to C 0 topology.
(2) If v P Gt 0 (f ), x, y P S(f ) and ind x = ind y + 1, then ÿ kPZ n k (x, y; v)t k is a rational function of t of the form P (t) t m Q(t) , where P (t) and Q(t) are polynomials with integral coefficients, m P N, and Q(0) = 1.
(3) Let v P Gt 0 (f ) and U be a neighborhood of S(f ). Then for every w P Gt 0 (f ) such that w = v in U and w is sufficiently close to v in C 0 topology we have: n k (x, y; v) = n k (x, y; w) for every x, y P S(f ), k P Z.
The present paper uses the terminology and definitions of the paper [15], and should be read as a follow up of [15].  (2) If v P Gt 1 (f ) then for every x, y P S(f ) with ind x = ind y + 1 we have r n(r x, r y; v) P Im ℓ.
(3) Let v P Gt 1 (f ). Let U be a neighborhood of S(f ). Then for every w P Gt 1 (f ) such that w = v in U and w is sufficiently close to v in C 0 topology we have: r n(r x, r y; v) = r n(r x, r y; w) for every x, y P S(f ).

Morse forms within arbitrary cohomology classes.
At present we can prove the analogue of Theorem A in the case of arbitrary Morse forms only for the incidence coefficients associated with free abelian coverings.
Let ω be a Morse form on a closed connected manifold M . If ϕ : x M Ñ M is any regular covering with the structure group G, such that ϕ˚([ω]) = 0, then the homomorphism tωu : π 1 M Ñ R factors as π 1 M Ñ G Ñ R and it is not difficult to see that the incidence coefficients p n(p x, p y; v) are defined for every v P Gt(ω) (here we suppose that ind x = ind y + 1, and that for every p P S(ω) a liftp of p to x M and an orientation of the stable manifold of p are chosen). In particular, it is the case for the maximal free abelian covering M P Ý Ñ M with the structure group H 1 (M, Z)/Tors « Z m . By abuse of notation we shall denote the corresponding homomorphism Z m Ñ R by the same symbol as the de Rham cohomology class [ω] of ω. Assume that [ω] = 0. Put Theorem B. There is a subset Gt 1 (ω) Ă Gt(ω) with the following properties: (1) Gt 1 (ω) is open and dense in Gt(ω) with respect to C 0 topology.
Incidence coefficients in the Novikov Complex

131
(3) Let v P Gt 1 (ω). Let U be a neighborhood of S(ω). Then for every w P Gt 1 (ω) such that w = v in U and w is sufficiently close to v in C 0 topology we have: n(x, y; v) = n(x, y; w) for every x, y P S(ω), k P Z.
1.3.3. An example. In Section 5 we construct a three-manifold M , a Morse map f : M Ñ S 1 and an f -gradient v such that n 0 (x,ȳ; v) = 0 and for k ě 0 we have Theorem C below asserts a much stronger property, pertaining to the universal covering of M and counting for every g P G = π 1 (M ) the absolute value of the algebraic number of trajectories joining x with y¨g. To state this theorem we need one more definition. Let G be a group. For an element a = ř n g g P ZG we denote by }a} the sum ř |n g |. Let ξ : G Ñ R be a homomorphism. For λ = ř g n g g P ZGξ and c P R we denote by λ[c] the element ř ξ(g)ěc n g g of ZG and we set N c (λ) = }λ[c]}. We shall say that λ is of exponential growth if there are A, B ě 0 such that for every c ă 0 we have N c (λ) ď Ae´c B . It is easy to prove that the elements of exponential growth form a subring of ZGξ containing ZG.

Theorem C.
Let v be an ω-gradient, belonging to Gt 1 (ω), and x, y P S(ω) with ind x = ind y+1. Then r n(r x, r y; v) P Z[π 1 M ]t ωu is of exponential growth.
In the paper [3, Theorem 2.21] D. Burghelea and S. Haller also prove that the property (1.1) holds for C 0 -generic gradients 2 . 132 A. Pajitnov 1.3.5. Exponential estimates of absolute number of trajectories: Morse maps M Ñ S 1 . The first published version of the Exponential Growth conjecture appeared in the paper of V. I. Arnold "Dynamics of intersections", 1989. This version is also the strongest one. Arnold writes in [1, p. 83]: "The author is indebted to S. P. Novikov who has communicated the following conjecture, which was the starting point of the present paper. Let p : Ă M Ñ M be a covering of a compact manifold M with fiber Z n , and let α be a closed 1-form on M such that p˚α = df , where f : M Ñ R is a Morse function. The Novikov conjecture states that, "generically" the number of the trajectories of the vector field´grad f on M starting at a critical point x of the function f of index k and connecting it with the critical points y having index k´1 and satisfying f (y) ě f (x)´n, grows in n more slowly than some exponential, e an ." In the present paper we prove this conjecture for the C 0 -dense subset in the set of all gradients (Theorems C and D below). We assume here the terminology of Subsection 1.3.1. The set of all f -gradients of class C 8 will be denoted by The set of all good f -gradients will be denoted by Gd(f ). For v P G(f ) we denote by the same letter v the t-invariant lift of v to M . Choose a lift F : M Ñ R of f to M . It is easy to prove that for p, q P S(F ), ind p = ind q + 1 and for for v P Gd(f ) the set of (´v)-trajectories, joining p to q is finite. The lifts of critical points of f to M being chosen, denote by N k (x, y; v) the number of (´v)-trajectories joiningx toȳt k (where ind x = ind y + 1).

Theorem D.
In the set G(f ) there is a subset G 0 (f ) with the following properties: (2) Let v P G 0 (f ). Then there are constants C, D ą 0 such that for every x, y P S(f ) with ind x = ind y + 1 and for every k P Z we have N k (x, y; v) ď C¨D k . of ω with the corresponding homomorphism Z m Ñ R. We denote by G(ω) the set of all ω-gradients of class C 8 and by Gd(ω) the set of all good ω-gradients of class C 8 . Let v P Gd(ω). For every zero x of ω choose a lift x of x to M and an orientation of the stable manifold of x. Then for every g P Z m and every x, y P S(ω) with ind x = ind y + 1 the set of (´v)-trajectories joining x to yg is finite and we denote its cardinality by N (x, y, g; v). For c P R we denote by N ěc (x, y; v) the sum ř g: [ω](g)ěc N (x, y; g; v).
Theorem E. In the set G(ω) there is a subset G 0 (ω) with the following properties: (2) Let v P G 0 (ω). There exist constants C, D ą 0 such that for every x, y P S(ω) with ind x = ind y + 1 and every λ ă 0 we have

Remark 1.3.7.
Observe that Theorem D follows from Theorem E. We still keep both statements of Theorem D since the essential of the proof of Theorem E is based on the proof of Theorem D.

RANGING SYSTEMS AND RANGING FLOWS (AN OVERVIEW OF RESULTS OF [15])
The results of the present paper are based on the author's paper [15]. For the convenience of the reader we recall in this section some of the basic definitions and theorems of [15]. The contents of this section falls into two parts. In the first part (Subsections 2.2-2.3) we present the main technical tool for the rationality theorem, namely ranging systems. We show that if an f -gradient admit a ranging system adapted to a pair of critical points p, q, then the incidence coefficient of these two points is a rational function of the type described in the statement of the Main Theorem of [15]. The final result of this part is Theorem 2.3.4.
In the second part (Subsections 2.4-2.6) we show that the set of fgradients that admit a ranging system adapted to p, q, is open and dense in the set of all transverse gradients (with respect to C 0 -topology). The basic instruments here are the notions of ranging flow and ranging pair.
The exposition is preceded by a short introductory Subsection 2.1 where we collected the necessary definitions.
where α i ă 0 for i ď ind p and α i ą 0 for i ą ind p. The domain U p is called standard coordinate neighborhood. Any such extension r Φ p of Φ p will be called standard extension of Φ p .
The set Φ´1 p (R kˆt 0u), resp. Φ´1 p (t0uˆR n´k ), where k stands for ind p, is called negative disc, resp. positive disc. If for every i we have α i =˘1, we shall say that the coordinate system tΦ p u is strongly standard.
A family of f -charts is called f -chart-system, if the family tU p u is disjoint. We denote min p r p by d(U), and max p r p by D(U). If all the r p are equal to r, we shall say that U is of radius r. The set Φ´1 p (B n (0, λ)), where λ ď r p will be denoted by where k = ind f p, and r Φ p is some standard extension of Φ p . We say that a vector field v is an f -gradient if there is an f -chart system U, such that v is an f -gradient with respect to U. Definition 2.1.2. Assume that M is Riemannian and denote by D r (x), resp. B r (x) the closed, resp. open ball of radius r centered in x. Let also f : M Ñ R be a Morse function, v be an f -gradient, p P S(f ). Set: We denote by K(v) the union of all subsets D(p, v), and by B δ (v) the union of all subsets B δ (p, v) where p ranges over critical points of f .
We also denote by B δ (indďs ; v) the union of subsets B δ (p, v) (where p ranges over critical points of f of index ď s). Similar notations like D δ (indďs ; v) or K(ind=s ; v) etc. are now clear without special definition.
We obtain thus a diffeomorphism Sometimes we denote it by v ù [b,a] . In the next section we take a closer look at this diffeomorphism.

Ranging systems.
We proceed to the definition of ranging system (see [15,Definition 4.6]). Let f : W Ñ [a, b] be a Morse function on a compact Riemannian cobordism, f´1(b) = V 1 , f´1(a) = V 0 , v be an fgradient.
Definition 2.2.1. Let Λ = tλ 0 , . . . , λ k u be a finite set of regular values of f , such that λ 0 = a, λ k = b, and for each 0 ď i ď k´1 we have λ i ă λ i+1 and there is exactly one critical value of f in [λ i , λ i+1 ]. The values λ i , λ i+1 will be called adjacent. The set of pairs t(A λ , B λ )u λPΛ is called ranging system for (f, v) if (RS1) For every λ P Λ the sets A λ and B λ are disjoint compacts in f´1(λ).
(RS2) Let λ, µ P Λ be adjacent. Then for every p P S(f ) X f´1([λ, µ]) one of the two following properties i), ii) holds: The existence of a ranging system allows us to endow the gradient flow with a structure that resembles to some extent a continuous map between compact spaces. The next proposition is proved in [15,Lemma 4.8 and Propositions 4.9,4.10].
Then the following statements hold.
(1) There exists a homomorphism such that (2) There is ϵ ą 0 such that for any f -gradient w with }v´w} ă ϵ the ranging system t(A λ , B λ )u λPΛ is also a ranging system for (f, w), and H(v) = H(w).
which is C 0 -stable with respect to small C 0 -perturbations of v, as in property (2)of Proposition 2.2.2.
Ranging systems provide a convenient tool for computation of incidence coefficients in homology terms. Let also p P S(f ) and µ(p) and σ(p) be the (uniquely determined) adjacent elements of Λ such that f (p) P (µ(p), σ(p)). We will use the following notations: ‚ S + (p, v) denotes the sphere D(p,´v) X f´1(σ(p)), and ‚ S´(p, v) denotes the sphere D(p, v) X f´1(µ(p)).
If this does not lead to confusion, we denote these spheres simply by S´(p) and S + (p). We say that a ranging system t(A λ , B λ )u λPΛ is adapted to a pair p, q of critical points in S(f ), where ind p = ind q + 1, if S´(p) X B µ(p) = ∅ and S + (q) X A σ(q) = ∅.
If t(A λ , B λ )u λPΛ is adapted to p, q, we denote by [S´(p)] also the image of this fundamental class in

Consider the inclusion
See the proof of the following two propositions in [15,Proposition 4.12].
Proposition 2.2.5. Let t(A λ , B λ )u λPΛ be a ranging system adapted to p, q.
If v is an f -gradient satisfying the transversality condition then This formula imply immediately a C 0 -stability property for the incidence coefficients.

Proposition 2.2.6.
(1) There is ϵ ą 0 such that for every f -gradient v with }w´v} ă ϵ the system t(A λ , B λ )u λPΛ is a ranging system for (f, w) adapted to p, q.
(2) Let U (p) be a neighborhood of p, and U (q) be a neighborhood of q.
There is ϵ ą 0 such that for every f -gradient w with w| U (p) = v| U (p) and w| U (q) = v| U (q) and }w´v} ă ϵ we have and n(p, q; v) = n(p, q; w) (2.1) In our applications the cobordism W will be obtained as the result of cutting a closed manifold M along a regular level surface of a Morse function f : M Ñ S 1 . Such cobordism is endowed naturally with a diffeomorphism of its lower boundary on its upper boundary.

Definition 2.2.7.
A Riemannian cobordism W with BW = V 0 \ V 1 is said to be cyclic if there is an isometry Φ : V 0 Ñ V 1 (such an isometry will be regarded as part of the structure of a cyclic cobordism).
These techniques will be applied to the case of circle-valued Morse functions.
2.3. Equivariant ranging systems. Now we will apply the techniques of the previous section to investigation of the incidence coefficients in the Novikov complex. Let f : M Ñ S 1 be a Morse function, and v be an fgradient. We will assume here that the class [f ] P H 1 (M, Z) is indivisible, so that for all z P M we have that F (z)´F (zt) = 1.
We assume also S(f ) = ∅, and that 0 is a regular value of F . Choose a Riemannian metric on M . Then M obtains a t-invariant Riemannian metric. ‚ if σ P Σ then σ + n P Σ for all n P Z; ‚ if λ, µ P Σ are adjacent, then there is only one critical value of F between λ and µ. A set t(A σ , B σ )u σPΣ is called t-equivariant ranging system for (F, u), if (1) For every µ, ν P Σ, µ ă ν we have: t(A σ , B σ )u σPΣ,µďσďν is a ranging system for (F | F´1([µ,ν]),u) .
(2) A σ´n = A σ¨t n and B σ´n = B σ¨t n for every n P Z.
Any cyclic ranging system on the cobordism W = F´1([0, 1]) determines in an obvious way a t-equivariant ranging system. Let p and q be critical points of f , such that ind p = ind q + 1. Let l = ind q. Let Assume that the ranging system is adapted to the pair (p, q). We lift p and q to M in such a way thatq P tW andp P W . Put

The cohomology class
The next three assertions form the first basic ingredient of the proof of the main theorem of [15] (see [15,Proposition 4.18] for the proof).

Lemma 2.3.3.
Let G be a finitely generated abelian group, A an endomorphism of G, and λ : G Ñ Z a homomorphism. Then for every p P G the series ÿ is a rational function of t of the form P (t) Q(t) , where P, Q are polynomials and Q(0) = 1. (1) Assume that there exists a cyclic ranging system on the cobordism where P, Q are polynomials and Q(0) = 1.
The set of transverse f -gradients admitting a cyclic ranging system adapted to p, q is C 0 -open in the set of all transverse f -gradients.

Almost transverse gradients.
In this subsection we gathered some preliminaries necessary for construction of ranging systems for C 0 -generic flows; this construction will be recalled in the following subsections.
] be a Morse function and v an fgradient. We say that a Morse function ϕ : W Ñ R is adjusted to the pair (2) v is also a ϕ-gradient, Observe that it is not clear whether there is a ranging system for every gradient. It turns out that ranging systems exist for C 0 -generic gradients. We outline the proof in the next subsections.
] be a Morse function on a compact cobordism W , and v be an f -gradient. We say that v satisfies the almost transversality assumption if for every pair x, y of critical points of f we have We say aslo that v is almost good or almost transverse.
, with an ordering sequence a 0 , . . . , a n+1 , such that (2) for every l : 0 ď l ď n and every q P S(f ) with ind q = l there is a Morse function , and a regular value µ P (a l , a l+1 ) of ψ, such that ‚ D δ (q) Ă ϕ´1((a l , µ)), ‚ and for every r P S(f ), ind r = l, r = q we have

Lemma 2.4.5.
If v is almost good, then v is δ-separated for some δ ą 0.
where ϕ is a Morse function on a cobordism and u is a δ-separated ϕ-gradient will be called AM-flow 3 .
For an AM-flow (ϕ, u) the sets D δ (indďs , u) form an increasing filtration of W . This is just one of versions of handle decomposition of W .

Stratified submanifolds and their thickenings.
In this subsection we recall briefly the notions of s-submanifold and ts-submanifold introduced in [15]. These techniques will be used in the following subsection. We omit all proofs referring the reader to [15, §2].

2.5.A. Stratified submanifolds.
Let A = tA 0 , . . . , A k u be a finite sequence of subsets of a topological space X. For 0 ď s ď k, we denote A s also by A (s) ; the set A 0 Y¨¨¨Y A s is denoted by A ďs and also by A (ďs) . We say that A is a compact family if A (ďs) is compact for every s with 0 ď s ď k.
(2) X is a compact family.
For an s-submanifold X = tX 1 , . . . , X s u, the largest k such that X k = ∅ is called the dimension of X and is denoted by dim X. For a diffeomorphism Φ : M Ñ N and an s-submanifold X of M , we denote by Φ(X) the ssubmanifold of N defined by Φ(X) (i) = Φ(X (i) ).
If V is a submanifold of M and X is an s-submanifold of M , then we say that V is transversal to X (in symbols: Lemma 2.5.3. Assume that v satisfies the almost transversality condition. Then: is an s-submanifold of f´1(λ).

2.5.B. Good fundamental systems of neighborhoods, and ts-submanifolds.
Definition 2.5.4. Let X be a topological space, A = tA 0 , . . . , A k u a compact family of subsets of X, I an open interval (0, δ 0 ). A good fundamental system of neighborhoods of A (briefly: a gfn-system for A) is a family A = tA s (δ)u δPI,0ďsďk of open subsets of X such that for any positive integer s ď k the following conditions are fulfilled: . The interval I is called the interval of subdefinition of the gfn-system and A is called the core of A. We shall denote A s (δ) also by A (s) (δ) and A ďi (δ) also by A (ďi) (δ).
Let M be a manifold without boundary, X an s-submanifold of M , and X a gfn-system for X. We say that X is a ts-submanifold of M with core X. For a ts-submanifold X = tX s (δ)u δPI, 0ďsďk , we shall also denote X i (δ) by X (i) (δ), and X ďi (δ) by X (ďi) (δ). The basic example of a gfn-system and a ts-manifold is given by the next lemma.
is a gfn-system for D(v). If M is a closed manifold, this family is a tssubmanifold with core D(v).
Observe that there is no canonical way for choosing an interval of definition for this system. We shall say that λ ą 0 is in an interval of definition If v is λ-separated, then λ is in an interval of definition of D(v).

2.5.C. Tracks of subsets, s-submanifolds, and ts-submanifolds.
Definition 2.5.6. Let f : W Ñ [a, b] be a Morse function on a compact Riemannian cobordism W , and v be an f -gradient. Let X Ă V 1 . The set t γ(x, t;´v) | t ě 0, x P X u is called the track of X (with respect to v) and is denoted by T (X, v).
(2) If X is compact, and every (´v)-trajectory starting at a point of X reaches V 0 , then T (X, v) is compact.
(4) For any X and any

Definition 2.5.8.
Let v be an f -gradient satisfying the almost transversality condition. Let A be an s-submanifold tA 0 , . . . , The family will be called the track of A, and the family Let us proceed to tracks of ts-submanifolds.
It may happen that in order to turn the family tT A s (δ, v)u into a gfnsystem, it is necessary to reduce its initial interval of definition.
Definition 2.5.11. Let I = (0, µ), and let Z = tZ(δ)u δPI be a family of subsets of some space X. Let 0 ă ν ă µ. The family tZ(δ)u δP(0,ν) will be called a restriction of Z to the interval (0, ν). It is proved in [15, Lemma 2.11 and Proposition 2.12] that there is ϵ P (0, min(δ 0 , δ 1 )) such that the restriction of the family T A s (δ, v) to the interval (0, ϵ) is a gfn-system with the core T(A, v). Definition 2.5.12. This gfn-system will be denoted by T(A, v); we call it the track of A.
Observe that there is no canonical choice of an interval of definition for this system. 2.6. Ranging pairs. In this subsection, W is a Riemannian cobordism, , v an f -gradient satisfying the almost transversality condition, n = dim W .
, and there is a number δ ą 0 such that for any 0 ď s ď n´1 the following conditions are fulfilled: (RP1) δ is in intervals of definition of T(D(u 1 ), v) and of T(D(´u 0 ),´v); (RP2) the gradients v, u 0 , and u 1 are δ-separated; We say that v satisfies condition (RP) if (f, v) has a ranging pair.
The next theorem ( [15,Theorem 4.3]) is one of the basic results of [15]. We will not comment on the proof (it occupies §3 and first 2 pages of §4 of [15]), since the techniques of the proof will not be used in the present paper.
Theorem 2.6.2. Let ϵ ą 0. Then there is an f -gradient w satisfying the almost transversality condition and a ranging pair Assume now that (f, v) has a ranging pair (V 0 , V 1 ). We will now show that this ranging pair generates a family of ranging systems for (f, v). Moreover, for every p, q P S(f ) with ind p = ind q + 1, in this family there is a ranging system adapted to p, q.
Let Λ = tλ 0 , . . . , λ k u be a set of regular values of f such that ] there is only one critical value of f . We choose some δ 1 ą δ in the interval of definition of T(D(u 1 ), v) and T(D(´u 0 ),´v) such that for every 0 ď s ď n´1 we have (such a number δ 1 exists because δ belongs to an interval of definition of T(D(u 1 ), v) and of T(D(´u 0 ),´v)). Also, we require that the gradients v, u 0 , u 1 be δ 1 -separated. Now, for each integer s with 0 ď s ď n we define compact subsets A be a sequence of real numbers, and let 1 ď s ď n. We put The following lemma is proved in [15,Lemma 4.14].
)u λPΛ is a ranging system for (f, v) adapted to every pair p, q of critical points of f with ind p = s + 1, ind q = s.
(3) For every 0 ď s ď n´1 the pair is homotopy equivalent to a finite CW-pair that has cells of dimension s only.
In the case when W is cyclic we can strengthen the results above so to construct cyclic ranging systems for C 0 -generic gradients.
We say that v satisfies condition (RF) if there is a ranging flow for (f, v). The proof of the following theorem is similar to the proof of Theorem 2.6.2.
There is an f -gradient w satisfying the almost transversality condition and a ranging flow Similarly to the above for every ranging flow and any pair p, q of critical points of adjacent indices there is a finite type cyclic ranging system adapted to p, q. We arrive therefore at the following result which together with theorem 2.3.4 proves the Main Theorem of [15]. Theorem 2.6.7. Let f : M Ñ S 1 be a Morse function. Let also p, q be critical points of f such that ind p = ind q + 1. Then the set of transverse f -gradients admitting a cyclic ranging system adapted to p, q is C 0 -dense in the set of all transverse f -gradients.
3. MORSE MAPS M Ñ S 1 3.1. Algebraic preliminaries. We will work here with the terminology of §1.3.1.
ij are the entries of A s . The next lemma follows from the definition of the homomorphism ℓ.  Proof. It suffices to prove that every matrix entry of the matrix series u = ř sě0 A s , where A = (a ij ) and a ij P ZG (´1) , is of exponential growth. For an (mˆm)-matrix B = (b ij ) we denote by }B} the number max It is easy to check that }BC} ď }B}¨}C}¨m. Let A be (mˆm)-matrix.
Therefore, in any case there are c, d ą 0 such that for k ă 0, k P Z we have ÿ ξ(g)ěk |n g | ď c¨d´k.
For k ě 0 it is true obviously and this implies that u is of exponential growth.  (2) If v P Gt 1 (f ) then for every x, y P S(f ) with ind x = ind y + 1 the incidence coefficient r n(r x, r y; v) is of type (L).
(3) Let v P Gt 1 (f ). Let U be a neighborhood of S(f ). Then for every w P Gt 1 (f ) such that w = v in U and w is sufficiently close to v in C 0 topology we have: r n(r x, r y; v) = r n(r x, r y; w) for every x, y P S(f ) such that ind x = ind y + 1.
Assume that X X L = ∅, N &L and n + l = m. Then p N is transversal to p L, the set p N X p L is finite, and the intersection index p . The next lemma is standard. x is the p v-trajectory γ(x,¨; p v). It is easy to define with the help of this lifting procedure a diffeomorphism For X Ă V 1 we denote by abuse of notation p v ù (X z K(´v)) by p v ù (X). Let N be a oriented-lifted submanifold of V 1 . Then it is easy to see that v ù (N ) is a oriented-lifted submanifold of V 0 .
of right ZH-modules, such that: (2) There is an ϵ ą 0 such that for every f -gradient wwith }w´v} ă ϵ we have p H(v) = p H(w).
Proof. An easy induction argument shows that it is sufficient to prove the proposition in the case cardΛ = 1. Split the set S(F ) of all critical points of f in a disjoint union of two subsets S1(f ) and S2(f ) in such a way that ‚ for every p P S1(f ) the condition (RS2) i) of Definition 2.2.1 holds, ‚ for every p P S2(f ) the condition (RS2) ii) of Definition 2.2.1 holds.
It is easy to prove that for δ ą 0 sufficiently small we have: Fix some δ ą 0 satisfying (D1) and (D2). Let 0 ď µ 1 ă µ ď δ and U be any subset of V 1 such that (for example U = ∆(δ,´v) will do). Consider the following sequence of homomorphisms ) .

A. Pajitnov
Here p I is the corresponding inclusion. Note that the last arrow is well defined since All the three arrows are homomorphisms of right ZH-modules. (This is obvious for the first two arrows, as for the last just observe that p v ù commutes with the right action of H.) The composition p v ù˝E xc´1˝p I˚of this sequence will be denoted by p H(v; µ 1 , µ; U). An argument similar to the one in the beginning of the page 999 of [15] shows that this homomorphism does not depend neither on the choices of U , µ 1 , µ, δ, nor on the choice of presentation (if there is more then one such presentation). Therefore this homomorphism is determined by v, the ranging system t(A λ , B λ )u λPΛ and the covering Q : x W Ñ W . We shall denote it by p H(v). The proof of properties (1) and (2) to itself. It follows from the construction that for every g P G with Then p h ν (u) is a θ-semilinear endomorphism of ) .
Lemma 3.5.1. Let µ, ν P Σ, ν ď µ, and k P N. Let also N be orientedlifted submanifold of F´1(µ) z B µ such that N z IntA µ is compact, and L be a cooriented-lifted compact submanifold of F´1(ν) z A ν . Assume that dim N + dim L = dim M´1. Then: ) .
(2) For every f -gradient w, sufficiently close to u in C 0 -topology, the family t(A σ , B σ )u σPΣ is also a t-equivariant ranging system for (F, w) and  Z) and we make this assumption up to the end of this subsection. Fix first two points x, y P S(f ), ind x = ind y + 1. Recall that we have chosen a lift r x P Ă M for every x P S(f ). Denote Q(r x) byx. We can assume that F (ȳ) ă F (x) ď F (ȳ) + 1. Denote dim M by n; denote ind x by l + 1, then ind y = l. Choose some set Σ of regular values of F , satisfying (S) of Definition 2.3.1.
Let Gt 1 (f ; x, y) be the subset of Gt(f ), consisting of all the f -gradients v, such that there is an equivariant ranging system t(A σ , B σ )u σPΣ for (F, v) satisfying  The lifts x Þ Ñ r x and y Þ Ñ r y define a lift p is free. Choose some basis e 1 , . . . , e m of this module. The homomorphism p h η (v) of this module is θ-semilinear. Denote by B = (b ij ) its matrix, and denote by A the matrix (b ij θ). Let a To prove it write r n(r x, r y; v) = ÿ sě0 ( ÿ hPH ν(r x, r yhθ s )¨hθ s ) (here ν(r x, r yhθ s ) stands for the algebraic number of (´v)-trajectories, joiningx withỹhθ s ; note that since F (x) ď F (ȳ) + 1, there are no (´v)trajectories joining r x to r yg if ξ(g) ą 0). To make the following computation more easy to comprehend, we introduce the following terminology conventions (valid only here). The homomorphism p h η (v) : H Ñ H will be denoted by µ. We identify the cohomology classes in with their images in H˚(Q´1(F´1(ν) z B ν ), Q´1(A ν ) ) (thus suppressing p ii n the notation). We have: (by Lemma 3.5.1). The latter expression equals To obtain from this expression the formula (3.3) we need only a lemma, allowing to calculate µ s (ξ) in terms of the coordinates of ξ and the matrix of µ (the expression differs slightly from the standard linear-algebraic one since µ is θ-semilinear). Proof. We will use induction in s. We have Now substitute the expression for µ s (ξ) into the above formula, and the proof of (3.3) is over.

3) Stability with respect to perturbations of the gradient.
Let v P Gt 1 (f ; x, y) and U be a neighborhood of S(f ). We are going to prove that there is ϵ ą 0 such that for every w P Gt 1 (f ; x, y) with }w´v} ă ϵ and w| U = v| U we have: r n(r x, r y; v) = r n(r x, r y; w). Let w be an f -gradient, sufficiently close to v. Then t(A σ , B σ )u σPΣ is still a t-equivariant ranging system for (F, w), satisfying the conditions (3.1) and (3.2). It is not difficult to see that

MORSE FORMS WITHIN ARBITRARY COHOMOLOGY CLASSES
Let ω be a Morse form of irrationality degree q, v be an ω-gradient. Let Q : N Ñ M be the minimal free abelian covering such that Q˚ω is an exact form, that is, Q˚ω = dF where F : N Ñ R is a Morse function. The structure group of Q is isomorphic to Z m . If m = 1, the level surfaces F´1(a) and the cobordisms F´1([a, b]) are compact. If m ą 1 these manifolds are non-compact, and this is the main reason why the results of [15] do not carry over directly to this case m ą 1.
The way round this difficulty consists in approximating the form ω by a rational 1-form ω 1 , so that ω 1 = dϕ where ϕ is a circle-valued Morse function. Then the gradient v is approximated (in C 0 -topology) by a vector field w which is a gradient for both ω and ϕ and such that its incidence coefficients are rational functions.
This program is realized in the present section.
4.1. Algebraic preliminaries. We will need some lemmas about the ring Z[Z m ] and its completions and localizations.
Definition 4.1.1. Let η : Z m Ñ R be a non-zero homomorphism. We extend it to a linear map R m Ñ R, which will be denoted by the same letter. We say that a set Z Ă R m is an η-cone, if there is a compact convex nonempty set K Ă η´1(´1) such that We say that Z is (ξ, η)-cone if Z is ξ-cone and η-cone. We say that a set Z Ă R m is an integral η-cone if there are e 1 , . . . , e k P Z m , such that (1) rk(e 1 , . . . , e k ) = m, (2) η(e i ) ă 0, We shall also write Z = Zxe 1 , . . . , e k y. We say, that Z is an integral (ξ, η)cone if the vectors e 1 , . . . , e k above satisfy ξ(e i ) ă 0, η(e i ) ă 0 for all i. Note that an integral (ξ, η)-cone is a (ξ, η)-cone. Proof. We assume that η 1 , η 2 are linearly independent; the other case is considered similarly. Denote η´1 1 (´1) by H, and the set H X tη 2 (x) ă 0u by H 0 . Then H 0 is an open halfspace of H, containing Z X H. Denote by L the set tλx | λ ě 0, x P Z m u, then L is everywhere dense in H and in H 0 . It is not difficult to prove that there is a finite subset L 0 Ă L such that L 0 Ă H 0 and xL 0 y Ą Z X H. We can choose L 0 so that rkL 0 = m and the lemma is proved. □ (2) There is an integral ξ-cone Γ 0 such that for every family tA i u iPI of real numbers there is b P Z m with the property: is non empty and compact. (Indeed, let x be a vector such that |x| = 1 and ξ(x) = }ξ}. Then a =´x ξ(x) P Z. Further, if Z is not bounded, then there is a sequence x n P Z such that |x n | Ñ 8. Consider the sequence x n /|x n |. We can assume that it converges to some v with }v} = 1. Since ξ(x n ) =´1, we have ξ(v) = 0. Further, for every i we have 0]) and the intersection Γ X ξ´1(0) consists of 0. Therefore x P Γ and x = 0 implies x = λy, y P Z.
(2) Choose a vector x 0 P Z m such that ξ(x 0 ) ă 0 and for every 1 ď i ď m we have (such a vector exists; it suffices to note that (ξ˘α i )(a) =´1¯α i (x) ξ(x) ă 0, and to approximate a by an element y 0 P Q m ). Denote Therefore if p is sufficiently big, we obtain (4. Then apply Lemma 4.1.2 to obtain an integral ξ-cone Γ 0 , such that Γ Ă Γ 0 . □ We will use some terminology and results from [11]. Let η : Z m Ñ R be a homomorphism and Z = Zxe 1 , . . . , e k y be an integral ξ-cone. Denote   The faithful flatness property cited above implies immediately that if P, Q P Z[Z m ] and there is x P σ´1 Z (Z[Z]p) such that P = Qx, then Let η be a linear form. Put Proof. It suffices to prove that every coefficient of the (kˆk)-matrix  be an ω-chart-system such that G(U p , Φ p ) ď C for every p P S(ω) and that rC ă δ 12 . This condition implies in particular that U p Ă D(p, δ 12 ). Choose some lifts r U p of neighborhoods U p , extending x Þ Ñ r x. Let D ą 0 be less than min Now let γ be a v-trajectory, joining r x with r yg, and let be the points in p´1(S(ω)) such that γ intersects r U A i . Then The length of the part of γ inside of n+1 Ť i=0 r U A i is not more than 2rC(n + 1).
Denote by t i and τ i the moments when γ enters and leaves r Therefore the total time which γ can spend outside Ť p r U A i is not more than | r F (r yg)´r F (r x)|/D, and the length of the corresponding part of the curve is at most E D | r F (r yg)´r F (r x)|. Since γ joins r x with r yg, the last expression is at most E D ( r F (r x)´r F (r y)´tωu(g)). Therefore where A is chosen so that A/2 ě | r F (r x)´r F (r y)| for every x, y P S(f ) and B = 2E/D. The inequality l(γ) ď A´Btωu(g) follows. □ Let ω be a Morse form, and tΦ p : U p Ñ B n (0, r p )u pPS(ω) be an ω-chartsystem. Choose a basis a 1 , . . . , a m in H 1 (M, Z)/Tors.
Definition 4.2.3. We shall say that ω ϵ is a Morse family, if for every ⃗ ϵ with |µ| ď ϵ the form ω µ is a Morse form and S(ω µ ) = S(ω). Let ω ϵ be a Morse family, and v be a vector field. We say, that v is an ω ϵ -gradient, if v is an ω ϵ -gradient for each ω µ P ω ϵ .

2)
Let v be an ω-gradient. Then there is ϵ ą 0 such that v is an ω ϵ -gradient.
therefore ω µ (x) ą 0. Finally, u has a standard form with respect to some ω-chart-system. A suitable restriction of this system will be an ω µ -chartsystem for any µ with |µ| ă ϵ. □ Now we can define the incidence coefficients with respect to the universal cover. The preceding lemma implies that v is anω-gradient for some 1formω which cohomology class is rational. Therefore (see [12]) for every g P π 1 (M ) there is at most finite set of (´v)-trajectories joining r x with r yg if ind x = ind y + 1. Choose orientations of descending discs. For each such trajectory we denote by ϵ(γ) the sign of intersection of D(r x, v) with D(r yg,´v) along γ.  We have chosen a Riemannian metric on M , whence the manifold M obtains a Riemannian metric, which is Z m -invariant. We have also chosen a basis (a 1 , . . . , a m ) in H 1 (M, Z)/Tors. Therefore this group is identified with Z m , and the vector space H 1 (M, R) with the dual space of linear forms R m Ñ R. Choose the L 1 -norm in R m . Then the dual space obtains the where tai u is the base dual to ta i u.)  Let v be an ω-gradient. There is ϵ ą 0, such that every linear form η : R m Ñ R with }[ω]´η} ď ϵ is a cohomology class of a Morse form ω(η) such that v is an ω-gradient.

Definition 4.3.2.
For two critical points x, y P S(ω) and an ω-gradient v we set I(x, y; v) = tg P π 1 (M ) | there is a (´v)-trajectory γ joiningx toỹ¨gu.
If the set of (´v)-trajectories joiningx toỹ¨g is finite, we denote by N (x,ỹ, g; v) its cardinality. (We identify here two trajectories which differ by a parameter change.)
Also there are integral [γ 1 ]-cone Γ 2 and a 2 P Z m such that such that Q(I(x, y; v)) Ă Γ 2 + a 2 . Adding to the generators of Γ 1 and Γ 2 some integral vectors we can assume that h P IntΓ 1 , h P IntΓ 2 . Then there exists N P N such that Γ 1 + a 1 Ă Γ 1´N h and Γ 2´N h Ą Γ 2 + a 2 . Thus [ ω] is defined. We shall assume that the lifts x of points x P S(ω) are chosen so that Q(r x) = x. Note that obviously supp (n(x, y; v)) Ă Q(supp r n(r x, r y; v)). We shall now prove the properties of Gt 1 (ω) required in the statement.

Proof of Theorem
Indeed, if v satisfies (C) then every ω-gradient u, sufficiently close to v, is also an ω ϵ -gradient (due to Lemma 4.2.4) and u P Gt 1 (f 0 xξy) since Gt 1 (f 0 xξy) is C 0 -open in Gt(f 0 xξy) (by Theorem 3.2.1).
, then there exists an ϵ ą 0 such that v is an ω ϵ -gradient. Choose any form ω 1 P ω ϵ with [ω 1 ] P H 1 (M, Q). Then by Theorem 3.2.1 arbitrarily close to v one can find an ω 1 -gradient u P Gt 1 (f 0 xω 1 y). By Lemma 4.2.4, u can be chosen so as to be an ω ϵ -gradient. Since n(x, y; v) P σ´1 ∆ Z[∆]p the faithful flatness property imply

AN EXAMPLE
In this section we shall construct a Morse map M Ñ S 1 on a closed 3-manifold M , having two critical points: x of index 2 and y of index 1, such that n(x,ȳ; v) = ÿ kě0 n k t k with n k " α¨β k with α ă 0, β ą 0. For any f -gradient w sufficiently close to v in C 0 topology, we shall have: n k (x,ȳ; w) = n k (x,ȳ; v).
We start with a torus T 2 , remove from it two open discs and obtain a surface S with two components of boundary. Choose and fix a parallel β and a meridian α of this twice punctured torus. The copies of this surface (resp. the copies of α, β etc.) will be denoted by the same letter S (resp. α, β, etc.) adding indices in order to distinguish between them. The corresponding discs will be denoted by D 1 , D 2 . We glue a copy of S, denoted by S(1, 1) to a copy of S, denoted by S(1, 2) and close the boundary by two discs D 1 (1), D 1 (2). The resulting surface is called N . We glue three copies of S successively, close the boundary and obtain a closed surface The surface L is depicted on Figure 5.1 below.
We can find an F 0 -gradient v 0 such that which identifies S( 1 2 , 0) with S(0, 1) and S( 1 2 , 2) with S(0, 2) slightly diminished from the right so that the image of D 2 ( 1 2 ) contains D 2 (0) in its interior, and therefore -image of a δ-tubular neighborhood of . Glue together W 0 and W 1 along L, denote the resulting cobordism by W . Then we have BW = K Y N . Glue the functions F 1 and F 0 to a Morse function F : W Ñ [0, 1] and the vector fields v 1 , v 0 to an F -gradient v. We shall now define a ranging system for (F, v). Consider the set Λ = t0, 1 2 , 1u of regular values, and set The fact that t(A λ , B λ )u λPΛ is a ranging system for (F, v) follows immediately from the properties of v 0 , v 1 cited above. It is not difficult to compute the homomorphism It preserves the intersection form, therefore it can be realized by a diffeomorphism Φ : K Ñ K and it is easy to see that we can assume that Φ(x) = x for x P D 1 (0) Y D 2 (0). Denote by Ψ the composition of Φ with the subsequent identification of K with N . Denote by M the 3-dimensional manifold obtained by gluing of K to N by means of Ψ, and let f : M Ñ S 1 be the Morse function obtained from F . The corresponding cyclic covering M is a union of countably many copies of W , denoted by W [i], i P Z, glued together by the diffeomorphisms of the components of boundaries.
The identification W [k] Ñ W will be denoted by

Set
‚ Σ = tn/2 | n P Zu; It is obvious that t(A σ , B σ )u σPΣ is a t-equivariant ranging system for (F , v). Putx = J´1 0 (x) andȳ = J´1 0 (y). Then we have: Denote by D the matrix of Ψ˚˝H(v). Then To find D k (b 2 ) assume by induction that Therefore the vectors (β k , γ k ) satisfy and ) .

5.1.
Incidence coefficients with respect to the universal cover: the exponential estimate.
Lemma 5.1.1. Let ξ, η : R m Ñ R be non-zero linear forms, and let Γ be a (ξ, η)-integral cone. Then there is A ą 0 such that for every b P R m there is B P R such that for every c P R we have: Proof. Abbreviate ξ´1 ([c, 8)) by tξ ě cu. It is sufficient to consider the case b = 0, that is, to prove that there is A ě 0 such that Γ X tξ ě cu Ă Γ X tη ě Acu

On the Novikov complex for a Morse form.
In this paper we do not use the notion of the Novikov complex, and work only with the incidence coefficients. The latter were introduced however in [9] as the matrix entries of the boundary operators in the Novikov complex. In this subsection we use the results of Subsection 4.2 and 5.2 to give a simple proof of the fact that the boundary operators above indeed turn the graded module into a chain complex (that is, B 2 = 0). We reduce the proof to the corresponding statement about rational Morse forms, proved in [12]. We assume here the terminology of Subsection 1.2 of the Introduction. Moreover, if ∆ is an integral [ω]-cone, we denote by λ ∆ the subset of Z[π 1 M ]t ωu , defined by λ ∆ = tλ | there exists b P Z m such that Q(supp λ) Ă ∆ + bu.
It is not difficult to see that λ ∆ is a subring of Z[π 1 M ]t ωu . Now let v be an ω-gradient satisfying transversality assumption and ϵ ą 0 so small that ω ϵ is a Morse family and v is an ω ϵ -gradient. Let ξ P ω ϵ be a Morse form such that [ξ] P H 1 (M, Q). Then it follows from Lemma 4.3.5 that there is an integral ([ξ], [ω])-cone ∆, such that for every x, y P S(ω) with ind x = ind y + 1 we have r n(r x, r y; v) Ă λ ∆ .
Therefore the homomorphism is defined actually over the ring λ ∆ and to verify that B 2 = 0 it is sufficient to verify it over the ring Z[π 1 M ]t ξu , and this is done in [12].
Let v be an f -gradient, Λ = tλ 0 , . . . , λ k u be a set of regular values of f , such that and between any two adjacent values λ i and λ i+1 there is exactly one critical value of f .
.  ii) v is an almost good f -gradient.
so small that for each ξ P B ind p (0, α) there is an isotopy H t xξy such that }v´Ψ˚(wxξy)} ă ϵ. (6.4) We perform this construction for every p P S(f ), and obtain a diffeomorphism Ψ p , a vector ξ p , and a vector field w p xξ p y. We assume that θ p , α p , τ p are chosen to be independent of p, and we denote them by θ, α, τ omitting p in the notation. The set tξ p u pPS(f ) will be denoted by ⃗ ξ. Define a new vector field u = vx ⃗ ξy setting It follows from the construction that for every ⃗ ξ the vector field vx ⃗ ξy is an f -gradient. Note also that for p P S(f ) we have D(p,´vx ⃗ ξy) X f´1(µ p ) = S p (ξ p ).
The condition (6.4) implies that t(A (s) λ , B (s) λ )u λPΛ,0ďsďn is a BRS for (f, vx ⃗ ξy). Therefore vx ⃗ ξy is an almost good f -gradient. It will be called regular perturbation of v corresponding to ⃗ ξ.
Proposition 6.1.3. 1) Let p, q P S(f ) with ind p = ind q + 1. The set of (´vx ⃗ ξy)-trajectories 6 joining p with q is in a bijective correspondence with the set (D(p, v) X f´1(µ q )) X S q (ξ q ).
2) The vector field vx ⃗ ξy is a good f -gradient if for every p, q P S(f ) with ind p = ind q+1 the submanifold D(p, v)Xf´1(µ q ) of f´1(µ q ) is transversal to S q (ξ q ).
Proof. 1) Let γ be a (´vx ⃗ ξy)-trajectory joining p with q and γ(t 0 ) inf´1(µ q ), where t 0 P R. I claim that for t ă t 0 we have Indeed, if the opposite is true, let t 1 ă t 0 be the first moment when γ intersect supp (v´vx ⃗ ξy). Then there is s P S(f ) such that γ(t 1 ) P Tub(S s , θ 2 ). Note that (n´ind s) µ s and ind p ě ind s + 1.
Let p, q P S(f ), ind p = ind q + 1. It suffices to prove that the intersection D(p, vx ⃗ ξy) X f´1(µ q ) is transversal to S q (ξ q ). Let x be a point in the intersection of these manifolds. In the part 1) we have proved that there is a (´v)-trajectory, joining p with x and not intersecting supp (v´vx ⃗ ξy). Then 174 A. Pajitnov a small neighborhood of this trajectory does not intersect supp (v´vx ⃗ ξy) and the transversality sought follows from (D(p, v) X f´1(µ q ))&S q (ξ q ). □ 6.2. Volume estimates. We will use here the terminology of the previous subsection. Let t(A (s) λ , B (s) λ )u λPΛ,0ďsďn be a BRS for (f, v). Assume that δ ą 0 satisfies (6.2) from Subsection 6.1. Fix an integer s : 0 ď s ď n. Let λ ă µ be adjacent elements of Λ. Denote by S 1 the subset of all critical points of f of index ď s and with critical values in (λ, µ). The set is a compact subset of the domain of definition of v ù [µ,λ] . Denote by N and by A the maximum of (N (s) µ ) k over all µ P Λ, 0 ď s ď n, and 0 ď k ď n. (where the manifold S pˆB ind p (0, θ) is endowed with the product Riemannian metric). Denote by B the maximum of B (p) over all p P S(f ). Lemma 6.2.1. Let λ, µ P Λ, λ ă µ, and s be an integer with 0 ď s ď n. Let also N be a submanifold of f´1(µ) z B (Σ1) σ P Σ ñ σ + n P Σ for every n P Z.
(Σ3) For every A, B P R the set Σ X [A, B] is finite. Assume that for every integer s : 0 ď s ď n and every σ P Σ there are given compacts A (1) For every µ, ν P Σ, µ ă ν the system t(A Up to the end of this subsection we assume that (F, v) has a BERS t(A (s) σ , B (s) σ )u σPΣ,0ďsďn . Choose any σ P Σ and denote by W the cobordism F´1([σ, σ +1]) and by Λ the set ΣX[σ, σ+1]. We apply the constructions of Subsections 6.1 and 6.2 to W and then extend the results to the whole of M in the t-invariant way, thus obtaining the sets Tub(S q , κ) Ă F´1(µ q ) for every q P S(F ). Let p, q P S(F ). Assume that F (p) ą F (q) and ind p = ind q + 1. Denote by N p,q the submanifold of dimension ind q of S qˆB ind q (0, θ 2 ), defined by ) ) .
The next lemma follows from Lemma 6.2.1.

Lemma 6.3.2.
There are constants C, D ą 0 such that for every p, q P S(F ) with ind p = ind q + 1 and F (p) ą F (q) we have: The next lemma is a direct consequence of Sard Theorem. such that for every ξ P Q and every p P S(F ) with ind p = ind q + 1 we have: D(p, v)&S q (ξ). □ The next proposition follows from (5.2) by arguments due to V. I. Arnold (see [1, p. 81]). Proposition 6.3.4. Let q P S(F ). Then there is a subset Q Ă B ind q (0, α) of full measure such that for every ξ P Q we have: ‚ For every p P S(F ) with ind p = ind q + 1 there are constants K, R ą 0, such that for every integer l ě 0 we have: 7(N pt´l,q X (S qˆt ξu)) ď K¨R l . Corollary 6.3.5. Let ν ą 0. Then there exists a good f -gradient u with }u´v} ă ν and constants K, L ą 0 such that for every p, q P S(f ) with ind p = ind q + 1 we have N l (p, q; v) ď L¨K l .
Proof. The construction of regular perturbation, described in §6.1, applied to v| W , gives for every ⃗ ξ = tξ p u pPS(F )XW , ξ p P B ind p (0, α), an F | W -gradient vx ⃗ ξy. Since supp (v´vx ⃗ ξy) does not intersect BW , we can extend vx ⃗ ξy tinvariantly to M and obtain a t-invariant F -gradient, which will be denoted by the same symbol vx ⃗ ξy. Note that since t(A (s) σ , B (s) σ )u σPΣ,0ďsďn is a BERS for (F, vx ⃗ ξy), the vector field vx ⃗ ξy is an almost good F -gradient. If all ξ q P B ind q (0, α) are sufficiently small, we will have }u´vx ⃗ ξy} ă ν. Lemma 6.3.3 and Proposition 6.3.4 imply that we can choose the components ξ q of ⃗ ξ in such a way that (1) For every p P S(F ), ind p = ind q + 1 we have: D(p, v) X F´1(µ q )&S q (ξ q ).
(2) There are K, R ą 0 such that for every p P S(F ) with ind p = ind q + 1 we have: 7(D(pt´l, v) X S q (ξ q )) ď K¨R l .
Apply Proposition 6.1.3 and deduce that u = vx ⃗ ξy satisfy the conclusions of the Corollary. □ 6.4. Proof of Theorem D. We can assume that S(f ) = 0 and that the homotopy class of f is indivisible. In view of Corollary 6.3.5 it is sufficient to prove that the set of f -gradients, having a BERS, is C 0 -dense in G(f ). To prove that, let v be any f -gradient and ϵ ą 0. Choose any regular value λ of F and denote by W the cobordism F´1([λ, λ+1]). Choose any set of regular values of F , satisfying (Σ1)-(Σ3) of Definition 6.3.1. Theorem 2.6.5 implies that there is an almost good F | W -gradient u and a ranging pair (V 0 , V 1 ) for (F | W , u) such that }v´u} ď ϵ and v = u near BW . We can also assume that V 0 t = V 1 . Then the procedure, described in [15,Construction 4.13], page 1000 defines a bunch of ranging systems for (F | W , u). Extend u to a t-invariant F -gradient. The BRS constructed is easily extended to a BERS for (F, u). □ We mention here an obvious corollary of Theorem D which will be of use in the proof of Theorem E. For a good f -gradient v, c P R and x, y P S(f ) with ind x = ind y + 1 we denote by N ěc (x, y; v) the sum ř kěc N k (x, y; v). Corollary 6.4.1. Let v P G 0 (f ). Then there are constants R, Q ą 0 such that for every critical points x, y P S(f ) with ind x = ind y + 1 and every c P R we have: N ěc (x, y; v) ď R¨Q´c. □ 6.5. Proof of the Theorem E. By Lemma 4.2.4 find ϵ ą 0 and δ ą 0 such that ω ϵ is a Morse family, v is an ω ϵ -gradient and every ω-gradient u with }u´v} ă δ is an Ω ϵ -gradient. Choose some ξ P Ω ϵ with [ξ] P H 1 (M, Q) and choose (by Theorem D) a good ξ-gradient u, satisfying the condition (2) of Theorem D and such that }u´v} ă δ. Then u is an ω ϵ -gradient. By

ACKNOWLEDGMENTS
The author thanks the anonymous referee for many helpful remarks, that have lead to a substantial improvement of the manuscript.