Zeros of closed 1-forms, homoclinic orbits, and Lusternik - Schnirelman theory

Michael Farber^{∗}

June 18, 2001

Abstract

In this paper we study topological lower bounds on the number of zeros of closed 1-forms without Morse type assumptions. We prove that one may always find a representing closed 1-form having at most one zero. We introduce and study a generalization cat(X, ξ) of the notion of Lusternik - Schnirelman category, depending on a topological space X and a 1-dimensional real cohomology class ξ ∈ H^{1}(X; R). We prove that any closed 1-form ω in class ξ has at least cat(X, ξ) zeros assuming that ω admits a gradient-like vector field with no homoclinic cycles. We show that the number cat(X, ξ) can be estimated from below in terms of the cup-products and higher Massey products.

This paper corrects some my statements made in [6], [7].

1991 Mathematics Subject Classification: 37Cxx, 58Exx, 53Dxx Keywords: Morse theory, Lusternik - Schnirelman theory, closed

1-form, Massey products, homoclinic orbits

1 Introduction

The Novikov inequalities [12, 13] estimate the numbers of zeros of different indices of closed 1-forms ω on manifolds lying in a given cohomology class, assuming that all the zeros are non-degenerate in the sense of Morse.

∗Partially supported by the US - Israel Binational Science Foundation; part of this work was done while the author visited Max-Planck Institute for Mathematics in Bonn

Applications of the Novikov inequalities in mechanics, in geometry and in symplectic topology are well-known, see [3], [9], [14], [15], [10].

In this paper we show that one may always realize a nonzero cohomology class by a closed 1-form with at most one (degenerate) zero. In the proof we use the technique of rearrangements of critical points and the result of F. Takens [18], which describes the conditions when several critical points of a function could be collided into one.

Central role in this paper plays a suitable generalization of the notion of Lusternik - Schnirelman category. For any pair (X, ξ), consisting of a topological space X and a real cohomology class ξ ∈ H^{1}(X; R), we define a nonnegative integer cat(X, ξ), the category of X with respect to the cohomology class ξ. The definition of cat(X, ξ) is similar in the spirit to the definition of cat(X); it deals with open covers of X with certain homotopy properties. We show that cat(X, ξ) depends only on the homotopy type of (X, ξ) and coincides with cat(X) in the case ξ = 0. If ξ ≠ 0 then cat(X, ξ) < cat(X); we show by examples that the difference cat(X) − cat(X, ξ) may be arbitrary.

The main Theorem of the paper (Theorem 4.1) states that any smooth closed 1-form ω on a smooth closed manifold X must have at least cat(X, ξ) geometrically distinct zeros, where ξ = [ω] ∈ H^{1}(X; R) denotes the cohomology class of ω, assuming that ω admits a gradient-like vector field with no homoclinic cycles. Recall that a homoclinic orbit is defined as a trajectory γ(t), t ∈ R, such that both limits lim_{t→+∞} γ(t) and lim_{t→−∞} γ(t) exist and are equal. More generally, a homoclinic cycle of length n is a sequence of orbits γ_{1}, γ_{2}, …, γ_{n} such that

lim_{t→+∞} γ_{i}(t) = lim_{t→−∞} γ_{i+1}(t)

for i = 1, …, n − 1 and

lim_{t→+∞} γ_{n}(t) = lim_{t→−∞} γ_{1}(t).

Viewed differently, the main Theorem of the paper claims that any gradient-like vector field of a closed 1-form has a homoclinic cycle assuming that the number of zeros is less than cat(M, ξ).

The homoclinic orbits were discovered by H. Poincaré and were studied by S. Smale. In the mathematical literature there are many results about existence of homoclinic orbits in Hamiltonian systems. Homoclinic orbits may not exist in the gradient systems for functions, i.e. in the case ξ = 0, corresponding to the classical Lusternik - Schnirelman theory.

In recent papers [5], [6], [7] we described cohomological cup-length type estimates on the number of zeros of closed 1-forms. Unfortunately, they are incorrect as stated. Purely algebraic Proposition 3 of [6] and a similar in character Lemma 6.6 of [7] are incorrect. These algebraic statements hold under slightly stronger assumptions, however, in oder to meet these assumptions in the main theorems, one has to make extra assumptions on closed 1-form. This paper gives a different (more geometric) approach to show that the main results of [7] hold for closed 1-forms having gradient-like vector field with no homoclinic cycles. The cohomological lower bounds of [6] require also some additional assumption, which will be discussed elsewhere.

I would like to thank Pierre Milman, Kaoru Ono and Shmuel Weinberger for useful discussions and help.

2 Colliding the critical points

In this section we will prove the following realization result.

Theorem 2.1 Let M be a closed connected n-dimensional smooth manifold, and let ξ ∈ H^{1}(M; Z) be a nonzero cohomology class. Then there exists a smooth closed 1-form ω in class ξ having at most one zero.

Proof. We will assume that the class ξ ∈ H^{1}(M; Z) is indivisible, i.e. is not a multiple of another integral class. Our statement clearly follows from this special case.

Our purpose is to show that we may find a smooth map ϕ : M → S^{1} with Morse critical points, having the following properties:

(A) The cohomology class of the closed 1-form ω̃ = ϕ^{∗}(dθ) coincides with ξ, where dθ denotes the standard angular form on the circle S^{1}.

(B) All fibers ϕ^{−1}(b) are connected, where b ∈ S^{1}.

(C) The map ϕ has at most one critical value b_{0} ∈ S^{1}. In other words, all critical points of ϕ lie in the same fiber f^{−1}(b_{0}).

Having achieved this, we may apply the technique of F. Takens [18], pages 203 - 206, which allows to collide the critical points of ϕ (equivalently, the zeros of closed 1-form ω̃) into a single degenerate critical point of a closed 1-form ω lying in the same cohomology class. Namely, firstly, we may find a piecewise smooth tree Γ ⊂ ϕ^{−1}(b_{0}) containing all the critical points of ϕ. Secondly, we may find a continuous map Ψ : M → M with the following properties:

Ψ(Γ) is a single point p ∈ Γ;

Ψ|_{M−Γ} is a diffeomorphism onto M − p;

Ψ is the identity map on the complement of a small neighborhood of Γ; in particular, Ψ is homotopic to the identity map M → M. The circle valued map ϕ ∘ Ψ^{−1} is well-defined and is continuous. Moreover, ϕ ∘ Ψ^{−1} is smooth on M − p. Applying Theorem 2.7 from [18] we see that we may replace the map ϕ ∘ Ψ^{−1} in a small neighborhood of p by a smooth map ψ : M → S^{1} having a single critical point. ψ is homotopic to ϕ and thus closed 1-form

ω = ψ^{∗}(dθ)

lies in the cohomology class ξ and has possibly a single zero.

In the rest of the proof we will show that we may find a smooth Morse map ψ : M → S^{1} with properties (A) - (C) above.

It is well known that any indivisible class ξ ≠ 0 may be realized by a connected codimension one submanifold V ⊂ M with oriented normal bundle. Cutting M along V produces a compact cobordism N with ∂N = ∂_{+}N ∪ ∂_{−}N, a disjoint union of two copies of V. Consider a Morse function f : N → [0, 1] having 0 and 1 as regular values and f^{−1}(0) = ∂_{+}N, f^{−1}(1) = ∂_{−}N. We may assume that f has no critical points of indices 0 and n = dim M. Moreover, we may construct f, such that all level sets f^{−1}(c), where c ∈ [0, 1], are connected and having the self-indexing property: all critical points of f having Morse index i lie in f^{−1}(i/n), where i = 1, …, n − 1. The map N → S^{1}, where x ↦ exp(2πif(x)), defines a smooth map ϕ_{1} : M → S^{1} in the cohomology class ξ with connected fibers having the following property: for any critical point m ∈ M, dϕ_{1}(m) = 0, with Morse index i, the image ϕ_{1}(x) equals exp(2πi/n). In other words, the critical points of ϕ_{1} with the same Morse index lie in the same fiber, and these critical fibers ϕ_{1}^{−1}(b) appear in the order of their Morse indices, while the point b moves in the positive direction along the circle S^{1}.

For points b_{1}, …, b_{r} ∈ S^{1} on the circle S^{1} we will write

b_{1} < b_{2} < ⋯ < b_{r} < b_{1}

to denote that moving from point b_{1} in the positive direction along the circle S^{1}, we first meet b_{2}, then b_{3}, …, until we again meet b_{1}.

Let us now formulate the following approximation to property (C):

(C_{j}) All critical points of a smooth Morse map ϕ : M → S^{1} with Morse index i lie in the same fiber ϕ^{−1}(b_{i}), where b_{i} ∈ S^{1}, i = 1, …, n − 1, and

b_{1} < b_{1} < ⋯ < b_{j} = b_{j+1} = ⋯ = b_{n−1} < b_{1}.

In particular, the critical values b_{j} = b_{j+1} = ⋯ = b_{n−1} coincide.

Note that (C_{1}) is equivalent to (C), which is our purpose.

We have found above a smooth map ϕ_{1} : M → S^{1}, satisfying (A), (B), and (C_{n−1}). On the next step we will show that we may replace ϕ_{1} by a smooth Morse map ϕ_{2} : M → S^{1} with properties (A), (B) and (C_{n−2}). Let b_{1} < b_{2} < ⋯ < b_{n−1} < b_{1} be the critical values of ϕ_{1}. Consider a point c ∈ S^{1}, lying between b_{n−2} and b_{n−1}. Cut M along the submanifold ϕ_{1}^{−1}(c) and consider the obtained cobordism N and a Morse function g from N to an interval, obtained by cutting the circle S^{1} at point c. All level sets of g are connected. Moving from the bottom of this cobordism to the top, we meet n − 1 critical levels; first we meet the level containing the critical points of Morse index n − 1, then the levels containing the critical points with Morse indices 1, 2, …, n − 2. We will use the theory of S. Smale of rearrangement of critical points. Choose a generic gradient like vector field v for g. Then all integral trajectories of ±v, which go out of the critical points of index n − 1, reach ∂N without interaction with the other critical points. Therefore we may slide the critical points of index n − 1 some distance up, putting them on the same level with the critical point of index n−2, see [11], Theorem 4.1. In other words, we may replace g by a Morse function g′, which coincides with g near ∂N, has the same critical points, but the value at the critical points of index n − 1 equals the value at the critical points of index n − 2. Note that the level sets of g′ are all connected:

(a) the bottom level g′^{−1}(0) is unchanged and so it is connected;

(b) passing the critical levels with Morse indices 1, 2, …, n − 3 may not create nonconnected level sets;

(c) in principle, nonconnected level sets may appear after passing the top critical value contained the critical points of indices n − 2 and n − 1; however in our situation all higher upper level sets are the same as for the previous function g, and so they are all connected.

Folding this cobordism back, gives a smooth map ϕ_{2} : M → S^{1} having properties (A), (B), and (C_{n−2}).

We may proceed similarly to find a smooth map ϕ_{3} : M → S^{1} with properties (A), (B), and (C_{n−3}). The critical values of ϕ_{2} are b_{1} < b_{2} < ⋯ < b_{n−2} = b_{n−1} < b_{1}. We find a point c ∈ S^{1} between b_{n−3} and b_{n−2} = b_{n−1} and cut M along ϕ_{2}^{−1}(c). The Morse function on the obtained cobordism will have n − 2 critical levels. The lowest will be the level containing all critical points of indices n − 2 and n − 1 and then the critical levels of points with the Morse indices 1, …, n − 3. Repeating the above procedure, we may slide the critical points of index n − 2 and n − 3 the same distance up putting them on the same level with the critical points of index n − 3.

Proceeding in this way inductively we arrive at a smooth map ϕ_{n−1} : M → S^{1} having properties (A), (B), and (C_{1}) = (C).

This completes the proof. □

According to a remark in [3], a statement in the spirit of Theorem 2.1 was made by Yu. Chekanov at a seminar talk in 1996. As far as I know, no written account of his work is available.

3 Category of a space with respect to a cohomology class

3.1 Definition of cat(X, ξ)

Let X be a finite CW-complex and let ξ ∈ H^{1}(X; R) be a real cohomology class. We will define below a numerical invariant cat(X, ξ), (the category of X with respect to class ξ), depending only on the homotopy type of the pair (X, ξ). It will turn into the classical Lusternik - Schnirelman category cat(X) in the case ξ = 0. The main property of cat(X, ξ) is that it gives a relation between the number of geometrically distinct zeros, which have closed 1forms realizing class ξ, and the homoclinic orbits of their gradient-like vector fields, see Theorem 4.1.

Fix a continuous closed 1-form ω on X representing the cohomology class ξ, see the Appendix for definitions.

Definition 3.1 We will define cat(X, ξ) to be the least integer k such that for any integer N > 0 there exists an open cover

X = F ∪ F_{1} ∪ ⋯ ∪ F_{k},  (3.1)

such that:

(a) Each inclusion F_{j} → X is null-homotopic, where j = 1, …, k.

(b) There exists a homotopy h_{t} : F → X, where t ∈ [0, 1], such that h_{0} is the inclusion F → X and for any point x ∈ F,

∫_{γ_{x}} ω ≤ −N,  (3.2)

where the curve γ_{x} : [0, 1] → X is given by γ_{x}(t) = h_{t}(x).

The meaning of the line integral ∫_{γ} ω, where γ : [0, 1] → X is a continuous curve, is explained in the Appendix.

Intuitively, condition (b) means that in the process of the homotopy h_{t} every point of F makes at least N full twists (in the negative direction) with respect to ω. We want to emphasize that when N tends to infinity we will obtain a sequence of different coverings (3.1) with the same number k and with the set F becoming more and more complicated, so that its limit is wild, looking like a fractal. This explains the approximative nature of our Definition 3.1, which allows to avoid these difficulties.

Observe that cat(X, ξ) does not depend on the choice of the continuous closed 1-form ω (which appears in Definition 3.1) and depends only on the cohomology class ξ = [ω]. Indeed, is ω′ is another continuous closed 1-form representing ξ then ω − ω′ = df, where f : X → R is a continuous function, and for any continuous curve γ : [0, 1] → X,

| ∫_{γ} ω − ∫_{γ} ω′| = |f(γ(1)) − f(γ(0))| ≤ C,

where the constant C is independent of γ. Here we have used compactness of X. This shows that if we may construct open covers (3.1) of X such that (b) is satisfied with an arbitrary large N > 0 then the same is true with ω′ replacing ω.

In general the following inequality holds

cat(X, ξ) ≤ cat(X),  (3.3)

since we may always consider covers (3.1) with F empty. Here cat(X) denotes the classical Lusternik - Schnirelman category of X, i.e. the least integer k such that there is an open cover X = F ∪ F_{1} ∪ ⋯ ∪ F_{k}, such that each inclusion F_{j} → X is null-homotopic, where j = 1, …, k.

Inequality (3.3) can be improved

cat(X, ξ) ≤ cat(X) − 1  (3.4)

assuming that X is connected and ξ ≠ 0. This follows because under the above assumptions any open subset F ⊂ X, which is contractible to a point in X, satisfies (b) of Definition 3.1 for any N. Hence, given a categorical open cover X = G_{1} ∪ ⋯ ∪ G_{r} we may set F = G_{1} and F_{j} = G_{j+1} for j = 1, …, r − 1, which gives a cover of X satisfying Definition 3.1.

Observe also that

cat(X, ξ) = cat(X, λξ),  (3.5)

for λ ∈ R, λ > 0, as follows clearly from the above definition.

3.2 A reformulation

Sometimes it will be convenient to use a different version of condition (b) of Definition 3.1, which we will now describe.

Let p : X̃ → X be the normal covering corresponding to the kernel of the homomorphism of periods (cf. Appendix)

π_{1}(X, x_{0}) → R, [γ] ↦ ∫_{γ} ω.  (3.6)

The group of covering transformations Γ of this covering equals the image of the homomorphism (3.6). The induced closed 1-form p^{∗}ω equals df, where f : X̃ → R is a continuous function. Denote p^{−1}(F) by F̃; it is an open subset invariant under Γ. Then condition (b) is equivalent to:

(b′) There exists a homotopy h̃_{t} : F̃ → X̃, where t ∈ [0, 1], such that h̃_{0} is the inclusion F̃ → X̃, each h̃_{t} is Γ-equivariant, and for any point x ∈ F̃,

f(h̃_{1}(x)) − f(x) ≤ −N.

Condition (b′) requires that under the homotopy h_{t} any point of F̃ descends at least N units down, measured by function f. The homotopy h̃_{t} is a lift of the homotopy h_{t} which exists because of the HLP (homotopy lifting property) of coverings. If γ̃_{y}, where y ∈ F̃, p(y) = x denotes the path γ̃_{y}(t) = h̃_{t}(y) in X̃ then γ_{x} = p_{∗}γ̃_{y} and

∫_{γ_{x}} ω = ∫_{p_{∗}γ̃_{y}} ω = ∫_{γ̃_{y}} p^{∗}ω = ∫_{γ̃_{y}} df = f(h̃_{1}(y)) − f(y).

This explains equivalence between (b) and (b′).

3.3 Examples

Example 3.2 Consider first the case when ξ = 0. Let us show that then (3.3) is an equality. ξ = 0 implies ω = df, where f : X → R is continuous. Then for any curve γ : [0, 1] → X the integral ∫_{γ} ω = f(γ(1)) − f(γ(0)) cannot become smaller than the variation of f on X. Therefore, for ξ = 0 inequality (3.2) may be satisfied for large N only if F = ∅. This proves cat(X, 0) = cat(X).

Example 3.3 Let X be a mapping torus, i.e. X is obtained from a cylinder Y × [0, 1], where Y is a compact complex, by identifying points (y, 0) and (ϕ(y), 1) for all y ∈ Y, where ϕ : Y → Y is a continuous map. Note that we do not assume that ϕ is a homeomorphism or a homotopy equivalence. We will denote points of X by pairs ⟨y, s⟩, where y ∈ Y, s ∈ [0, 1], understanding that ⟨y, 0⟩ = ⟨ϕ(y), 1⟩. X admits a natural projection q : X → S^{1}, where q⟨y, s⟩ = exp(2πis), and we will denote by ω = q^{∗}(dθ) the pullback of the standard angular form dθ of S^{1}; ω is a closed 1-from on X.

Let us show that cat(X, ξ) = 0, where ξ = [ω] ∈ H^{1}(X; R). Given a number N > 0, define a homotopy h_{t} : X → X, where for t ∈ [0, 1]

h_{t}⟨y, s⟩ = ⟨ϕ^{n(t,s)}(y), s − Nt + n(t, s)⟩.

Here n(t, s) denotes the number of integers contained in the semi-open interval (s − Nt, s]. It is easy to see that this formula defines a continuous homotopy, h_{0} = id and ∫_{γ_{x}} ω = −N for any point x ∈ X, where γ_{x}(t) = h_{t}(x). This shows that cat(X, ξ) = 0.

Note that the Lusternik - Schnirelman category cat(X) of a mapping torus may be arbitrary large (for example, cat(T^{n}) = n + 1). Hence the above example shows that the difference

cat(X) − cat(X, ξ)

may be arbitrarily large.

Example 3.4 In Definition 3.1 the notions “up”and “down”appear non symmetrically. Hence it may happen that cat(X, ξ) ≠ cat(X, −ξ). We will see such example now.

Consider the mapping torus X, as described in the previous example, with Y being the sphere S^{2} and with ϕ : Y → Y a map of degree 2. We have seen that cat(X, ξ) = 0, where ξ ∈ H^{1}(X; R) denotes the cohomology class described above. Let us show that cat(X, −ξ) ≥ 1.

The universal covering p : X̃ → X is Milnor’s telescope; it can be described as follows. X̃ is obtained from the disjoint union ⨿_{n∈Z}X_{n}, where X_{n} denotes Y × [n × n + 1], by identifying any point (y, n) ∈ X_{n} with (ϕ(y), n) ∈ X_{n−1}. The projection p : X̃ → X maps any point (y, t) ∈ X_{n} to ⟨ϕ^{[t−n]}(y), {t}⟩. If ω is the closed 1-form on X described in the previous example then p^{∗}ω = df where f : X̃ → R is the continuous function given by f(y, t) = t.

[figure: X̃; R; ↓ f]

Figure 1: Covering of mapping torus

Let X^{k} ⊂ X̃ denote ⋃_{n≥k} X_{n}. It is easy to see that H_{2}(X^{k}; Z) is isomorphic to Z with the fundamental class of the sphere Y × (k + 1) ⊂ X_{k} as a generator. The inclusion X^{k} → X^{k−1} induces on the homology the homomorphism H_{2}(X^{k}; Z) → H_{2}(X^{k−1}; Z) of multiplication by 2. Hence we see that H_{2}(X̃; Z) is isomorphic to the abelian group Z_{(2)} of rational numbers with denominators powers of 2 and the image of H_{2}(X^{k}; Z) can be identified with 2^{k}Z ⊂ Z_{(2)}. It follows that the sphere Y ×k ⊂ X_{k} cannot be homotoped into X^{k} by a homotopy in X̃. Using the condition (b′) of subsection 3.2 we see that there is no deformation taking the sphere Y × k up into X^{k} and this proves that cat(X, −ξ) > 0.

On the contrary, there is a deformation of X̃ taking all points arbitrarily far down (with respect to f), as we have seen in the previous example.

Example 3.5 Consider a bouquet X = Y ∨ S^{1} and assume that the class ξ ∈ H^{1}(X, R) satisfies ξ|_{Y} = 0 and ξ|_{S^{1}} ≠ 0. Let us show that then

cat(X, ξ) = cat(Y) − 1.

Consider an open cover X = F ∪ F_{1} ∪ ⋯ ∪ F_{k} satisfying conditions (a) and (b) of Definition 3.1. Let F′ denote F ∩Y and F′_{j} = F_{j} ∩Y. We want to show that F′ is contractible in Y. This would imply that Y = F′ ∪F′_{1} ∪⋯∪F′_{k} is a categorical cover of Y, and hence cat(X, ξ) + 1 ≤ cat(Y). Let h_{t} : F → X be a homotopy as in (b) with N large enough. For any x ∈ F′, the curve h_{t}(x) has to pass along the circle S^{1} many times and hence it arrives to the point q of intersection S^{1} ∩ Y for some t ∈ [0, 1]. For x ∈ F′ denote by σ(x) ∈ [0, 1] the minimal number such that h_{σ(x)}(x) = q. The function σ : F′ → R is continuous. Therefore h′_{t}(x) = h_{tσ(x)}(x) defines a homotopy F′ → Y with h′_{0}(x) = x and h′_{1}(x) = q.

The opposite inequality cat(X, ξ) + 1 ≥ cat(Y) is clear since for any categorical open cover Y = G_{0} ∪ ⋯ ∪ G_{r} we may set F = S^{1} ∪ G_{0} and F_{j} = G_{j}, where j = 1, …, r. The set F satisfies (b) of Definition 3.1 for any integer N > 0.

This example shows that the integer cat(X, ξ) may assume arbitrary nonnegative values.

3.4 Homotopy invariance

Lemma 3.6 Let ϕ : X_{1} → X_{2} be a homotopy equivalence, ξ_{i} ∈ H^{1}(X_{i}; R), where i = 1, 2, and ξ_{1} = ϕ^{∗}(ξ_{2}) ∈ H^{1}(X_{1}; R). Then

cat(X_{1}, ξ_{1}) = cat(X_{2}, ξ_{2})  (3.7)

Proof. Let ψ : X_{2} → X_{1} be a homotopy inverse of ϕ. Choose a closed 1-form ω_{1} on X_{1} in the cohomology class ξ_{1}. Then ω_{2} = ψ^{∗}ω_{1} is a closed 1-form on X_{2} lying in cohomology class ξ_{2}.

Fix a homotopy r_{t} : X_{1} → X_{1}, where t ∈ [0, 1], such that r_{0} = id_{X_{1}} and r_{1} = ψ ∘ ϕ. Compactness of X_{1} implies that there is a constant C > 0 such that | ∫_{α_{x}} ω_{1}| < C for any point x ∈ X_{1}, where α_{x} is the track of the point x under homotopy r_{t}, i.e. α_{x}(t) = r_{t}(x).

Suppose that cat(X_{2}, ξ_{2}) ≤ k. Given any N > 0, there is an open covering X_{2} = F ∪ F_{1} ∪ ⋯ ∪ F_{k}, such that F_{1}, …, F_{k} are contractible in X_{2} and there exists a homotopy h_{t} : F → X_{2}, where t ∈ [0, 1], such that ∫_{γ_{x}} ω_{2} ≤ −N − C for any x ∈ F, where γ_{x}(t) = h_{t}(x). Define

G = ϕ^{−1}(F), G_{j} = ϕ^{−1}(F_{j}), j = 1, …, k.

These sets form an open cover of X_{1} = G ∪ G_{1} ∪ ⋯ ∪ G_{k}. Let us show that the set G ⊂ X_{1} satisfies condition (b) of Definition 3.1. Define a homotopy h′_{t} : G → X_{1}, where t ∈ [0, 1], by

h′_{t}(x) = \{ \begin{matrix}r_{2t}(x), & for 0 ≤ t ≤ 1/2, \\ ψ(h_{2t−1}(ϕ(x))), & for 1/2 ≤ t ≤ 1. \end{matrix}  (3.8)

Then h′_{0} is the inclusion G → X_{1} and for any point x ∈ G holds

∫_{γ′_{x}} ω_{1} ≤ −N,

where γ′_{x}(t) = h′_{t}(x) is the track of x under homotopy h′_{t}.

The following diagram

\begin{aligned}G_{j} & →^{⊂} & X_{1} \\ ϕ ↓ & & ↑ ψ \\ F_{j} & →^{⊂} & X_{2} \end{aligned}

is homotopy commutative and the horizontal map below is null-homotopic. This shows that the inclusion G_{j} → X_{1} is null-homotopic, where j = 1, …, k.

The above argument proves that cat(X_{1}, ξ_{1}) ≤ cat(X_{2}, ξ_{2}).

The inverse inequality follows similarly. □.

4 An estimate on the number of zeros

In this section we will use invariant cat(X, ξ) to obtain a lower bound on the number of zeros of vector fields having no homoclinic orbits.

Let ω be a smooth closed 1-form on a connected closed smooth manifold M. We will assume that ω has finitely many zeros. A smooth vector field v is a gradient-like vector field for on M if (1) ω(v) > 0 on the complement of the set of zeros of v, and (2) in a neighborhood U_{p} ⊂ M of any zero p, the field v|_{U_{p}} coincides with the gradient of 1-form ω with respect to a Riemannian metric on U_{p}.

An integral trajectory γ(t) of vector field v is a homoclinic orbit if both limits lim_{t→±∞} γ(t) = γ(±∞) exist and are equal. The point p = γ(±∞) is then a zero of v. Note that for any homoclinic orbit γ holds ∫_{γ} ω > 0. Hence, homoclinic orbits do not exist in the case ξ = 0, i.e. in the usual gradient systems.

More generally, a homoclinic cycle of length n is a sequence of trajectories γ_{1}, …, γ_{n} of the field v such that all the limits lim_{t→±∞} γ_{i}(t) exist, where i = 1, …, n, and

lim_{t→+∞} γ_{i}(t) = lim_{t→−∞} γ_{i+1}(t), i = 1, …, n.

For i = n this means that lim_{t→+∞} γ_{n}(t) = lim_{t→−∞} γ_{1}(t), and so the union of the curves γ_{i} form a closed cycle.

Theorem 4.1 Let ω be a smooth closed 1-form on a closed manifold M and let ξ = [ω] ∈ H^{1}(M; R) denote the cohomology class of ω. If ω admits a gradient-like vector field v with no homoclinic cycles, then ω has at least cat(M, ξ) geometrically distinct zeros.

Here is a different formulation of the above Theorem:

Theorem 4.2 If the number of zeros of a smooth closed 1-form ω is less than cat(M, ξ), where ξ = [ω] ∈ H^{1}(M; R) denotes the cohomology class of ω, then any gradient-like vector field for ω has a homoclinic cycle.

Combined with Theorem 2.1, this shows that there may exist homoclinic cycles which cannot be destroyed while perturbing the gradient-like vector field. This “focusing effect” starts when the number of zeros of a closed 1form becomes less than cat(M, ξ). It is a new phenomenon, not occuring in the Novikov theory: assuming that the zeros of ω are all Morse it is always possible to find a gradient-like vector fields v for ω such that any integral trajectory connects a zero with higher Morse index with a zero with lower Morse index (by the Kupka - Smale Theorem [16]).

4.1 Proof of Theorem 4.1

Let p_{1}, …, p_{k} ∈ M denote all the zeros of ω. Assume that there exists a gradient-like vector field v for ω with no homoclinic orbits. Our purpose is to show that then cat(M, ξ) ≤ k.

Fix a number N > 0.

Consider the flow M × R → R, where (m, t) ↦ m · t, generated by the field −v.

Choose small closed disks U_{j} around each point p_{j}, where j = 1, …, k. We will assume that for i ≠ j the disks U_{i} and U_{j} are disjoint. Also, we will fix a Riemannian metric on M such that in the disks U_{i}, where i = 1, …, k, the field v is the gradient of ω with respect to this metric.

We claim that: one may choose closed disks V_{i}, where i = 1, …, k, such that:

(a) p_{i} ∈ Int V_{i} and V_{i} ⊂ Int U_{i},

(b) the disk V_{i} is gradient-convex (see Appendix B) in the following sense. Consider the covering π : M̃ → M corresponding to the kernel of the homomorphism of periods (3.6). Then the form ω lifts to M̃ as a smooth function f : M̃ → M, i.e. π^{∗}ω = df, and the field v lifts to a gradient-like vector field ṽ of function −f. We require any lift to M̃ of the disk V_{i} to be gradient-convex, see Appendix B.

(c) Let ∂_{−}V_{i} denote the set of points p ∈ ∂V_{i}, such that for all sufficiently small τ > 0, holds p · τ ∉ V_{i}. Then we require that for any p ∈ ∂_{−}V_{i} there exists no real number t_{p} > 0, such that: the point p · t_{p} belongs to Int V_{i}, and ∫_{p}^{p·t_{p}} ω ≥ −N, where the integral is calculated along the integral trajectory σ_{p} : [0, t_{p}] → M, σ_{p}(t) = p · t.

Assume first that V_{i} is any neighborhood satisfying (a) and (b). Then any trajectory p · t, where p ∈ ∂_{−}V_{i}, leaves U_{i} before it may re-enter V_{i}. This follows from gradient convexity of V_{i}, since the disk U_{i} also lifts to the covering M̃ → M. Hence:

(i) There exists a > 0, such that for any p ∈ ∂_{−}V_{i} and t > 0 with p·t ∈ V_{j} holds t ≥ a.

We may take a = min{l_{i}v_{i}^{−1}; i = 1, …, k}, where l_{i} > 0 denotes the distance between V_{i} and M − Int U_{i}, and v_{i} = max |v(x)| for x ∈ U_{i} − Int V_{i}.

Note that if we shrink the disks V_{i} the number a > 0 may only increase, assuming that V_{i} are sufficiently small.

(ii) There exists b > 0, such that for any p ∈ ∂_{−}V_{i} and t > 0 with p·t ∈ V_{j} holds

∫_{p}^{p·t} ω < −b.

We may take b = min{ϵ_{i}a_{i}; i = 1, …, k}, where −ϵ_{i} = max ω(v)(x) for x ∈ U_{i} − Int V_{i}.

Suppose that we may never achieve (c) by shrinking the disk V_{i} satisfying conditions (a) and (b). Then there exists a sequence of points p_{i,n} ∈ ∂_{−}V_{i} and two sequences of real numbers t_{i,n} > 0 and s_{i,n} < 0 such that:

(1) the set p_{i,n} · [s_{i,n}, 0] is contained in the disk V_{i} and the point p_{i,n} · s_{i,n} converges to p_{i} as n tends to ∞;

(2) p_{i,n} · t_{i,n} converges to p_{i};

(3)

∫_{p_{i,n}}^{p_{i,n}·t_{i,n}}ω ≥ −N.

Let γ_{i,n} ∈ H_{1}(M) denote the homology class of the loop obtained as follows. Start at the point p_{i,n} ∈ ∂_{−}V_{i}, follow the trajectory p_{i,n} · [0, t_{i,n}], and then connect the endpoint p_{i,n} · t_{i,n} with p_{i,n} by a path inside V_{i}. We claim that the set {γ_{i,n}}_{n≥1} ⊂ H_{1}(M) of the obtained homology classes is finite. This would follow once we show that the total length L_{i,n} of the parts of any trajectory p_{i,n} ·[0, t_{i,n}] spent outside the set U_{1} ∪⋯∪U_{k}, is bounded. Writing

∫_{γ} ω = ∫_{a}^{b} ω(γ̇(t))dt = ∫_{a}^{b} \frac{ω(γ̇(t))}{|γ̇(t)|} · |γ̇(t)|dt

and using (3) above, it is easy to see that

L_{i,n} ≤ N · c^{−1},

where c > 0 is given by

c = − max {ω(v_{x}) · |v_{x}|^{−1} ; x ∈ M − ⋃_{i=1}^{k} Int U_{i}}.

Passing to a subsequence, we may assume that p_{i,n} converges to a point q_{i} ∈ ∂_{−}V_{i} and the sequences s_{i,n} and t_{i,n} have finite or infinite limits s_{i} and t_{i} correspondingly. We may also assume that the homology class γ_{i,n} ∈ H_{1}(M) is independent of n.

(1) and (2) imply t_{i} = +∞ and s_{i} = −∞.

First we want to show that lim_{t→−∞} q_{i}·t = p_{i}. We will identify V_{i} with one of its lifts to the covering M̃. If there exists s < 0 such that f(q_{i}·s) > f(p_{i})+ϵ for some ϵ > 0, then the set {(x, s) ∈ M̃ × R; f(x · s) > f(p_{i}) + ϵ} (which is open) contains (p_{i,n}, s_{i,n}) for all n sufficiently large, which contradicts (1). This shows that on the covering M̃ holds q_{i} · s → p_{i} for s → −∞. Hence the same relation holds on the initial manifold M, as well.

Now we want to understand the limit lim_{t→+∞} q_{i} · t. Consider the above lift of V_{i} to the covering M̃, such that the flow becomes the gradient-like flow ṽ of function −f : M̃ → R. The points p_{i,n} · t_{i. n} all belong to a translate gV_{i} of the disk V_{i}, where g is independent of n. The trajectory q_{i} · t in M̃ for large t may either reach the neighborhood gV_{i}, or it may be ”caught” by some other critical point of f on the way.

Let us show that in the first case the point q_{i} · t tends to gp_{i}, as t tends to +∞ and hence the vector field v on M has a homoclinic orbit, starting and ending at p_{i}. If it is not true, then f(q_{i} · t) < f(gp_{i}) − ϵ for some t and ϵ > 0. Then the open set {(x, t) ∈ M̃ × R; f(x · s) < f(gp_{i}) − ϵ} would contain (p_{i,n}, t_{i,n}) for all n sufficiently large, which contradicts (2). This shows that on the covering M̃ holds q_{i}·t → gp_{i} for t → +∞. Hence on M, the trajectory q_{i} · t tends to p_{i} for t → +∞.

Consider now the second possibility; we will show that in this case v has a homoclinic cycle of length greater than 1. Assume that the limit q_{i} · t for t → ∞ equals hp_{j}, where h is an element of the group of periods of ω, and

f(gp_{i}) + b < f(hp_{j}) and f(hp_{j}) < f(p_{i}) − b.

Then for large n, the trajectory starting at p_{i,n} enters the neighborhood hV_{j} and leaves it at some point p′_{i,n} ∈ ∂_{−}(hV_{j}). Passing to a subsequence, we may assume that the sequence p′_{i,n} converges to a point q′_{i} ∈ ∂_{−}(hV_{j}). Then, as above, lim_{t→−∞} q′_{i} · t equals hp_{j} and the limit lim_{t→+∞} q′_{i} · t either equals gp_{i}, or it equals h_{1}p_{j_{1}} ∈ M̃, where

f(gp_{i}) + b < f(h_{1}p_{j_{1}}) and f(h_{1}p_{j_{1}}) < f(hp_{j}) − b.

Continuing these arguments by induction, we obtain a homoclinic cycle on the manifold M, “starting and ending”at p_{i}. The number of steps in the above process is finite (at most [N/b]).

The above argument proves existence of the disks V_{i} with properties (a), (b), (c), assuming that vector field v has no homoclinic cycles.

Next we will construct an open cover F ∪ F_{1} ∪ F_{2} ∪ ⋯ ∪ F_{k} = M.

We will define F as the set of all points p ∈ M such that there exists a positive number t_{p} > 0, so that the integral curve σ_{p} : [0, t_{p}] → M, where σ_{p}(t) = p · t, satisfies ∫_{σ_{p}} ω = −N. It is clear that F is open, and p ↦ t_{p} is a continuous real valued function on F.

We may define a homotopy

h_{τ} : F → F, by h_{τ}(p) = p · (τt_{p}), τ ∈ [0, 1]

This homotopy satisfies condition (b) of Definition 3.1.

Now we will define the sets F_{j}, where j = 1, …, k. We say that p ∈ F_{j} if for some t_{p} > 0 the point p · t_{p} belongs to the interior of V_{j}, and

∫_{σ_{p}} ω > −N,

where σ_{p} : [0, t_{p}] → M is given by σ_{p}(t) = p · t. It is clear that F_{j} is open. The sets F, F_{1}, …, F_{k} cover M. Indeed, for any point p ∈ M either

∫_{p}^{p·t}ω < −N

for some t > 0, or the trajectory γ(t) = p · t “enters” a zero p_{j}, so that

lim_{t→∞} ∫_{p}^{p·t}ω ≥ −N.

In the first case p belongs to F, and in the second case p belongs to F_{j}.

Now we will show that the set F_{j}, where j = 1, …, k, is contractible in M. For any point p ∈ M let J_{p} ⊂ R denotes the set J_{p} = {t ≥ 0; p · t ∈ V_{j}}. Because of our assumption about gradient-convexity of V_{j}, the set J_{p} is a union of disjoint closed intervals, and some of these intervals may degenerate to a point. Consider the first interval [a_{p}, b_{p}] ⊂ J_{p}. If this interval degenerates to a point (i.e. the trajectory through p touches V_{j}), then p does not belong to the set F_{j}, according to our assumption (c). By the same reason, points of ∂_{−}V_{j} do not belong to F_{j}.

Assume now p ∈ F_{j} and p ∉ Int V_{j}. Then the point p·t lies in the interior of V_{j} for a_{p} < t < b_{p}. Also, we have

∫_{p}^{p·a_{p}}ω > −N.

The function ϕ_{j} : F_{j} → R, given by

ϕ_{j}(p) = \{ \begin{matrix}0, & for & p ∈ Int V_{j}, \\ a_{p}, & for & p ∈ F_{j} − Int V_{j} \end{matrix}

is continuous. To show this, suppose that a sequence of points x_{n} ∈ F_{j}, where n = 1, 2, …, converges to x_{0} ∈ F_{j}. First consider the case x_{0} ∈ V_{j}. Since x_{0} ∈ F_{j}, we conclude that x_{0} ∉ ∂_{−}V_{j} (because of condition (c)), i.e. x_{0} · τ belongs to Int V_{j} for all τ ∈ (0, ϵ). It follows that for any τ ∈ (0, ϵ) holds x_{n} · τ ∈ Int V_{j} for all large n. Therefore, a_{x_{n}} < τ for all large n. Hence, the sequence a_{x_{n}} converges to 0. Consider now the case when x_{0} does not belong to V_{j}. Then the trajectory x_{0} · t does not touch V_{j} for t < a_{x_{0}} (again, because of condition (c)) and t = a_{x_{0}} is the first moment when the trajectory penetrates V_{j}. We know also that the velocity vector v_{x} is transversal to the boundary ∂V_{j}, where x = x_{0} · a_{x_{0}}. Then the sequence a_{x_{n}} converges to a_{x_{0}}, as follows from continuity of solutions of ordinary differential equation with respect to the initial conditions.

We may define a homotopy

h_{τ} : F_{j} → M, h_{τ}(p) = p · (τϕ_{j}(p)), p ∈ F_{j}, τ ∈ [0, 1].

Here h_{0} is the inclusion F_{j} → M and h_{1} maps F_{j} into the disk V_{j}. This completes the proof. □

5 Moving homology classes

Here we study the effect of condition (b′) of subsection 3.2 on homology classes. For simplicity we assume that the group of periods Γ ⊂ R is infinite cyclic.

k will denote a field.

Let X̃ → X be an infinite cyclic covering, i.e. a regular covering of a finite CW-complex having an infinite cyclic group of covering transformations. In this section we will denote by τ : X̃ → X̃ a fixed generator of this group.

Let K ⊂ X̃ be a compact subset such that X̃ is the union of the translates τ^{i}(K), where i ∈ Z.

Definition 5.1 We will say that a homology class z ∈ H_{q}(X̃; k) is movable to +∞ if for any integer N ∈ Z there exists a cycle in ⋃_{i>N} τ^{i}(K) representing z. Similarly, a homology class z ∈ H_{q}(X̃; k) is movable to −∞ if for any integer N ∈ Z there exists a cycle in ⋃_{i<N} τ^{i}(K) representing z.

It is clear that the above properties of homology class z ∈ H_{q}(X̃; k) do not depend on the choice of compact K.

The following Lemma gives an approximative condition of movability. Roughly, it claims: if a cycle can be moved sufficiently large distance away then it may be moved arbitrarily far away. The word “cycle”means “singular cycle with coefficients in k”.

Lemma 5.2 Let K ⊂ X̃ be a compact subset such that ⋃_{j∈Z} τ^{j}(K) coincides with X̃. Then there exists an integer N > 0 (depending on K), such that the following properties (a) and (b) hold:

(a) Let a cycle c in K be homologous in X̃ to a cycle in ⋃_{j≥N} τ^{j}(K). Then the homology class [c] ∈ H_{q}(X̃; k) is movable to +∞;

(b) Let a cycle c in K be homologous in X̃ to a cycle in ⋃_{j≤−N} τ^{j}(K). Then the homology class [c] ∈ H_{q}(X̃; k) is movable to −∞.

Proof. Denote

V_{r} = im[H_{q}(K; k) → H_{q}(X̃; k)] ∩ im[H_{q}((⋃_{j≥r} τ^{j}K); k) → H_{q}(X̃; k)].

where r = 1, 2, . . .. Then V_{1} ⊃ V_{2} ⊃ V_{3} ⊃ … is a decreasing sequence of finite-dimensional vector spaces. Hence, there exists an integer N, such that V_{N} coincides with V_{∞} = ∩_{r>0}V_{r}. Any homology class z ∈ V_{∞} is movable to +∞. Thus, this N satisfies (a). By the similar reasons, we may increase N, if necessary, such that (b) is satisfied as well. □

Lemma 5.3 Given a homology class z ∈ H_{q}(X̃; k), the following conditions are equivalent:

(i) z is movable to +∞; (ii) z is movable to −∞;

(iii) z is a torsion element of k[τ, τ^{−1}]-module H_{q}(X̃; k), i.e. there exists a nontrivial Laurent polynomial p(τ) ∈ k[τ, τ^{−1}], such that p(τ)z = 0.

Proof. We will show that (i) implies (iii) and that (iii) implies both (i) and (ii). The implication (ii) ⇒ (iii) follows similarly.

Assume that a class z ∈ H_{q}(X̃; k) is movable to +∞. Realize z by a cycle c in X̃ and specify a compact subset K ⊂ X̃ containing c and such that X̃ = ⋃_{i∈Z} τ^{i}(K). Assume that N > 0 is large enough, so that it satisfies Lemma 5.3 and the subset τ^{N}(K) is disjoint from K. Let us show that for any integer r ≥ N the homology class z may be realized by a cycle in τ^{r}(K). Since z is movable to +∞, there exists a cycle c′ in U = ⋃_{j≥2r} τ^{j}(K) representing z. Write X̃ = B ∪ C, where B contains K, C contains U and B ∩ C = τ^{r}K. In the Mayer - Vietoris sequence

H_{q}(B ∩ C; k) → H_{q}(B; k) ⊕ H_{q}(C; k) → H_{q}(X̃; k),

the difference [c] − [c′] goes to zero. Hence there is a cycle in B ∩ C = τ^{r}K, which is homologous to c in B and homologous to c′ in C. This proves our claim.

Consider

V = ⋂_{r≥N} im[H_{q}(τ^{r}K; k) → H_{q}(X̃; k)] ⊂ H_{q}(X̃; k).

It is a finite dimensional k-linear subspace; we observe that V is invariant under τ^{−1}. Hence, by the Caley - Hamilton Theorem, there exists a polynomial p(τ) ∈ k[τ], such that p(τ^{−1}) acts trivially on V. Since z belongs to V, we obtain p(τ^{−1})z = 0. This shows that (i) implies (iii).

Let us show that (iii) implies (i) and (ii). Suppose that z ∈ H_{q}(X̃; k) is such that p(τ)z = 0 for a Laurent polynomial

p(τ) = ∑_{i=r}^{r+ℓ} a_{i}τ^{i}, a_{i} ∈ k, where a_{r} ≠ 0, a_{r+ℓ} ≠ 0.

Consider the ring R = k[τ, τ^{−1}]/J, where J is the ideal generated by p(τ). The powers τ^{i}, where i = r, r + 1, …, r + ℓ − 1, form an additive basis of R. Since multiplication by τ is an automorphism of R, we obtain that for any integer N, the powers τ^{N}, τ^{N+1}, …, τ^{N+ℓ−1} form a linear basis of R as well. In particular, we may express 1 ∈ R as a linear combination of τ^{N}, τ^{N+1}, …, τ^{N+ℓ−1} in R. This means that for any N we may find numbers b_{j} ∈ k, where j = N, …, N + ℓ − 1, such that

z = ∑_{j=N}^{N+ℓ−1} b_{j}τ^{j}z in H_{q}(X̃; k).

Assume that K ⊂ X̃ is a compact subset such that X̃ = ⋃_{j∈Z} τ^{j}K and class z can be realized by a cycle c in K. Then for any integer N class z may be realized by the cycle

∑_{j=N}^{N+ℓ−1} b_{j}τ^{j}c lying in ⋃_{N≤j<N+ℓ} τ^{j}K.

Hence z is movable to both ends ±∞ of X̃. □

6 Cohomological lower bound for cat(X, ξ)

In this section we will give cohomological lower bounds on cat(X, ξ). For simplicity we assume that ξ is integral, i.e. ξ ∈ H^{1}(X; Z).

6.1 Statement of the result

Let k be a field.

Given a finite CW-complex X and an integral cohomology class ξ ∈ H^{1}(X; Z). For any nonzero a ∈ k there is a local system over X with fiber k such that the monodromy along any loop γ in X is multiplication by a^{⟨ξ,γ⟩}

a^{⟨ξ,γ⟩} : k → k, q ↦ a^{⟨ξ,γ⟩}q, for q ∈ k.

Here ⟨ξ, γ⟩ ∈ Z denotes the value of the class ξ on the loop γ. This local system will be denoted a^{ξ}. The cohomology of this local system H^{q}(X; a^{ξ}) is a vector space over k of finite dimension. Note that for a = 1 the local system a^{ξ} is the constant local system k. If a, b ∈ k^{∗} are two nonzero numbers, there is an isomorphism of local systems

a^{ξ} ⊗ b^{ξ} ≃ (ab)^{ξ}.

Hence we have well-defined cup-product pairing

∪ : H^{q}(X; a^{ξ}) ⊗ H^{q′}(X; b^{ξ}) → H^{q+q′}(X; (ab)^{ξ}).  (6.1)

The following Theorem is one of the main results of this section.

Theorem 6.1 Assume that there exist cohomology classes

u ∈ H^{q}(X; a^{ξ}), v ∈ H^{q′}(X; b^{ξ}), w_{j} ∈ H^{d_{j}}(X; k), where j = 1, …, r,

such that (i) d_{1} > 0, …, d_{r} > 0; (ii) the cup product

u ∪ v ∪ w_{1} ∪ w_{2} ∪ ⋯ ∪ w_{r} ∈ H^{d}(X; (ab)^{ξ}),  (6.2)

is nontrivial, where d = q + q′ + d_{1} + ⋯ + d_{r};

(iii) the numbers a, b ∈ k^{∗} do not belong to a finite subset Supp(X, ξ) ⊂ k^{∗} depending on the pair (X, ξ), see below.

Then cat(X, ξ) > r.

Property (iii) may be also expressed by saying that the numbers a ∈ k^{∗} and b ∈ k^{∗} are generic.

For ξ ≠ 0 and a ≠ 1 the bundle a^{ξ} admits no globally defined flat sections and so the zero-dimensional cohomology H^{0}(X; a^{ξ}) = 0 vanishes. Hence for ξ ≠ 0 the classes u and v in Theorem 6.1 must automatically have positive degrees, i.e. q > 0 and q′ > 0.

6.2 The set Supp(X, ξ)

For ξ = 0 we will define Supp(X, ξ) ⊂ k^{∗} to be the empty set.

Now we will define the set Supp(X, ξ) ⊂ k^{∗} assuming that ξ ∈ H^{1}(X; Z) is nonzero and indivisible. Let p : X̃ → X be the infinite cyclic covering corresponding to ξ. Fix a generator τ : X̃ → X̃ of the group of covering translations such that f(τx) = f(x) + 1 for all x ∈ X̃, where f : X̃ → R is a continuous function satisfying df = p^{∗}ω. Here ω is a closed 1-form on X in the cohomology class ξ. Since the translate τ : X̃ → X̃ acts on the homology, we obtain that H_{∗}(X̃; k) is a graded module over the ring Λ = k[τ, τ^{−1}] of Laurent polynomials. The ring Λ is Noetherian, hence H_{∗}(X̃; k) is finitely generated over Λ. Its Λ-torsion submodule

T = Tor_{Λ}(H_{∗}(X̃; k))

(which, according to the previous section, coincides with the set of homology classes movable to ±∞) is a finite-dimensional k-vector space. Multiplication by τ is an invertible k-linear operator τ : T → T. We will define the set Supp(X, ξ) ⊂ k^{∗} as the set of eigenvalues of τ^{−1} : T → T.

In the case when ξ = λη, where η ∈ H^{1}(X; Z) is indivisible and λ ∈ Z, λ > 0, we will define the set Supp(X, ξ) as {a ∈ k^{∗}; a^{λ} ∈ Supp(X, η)}.

Note that for k = C the set Supp(X, ξ) ⊂ C consists of finitely many algebraic numbers. Hence for k = C in Theorem 6.1 one may always take for a, b ∈ C^{∗} arbitrary transcendental numbers.

6.3 Lifting property

The proof of Theorem 6.1 is based on the following lifting property of cohomology classes:

Theorem 6.2 Let X be a finite CW-complex, ξ ∈ H^{1}(X; Z), a ∈ k^{∗}, a ∉ Supp(X, ξ). Fix a continuous closed 1-form ω in cohomology class ξ and a compact K ⊂ X, such that ξ|_{K} = 0. Then for any open subset F ⊂ K satisfying condition (b) of Definition 3.1 with respect to closed 1-form ω with integer N > 0 large enough, the homomorphism

H^{q}(X, F; a^{ξ}) → H^{q}(X; a^{ξ})  (6.3)

is an epimorphism.

Proof of Theorem 6.2. We already observed in Example 3.2 that for ξ = 0 condition (b) of Definition 3.1 may be satisfied for large N only if F = ∅. Therefore Theorem 6.2 is true for ξ = 0.

We will assume below that ξ ≠ 0. Moreover, without loss of generality we may assume that class ξ ∈ H^{1}(X; Z) is indivisible.

Let f : X̃ → R be a continuous function (unique up to a constant) satisfying df = p^{∗}ω. Then f(τx) = f(x) + 1 for all x ∈ X̃, where τ : X̃ → X̃ a generator of the group of translations. Since ξ|_{K} = 0, the compact K ⊂ X may be lifted to X̃. Fix such lift K ⊂ X̃. We will assume that the function f|_{K} assumes values in [0, c] ⊂ R for some integer c > 0; this may be always achieved by adding a constant to f.

Let N′ > 0 be the number given by Lemma 5.2 applied to a lift of K to X̃. Then any cycle in K ⊂ X̃, which is homologous in X̃ to a cycle in ⋃_{j≤−N′} τ^{j}(K), is movable to −∞, and hence (according to Lemma 5.3) represents a Λ-torsion homology class in H_{∗}(X̃; k).

It follows, that for any subset F ⊂ K, satisfying condition (b′) of subsection 3.2 with N > N′, the homomorphism H_{∗}(F̃; k) → H_{∗}(X̃; k) induced by the inclusion F̃ → X̃, takes values in Λ-torsion submodule T of H_{∗}(X̃; k). The set F̃ is a disjoint union of infinitely many copies of F and hence the homology of F̃ is H_{∗}(F̃; k) ≃ H_{∗}(F; k) ⊗_{k} Λ.

The claim that (6.3) is an epimorphism is equivalent to the claim that H^{∗}(X; a^{ξ}) → H^{∗}(F; a^{ξ}) is the zero map. Using duality between homology and cohomology we see the later is equivalent to the statement that the homomorphism H_{∗}(F; a^{−ξ}) → H_{∗}(X; a^{−ξ}), induced by the inclusion F → X on the homology of the dual local system a^{−ξ}, is zero. Since ξ|_{F} = 0, the local system a^{−ξ}|_{F} ≃ k is trivial. Hence we want to show that the homomorphism H_{∗}(F; k) → H_{∗}(X̃; a^{−ξ}) is zero. The inclusion F → X equals the composition F →^{⊂} K → X̃ →^{p} X, and we know that H_{∗}(F; k) → H_{∗}(X̃; k) takes values in the Λ-torsion submodule T.

To complete the proof it is enough to show that for a ∉ Supp(X, ξ) holds p_{∗}(T) = 0, where the homomorphism p_{∗} : H_{∗}(X̃; k) → H_{∗}(X; a^{−ξ}) is induced by the covering projection p : X̃ → X. Consider the following well-known exact sequence

⋯ → H_{i}(X̃; k) ⟶^{τ−b} H_{i}(X̃; k) →^{p_{∗}} H_{i}(X; b^{ξ}) → …,

where b = a^{−1}. The linear map τ − b : T → T is an isomorphism and hence the submodule T ⊂ H_{∗}(X̃; k) is contained in the image of τ −b and therefore p_{∗}(T) = 0. This completes the proof. □

6.4 Proof of Theorem 6.1

If ξ = 0 under the conditions of Theorem 6.1 the classical Lusternik - Schnirelman theory gives cat(X) > r +2; hence Theorem 6.1 holds for ξ = 0.

We will assume below that the class ξ ≠ 0 is nonzero and indivisible. Suppose that cat(X, ξ) ≤ r. Let us show that any cup-product (6.2)

satisfying conditions (i), (ii), (iii) of Theorem 6.1 must vanish. Since ξ is an integral class ξ ∈ H^{1}(X; Z), we may cover X by two open subsets

X = U ∪ W, such that ξ|Ū = 0, ξ|W̄ = 0.

Choose a continuous closed 1-form ω in cohomology class ξ. Our assumption cat(X, ξ) ≤ r implies that for any N > 0 there is an open cover X = F ∪ F_{1} ∪ ⋯ ∪ F_{r} satisfying properties (a) and (b) of Definition 3.1.

Each cohomology class w_{j} ∈ H^{d_{j}}(X; k) may be lifted to a relative cohomology class w̃_{j} ∈ H^{d_{j}}(X, F_{j}; k), where j = 1, …, r. This follows from the cohomological exact sequence

H^{d_{j}}(X, F_{j}; k) → H^{d_{j}}(X; k) → H^{d_{j}}(F_{j}; k),

since d_{j} > 0 and the second map vanishes as a consequence of the fact that the inclusion F_{j} → X is null-homotopic.

Assuming that N > 0 is large enough we may use the lifting property of Theorem 6.2, applied to compact K = Ū, to find a lift of the cohomology class u ∈ H^{q}(X; a^{ξ}) to a relative cohomology class ũ ∈ H^{q}(X, F ∩ U; a^{ξ}). Similarly, by Theorem 6.2 applied to the compact W̄, we may lift the class v ∈ H^{q′}(X; b^{ξ}) to a relative cohomology class ṽ ∈ H^{q′}(X, F ∩ W; a^{ξ}).

Therefore the product (6.2) is obtained from a product

ũ ∪ ṽ ∪ w̃_{1} ∪ ⋯ ∪ w̃_{r} ∈ H^{∗}(X, X; (ab)^{ξ}) = 0

(lying in the trivial group) by restricting onto X. Hence any cup-product (6.2) must vanish. □

Remark. Lemma 6.6 in [7] is incorrect as stated. As a consequence Proposition 6.5 and Corollary 6.7 of [7] are correct only with some extra conditions. The lifting property of Theorem 6.2 of the present paper replaces Corollary 6.7 of [7].

6.5 Cohomological estimate using Massey products

Let X be a finite CW-complex and let ξ ∈ H^{1}(X; Z).

Definition 6.3 A cohomology class v ∈ H^{q}(X; k) is called ξ-survivor if vanishes the cup-product v ∪ ξ = 0 and vanish all higher Massey products of the form

⟨v, \underbrace{ξ, ξ, …, ξ}_{r times}⟩ ∈ H^{q+1}(X; k)

for any r > 1.

We refer to sections 5 and 9 of paper [7], where these Massey products were described in detail.

Theorem 6.4 Let X be a finite CW-complex and let ξ ∈ H^{1}(X; Z) be indivisible. Assume that there exist cohomology classes w_{j} ∈ H^{d_{j}}(X; k) of positive degree d_{j} > 0, where j = 1, …, r, having a nonzero cup-product

0 ≠ w_{1} ∪ w_{2} ∪ ⋯ ∪ w_{r} ∈ H^{∗}(X; k).

If among the classes w_{1}, …, w_{r} at least two are ξ-survivors, then

cat(X, ξ) ≥ r − 1.

Proof. We may assume in the proof that the coefficient field k is algebraically closed; otherwise we could replace k by its algebraic closure.

Suppose that two first cohomology classes w_{1}, w_{2} are ξ-survivors. It is shown in section 9.5 of [7], that one may deform w_{1} and w_{2} to cohomology classes w′_{1} ∈ H^{d_{1}}(X; a^{ξ}) and w′_{2} ∈ H^{d_{2}}(X; a^{−ξ}), where a ∈ k is generic, such that the cup-product

0 ≠ w′_{1} ∪ w′_{2} ∪ w_{3} ∪ ⋯ ∪ w_{r} ∈ H^{∗}(X; k)

is still nonzero. Now Theorem 6.4 follows from Theorem 6.1. □

Example 6.5 Let T^{n} = S^{1} × ⋯ × S^{1} be n-dimensional torus. Fix a point x = (x_{1}, …, x_{n}) ∈ T^{n} and for i = 1, …, n consider (n − 1)-dimensional subtorus T_{i}^{n−1} ⊂ T^{n} consisting of points with i-th coordinate equal to x_{i}. Let X = T^{n}#(S^{1} × S^{n−1}) be obtained from T^{n} by adding a handle of index 1. In other words, we remove from T^{n} interiors of two small disjoint disks D_{1} and D_{2} and connect their boundaries by a tube S^{n−1} ×[0, 1]. We will assume that the disks D_{1} and D_{2} do not meet the subtori T_{i}^{n−1} for i = 1, …, n. Let ξ ∈ H^{1}(X; Z) be the cohomology class Poincaré dual to ∂D_{1}.

Each torus T_{i} ⊂ X has trivial normal bundle and hence determines a cohomology class w_{i} ∈ H^{1}(X; Z). It is clear that the cup-product

w_{1} ∪ ⋯ ∪ w_{n} ∈ H^{n}(X; Z)

is nonzero since T_{1}, …, T_{n} are mutually transversal and their intersection consists of one point. As in §5 of [7] one may show that the classes w_{i} are all ξ-survivors. Theorem 6.4 applies and gives cat(X, ξ) ≥ n − 1. By Theorem 4.1 any smooth closed 1-form in class ξ has at least n − 1 geometrically distinct zeros.

This estimate is sharp and cannot be improved. Indeed, since dim X = n Theorem 3.2 of [7] claims that there always exists a closed 1-form ω in class ξ with n − 1 zeros.

Example 6.6 Let X be bouquet X = Y ∨ S^{1} as in Example 3.5. We will assume that the cohomology class ξ ∈ H^{1}(X; Z) is such that ξ|_{Y} = 0 and ξ|_{S^{1}} ≠ 0 is the generator. We want to find the estimate given by Theorem 6.4 in this example. We have H^{q}(X; k) = H^{q}(Y; k) for q > 1 and H^{1}(X; k) = H^{1}(Y; k)⊕H^{1}(S^{1}; k). The last summand is generated by class ξ. It has trivial cup-products and Massey products with all other classes. If cohomology classes w_{j} ∈ H^{d_{j}}(Y; k), where j = 1, …, r, are such that d_{j} > 0 and

0 ≠ w_{1} ∪ ⋯ ∪ w_{r} ∈ H^{∗}(Y; k)

then Theorem 6.4 applies and gives cat(X, ξ) ≥ r − 1. In Example 3.5 we have shown that cat(X, ξ) = cat(Y) − 1. Hence we obtain cat(Y) ≥ r, which is weaker by 1 than the well-known inequality claiming that cat(Y) is greater than the cup-length of Y.

This example shows that our cohomological estimates may be slightly improved.

A Appendix: Closed 1-forms on topological spaces

Differential 1-forms are defined only for smooth manifolds. Closed 1-forms may be defined for general topological spaces, as we show in this appendix.

A continuous closed 1-form ω on a topological space X is defined as a collection {f_{U}}_{U∈U} of continuous real valued functions f_{U} : U → R, where U = {U} is an open cover of X, such that for any pair U, V ∈ U the difference

f_{U}|_{U∩V} − f_{V} |_{U∩V} : U ∩ V → R

is a locally constant function. Another such collection {g_{V} }_{V ∈V} (where V is another open cover of X) defines an equivalent closed 1-form if for any point x ∈ X there is an open neighborhood W such that for some open sets U ∈ U and V ∈ V containing W the difference f_{U}|_{W} − g_{V} |_{W} is locally constant. The set of all continuous closed 1-forms on X is a real vector space.

As an example consider an open cover U = {X} consisting of the whole space X. Then any continuous function f : X → R defines a closed 1-form on X, which is denoted df.

For two continuous functions f, g : X → R holds df = dg if and only if the difference f − g : X → R is locally constant.

One may integrate continuous closed 1-forms along continuous paths. Let ω be a continuous closed 1-form on X given by a collection of continuous real valued functions {f_{U}}_{U∈U} with respect to an open cover U of X. Let γ : [0, 1] → X be a continuous path. The line integral ∫_{γ} ω is defined as follows. Find a subdivision t_{0} = 0 < t_{1} < ⋯ < t_{N} = 1 of the interval [0, 1] such that for any i the image γ[t_{i}, t_{i+1}] is contained in a single open set U_{i} ∈ U. Then we define

∫_{γ} ω = ∑_{i=0}^{N−1} [f_{U_{i}}(γ(t_{i+1})) − f_{U_{i}}(γ(t_{i}))].  (A.1)

The standard argument shows that the integral (A.1) does not depend on the choice of the subdivision and the open cover U.

Lemma A.1 Let ω be a continuous closed 1-form on topological spaces X. For any pair of continuous paths γ, γ′ : [0, 1] → X with the common beginning γ(0) = γ′(0) and the common end points γ(1) = γ′(1), holds

∫_{γ} ω = ∫_{γ′} ω,

provided that γ and γ′ are homotopic relative to the boundary.

Proof. It is standard. □

Any closed 1-form defines homomorphism of periods

π_{1}(X, x_{0}) → R, [γ] ↦ ∫_{γ} ω ∈ R  (A.2)

given by integration of 1-form ω along closed loops γ : [0, 1] → X with γ(0) = x_{0} = γ(1).

Lemma A.2 The homomorphism of periods (A.2) is a group homomorphism.

Lemma A.3 Let X be a path connected topological space. A continuous closed 1-form ω on X equals df for a continuous function f : X → R if and only if ω defines a trivial homomorphism of periods (A.2).

Proof. If ω = df then for any path γ in X holds ∫_{γ} ω = f(q) − f(p), where q = γ(1), p = γ(0). Hence ∫_{γ} ω = 0 if γ is a closed loop.

Conversely, assume that the homomorphism of periods (A.2) is trivial. One defines a continuous function f : X → R by

f(x) = ∫_{x_{0}}^{x} ω.

Here the integration is taken over an arbitrary path connecting x_{0} to x. Assume that ω is given by a collection of continuous functions f_{U} : U → R with respect to an open cover {U} of X. Then for any two points x, y lying in the same path-connected component of U,

f(y) − f(x) = ∫_{x}^{y} ω = f_{U}(y) − f_{U}(x).

This shows that the function f − f_{U} is locally constant on U. Hence df = ω. □

Any continuous closed 1-form ω on a topological space X defines a (singular) cohomology class [ω] ∈ H^{1}(X; R). It is defined by the homomorphism of periods (A.2) viewed as an element of Hom(H_{1}(X); R) = H^{1}(X; R). As follows from the above Lemma, two continuous closed 1-forms ω and ω′ on X have the same cohomology class [ω] = [ω′] if and only if their difference ω − ω′ equals df, where f : X → R is a continuous function.

Recall that a topological space X is homologically locally connected if for every point x ∈ X and a neighborhood U of x there exists a neighborhood V of x in U such that H̃_{q}(V) → H̃_{q}(U) is trivial for all q.

Lemma A.4 Let X be a paracompact Hausdorff homologically locally connected topological space. Then any singular cohomology class ξ ∈ H^{1}(X; R) may be realized by a continuous closed 1-form on X.

Proof. Consider the following exact sequence of sheaves over X

0 → R_{X} → C_{X} → B_{X} → 0.  (A.3)

Here R_{X} denotes the sheaf of locally constant functions, C_{X} denotes the sheaf of real valued continuous functions, and B_{X} denotes the sheaf of germs of continuous functions modulo locally constant. More precisely, B_{X} is the sheaf corresponding to the presheaf U ↦ C_{X}(U)/R_{X}(U). Comparing with our definition of continuous closed 1-form, we find that the space of global sections of B_{X},

H^{0}(X; B_{X}),

coincides with the space of continuous closed 1-forms on X.

From (A.3), using that C_{X} is a fine sheaf, we obtain an exact sequence

0 → H^{0}(X; R) → H^{0}(X; C_{X}) →^{d} H^{0}(X; B_{X}) →^{[ ]} H^{1}(X; R_{X}) → 0.

Here H^{0}(X; C_{X}) = C(X) is the set of all continuous functions on X, and the map d acts by assigning to a continuous function f : X → R the closed 1-form df ∈ H^{0}(X; B_{X}). The group H^{1}(X; R_{X}) is the Čech cohomology Ȟ^{1}(X; R) and the map [ ] assigns to a closed 1-form ω its Čech cohomology class [ω] ∈ Ȟ^{1}(X; R). The natural map Ȟ^{1}(X; R) → H^{1}(X; R) is an isomorphism assuming that X is paracompact, Hausdorff, and homologically locally connected, cf. [17].

This implies our statement. □

As we have shown in the proof, the space of continuous closed 1-forms H^{0}(X; B_{X}) on a connected topological space X can be described by the following short exact sequence

0 → C(X)/R →^{d} H^{0}(X; B_{X}) →^{[ ]} Ȟ^{1}(X; R) → 0.  (A.4)

B Appendix: Gradient-Convex Neighborhoods

Let M be a smooth manifold and let f : M → R be a C^{2}-smooth function with isolated critical points. Let v be a gradient-like vector field for f. This means that v(f) > 0 on the complement of the set of critical points, and v coincides with the gradient of f on an open neighborhood of the critical points with respect to a Riemannian metric.

We will denote by M × R → M, (m, t) ↦ m · t, the flow of the field v; we will assume that it is defined for all t ∈ R.

Lemma B.1 Any open neighborhood V ⊂ M of a critical point p ∈ M, df_{p} = 0, contains a compact neighborhood U of p, such that

(1) for any point m ∈ M the set J_{m} = {t ∈ R; m · t ∈ U} ⊂ R is either empty, or a closed interval [a_{m}, b_{m}], possibly degenerated to a point, i.e. with a_{m} = b_{m},

(2) the function {m ∈ M; J_{m} ≠ ∅} → R, where m ↦ a_{m} ∈ R, is continuous.

The proof below was essentially suggested by P. Millman.

Proof. We will assume that f(p) = 0, that V̄ is compact, and that point p is the only critical point of function f in V̄. Let V_{0} be an open neighborhood of p with compact closure V̄_{0} ⊂ V and such that in V_{0} there exist local coordinates x_{1}, …, x_{n} and v|_{V_{0}} is the gradient of f with respect to a Riemannian metric g_{ij}. Fix a smooth function ψ : M → [0, 1], such that ψ|_{V_{0}} ≡ 0 and ψ|_{(M−V)} ≡ 1.

We want to show that there is a constant λ > 0 such that the derivative of the function

Ψ = ⟨gradf, gradf⟩ : M → R_{+}

along the gradient flow of f satisfies in V_{0} the inequalities:

−λΨ ≤ \frac{dΨ}{dt} ≤ λΨ.  (B.1)

In the local coordinates

Ψ = ∑_{ij} g^{ij} · \frac{∂f}{∂x_{i}} · \frac{∂f}{∂x_{j}}

and

\frac{dΨ}{dt} = ∑_{ij}⟨grad g^{ij}, gradf⟩ · \frac{∂f}{∂x_{i}} · \frac{∂f}{∂x_{j}} +

+2 ⟨gradf, ∑_{ij} g^{ij} · \frac{∂f}{∂x_{j}} · grad\frac{∂f}{∂x_{i}} ⟩.

Both the first and the second terms in this sum can be viewed as symmetric bilinear forms in the partial derivatives of f with continuous coefficients, and hence our statement (B.1) follows.

We will define now two smooth functions F_{+}, F_{−} : M → R by

F_{±} = ±2f + λ^{−1} · Ψ + ψ.  (B.2)

In V_{0} − {p} using (B.1) we have

\begin{matrix}\frac{dF_{+}}{dt} = 2Ψ + λ^{−1} · \frac{dΨ}{dt} ≥ Ψ > 0, \\ \frac{dF_{−}}{dt} = −2Ψ + λ^{−1} · \frac{dΨ}{dt} ≤ −Ψ < 0. \end{matrix}

Hence we conclude that F_{+} increases and F_{−} decreases along the gradient flow of f in V_{0} − {p}.

We set

g = max{F_{+}, F_{−}} : M → R_{+}.

It is a continuous function with g(p) = 0. For any number 0 < c small enough, the set U_{c} = {x ∈ M; g(x) < c} is an open neighborhood of p, contained in V_{0}.

Let V_{1} be an open neighborhood of p with V̄_{1} ⊂ V_{0}. Let ϵ > 0 be such that any trajectory γ(t) ∈ M of the flow of v with γ(t_{1}) ∈ V̄_{1}, and γ(t_{2}) ∈ \overline{M − V_{0}}, where t_{1} < t_{2}, satisfies

f(γ(t_{2})) − f(γ(t_{1})) > ϵ.  (B.3)

We will show that Ū_{c} satisfies the conditions of the Lemma assuming that 0 < c < ϵ and c is small enough, so that U_{c} ⊂ V_{1}. Since ∂Ū_{c} = g^{−1}(c), a trajectory γ(t) = m·t enters the set Ū_{c} at t = a if and only if F_{−}(γ(a)) = c and F_{+}(γ(a)) ≤ c. Moreover, if F_{−}(γ(a)) = c and F_{+}(γ(a)) = c, the trajectory leaves Ū_{c} immediately (i.e. the trajectory γ(t) is tangent to the boundary ∂U_{c}), and if F_{−}(γ(a)) < c, the trajectory penetrates the interior of Ū_{c}.

Similarly, if t = b is such that F_{−}(γ(b)) ≤ c and F_{+}(γ(b)) = c, the trajectory γ(t) leaves the set Ū_{c} for t > b. We know that, while γ(t) stays in V_{1}, the function F_{+} increases and so the trajectory remains away from Ū_{c}. May this trajectory come back to Ū_{c} for some large time t = c > b? If this happens, then f(γ(c)) > ϵ (because of (B.3) and using f(γ(b)) ≥ 0) and hence g(γ(c)) > ϵ.

This proves that under our assumptions on c the set {t; γ(t) ∈ Ū_{c}} coincides with the interval [a, b].

Now we are left to prove statement (2) of the Lemma. Let A ⊂ M denote the set of points m ∈ M, such that m · t belongs to V_{1} for some t. Since

\frac{dF_{−}}{dt} < 0, the equation F_{−}(m·a_{m}) = c defines a continuous function of m ∈ A.

Similarly, the equation F_{+}(m · b_{m}) = c defines a continuous function A → R, where m ↦ b_{m}. The set {m ∈ M; J_{m} ≠ ∅} equals {m ∈ A; a_{m} ≤ b_{m}}. This implies our claim (2). □

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Address:

Michael Farber,

Department of Mathematics,

Tel Aviv University, Ramat Aviv 69978, Israel farber@math.tau.ac.il