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

DOI: http://dx.doi.org/10.12775/TMNA.2002.008

#### 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 ${\rm cat}(X,\xi)$ of the notion of the

Lusternik-Schnirelman category,

depending on a topological space $X$ and a $1$-dimensional real cohomology

class $\xi\in H^1(X;\mathbb R)$.

We prove that any closed $1$-form $\omega$ in class $\xi$

has at least ${\rm cat}(X,\xi)$ zeros assuming that $\omega$ admits a gradient-like

vector field with no homoclinic cycles.

We show that the number ${\rm cat}(X,\xi)$ can be estimated from below in terms

of the cup-products and higher

Massey products.

< p> This paper corrects some my statements made in

[< i> Lusternik–Schnirelman theory for closed $1$-forms< /i> , Comment. Math. Helv. < b> 75< /b>

(2000), 156–170] and [< i> Topology of closed $1$-forms and their critical points,

Topology < b> 40< /b> (2001), 235–258].< /p>

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 ${\rm cat}(X,\xi)$ of the notion of the

Lusternik-Schnirelman category,

depending on a topological space $X$ and a $1$-dimensional real cohomology

class $\xi\in H^1(X;\mathbb R)$.

We prove that any closed $1$-form $\omega$ in class $\xi$

has at least ${\rm cat}(X,\xi)$ zeros assuming that $\omega$ admits a gradient-like

vector field with no homoclinic cycles.

We show that the number ${\rm cat}(X,\xi)$ can be estimated from below in terms

of the cup-products and higher

Massey products.

< p> This paper corrects some my statements made in

[< i> Lusternik–Schnirelman theory for closed $1$-forms< /i> , Comment. Math. Helv. < b> 75< /b>

(2000), 156–170] and [< i> Topology of closed $1$-forms and their critical points,

Topology < b> 40< /b> (2001), 235–258].< /p>

#### Keywords

Morse theory; Lusternik-Schnirelman theory; closed 1-form; Massey products; homoclinic orbits

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