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

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>