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

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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. This paper corrects some my statements made in [ Lusternik–Schnirelman theory for closed $1$-forms , Comment. Math. Helv. 75 (2000), 156–170] and [ Topology of closed $1$-forms and their critical points, Topology 40 (2001), 235–258].