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

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 ∈ H1(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