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 cohomology class ξ∈H1(X;R). We prove that any closed 1-form has at least cat(X,ξ) zeros
assuming that it 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
statements made in my previous papers on this subject.Comment: 34 pages. A refernce adde