  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 $cat(X,\xi)$ of the notion of Lusternik - Schnirelman
category, depending on a topological space $X$ and a cohomology class $\xi\in
H^1(X;\R)$. We prove that any closed 1-form has at least $cat(X,\xi)$ zeros
assuming that it admits a gradient-like vector field with no homoclinic cycles.
We show that the number $cat(X,\xi)$ 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.
