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

الأصل · EN

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;). 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.