Zeros of closed 1-forms, homoclinic orbits and Lusternik-Schnirelman theory - Farber, Michael (Topological Methods in Nonlinear Analysis, 2002)
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 the Lusternik-Schnirelman category, depending on a topological space X and a 1-dimensional real cohomology class ξH^1(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. < p> This paper corrects some my statements made in [< i> Lusternik–Schnirelman theory for closed 1-forms< /i> , Comment. Math. Helv. < b> 75< /b> (2000), 156–170] and [< i> Topology of closed 1-forms and their critical points, Topology < b> 40< /b> (2001), 235–258].< /p>