Cohomological estimates for cat(X,xi).

Abstract

This paper studies the homotopy invariant cat(X,ξ) introduced in [1: Michael Farber, `Zeros of closed 1-forms, homoclinic orbits and Lusternik–Schnirelman theory', Topol. Methods Nonlinear Anal. 19 (2002) 123–152]. Given a finite cell-complex X, we study the function ξ→cat(X,ξ) where ξ varies in the cohomology space H1(X;R). Note that cat(X,ξ) turns into the classical Lusternik–Schnirelmann category cat(X) in the case ξ=0. Interest in cat(X,ξ) is based on its applications in dynamics where it enters estimates of complexity of the chain recurrent set of a flow admitting Lyapunov closed 1–forms, see [1] and [2: Michael Farber, ‘Topology of closed one-forms’, Mathematical Surveys and Monographs 108 (2004)]. In this paper we significantly improve earlier cohomological lower bounds for cat(X,ξ) suggested in [1] and [2]. The advantages of the current results are twofold: firstly, we allow cohomology classes ξ of arbitrary rank (while in [1] the case of rank one classes was studied), and secondly, the theorems of the present paper are based on a different principle and give slightly better estimates even in the case of rank one classes. We introduce in this paper a new controlled version of cat(X,ξ) and find upper bounds for it. We apply these upper and lower bounds in a number of specific examples where we explicitly compute cat(X,ξ) as a function of the cohomology class ξ∈ H1(X;R).