Cohomological estimates for $\mathrm{cat}(X,\xi)$

Abstract

This paper studies the homotopy invariant $cat\left(X,\xi \right)$ 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 $\xi \mapsto cat\left(X,\xi \right)$ where $\xi$ varies in the cohomology space $H^1\left(X;R\right)$. Note that $cat\left(X,\xi \right)$ turns into the classical Lusternik–Schnirelmann category $cat\left(X\right)$ in the case $\xi =0$. Interest in $cat\left(X,\xi \right)$ 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 $cat\left(X,\xi \right)$ suggested in [1] and [2]. The advantages of the current results are twofold: firstly, we allow cohomology classes $\xi$ 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 $cat\left(X,\xi \right)$ and find upper bounds for it. We apply these upper and lower bounds in a number of specific examples where we explicitly compute $cat\left(X,\xi \right)$ as a function of the cohomology class $\xi \in H^1\left(X;R\right)$.