This paper studies the homotopy invariant
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
, we study the
function where
varies in the
cohomology space .
Note that
turns into the classical Lusternik–Schnirelmann category
in the
case .
Interest in
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
suggested in
[1] and [2]. The advantages of the current results are twofold: firstly, we allow cohomology
classes
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
and find upper bounds for it. We apply these upper and lower bounds
in a number of specific examples where we explicitly compute
as a function of the
cohomology class .
