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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 in H1(X;R).
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