Abstract
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 .
Citation
Michael Farber. Dirk Schütz. "Cohomological estimates for $\mathrm{cat}(X,\xi)$." Geom. Topol. 11 (3) 1255 - 1288, 2007. https://doi.org/10.2140/gt.2007.11.1255
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