Volume 11, issue 3 (2007)

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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Cohomological estimates for $\mathrm{cat}(X,\xi)$

Michael Farber and Dirk Schütz

Geometry & Topology 11 (2007) 1255–1288

arXiv: math.AT/0609005

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 ξcat(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).

Keywords
Lusternik–Schnirelmann theory, closed 1-form, cup-length
Mathematical Subject Classification 2000
Primary: 58E05
Secondary: 55N25, 55U99
References
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Publication
Received: 15 November 2005
Accepted: 8 May 2007
Published: 20 June 2007
Proposed: Steve Ferry
Seconded: Walter Neumann, Wolfgang Lueck
Authors
Michael Farber
Department of Mathematics
University of Durham
Durham DH1 3LE
UK
http://maths.dur.ac.uk/~dma0mf/
Dirk Schütz
Department of Mathematics
University of Durham
Durham DH1 3LE
UK
http://maths.dur.ac.uk/~dma0ds/