Glasgow Mathematical Journal

Table of Contents - 2011 - Volume 53, Issue 01  

Research Article

ON THE RELATIVE LUSTERNIK–SCHNIRELMANN CATEGORY WITH RESPECT TO A REAL COHOMOLOGY CLASS

TIEQIANG LIa1 and DIRK SCHÜTZa1

a1 Department of Mathematical Sciences, Durham University, South Road, Durham DH1 3LE, UK e-mail: tieqiang.li@durham.ac.uk, dirk.schuetz@durham.ac.uk

Abstract

In this paper, we study a homotopy invariant cat(X, B, [ω]) on a pair (X, B) of finite CW complexes with respect to the cohomology class of a continuous closed 1-form ω. This is a generalisation of a Lusternik–Schnirelmann-category-type cat(X, [ω]), developed by Farber in [3, 4], studying the topology of a closed 1-form. This paper establishes the connection with the original notion cat(X, [ω]) and obtains analogous results on critical points and homoclinic cycles. We also provide a similar ‘cuplength’ lower bound for cat(X, B, [ω]).

(Received August 10 2009)

(Revised January 22 2010)

(Accepted February 21 2010)

(Online publication August 25 2010)

2010 Mathematics Subject Classification

  • Primary: 55M30;
  • Secondary: 58E05;
  • 37C29