Discrete Morse theory and Hopf bundles

Dmitry N. Kozlov

Vol. 249 (2011), No. 2, 371–376

DOI: 10.2140/pjm.2011.249.371

Abstract

We use Hopf bundles to give an example of a regular CW complex X and an acyclic matching M on the face poset of X, such that there are no critical cells in neighboring dimensions but the complex X is not homotopy equivalent to the corresponding wedge of spheres. The key fact here is that the higher homotopy groups of spheres are nontrivial. We also give a sufficient condition on an acyclic matching M for concluding that X is homotopy equivalent to a wedge of spheres indexed by the critical cells.

Keywords

homotopy group, fibrations, gluing map, acyclic matching, long exact sequence for homotopy

Mathematical Subject Classification 2000

Primary: 57Q05

Secondary: 54G20

Milestones

Received: 12 November 2009

Accepted: 4 October 2010

Published: 1 February 2011