Vol. 249, No. 2, 2011

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Discrete Morse theory and Hopf bundles

Dmitry N. Kozlov

Vol. 249 (2011), No. 2, 371–376

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.

homotopy group, fibrations, gluing map, acyclic matching, long exact sequence for homotopy
Mathematical Subject Classification 2000
Primary: 57Q05
Secondary: 54G20
Received: 12 November 2009
Accepted: 4 October 2010
Published: 1 February 2011
Dmitry N. Kozlov
Department of Mathematics
Bremen University
Bibliotheksstrasse 1
D-28334 Bremen