Discrete Morse theory and Hopf bundles
Dmitry N. Kozlov · Pacific Journal of Mathematics · 2011
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.This research was