Discrete Morse theory and localization

Abstract

Incidence relations among the cells of a regular CW complex produce a posetenriched category of entrance paths whose classifying space is homotopy-equivalent to that complex. We show here that each acyclic partial matching (in the sense of discrete Morse theory) of the cells corresponds precisely to a homotopy-preserving localization of the associated entrance path category. Restricting attention further to the full localized subcategory spanned by critical cells, we obtain the discrete flow category whose classifying space is also shown to lie in the homotopy class of the original CW complex. This flow category forms a combinatorial and computable counterpart to the one described by Cohen, Jones and Segal in the context of smooth Morse theory.

Morse Theory without Non-Degeneracy

“…These approaches to the Morse inequalities have been extended in different ways from classical Morse (and Morse-Smale) functions to Morse-Bott and minimally degenerate functions [5,7,13,14,36,48,81], to Novikov inequalities for closed 1-forms [15,16,17,18,60,61,62], to allow to have boundary [2,11,23,50,51] and in suitable circumstances to be non-compact or infinite-dimensional [1,4,15,16,17,18,20,28,30,32,33,34,37,65]; some approaches using stratifications do not require to be a manifold [38,63,76,80], and discrete versions of Morse theory (cf. [35,49,57,58]) have also been studied. For the main results in this article we will assume that is a (finite-dimensional) compact Riemannian manifold without boundary, though we will briefly consider other situations.…”

Morse Theory for Complexes of Groups

“…i.e., the canonical functor from the entrance path category to its localisation about Σ. Here is the main result of [17].…”

Morse theory without nondegeneracy

“…for example [38] and more recently [74]) and discrete versions of Morse theory (cf. [35,49,56,57]) have also been studied.…”