Discrete Morse theory and localization

TL;DR

This paper connects discrete Morse theory with localization of entrance path categories, showing how acyclic matchings correspond to homotopy-preserving localizations and leading to a combinatorial flow category equivalent to the original complex.

Contribution

It establishes a precise correspondence between acyclic partial matchings and localizations of entrance path categories, creating a combinatorial flow category analogous to smooth Morse theory.

Findings

Acyclic matchings correspond to homotopy-preserving localizations.

The discrete flow category is homotopy equivalent to the original CW complex.

Provides a combinatorial framework similar to smooth Morse theory.

Abstract

Incidence relations among the cells of a regular CW complex produce a poset-enriched 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) on 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.