Discrete Morse theory and classifying spaces

TL;DR

This paper refines Forman's discrete Morse theory by introducing flow paths and a 2-category framework, enabling a combinatorial reconstruction of the homotopy type of CW complexes.

Contribution

It develops a new combinatorial approach using flow paths and 2-categories to recover the homotopy type, extending Forman's discrete Morse theory.

Findings

Classifying space of the 2-category is homotopy equivalent to the original complex.

Flow paths contain sufficient information to reconstruct homotopy types.

Provides a combinatorial description of homotopy equivalences in discrete Morse theory.

Abstract

The aim of this paper is to develop a refinement of Forman's discrete Morse theory. To an acyclic partial matching $\mu$ on a finite regular CW complex $X$, Forman introduced a discrete analogue of gradient flows. Although Forman's gradient flow has been proved to be useful in practical computations of homology groups, it is not sufficient to recover the homotopy type of $X$. Forman also proved the existence of a CW complex which is homotopy equivalent to $X$ and whose cells are in one-to-one correspondence with the critical cells of $\mu$, but the construction is ad hoc and does not have a combinatorial description. By relaxing the definition of Forman's gradient flows, we introduce the notion of flow paths, which contains enough information to reconstruct the homotopy type of $X$, while retaining a combinatorial description. The critical difference from Forman's gradient flows is the existence of a partial order on the set of flow paths, from which a $2$-category $C(\mu)$ is constructed. It is shown that the classifying space of $C(\mu)$ is homotopy equivalent to $X$ by using homotopy theory of $2$-categories. This result can be also regarded as a discrete analogue of the unpublished work of Cohen, Jones, and Segal on Morse theory in early 90's.