Directed path spaces via discrete vector fields

TL;DR

This paper uses Discrete Morse Theory to construct minimal CW-complexes homotopy equivalent to spaces of directed paths in semi-cubical sets, simplifying their topological analysis.

Contribution

It introduces a method to build minimal CW-complexes for directed path spaces in semi-cubical sets using discrete vector fields.

Findings

Constructed CW-complexes are homotopy equivalent to directed path spaces.

In many cases, the CW-complex cells correspond to homology generators.

The method simplifies the topological study of directed paths in semi-cubical sets.

Abstract

Let $K$ be an arbitrary semi-cubical set that can be embedded in a standard cube. Using Discrete Morse Theory, we construct a CW-complex that is homotopy equivalent to the space $\vec{P}(K)_v^w$ of directed paths between two given vertices $v,w$ of $K$. In many cases, this construction is minimal: the cells of the constructed CW-complex are in 1--1 correspondence with the generators of the homology of $\vec{P}(K)_v^w$.