Morse matchings on polytopes

TL;DR

This paper develops a method using discrete Morse theory to construct homology bases for CW complexes, applying it to subcomplexes of the half cube polytope and providing explicit acyclic matchings.

Contribution

It introduces a novel approach for constructing homology bases via Morse matchings on polytopes, with explicit constructions for the half cube.

Findings

Explicit acyclic Morse matchings on the half cube face lattice

Homology bases constructed for certain CW complexes

Potential applicability to other symmetric polytopes

Abstract

We show how to construct homology bases for certain CW complexes in terms of discrete Morse theory and cellular homology. We apply this technique to study certain subcomplexes of the half cube polytope studied in previous works. This involves constructing explicit complete acyclic Morse matchings on the face lattice of the half cube; this procedure may be of independent interest for other highly symmetric polytopes.