Morse Matchings on a Hypersimplex

TL;DR

This paper introduces a new family of acyclic Morse matchings on the face lattice of hypersimplices, providing tools for classifying subcomplexes with specific homological properties.

Contribution

It presents a novel construction of Morse matchings on hypersimplices, enabling future classification of subcomplexes based on homology.

Findings

Complete acyclic Morse matchings on hypersimplex face lattice

Framework for classifying subcomplexes with homology in a single degree

Foundation for describing homology bases of these subcomplexes

Abstract

We present a family of complete acyclic Morse matchings on the face lattice of a hypersimplex. Since a hypersimplex is a convex polytope, there is a natural way to form a CW complex from its faces. In a future paper we will utilize these matchings to classify every subcomplex whose reduced homology groups are concentrated in a single degree and describe a homology basis for each of them.