Morse Theoretic Templates for High Dimensional Homology Computation

TL;DR

This paper introduces templates in discrete Morse theory to efficiently compute homological invariants of high-dimensional cell complexes, demonstrating their effectiveness in various topological computations.

Contribution

It proposes the concept of templates for discrete Morse theory, enabling memory-efficient homology computations in high dimensions and complex spaces.

Findings

Successfully computed homology of cubical complexes homotopy equivalent to spheres up to dimension 20

Computed Conley complexes and connection matrices for braid space examples

Demonstrated efficiency and effectiveness of templates in high-dimensional homology calculations

Abstract

We introduce the notion of a template for discrete Morse theory. Templates provide a memory efficient approach to the computation of homological invariants (e.g., homology, persistent homology, Conley complexes) of cell complexes. We demonstrate the effectiveness of templates in two settings: first, by computing the homology of certain cubical complexes which are homotopy equivalent to $\mathbb{S}^d$ for $1\leq d \leq 20$, and second, by computing Conley complexes and connection matrices for a collection of examples arising from a Conley-Morse theory on spaces of braids diagrams.