Morse Theoretic Templates for High Dimensional Homology Computation

2021/05/20 by Shaun Harker, Harker, Shaun, Konstantin Mischaikow +3 · 1 citation
Computer Science · Mathematics · Medicine · #Advanced Neuroimaging Techniques and Applications #Algebraic Topology (math.AT) #Dynamical Systems (math.DS) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.2105.09870

openalex publication_date 2021/05/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 \mathbbS^d for 1≤ d ≤ 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.

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