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

Morse Theoretic Templates for High Dimensional Homology Computation