Discrete Morse theory for open complexes

2024/02/19 by Knudson, Kevin P., Scoville, Nicholas A. · 1 citation
#55N35 #55U10 #57Q70 #Algebraic Topology (math.AT) #FOS: Mathematics

paper · doi:10.48550/arxiv.2402.12116

Abstract

We develop a discrete Morse theory for open simplicial complexes K=X∖ T where X is a simplicial complex and T a subcomplex of X. A discrete Morse function f on K gives rise to a discrete Morse function on the order complex S_K of K, and the topology change determined by f on K can be understood by analyzing the the topology change determined by the discrete Morse function on S_K. This topology change is given by a structure theorem on the level subcomplexes of S_K. Finally, we show that the Borel-Moore homology of K, a homology theory for locally compact spaces, is isomorphic to the homology induced by a gradient vector field on K and deduce corresponding weak Morse inequalities.

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