Discrete Morse theory for open complexes
Kevin P. Knudson, Nicholas A. Scoville · Topology and its Applications · 2026
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 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. The gradient vector field on K provides a novel alternative to compute Borel-Moore homology.