Connecting Discrete Morse Theory and Persistence: Wrap Complexes and Lexicographic Optimal Cycles

We study the connection between discrete Morse theory and persistent homology in the context of shape reconstruction methods. Specifically, we compare the Wrap complex, introduced by Edelsbrunner as a subcomplex of the Delaunay complex, and the lexicographic optimal homologous chains, considered by Cohen-Steiner, Lieutier, and Vuillamy. We show that given any cycle in a Delaunay complex at some radius threshold, the lexicographically optimal homologous cycle is supported on the Wrap complex at the same threshold, thereby establishing a close connection between the two methods. This result is obtained as a consequence of a general connection between reduction of cycles in the computation of persistent homology and gradient flows in the algebraic generalization of discrete Morse theory, which is of independent interest.