Birth and death in discrete Morse theory

Suppose M is a finite cell decomposition of a space X and that for 0 = t(0) < t(1) < ... < t(r) = 1 we have a discrete Morse function Ft(i), :M -> R It In this paper, we study the births and deaths of critical cells for the functions Ft(i), and present an algorithm for pairing the cells that occur in adjacent slices. We first study the case where the cell decomposition of X is the same for each and then generalize to the case where they may differ. This has potential applications in topological data analysis, where one has function values at a sample of points in some region in space at several different times or at different levels in an object. (C) 2016 Elsevier Ltd. All rights reserved.