Birth and death in discrete Morse theory. (January 2017) - King, Henry; Knudson, Kevin; Mramor Kosta, Neža
Abstract: 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 F t i : M → R . In this paper, we study the births and deaths of critical cells for the functions F t 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 t i, 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.