Birth and death in discrete Morse theory

SupposeM is a finite simplicial complex and that for 0 = t0, t1, . . . , tr = 1 we have a discrete Morse function Fti : M → R. In this paper, we study the births and deaths of critical cells for the functions Fti and present an algorithm for pairing the cells that occur in adjacent slices. We first study the case where the triangulation of M is the same for each ti, and then generalize to the case where the triangulations may differ. This has potential applications in data imaging, 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.