The Morse Theory of Čech and Delaunay Filtrations
Ulrich Bauer, Herbert Edelsbrunner · 2014
Given a finite set of points in Rn and a positive radius, we study the Čech, Delaunay--Čech, alpha, and wrap complexes as instances of a generalized discrete Morse theory. We prove that the latter three complexes are simple-homotopy equivalent. Our results have applications in topological data analysis and in the reconstruction of shapes from sampled data.