Discrete Stratified Morse Theory: A User's Guide

Inspired by the works of Forman on discrete Morse theory, which is a combinatorial adaptation to cell complexes of classical Morse theory on manifolds, we introduce a discrete analogue of the stratified Morse theory of Goresky and MacPherson [17]. We describe the basics of this theory and prove fundamental theorems relating the topology of a general simplicial complex with the critical simplices of a discrete stratified Morse function on the complex. We also provide an algorithm that constructs a discrete stratified Morse function out of an arbitrary function defined on a finite simplicial complex; this is different from simply constructing a discrete Morse function on such a complex. We borrow Forman’s idea of a “user’s guide,” where we give simple examples to convey the utility of our theory. ∗E-mail: [email protected]. †E-mail: [email protected]. ar X iv :1 80 1. 03 18 3v 1 [ cs .C G ] 9 J an 2 01 8