Discrete Stratified Morse Theory: Algorithms and A User's Guide
Abstract
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. 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 then give simple examples to convey the utility of our theory. Finally, we relate our theory with the classical stratified Morse theory in terms of triangulated Whitney stratified spaces.
 Publication:

arXiv eprints
 Pub Date:
 January 2018
 arXiv:
 arXiv:1801.03183
 Bibcode:
 2018arXiv180103183K
 Keywords:

 Computer Science  Computational Geometry;
 Mathematics  Algebraic Topology
 EPrint:
 Full and updated version of an extended abstract previously published at International Symposium on Computational Geometry (SOCG), 2018