Towards Discrete Stratified Morse Theory

Forman's discrete Morse theory is a combinatorial adaptation of classical Morse theory on manifolds to cell complexes. 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. Furthermore, we relate our theory with the classical stratified Morse theory in terms of triangulated Whitney stratified spaces. This is a joint work with Kevin Knudson. If time permits, we will discuss some recent efforts in expanding the above theory.