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Inspired by the works of Forman on discrete Morse theory, which is acombinatorial adaptation to cell complexes of classical Morse theory onmanifolds, we introduce a discrete analogue of the stratified Morse theory ofGoresky and MacPherson. We describe the basics of this theory and provefundamental theorems relating the topology of a general simplicial complex withthe critical simplices of a discrete stratified Morse function on the complex.We also provide an algorithm that constructs a discrete stratified Morsefunction out of an arbitrary function defined on a finite simplicial complex;this is different from simply constructing a discrete Morse function on such acomplex. We then give simple examples to convey the utility of our theory.Finally, we relate our theory with the classical stratified Morse theory interms of triangulated Whitney stratified spaces.

Discrete Stratified Morse Theory: Algorithms and A User's Guide