Francesca Tombari: Stratified Morse theory for cell complexes

Abstract.

Morse theory has been adapted to the cellular and stratified settings by Forman and Goresky-MacPherson, respectively. While the first takes advantage of the combinatorial nature of regular CW complexes, the second exhibits the local structure of an attachment geometrically as a product of a tangential part and a normal part which respect the underlying stratification. Our goal in this talk is to propose a new approach to stratified discrete Morse theory.
Our first novelty is to restrict the category of Forman-critical cells for a stratified discrete Morse function, excluding some of those which certainly do not correspond to changes in sublevel set homology. We are then able to prove the standard Theorems A and B of Morse theory, which respectively establish that the topology only changes when crossing critical values. This approach does not retain the desired local representation of Morse data into a tangential and a normal part. We finally describe how to restore this product structure by passing to the (stratified) barycentric subdivision (Joint work with Vidit Nanda).