Multiparameter Discrete Morse Theory

TL;DR

This paper extends discrete Morse theory to vector-valued functions to facilitate the computation of multiparameter persistence, introducing new theoretical tools and results for analyzing multi-parameter topological data.

Contribution

It generalizes Forman's Morse theory to multiparameter functions, establishing new results on sublevel sets, Morse decompositions, and inequalities in the multiparameter context.

Findings

Generalizes Morse theory for vector-valued functions

Establishes a more general sublevel set result

Derives Morse inequalities for multiparameter functions

Abstract

The main objective of this paper is to extend Morse-Forman theory to vector-valued functions. This is mostly motivated by the need to develop new tools and methods to compute multiparameter persistence. To generalize the theory, in addition to adapting the main definitions and results of Forman to this vectorial setting, we use concepts of combinatorial topological dynamics studied in recent years. This approach proves to be successful in the following ways. First, we establish a result which is more general than that of Forman regarding the sublevel sets of a multidimensional discrete Morse function. Second, we find a way to induce a Morse decomposition in critical components from the critical points of such a function. Finally, we deduce a set of Morse equation and inequalities specific to the multiparameter setting.