On Discrete Morse-Bott Theory

TL;DR

This paper extends discrete Morse theory to Morse-Bott theory, introducing a refined definition of critical sets and establishing inequalities that unify discrete and continuous Morse theories.

Contribution

It develops a natural extension of discrete Morse theory to Morse-Bott theory, improving the critical cell structure and proving related inequalities.

Findings

Extended the definition of critical sets to include critical sets

Established discrete Morse-Bott inequalities

Unified discrete and continuous Morse theories

Abstract

This paper shows that discrete Morse-Bott theory can be developed as a natural extension of R. Forman's discrete Morse theory by improving the definition of the discrete Morse-Bott function introduced by S. Yaptieu. To this end, we demonstrate that the combinatorial structure of critical cells can be extended to critical sets intuitively. Furthermore, we establish the discrete Morse-Bott inequalities, providing a unified view that extends both the discrete Morse inequalities and the continuous Morse-Bott inequalities.