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On Discrete Morse-Bott Theory

Yuto Nishikawa, Tomoo Yokoyama

TL;DR

This work extends discrete Morse theory to a discrete Morse-Bott setting by refining the original definition to allow critical sets and by introducing a collection-based decomposition of the CW complex. It defines r-paths, collections, and reduced collections to capture symmetries and higher-dimensional critical structures, and establishes a Morse-Bott inequality that generalizes both discrete Morse inequalities and continuous Morse-Bott inequalities. The authors prove that the framework recovers the classical discrete Morse theory as a special case when collections are small and show how the resulting Poincaré-type relationships bound Betti numbers via the reduced critical data. This provides a combinatorial toolkit for topological analysis of data with symmetry or group actions, enabling more flexible homological computations on discrete structures.

Abstract

This paper shows that discrete Morse-Bott theory can be developed in an intuitive way, which is achieved by improving the definition of the discrete Morse-Bott function originally introduced by S. Yaptieu. In fact, we demonstrate that various natural properties hold for discrete Morse-Bott functions and, in particular, establish the discrete Morse-Bott inequality, which can be regarded as an extension of both the discrete Morse inequalities and the continuous Morse-Bott inequalities.

On Discrete Morse-Bott Theory

TL;DR

This work extends discrete Morse theory to a discrete Morse-Bott setting by refining the original definition to allow critical sets and by introducing a collection-based decomposition of the CW complex. It defines r-paths, collections, and reduced collections to capture symmetries and higher-dimensional critical structures, and establishes a Morse-Bott inequality that generalizes both discrete Morse inequalities and continuous Morse-Bott inequalities. The authors prove that the framework recovers the classical discrete Morse theory as a special case when collections are small and show how the resulting Poincaré-type relationships bound Betti numbers via the reduced critical data. This provides a combinatorial toolkit for topological analysis of data with symmetry or group actions, enabling more flexible homological computations on discrete structures.

Abstract

This paper shows that discrete Morse-Bott theory can be developed in an intuitive way, which is achieved by improving the definition of the discrete Morse-Bott function originally introduced by S. Yaptieu. In fact, we demonstrate that various natural properties hold for discrete Morse-Bott functions and, in particular, establish the discrete Morse-Bott inequality, which can be regarded as an extension of both the discrete Morse inequalities and the continuous Morse-Bott inequalities.

Paper Structure

This paper contains 11 sections, 20 theorems, 73 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Fundamental concepts
  4. Fundamental properties
  5. Discrete Morse function
  6. Fundamental concepts
  7. Discrete Morse-Bott function
  8. Definition of discrete Morse-Bott function
  9. Reduced collection
  10. Generalization of a discrete Morse function
  11. Natural properties of discrete Morse-Bott functions

Key Result

Lemma 2.1

For any cells $\nu, \sigma, \tau \in \mathbb{K}$ with $\nu\overset{\mathrm{reg}}{\prec}\sigma\overset{\mathrm{reg}}{\prec}\tau$, there is a cell $\widetilde{\sigma}\ne\sigma$ such that $\nu\prec\widetilde{\sigma}\prec\tau$.

Figures (1)

  • Figure 1: An example of a CW complex without an irregular cell of one higher dimension for any zero cell

Theorems & Definitions (85)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Definition 8
  • Example 1
  • Definition 9
  • ...and 75 more