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The space of discrete Morse functions

Julian Brüggemann

TL;DR

For a finite CW complex $X$, this work constructs the space $\mathcal{M}(X)$ of discrete Morse functions as the union of Morse regions in the Morse arrangement $\mathcal{A}(X)\subset \mathbb{R}^X$, providing a combinatorial-geometric framework for parametric filtrations via the face poset $F(X)$. It develops three linked constructs—the spaces of discrete Morse matchings, merge trees, and barcodes—each given by corresponding poset-types and encodings, and introduces a path metric $d$ on $\mathcal{M}(X)$ that accounts for hyperplane-crossing transitions. The paper establishes fundamental properties: $\mathcal{M}(X)$ is contractible (indeed a star domain with $p_\sigma=\dim(\sigma)$), proves density of Morse–Benedetti subspaces, and proves Lipschitz continuity of maps from discrete Morse functions to merge trees and barcodes. It forges deep connections with smooth Morse theory via Cerf theory, including a discrete Cerf framework, and develops a comprehensive machinery to relate spaces of discrete Morse functions across morphisms, collapses, and extensions, with potential implications for stability and inverse persistence problems. Overall, the work lays a unifying, geometrically flavored foundation for studying discrete Morse functions and their filtrations through moduli-space–style constructions and interrelated combinatorial-geometric objects.

Abstract

In this work, we introduce a combinatorial-geometric model for the space of discrete Morse functions on any CW complex $X$. We relate this version of a space of discrete Morse functions to the space of cellular filtrations of $X$ and discuss its relationship to various concepts such as smooth Morse theory, Cerf theory, complexes of discrete Morse matchings, and induced merge trees and barcodes.

The space of discrete Morse functions

TL;DR

For a finite CW complex , this work constructs the space of discrete Morse functions as the union of Morse regions in the Morse arrangement , providing a combinatorial-geometric framework for parametric filtrations via the face poset . It develops three linked constructs—the spaces of discrete Morse matchings, merge trees, and barcodes—each given by corresponding poset-types and encodings, and introduces a path metric on that accounts for hyperplane-crossing transitions. The paper establishes fundamental properties: is contractible (indeed a star domain with ), proves density of Morse–Benedetti subspaces, and proves Lipschitz continuity of maps from discrete Morse functions to merge trees and barcodes. It forges deep connections with smooth Morse theory via Cerf theory, including a discrete Cerf framework, and develops a comprehensive machinery to relate spaces of discrete Morse functions across morphisms, collapses, and extensions, with potential implications for stability and inverse persistence problems. Overall, the work lays a unifying, geometrically flavored foundation for studying discrete Morse functions and their filtrations through moduli-space–style constructions and interrelated combinatorial-geometric objects.

Abstract

In this work, we introduce a combinatorial-geometric model for the space of discrete Morse functions on any CW complex . We relate this version of a space of discrete Morse functions to the space of cellular filtrations of and discuss its relationship to various concepts such as smooth Morse theory, Cerf theory, complexes of discrete Morse matchings, and induced merge trees and barcodes.
Paper Structure (11 sections, 32 theorems, 18 equations, 7 figures)

This paper contains 11 sections, 32 theorems, 18 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Hyperplane Arrangements and Their Associated Spaces
  3. The Space of Discrete Morse Functions
  4. The path metric on spaces of discrete Morse functions
  5. Relationship to other concepts in mathematics
  6. Smooth Morse Theory and Cerf Theory
  7. Discrete Cerf theory
  8. The space of discrete Morse matchings
  9. The space of merge trees
  10. The space of barcodes
  11. Future directions

Key Result

Theorem 1

It follows from maintheosmooth that Cerf's map from CerfLemma extends to a map $\eta \colon \mathcal{N}\rightarrow \mathcal{M}(M_{\mathcal{N}})$, where $M_{\mathcal{N}}$ denotes the CW decomposition of $M$ induced by any Morse function in the path component $\mathcal{N}$ of $M$. Moreover, Cerf's pro

Figures (7)

  • Figure 1: An example of a merge tree induced by a height function on a manifold and the corresponding poset of merge tree type.
  • Figure 2: Three different dMfs that induce the same matching (indicated in red). The one in the middle is weakly Morse--Benedetti but not Morse--Benedetti. The one on the right is Morse--Benedetti.
  • Figure 3: The Morse arrangement in the space of discrete functions on $I$
  • Figure 4: A simplicial map between regular CW complexes
  • Figure 5: Comparison between smooth and discrete Cerf theory
  • ...and 2 more figures

Theorems & Definitions (134)

  • Theorem 1: \ref{['CorCerf']}
  • Proposition 1: \ref{['PropSpaceOfMorseMatchings']}
  • Proposition 2: \ref{['PropEuclideanEditDistanceIsMetric']}
  • Theorem 2: \ref{['TheoInducedMergeTreeIsContinuous']}
  • Proposition 3: \ref{['PropMapInducedBarcode']}
  • Definition 2.1: compare OrMa and AguMaha
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • ...and 124 more