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Cofiltrations of spanning trees in multiparameter persistent homology

Fritz Grimpen, Anastasios Stefanou

TL;DR

The paper tackles the problem of generating multiparameter persistent homology features with coefficients in a PID by leveraging spanning-tree concepts. It develops a framework where a cofiltration of order-minimal spanning trees yields an upper-set module epimorphic cover of the first cycle module $Z_1(X)$ and thus of $H_1(X)$ for a finite multifiltration $X$. Key contributions include the existence of such cofiltrations (Theorem main) and a higher-dimensional generalization to $n$-spanning complexes, establishing a path toward generating higher-dimensional invariants. The approach combines algebraic topology, order-theoretic constructions, and persistence-module theory to provide concrete generators and a structural decomposition that can aid computations in multiparameter settings with arbitrary PID coefficients.

Abstract

Given a multiparameter filtration of simplicial complexes, we consider the problem of explicitly constructing generators for the multipersistent homology groups with arbitrary PID coefficients. We propose the use of spanning trees as a tool to identify such generators by introducing a condition for persistent spanning trees, which is accompanied by an existence result for cofiltrations consisting of spanning trees. We also introduce a generalization of spanning trees, called spanning complexes, for dimensions higher than one, and we establish their existence as a first step towards this direction.

Cofiltrations of spanning trees in multiparameter persistent homology

TL;DR

The paper tackles the problem of generating multiparameter persistent homology features with coefficients in a PID by leveraging spanning-tree concepts. It develops a framework where a cofiltration of order-minimal spanning trees yields an upper-set module epimorphic cover of the first cycle module and thus of for a finite multifiltration . Key contributions include the existence of such cofiltrations (Theorem main) and a higher-dimensional generalization to -spanning complexes, establishing a path toward generating higher-dimensional invariants. The approach combines algebraic topology, order-theoretic constructions, and persistence-module theory to provide concrete generators and a structural decomposition that can aid computations in multiparameter settings with arbitrary PID coefficients.

Abstract

Given a multiparameter filtration of simplicial complexes, we consider the problem of explicitly constructing generators for the multipersistent homology groups with arbitrary PID coefficients. We propose the use of spanning trees as a tool to identify such generators by introducing a condition for persistent spanning trees, which is accompanied by an existence result for cofiltrations consisting of spanning trees. We also introduce a generalization of spanning trees, called spanning complexes, for dimensions higher than one, and we establish their existence as a first step towards this direction.
Paper Structure (23 sections, 21 theorems, 25 equations, 3 figures)

This paper contains 23 sections, 21 theorems, 25 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Multiparameter persistence
  3. Minimal presentations
  4. Related work
  5. Motivation
  6. Our contributions
  7. Organization
  8. Preliminaries
  9. Simplicial complexes and simplicial homology
  10. Ordered simplicial complexes
  11. Persistence modules and persistent homology
  12. Spanning trees of simplicial complexes
  13. Homological spanning trees
  14. Operations on spanning trees
  15. Homology of spanning trees
  16. ...and 8 more sections

Key Result

Theorem 1.1

If $X$ is a 1-dimensional simplicial complex and $T$ a spanning tree of $X$, then $H_1(X) \cong C_1(X, T)$.

Figures (3)

  • Figure 1: The relations of the lexicographic order on $2^X$ for $X = \{ 1 < 2 < 3 \}$, where the entry "$\geq$" denotes "$\text{row label} \geq \text{column label}$" and the entry "$\leq$" denotes "$\text{row label} \leq \text{column label}$".
  • Figure 2: The spanning trees $T_1$, $T_2$, and $T_3$ of the triangle complex $F_3$. The edges not contained in the spanning trees are indicated as dashed lines.
  • Figure 3: A $3\times 3$-filtration of simplicial complexes, which does not admit a subfiltration of spanning trees.

Theorems & Definitions (71)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • Remark 2.6
  • ...and 61 more