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Notes on Pointwise Finite-Dimensional $2$-Parameter Persistence Modules

Wenwen Li, Murad Ozaydin

TL;DR

The paper analyzes pointwise finite-dimensional $2$-parameter persistence modules that admit a finite convex isotopy subdivision, proving such $M$ are isomorphic to a chamber-constant module $N$ with explicit inter-chamber morphisms $\phi_{J_1J_2}$ and that the subdivision yields a finite encoding of $M$. It shows that finite convex isotopy subdivisions are effectively finite constant subdivisions and provides a fully-faithful encoding via a chamber-poset construction, reducing analysis to chamberwise data. A Krull–Schmidt perspective links indecomposables of $M$ to those of $N$, enabling decomposition via these reduced representations. Finally, the authors establish a complete classification of indecomposable thin $2$-parameter modules as polytope modules, clarifying the structure in dimension two (with caveats for higher parameters).

Abstract

In this paper, we study pointwise finite-dimensional (p.f.d.) $2$-parameter persistence modules where each module admits a finite convex isotopy subdivision. We show that a p.f.d. $2$-parameter persistence module $M$ (with a finite convex isotopy subdivision) is isomorphic to a $2$-parameter persistence module $N$ where the restriction of $N$ to each chamber of the parameter space $(\mathbb{R},\leq)^2$ is a constant functor. Moreover, we show that the convex isotopy subdivision of $M$ induces a finite encoding of $M$. Finally, we prove that every indecomposable thin $2$-parameter persistence module is isomorphic to a polytope module.

Notes on Pointwise Finite-Dimensional $2$-Parameter Persistence Modules

TL;DR

The paper analyzes pointwise finite-dimensional -parameter persistence modules that admit a finite convex isotopy subdivision, proving such are isomorphic to a chamber-constant module with explicit inter-chamber morphisms and that the subdivision yields a finite encoding of . It shows that finite convex isotopy subdivisions are effectively finite constant subdivisions and provides a fully-faithful encoding via a chamber-poset construction, reducing analysis to chamberwise data. A Krull–Schmidt perspective links indecomposables of to those of , enabling decomposition via these reduced representations. Finally, the authors establish a complete classification of indecomposable thin -parameter modules as polytope modules, clarifying the structure in dimension two (with caveats for higher parameters).

Abstract

In this paper, we study pointwise finite-dimensional (p.f.d.) -parameter persistence modules where each module admits a finite convex isotopy subdivision. We show that a p.f.d. -parameter persistence module (with a finite convex isotopy subdivision) is isomorphic to a -parameter persistence module where the restriction of to each chamber of the parameter space is a constant functor. Moreover, we show that the convex isotopy subdivision of induces a finite encoding of . Finally, we prove that every indecomposable thin -parameter persistence module is isomorphic to a polytope module.
Paper Structure (5 sections, 17 theorems, 26 equations, 1 figure)

This paper contains 5 sections, 17 theorems, 26 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries: Persistence Modules Over Posets
  3. $2$-Parameter p.f.d. Persistence Modules with Convex Isotopy Subdivisions
  4. Finite Encodings of the $2$-Parameter p.f.d. Persistence Modules with Finite Convex Isotopy Subdivisions
  5. Thin Polycode Modules

Key Result

Theorem 1

Given $M\in\mathbf{\boldsymbol{\mathsf{vect}}}_{\mathbb{F}}^{(\mathbb{R},\leq)^{2} }$. Assume there exists a finite convex isotopy subdivision of $(\mathbb{R},\leq)^{2}$ subordinate to $M$. Then $M\cong N$, where $N:(\mathbb{R},\leq)^{2}\rightarrow \mathbf{\boldsymbol{\mathsf{vect}}}_{\mathbb{F}}$ i

Figures (1)

  • Figure 1: The Hasse diagram of $(P,\leq)$ associated to the hyperplane arrangement in the parameter space of $PH_0(Y^2_{r,L_{e_1}};\mathbb{F})$li2023persistent

Theorems & Definitions (39)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 1
  • proof
  • Definition 1: Isotopy Subdivision
  • Definition 2
  • Lemma 2
  • proof
  • Lemma 3
  • ...and 29 more