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Decomposition of Persistent Homology and Spectral Sequences

Peiqi Yang, Yingfeng Hu, Hao Wu

Abstract

We study the relation between the persistent homology and the spectral sequence of a filtered chain complex over a field. Our method is based on a decomposition of the persistent homology. We demonstrate that, under fairly general assumptions, these two algebraic structures capture the same information from the filtered chain complex.

Decomposition of Persistent Homology and Spectral Sequences

Abstract

We study the relation between the persistent homology and the spectral sequence of a filtered chain complex over a field. Our method is based on a decomposition of the persistent homology. We demonstrate that, under fairly general assumptions, these two algebraic structures capture the same information from the filtered chain complex.
Paper Structure (9 sections, 15 theorems, 64 equations)

This paper contains 9 sections, 15 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Statement of results
  4. Decomposition of Persistent Homology
  5. Persistent chain complexes
  6. Existence of the decomposition
  7. Uniqueness of the decomposition
  8. Spectral Sequences
  9. Relations Between $E^{(r)}$ and $PH$

Key Result

Theorem 1.4

Let $(PC, d_x)$ be the persistent chain complex of a filtered chain complex $(C,d, \mathcal{F})$ over $\mathbb{F}$ satisfying: Then, up to chain homotopy and permutation of factors, $(PC,d_x)$ can be uniquely decomposed into a direct sum of graded chain complexes of types eq:UF and eq:UT. More precisely, for each $n \in \mathbb{Z}$, there exist $K_n \in \mathbb{Z}_{+, \infty}$ and a unique sequen

Theorems & Definitions (37)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 1.6
  • Definition 1.7
  • Theorem 1.8
  • Corollary 1.9
  • Definition 2.1
  • ...and 27 more