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K-Theory of Multi-parameter Persistence Modules: Additivity

Ryan E. Grady, Anna Schenfisch

Abstract

Persistence modules stratify their underlying parameter space, a quality that make persistence modules amenable to study via invariants of stratified spaces. In this article, we extend a result previously known only for one-parameter persistence modules to grid multi-parameter persistence modules. Namely, we show the $K$-theory of grid multi-parameter persistence modules is additive over strata. This is true for both standard monotone multi-parameter persistence as well as multi-parameter notions of zig-zag persistence. We compare our calculations for the specific group $K_0$ with the recent work of Botnan, Oppermann, and Oudot, highlighting and explaining the differences between our results through an explicit projection map between computed groups.

K-Theory of Multi-parameter Persistence Modules: Additivity

Abstract

Persistence modules stratify their underlying parameter space, a quality that make persistence modules amenable to study via invariants of stratified spaces. In this article, we extend a result previously known only for one-parameter persistence modules to grid multi-parameter persistence modules. Namely, we show the -theory of grid multi-parameter persistence modules is additive over strata. This is true for both standard monotone multi-parameter persistence as well as multi-parameter notions of zig-zag persistence. We compare our calculations for the specific group with the recent work of Botnan, Oppermann, and Oudot, highlighting and explaining the differences between our results through an explicit projection map between computed groups.
Paper Structure (8 sections, 10 theorems, 20 equations, 2 figures)

This paper contains 8 sections, 10 theorems, 20 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background and Conventions
  3. Stratified Spaces and Entrance Paths
  4. Grid Modules
  5. Waldhausen $K$-Theory
  6. $K$-Theory of Grid Modules
  7. Connections: Euler Manifolds and Rank Exact $K$-Theory
  8. Rank Exact $K$-Theory

Key Result

Theorem 2.3.3

Let be a standard split SES of Waldhausen categories. Then the functors $i$ and $g$ induce an equivalence of spectra

Figures (2)

  • Figure 2.1: A cubical two-manifold (shaded) appearing as a substratified space of the stratified parameter space $(\mathbb{R}^2; \{\{0, 1, \dots, 6 \},\{0,1, \dots, 4\}\})$.
  • Figure 3.1: The unfolding of a three-dimensional cube $C$ with height two in each parameter (an instance of the cube $X' \setminus \mathfrak{c}$ discussed in the induction on height in the proof of Theorem \ref{['thm:multistratadd']}). The thick blue submodule $U$ is a closed connected collection of codimension-two faces of the cube along which we can unfold (right). Since the unfolding (left) is a net, connected components of both the $C \setminus U$ and $U$ cannot be more than two-parameter modules.

Theorems & Definitions (25)

  • Definition 2.1.1
  • Definition 2.1.2
  • Definition 2.1.3
  • Definition 2.1.4
  • Definition 2.2.1
  • Definition 2.2.2: Stratified $d$-Parameter Space and its Entrance Path Category
  • Example 2.2.3: Stratified Two-Parameter Space and its Stratifying Set
  • Definition 2.2.4: Cubical Grid Manifold
  • Definition 2.3.1
  • Definition 2.3.2
  • ...and 15 more