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Filtered Topology and Persistence in Stable Homotopy

John Miller

TL;DR

This work develops a rigorous filtered-topology framework that integrates persistence with homotopy theory. By defining the category $\mathcal{F}\text{Top}_*$ and proving a filtered CW-approximation alongside a weighted Euler characteristic valued in $\Lambda_P$, it establishes filtered-homotopy invariants and algebraic tools for comparing filtrations. It then constructs a persistence Spanier-Whitehead category $\mathcal{P}\text{SW}$, proves its triangulated structure, and introduces fragmentation metrics that quantify filtration complexity via cone weights. The authors show a ring isomorphism $K(\mathcal{P}\text{SW}_0) \cong \Lambda_P$ induced by the weighted Euler characteristic, and outline extensions to filtered spectra and generalized persistence homology theories, linking to Novikov polynomials and stable/persistence homologies. The framework opens a path to persistent stable homotopy theory with potential applications to filtered invariants and algebraic K-theory in a persistence setting.

Abstract

We define a category of filtered topological spaces and explore some of its homotopy theoretic properties, including a filtered analogue of CW approximation. With this, we define and study a filtered (weighted) variant of the Euler characteristic and show this is a `filtered homotopy invariant'. We then go on to use the recent work of Biran, Cornea and Zhang by considering a persistence Spanier-Whitehead category of filtered CW complexes and show this is a triangulated persistence category and discuss the fragmentation metrics induced by this structure. We go on to show that the K-group of this persistence category is isomorphic to the ring of Novikov polynomials and this isomorphism is induced by the weighted Euler characteristic. Finally, we discuss how these constructions extend to a filtered stable homotopy category and its relation to filtered/persistence homologies.

Filtered Topology and Persistence in Stable Homotopy

TL;DR

This work develops a rigorous filtered-topology framework that integrates persistence with homotopy theory. By defining the category and proving a filtered CW-approximation alongside a weighted Euler characteristic valued in , it establishes filtered-homotopy invariants and algebraic tools for comparing filtrations. It then constructs a persistence Spanier-Whitehead category , proves its triangulated structure, and introduces fragmentation metrics that quantify filtration complexity via cone weights. The authors show a ring isomorphism induced by the weighted Euler characteristic, and outline extensions to filtered spectra and generalized persistence homology theories, linking to Novikov polynomials and stable/persistence homologies. The framework opens a path to persistent stable homotopy theory with potential applications to filtered invariants and algebraic K-theory in a persistence setting.

Abstract

We define a category of filtered topological spaces and explore some of its homotopy theoretic properties, including a filtered analogue of CW approximation. With this, we define and study a filtered (weighted) variant of the Euler characteristic and show this is a `filtered homotopy invariant'. We then go on to use the recent work of Biran, Cornea and Zhang by considering a persistence Spanier-Whitehead category of filtered CW complexes and show this is a triangulated persistence category and discuss the fragmentation metrics induced by this structure. We go on to show that the K-group of this persistence category is isomorphic to the ring of Novikov polynomials and this isomorphism is induced by the weighted Euler characteristic. Finally, we discuss how these constructions extend to a filtered stable homotopy category and its relation to filtered/persistence homologies.
Paper Structure (14 sections, 32 theorems, 182 equations, 3 figures)

This paper contains 14 sections, 32 theorems, 182 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Review of persistence theory
  3. Filtered topological spaces
  4. Definitions
  5. Products and Wedges
  6. Homotopy
  7. Filtered CW-complexes and weighted Euler characteristic
  8. A filtered loop-suspension adjunction
  9. Filtered stable homotopy
  10. A persistence Spanier-Whitehead category
  11. Strict exact triangles
  12. Fragmentation distances and sizes
  13. K-group
  14. Filtered Spectra

Key Result

Lemma 1.1

Given any filtered space $X\in \mathcal{F} \text{Top}_*$ there exists a filtered CW complex $\bar{X}$ and a map $f: \bar{X} \to X$ of shift zero, such that $f(r): \bar{X}( r) \to X(r)$ is a weak equivalence for all $r$, and moreover the following homotopy commutes for all $r \leq s$\begin{tikzcd}

Figures (3)

  • Figure 1: Diagrams representing the naive product $X \times Y$ (left) and the product $X\bar{\times}Y$ (right).
  • Figure 2: A depiction of $(S^1_0 \bar{\times} X)(r)$ showing the eternal subcomplex $S^1 \vee X$.
  • Figure 3: A generic picture of $T(r)$ in blue for various values of $r$ with a deformation retract to a corresponding CW complex coloured in red/orange (left). Along with a depiction of a filtered CW complex with cell weights labeled which is filtered weak equivalent (right).

Theorems & Definitions (105)

  • Lemma 1.1: Filtered CW approximation
  • Lemma 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Proposition 1.6
  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Definition 2.4
  • ...and 95 more