Filtered Topology and Persistence in Stable Homotopy

TL;DR

This paper develops a filtered homotopy theory framework, introducing filtered invariants and categories, and connects these to persistence modules, K-theory, and stable homotopy, advancing the mathematical understanding of persistent topological structures.

Contribution

It introduces a filtered homotopy theoretic framework, including a filtered Euler characteristic and persistence categories, linking stable homotopy and persistence modules in a novel way.

Findings

Filtered homotopy invariants are established.

The K-group is isomorphic to Novikov polynomials.

Filtered stable homotopy categories relate to persistence homologies.

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.