Persistent Simple-homotopy invariants via discrete Morse theory

TL;DR

This paper presents a refined approach to persistent homology that captures simple-homotopy-theoretic features, introduces the Morse complexity profile, and develops invariants like persistent Whitehead torsion, enhancing topological data analysis.

Contribution

It introduces the Morse complexity profile and persistent Whitehead torsion as new invariants that detect subtle topological features beyond traditional persistent homology.

Findings

Morse complexity profile is invariant under simple-homotopy equivalence.

Persistent Whitehead torsion is invariant under interleaving and simple-homotopy.

Algorithm for computing these invariants in Vietoris-Rips filtrations is provided.

Abstract

We introduce a refinement of persistent homology that detects simple-homotopy-theoretic phenomena invisible to homology. Given a filtered simplicial complex, we define the Morse complexity profile as the minimal number of critical simplices at each filtration level. We prove that this profile is invariant under levelwise simple-homotopy equivalence and detects filtrations indistinguishable by persistent homology. We establish conditional stability under simple interleavings and provide an efficient algorithm for Vietoris-Rips filtrations. We also introduce a persistent version of Whitehead torsion and show that it is invariant under both levelwise simple-homotopy equivalence and interleaving equivalence of filtrations.