Persistent Simple-homotopy invariants via discrete Morse theory
Algebraic Topology
2026-04-14 v1
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.
Keywords
Cite
@article{arxiv.2604.10192,
title = {Persistent Simple-homotopy invariants via discrete Morse theory},
author = {Divya Ahuja and Jaya NN Iyer},
journal= {arXiv preprint arXiv:2604.10192},
year = {2026}
}