Year: 2026
Type: preprint
Abstract: Let $G$ be a finite group. In this article, we develop some results of equivariant persistent homotopy theory for persistent $G$-spaces. We introduce the notions of $G$-stable and $G$-homotopy invariant distances and define the $G$-homotopy interleaving distance, an equivariant analogue $d^G_{HI}$ of the homotopy interleaving distance of Blumberg and Lesnick. We prove that this distance $d^G_{HI}$ is $G$-stable and $G$-homotopy invariant and establish its universality by showing that it dominates any such distance. We further prove a version of equivariant persistent Whitehead theorem relating interleavings of equivariant persistent homotopy groups to $G$-homotopy interleavings of persistent $G$-spaces. We also prove an equivariant persistent nerve lemma for equivariant good covers of persistent $G$-spaces. As a consequence, we obtain an equivariant weak law of large numbers for filtrations, providing a homotopy-theoretic consistency result for random filtrations equipped with finite group of symmetries.
Source: arXiv (Cornell University)
Authors: Halder, Kushal, Sau, Subhankar, Sen, Debasis
Institution: Indian Institute of Technology Kanpur
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214050431
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