Equivariant Homotopy Interleaving and Persistent Whitehead Theorem
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 dGHI of the homotopy interleaving distance of Blumberg and Lesnick. We prove that this distance dGHI 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.
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BibTeX
@misc{halder2026equivariant,
title = {{Equivariant Homotopy Interleaving and Persistent Whitehead Theorem}},
author = {Kushal Halder and Subhankar Sau and Debasis Sen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.23440},
eprint = {2609.23440},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23440},
}
RIS
TY - UNPB
TI - Equivariant Homotopy Interleaving and Persistent Whitehead Theorem
AU - Kushal Halder
AU - Subhankar Sau
AU - Debasis Sen
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2609.23440
UR - https://arxiv.org/abs/2609.23440
AB - 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.
ER -