Persistent equivariant cohomology

TL;DR

This paper introduces persistent equivariant cohomology, explaining its concepts and providing explicit calculations for circle actions on Vietoris-Rips complexes, revealing how group actions influence topological features.

Contribution

It offers an accessible introduction to persistent equivariant cohomology and explicitly computes it for circle actions on Vietoris-Rips complexes using spectral sequences.

Findings

Explicit description of equivariant cohomology for circle actions

Identification of cohomology ring structure depending on scale parameter

Application of spectral sequences and Gysin homomorphism in computations

Abstract

This article has two goals. First, we hope to give an accessible introduction to persistent equivariant cohomology. Given a topological group $G$ acting on a filtered space, persistent Borel equivariant cohomology measures not only the shape of the filtration, but also attributes of the group action on the filtration, including in particular its fixed points. Second, we give an explicit description of the persistent equivariant cohomology of the circle action on the Vietoris-Rips metric thickenings of the circle, using the Serre spectral sequence and the Gysin homomorphism. Indeed, if $\frac{2\pi k}{2k+1} \le r < \frac{2\pi(k+1)}{2k+3}$, then $H^*_{S^1}(\mathrm{VR}^\mathrm{m}(S^1;r))\cong \mathbb{Z}[u]/(1\cdot3\cdot5\cdot\ldots \cdot (2k+1)\, u^{k+1})$ where $\mathrm{deg}(u)=2$.