A Guide to Equivariant Parametrized Cohomology

TL;DR

This paper provides a comprehensive guide to equivariant parametrized cellular cohomology, extending existing theories to new group actions, explaining key constructions, and presenting new computations for spaces with finite group actions.

Contribution

It offers a detailed explanation of the theory for finite groups, especially cyclic groups, and includes new explicit computations demonstrating the theory's application.

Findings

RO(Π B) is not always free

Equivariant parametrized cohomology aligns with cellular cohomology in local coefficients when G is trivial

New computations for spaces with G = C_2 and G = C_4

Abstract

This article investigates equivariant parametrized cellular cohomology, a cohomology theory introduced by Costenoble-Waner for spaces with an action by a compact Lie group $G$. The theory extends the $RO(G)$-graded cohomology of a $G$-space $B$ to a cohomology graded by $RO(\Pi B)$, the representations of the equivariant fundamental groupoid of $B$. This paper is meant to serve as a guide to this theory and contains some new computations. We explain the key ingredients for defining parametrized cellular cohomology when $G$ is a finite group, with particular attention to the case of the cyclic group $G=C_2$. We compute some examples and observe that $RO(\Pi B)$ is not always free. When $G$ is the trivial group, we explain how to identify equivariant parametrized cellular cohomology with cellular cohomology in local coefficients. Finally, we illustrate the theory with some new computations of parametrized cellular cohomology for several spaces with $G = C_2$ and $G=C_4$.