A genuine equivariant recognition principle for finite groups

TL;DR

This paper establishes a comprehensive equivariant recognition principle for finite groups, generalizing previous results by removing the trivial summand assumption, and proves an equivariant approximation theorem for all representations and spaces.

Contribution

It proves that an $ ext{E}_V$-algebra is equivalent to a $V$-fold loop space if fixed points are group-like, extending prior work to all finite group representations.

Findings

Proves the equivariant approximation theorem for all $G$-representations and spaces.

Generalizes recognition principle without the trivial summand assumption.

Shows fixed points of $ ext{E}_V$-algebras are group-like under certain conditions.

Abstract

For $G$ a finite group and $V$ a finite dimensional real $G$-representation, there is a $G$-operad $\mathbb{E}_{V}$ defined using embeddings of $V$-framed $G$-disks such that for any based $G$-space $X$, there is a naturally defined $\mathbb{E}_{V}$-algebra structure on the $V$-fold space $\Omega^V X$. Given an $\mathbb{E}_{V}$-algebra in $G$-spaces and a subgroup $H$ of $G$, the fixed points $A^H$ carry the structure of an $\mathbb{E}_{\dim V^H}$-algebra in spaces. We prove that an $\mathbb{E}_{V}$-algebra is equivalent to a $V$-fold loop space if and only if $A^H$ is group-like for all $H$ such that $\dim V^H \ge 1$. This generalizes a result by Guillou and May by removing the assumption that $V$ contains a trivial summand. They observed that equivariant recognition principle follows from an equivariant version of the approximation theorem, stating that $\Omega^V \Sigma^V X$ is the free group-like $\mathbb{E}_{V}$-algebra on a based $G$-space $X$. This has been proven by Hauschild in the case that $V$ contains a trivial summand and by Rourke and Sanderson in the case that $X$ is $G$-connected. Our proof proceeds by showing that the equivariant approximation theorem holds for all $G$-representations $V$ and all based $G$-spaces $X$.