A genuine equivariant recognition principle for finite groups

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 $Ω^V X$. 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 $Ω^V Σ^V X$ 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 $X$.