2025/08/06 by Juran, Branko · 1 citation
#55P48 #55P91 #Algebraic Topology (math.AT) #FOS: Mathematics
paper · doi:10.48550/arxiv.2508.04421
For G a finite group and V a finite dimensional real G-representation, there is a G-operad 𝔼V defined using embeddings of V-framed G-disks such that for any based G-space X, there is a naturally defined 𝔼V-algebra structure on the V-fold space ΩV X. Given an 𝔼V-algebra in G-spaces and a subgroup H of G, the fixed points AH carry the structure of an 𝔼dim VH-algebra in spaces. We prove that an 𝔼V-algebra is equivalent to a V-fold loop space if and only if AH is group-like for all H such that dim VH ≥ 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 ΩV ΣV X is the free group-like 𝔼V-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.