Equivariant Splitting of Certain Spaces CX

This dissertation generalizes the classical stable splitting theorem of Cohen, Taylor, and May to the genuine equivariant setting for a finite group $G$. We establish that for an $\Sigma$-free $G$-coefficient system $\mathscr{C}$ and a $G$-space $X$, the suspension $G$-spectrum of $F_r CX$ is isomorphic to a wedge sum of the suspension $G$-spectra of the extended powers $D_q X$ in the equivariant stable category. The proof relies on a reduction to separated $G$-coefficient systems and utilizes the equivariant approximation theorem of Guillou and May relating the Steiner operad to infinite loop $G$-spaces. We construct equivariant James maps and demonstrate that specific comparison maps involving configuration $G$-spaces induce isomorphisms on homology groups. We then discuss the implications of this splitting for computing $RO(G)$-graded homology. Finally, we observe that the restriction to finite groups arises solely from the use of the equivariant approximation theorem. Since the main result is not inherently a theorem about infinite loop spaces, we conjecture that it generalizes to compact Lie groups via proof techniques independent of this theorem.