Homotopy orbit spaces of representation spheres

Question

Let $G$ be a finite group and $V$ be finite-dimensional real representation of $G$. Write $S^V$ for the one-point compactification of $V$, with induced $G$-action, viewed as a pointed space, and consider the homotopy orbit space $(S^V)_{hG}$. For instance, if $V$ is the regular representation of $G=\mathbb{Z}/2$, $(S^V)_{hG}$ works out to be the suspension of $B\mathbb{Z}/2$. Is there some kind of general description of this space, presumably built out of classifying spaces of subgroups of $G$? What if I only care about the stable homotopy type? I'm most interested in the permutation representations of the symmetric groups.

Answer 1

The most salient fact about (S^V)_hG is that it is the Thom space of a vector bundle over BG. So this gives you a lot of control over the sorts of things seen by homology: e.g., a Thom isomorphism if the bundle is oriented with respect to your homology theory; if you put a CW-structure on BG, then (S^V)_hG has a CW-structure with cells in the same dimensions (shifted up by dim V). Morally, it's a "twisted suspension" of BG, and that fact encodes most of what I know about it.

For G=Z/2, and L=sign rep, you have (S^nL)_hG = RP^∞/RP^n-1, and something similar holds for Z/p, and for the symmetric group Σ_p (and perhaps for every group with periodic tate cohomology?)

If you have a decomposition of the form BG = hocolim BH, then you can pull your bundle over BG back to all the terms in the hocolim to get (S^V)_hG = hocolim (S^V)_hH.