I have been reading the paper "A generalization of the Atiyah-Segal completion theorem" by Adams, Haeberly, Jackowski and May, where I found the Following claim:
Let $G$ be finite group, $H\subset G$ a subgroup. Let $X$ be a compact pointed space, with $G$ acting on it. Then there is an isomorphism $$\tilde{K}^*_G((G/H)_+\bigwedge X)\cong \tilde{K}^*_H(X) $$
I know a similiar statement to be true from Segals paper on equivariant $K$-theory, namely that $K_G(G/H\times X)\cong K_H(X)$, but that is only in degree zero.
I suspect the more general version can be proven by noticing that $(/)_+⋀\cong (G/H\times X)/(G/H)$ and using the LES of $K$-theory, but it did not work out so far.
Can anyone give me reference where this more general version is proven, or provide some hints how to deduce it from the version I know to be true?
It suffices to prove this degree 0 since $\tilde{K}^{-q}_G((G/H)_+\bigwedge X):=\tilde{K}_G((G/H)_+\bigwedge (X\bigwedge S^q))$.
The crucial observation is that $G/H\to G/H\times X$ is an equivariant retract, so that the LES for the pair $(G/H\times X,\,G/H)$ falls apart into short exact sequences.
Then the "obvious" map $\tilde{K}_G((G/H)_+\bigwedge X)\to\tilde{K}_H(X)$ can be seen to be an isomorphism from the snake lemma and the already known isomorphisms $K_G(G/H\times X)\to K_H(X)$ and $K_G(G/H)\to K_H(*)$.