A relation between equivariant and non-equivariant stable cohomotopy (Q1095448)

We describe the stable G-cohomotopy group \(\pi ^ V_ G(X)\), G a finite group and V a G-module, in nonequivariant terms. Specifically, we construct a finite CW-complex \~T(X,V), an integer \(t=t(X,V)\) and for each G-module W, a homomorphism \(\psi _ W: \pi ^ t(\tilde T(S^ WX,S^{W\oplus V}))\to \pi ^ V_ G(X),\) \(t=t(S^ WX,S^{W\oplus V})\), which is an isomorphism for sufficiently large W. This isomorphism contains the known results: \(\pi ^ n_ G(X^ +)\simeq \pi ^ n(X/G^ +)\), \(n\in Z\), when X is free, and \(\pi ^ V_ G(X)\simeq \sum _{(H)}\{X;EW(H)^ +\wedge _{W(H)}S^{V^ H}\}\) when X is a trivial G-space. The key step in going from equivariant to nonequivariant homotopy is the construction of a one-one correspondence between homotopy classes of G-maps \(X\to Y\) and homotopy classes of collections of cross sections to certain bundles which lie inside the balanced product \(X\times _ GV\to X/G\).