Equivariant maps for homology representations

Abstract:

If Y is a homotopy representation of the finite group G of order n and X is a finite G-CW complex such that, for each subgroup H of G, ${H_ \ast }({X^H};{\mathbb {Z}_n}) = {H_ \ast }({Y^H};{\mathbb {Z}_n})$ then there exists a G-map $f:X \to Y$ such that $f_ \ast ^H:{H_ \ast }({X^H};{\mathbb {Z}_n}) \to {H_ \ast }({Y^H};{\mathbb {Z}_n})$ is an isomorphism for each subgroup H.