What represents equivariant cohomology? That is, given a $G$-space $X$, is there a $G$-space $Z$ such that $H^n_G(X; \Bbb Z) := H^n(X\times_G EG; \Bbb Z) = [X, Z]_G$ naturally? Looking around tells me about "equivariant Eilenberg MacLane spaces" but these seem to represent Bredon cohomology, not Borel equivariant cohomology.

@BalarkaSen I think Map(EG,K(Z,n)), viewed as a G-space by the action of G on EG, does this.

You're right, I just realized.

Where was the equivalence KO/(\eta) = KU first shown? This gets attributed to Wood, but I can't find a reference.