The Borel construction of equivariant cobordism

Question

In $K$ theory, the Borel construction of equivariant cohomology is somehow not the right one. The $G$-equivaraint $K$ theory of a point should be the representation ring of $G$, but $K(BG)$ is this ring completed at the ideal of zero dimensional representations (this is the Atiyah-Segal completion theorem).

I am wondering if a similar problem occurs with the Borel construction applied to equivariant cobordism (I am interested in unoriented, oriented, and spin cobordism). That is, we have a map $$ \Omega^*_G(*) \to \Omega^*(BG). $$ Is this map an isomorphism?

Answer 1

No, that map is not an isomorphism.

First, there are a couple of different things that you might mean by $\Omega^*_G(*)$, and if you want the full story then you need to distinguish between them, but there is no standard interpretation under which your map is iso. The basic point is that, just as in the Atiyah-Segal theorem, the codomain is uncountable and complete with respect to the augmentation ideal, but the domain is countable and not complete.

You can find much more in the paper "Localization and completion theorems for MU-module spectra" by Greenlees and May.

Answer 2

No, but for some G like $Z_2$ for unoriented, $T^k$ for the unitary case this map is injective. And u can read tom Dieck's paper for detail.