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?