Universal Embedding spaces for $G$-manifolds

For any compact Lie group $G$ and any $n$ we construct a smooth $G$-manifold $U_n(G)$ such that any smooth $n$-dimensional $G$-manifold can be embedded in $U_n(G)$ with a trivial normal bundle. Furthermore, we show that such embeddings are unique up to equivariant isotopy It is shown that the (inverse limit) of the cohomology of such spaces gives rise to natural classes which are the analogue for $G$-manifolds of characteristic classes for ordinary manifolds. The cohomotopy groups of $U_n(G)$ are shown to be equal to equivariant bordism groups.