Universal Embedding spaces for $G$-manifolds

TL;DR

This paper constructs universal embedding spaces for compact Lie group actions on manifolds, providing a framework for embedding, classifying, and understanding $G$-manifolds through cohomology and bordism groups.

Contribution

It introduces universal $G$-manifold embedding spaces that facilitate embeddings with trivial normal bundles and links their cohomology to characteristic classes and bordism groups.

Findings

Universal $G$-manifold embedding spaces are constructed.

Embeddings are unique up to equivariant isotopy.

Cohomology of these spaces relates to characteristic classes and equivariant bordism.

Abstract

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.