On the existence of equivariant embeddings of principal bundles into vector bundles

Abstract:

Let $G$ be a finite group and let $X$ be, say, a connected CW-complex of dimension $k \geqslant 1$. Let $\pi :E \to X$ be a principal $G$-bundle and $p:V \to X$ an $m$-dimensional $G$-vector-bundle with trivial action of $G$ on $X$. By an equivariant embedding of $\pi$ into $p$ we understand an equivariant embedding $h:E \to V$ commuting with projections. We prove a general embedding theorem, a main special case of which is the following Theorem. If $k < m$ and if the action of $G$ on $V$ is free outside the zero section for $p$, then any principal $G$-bundle $\pi :E \to X$ can be embedded equivariantly into $p:V \to X$.