Equivariant control data and neighborhood deformation retractions

Abstract

In this article we study Whitney (B) regular stratified spaces with the action of a compact Lie group $G$ which preserves the strata. We prove an equivariant submersion theorem and use it to show that such a $G$-stratified space carries a system of $G$-equivariant control data. As an application, we show that if $A\subset X$ is a closed $G$-stratified subspace which is a union of strata of $X$, then the inclusion $i:A\hookrightarrow X$ is a $G$-equivariant cofibration. In particular, this theorem applies whenever $X$ is a $G$-invariant analytic subspace of an analytic $G$-manifold $M$ and $A\hookrightarrow X$ is a closed $G$-invariant analytic subspace of $X$.

24 pages, to appear in Methods and Applications of Analysis