Equivariant control data and neighbourhood deformation retractions

n 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⊂X is a closed G-stratified subspace which is a union of strata of X, then the inclusion i:A↪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↪X is a closed G-invariant analytic subspace of X.