Commutative Control Data for Smoothly Locally Trivial Stratified Spaces

Abstract

For a compact Lie group G and a Hamiltonian G-space M with momentum map $μ:M \to g^*$, we prove that the zero level set $μ^{-1}(0)$ and the critical set of the norm-squared momentum map are neighbourhood smooth weak deformation retracts. To this end we show that these subsets, stratified by orbit types, satisfy a condition stronger than Whitney (B) regularity - smooth local triviality with conical fibers. Using this condition we construct control data in the sense of Mather with the additional properties that the fiber-wise multiplications by scalars, coming from the tubular neighbourhood structures, preserve strata and commute with each other. We use that control data to obtain the neighbourhood smooth weak deformation retraction. Finally, such structures for the zero level set $μ^{-1}(0)$ reduce to similar structures for the reduced space $μ^{-1}(0)/G$, yielding a similar result for the reduced space and its stratified subspaces.