Commutative Control Data for Smoothly Locally Trivial Stratified Spaces

Yoav Zimhony

For a compact Lie group G and a Hamiltonian G-space M, can we find a smooth weak deformation retraction from a neighbourhood of the zero level set of the momentum map onto it? If we do not require smoothness then this is already known, in fact one can obtain a strong deformation retraction. We will outline the construction of such smooth weak deformation retraction with the following steps.First we show that the zero level set, stratified by orbit types, satisfies a condition stronger than Whitney (B) regularity - smooth local triviality with conical fibers. Using this local 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. Finally we use this control data to obtain the neighbourhood smooth weak deformation retraction.We will also discuss a key technical tool used in the construction of the control data - Euler-like vector fields.

https://www.ias.edu/video/commutative-control-data-smoothly-locally-trivial-stratified-spaces