Commutative Control Data for Smoothly Locally Trivial Stratified Spaces

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AbstractFor a compact Lie group G and a Hamiltonian G-space M with momentum map $$\mu :M\rightarrow \mathfrak {g}^*$$ μ : M → g ∗ , we prove that the zero level set $$\mu ^{-1}(0)$$ μ - 1 ( 0 ) and the critical set $$\text {Crit}||{\mu }||^2$$ Crit | | μ | | 2 of the norm squared momentum map are neighbourhood smooth weak deformation retracts. This confirms a conjecture by Harada and Karshon, and implies that their localization theorem for Duistermaat-Heckmann distributions applies to these subsets. We construct these smooth weak deformation retractions in three steps. First, we prove that the stratifications of these subsets by orbit types satisfy a local condition stronger than Whitney (B) regularity — smooth local triviality with conical fibers. Then, using this local condition we construct control data in the sense of Mather with additional compatibility properties - tangentiality and commutativity. Finally we use this control data to obtain neighbourhood smooth weak deformation retractions.