On Diffeology of Orbit Spaces
Year: 2025
Type: preprint
Abstract: We investigate the correspondence between the geometry of a smooth compact Lie group action on a manifold $\mathrm{M}$ and the intrinsic smooth structure of the orbit space $\mathrm{M}/\mathrm{G}$. While the action on $\mathrm{M}$ is classically organized by the orbit-type stratification, we show this structure fails to predict the intrinsic $Klein\ stratif\!ication$ of the quotient, which partitions the space into the orbits of local diffeomorphisms, thereby classifying the space by its intrinsic singularity types. The correct correspondence, we prove, is governed by a finer partition on $\mathrm{M}$: the $isostabilizer\ decomposition$. We establish a surjective map from the components of this partition to the Klein strata of $\mathrm{M}/\mathrm{G}$. As a corollary, we obtain by pullback a new canonical stratification on $\mathrm{M}$, the $Inverse\ Klein\ Stratif\!ication$, and clarify its relationship with classical structures.
Source: arXiv (Cornell University)
Authors: Gürer, Serap, Iglesias-Zemmour, Patrick
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7077901773
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