On Diffeology of Orbit Spaces

2025/08/22 by Gürer, Serap, Iglesias-Zemmour, Patrick
#57S15 #58A35 #58A40 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2508.16743

Abstract

We investigate the correspondence between the geometry of a smooth compact Lie group action on a manifold M and the intrinsic smooth structure of the orbit space M/G. While the action on 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 M: the isostabilizer decomposition. We establish a surjective map from the components of this partition to the Klein strata of M/G. As a corollary, we obtain by pullback a new canonical stratification on M, the Inverse Klein Stratif ication, and clarify its relationship with classical structures.

Cited by

• Geometric Quantization by Paths -- Part I: The Simply Connected Case
• Why Diffeology?

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