On Diffeology of Orbit Spaces

TL;DR

This paper explores the intrinsic smooth structure of orbit spaces resulting from compact Lie group actions, revealing that the isostabilizer decomposition better predicts the space's stratification than classical orbit-type stratification.

Contribution

It introduces the isostabilizer decomposition as a finer partition that accurately reflects the intrinsic stratification of orbit spaces, linking it to Klein stratification.

Findings

The orbit-type stratification does not predict the intrinsic Klein stratification.

The isostabilizer decomposition provides a finer, more accurate partition.

A new canonical stratification on the manifold is established via pullback.

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.