# Local structure of the quotient of a Lie group action

Suppose $M$ is a smooth manifold and a compact Lie group $G$ acts freely on $M$, then it is well known that $M/G$ has a manifold structure.

Are there any results for the general case? (a) If the action is not free, what can we say about the local structure of the quotient? Can we define a stratification on the space? How regular are the strata? (b) Moreover, what can we say if $M$ and $G$ are Hilbert manifolds?

Thanks a lot!

• My favorite reference for the finite-dimensional manifold case is Bredon's "Introduction to compact transformation groups". The key word is slice theorem. When $G$ or $M$ is infinite dimensional it matters very much what they are. Sometimes the slice theorem is known and sometimes it it is false. Searching on "slice theorem" "infinite dimensional" will get you started. Jan 3, 2015 at 2:17
• @IgorBelegradek Thanks a lot! This is exactly what I'm looking for. Jan 3, 2015 at 3:23

Suppose one relaxes the condition that the action is free, replacing it with the condition that every point in $M$ has a finite $G$-stabilizer. In this case, the topological quotient $M/G$ carries a natural orbifold structure. The local model is therefore the quotient of $\mathbb{R}^n$ by a finite group acting linearly.

In general, $M/G$ is a stratified space, and the stratification is indexed by conjugacy classes of $G$-stabilizers.

• Thank you very much! Could you recommend a reference for this? Jan 2, 2015 at 19:46
• Bourbaki Lie groups and Lie algebras, chap. 9, gives very precise statements, most of them with no hypotheses on the stabilizers.
– abx
Jan 2, 2015 at 19:52
• There is an excellent paper on Stratified Symplectic Spaces by E. Lerman and R. Sjamaar. I think it probably contains some of these details. Jan 2, 2015 at 19:55

If you know German, then I strongly recommend you the Habilitation thesis of Markus Pflaum: "Ein Beitrag zur Geometrie und Analysis auf stratifizierten Räumen" for the stratification of the orbit space. If I remember correctly, then Ortega & Ratiu "Momentum Maps and Hamiltonian Reduction" also contains some general results about the orbit space although the main focus of this book lies on symplectic reduction.

• Pflaum's book is available in English as a Lecture Notes in Mathematics volume. The English title is "Analytic and Geometric Study of Stratified Spaces". Link to book. Apr 14, 2015 at 20:00