Let $G$ be a compact Lie group acting effectively on a compact, Hausdorff topological space $X$. I am looking for results of the type
If $X$ is a … and the action is … then $\dim(X/G)\leq \dim(X)-\dim(G)$.
Here $\dim$ denotes the covering dimension.
For example, $X$ is a smooth manifold and $G$ acts smoothly and almost-freely (all isotropy groups finite).
Can anyone point me to a more general theorem of this type?
Best Answer
I'm loathe to answer my own question, but...
Theorem IV.3.8 of Bredon's book Introduction to compact transformation groups (which sadly is on the page google books decided not to include!) seems satisfying.
Let $G$ be a compact Lie group acting locally smoothly on the manifold $M$, such that $M/G$ is connected. If $P$ is a principal orbit (an orbit of maximum dimension) then
$$\dim(M/G)=\dim(M)-\dim(P).$$