6
$\begingroup$

Let $X$ be a manifold and $R\subset X\times X$ be an equivalence relation. Let $X/R$ be the set of equivalence classes and let $p:X\rightarrow X/R$ be the projection. Give $X/R$ the quotient topology. It is known that there is at most one manifold structure on $X/R$ such that $p$ is a submersion. In page 92, of Serre's book "Lie algebras and Lie groups" Springer LNM 1500, there is a criterion for $X/R$ to be a manifold, which is named after Godement and says the following. The quotient $X/R$ is a manifold if and only if

1, $R$ is a sub manifold of $X\times X$ (which means $R\rightarrow X\times X$ is a closed embedding), and

2, the projection map $pr_2:R\rightarrow X$ is a submersion.

This criterion looks very powerful since it gives an iff condition. On the other hand, it also works for non-archimedean local fields. In the usual textbook on manifolds, there is a quotient theorem which requires the action is free and proper. The freeness seems too restrictive. For example, the modular curve $H/\textrm{SL}_2(\textbf{Z})$ exists, where $H$ is upper half plane, but the action of $\textrm{SL}_2(\textbf{Z})$ is not free.

Since this criterion is not very common in the standard textbook on manifolds, I really would like to see how we apply this criterion to solve problems. Does anybody know a reference which contains a lot of examples on the application of this criterion?

I am also very interested in the following special case. Let $G$ be a compact Lie group which acts on a manifold $X$. Do we know that there exists a $G$-invariant open subset $X_o\subset X$ such that $X-X_o$ has lower dimension and $X_o/G$ exists as a manifold? Is this something standard from the Godement's criterion? How about when the field is non-archimedean?

In my other post, Quotient of $\textrm{GL}(2,\textbf{R})$ by the conjugate action of $\textrm{SO}(2,\textbf{R})$ , I asked if the quotient $GL_2(R)/SO_2(R)$ exists as a manifold where $SO_2(R)$ acts on $GL_2(R)$ by conjugation. I got a nice answer from @YCor. But I also would like to know if it is possible to solve that problem using Godement's criterion.

The Godement's criterion is also mentioned in one answer of this question Under what conditions the quotient space of a manifold is a manifold?

Thanks in advance.

$\endgroup$
1
$\begingroup$

Here is an answer to your last question. Let $X$ be a manifold on which acts the compact Lie group $G$, the orbits $G.x$ of $x$ and $G.y$ of $y$ have the same type if the stabilizers of $x$ is conjugated to the stabilizer of $y$. There exists a notion of principal orbit type. The union of principal orbits is an open subset of $U$ of $X$ and the quotient of $U$ by $G$ is a manifold.

See p.18 of https://www.math.upenn.edu/~wziller/math661/LectureNotesLee.pdf

The proof here assume that $G$ preserves a differentiable metric a fact which is always true if $G$ is compact.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.