# Homotopy type of stabilizers

Let X be a contractible metric space and G a topological group acting transitively on X (i.e. given any two points x,y \in X, there exists g \in G such that gx=y).

My question is the following: is it true that given any x \in X its stabilizer Stab(x)={ g \in G : gx=x } and the whole group G have the same homotopy type?

If the answer is "no", I'd like to know some "mild" hypothesis that could be add to have an affirmative response.

For instance, I know that whenever G is a Lie group and H < G is a closed subgroup such that G/H is contractible, then G and H are homotopically equivalent (in this case H can be seen as the stabilizer of the coset H under the natural G-action on G/H). However, to assume that G is a Lie group seems to be too restrictive. In fact, I'd like to apply this "result" to some groups which are not locally compact.

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## 1 Answer

The short: no. E.g. let X be a topological group and G be the underlying discrete group of X, acting on X by left translation.

One standard hypothesis is the existence of slices. For some (hence any) point x in X, if there is an open neighborhood U of x and a section s:U -> G such that s(u)*x = u for all u in U, then the map p:G -> X given by p(g) = g*x is a fiber bundle with fiber Stab(x).

This hypothesis automatically implies that Stab(x) is weakly equivalent to G by the long exact sequence of the fibration. If G and Stab(x) have the homotopy type of CW-complexes then they are homotopy equivalent by the Whitehead theorem.

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Tyler, thanks for the ideas. About the counterexample, I didn't say it explicitly (sorry), but I'm interested in the case where G is connected. About the second and third paragraphs, you're completely right. But, is there any chance of getting an affirmative answer imposing extra hypothesis on G and/or on X rather than on the action itself? Thanks again. –  Alejandro Oct 22 '09 at 12:18
If G is abelian, you can apply the standard simplicial "classifying space" functor, which takes topological abelian groups to topological abelian groups, and takes an example of the above type to a connected example. So far as the further question, did you have specific types of less-tractable groups in mind? –  Tyler Lawson Oct 23 '09 at 4:45
I'm particularly interested in the case G < Diff(M) and X is some manifold, in general, different of M. So, the CW-complex-hypotheses seems reasonable, but the existence of slices seems to be too strong. –  Alejandro Oct 23 '09 at 13:43