Faithful actions of finite groups on topological spaces

2011-05-10T17:37:54Z

Suppose that $G$ is a finite group acting faithfully on a topological space $X$. In the smooth setting, one can deduce that for each $x$ in $M$, the induced map $$G_x \to Diff_x\left(M\right)$$ from the isotropy group of $x$ to the group of germs of locally defined diffeomorphisms is a monomorphism. Does this result hold true in the topological setting? (Replacing diffeomorphisms with homeomorphisms)

In the smooth setting, the result follows from the following standard lemma:

Lemma: *Let $M$ be a manifold and $G$ a finite subgroup of $\mathit{Diff}\left(M\right)$. Then for any smooth map $f:V \to M$ defined on a non-empty open connected submanifold of $M,$ such that $f\left(x\right) \in G \cdot x$ for all $x,$ there exists a unique element $g \in G$ such that $f=g|_V.$ *

I'm guessing the answer is NO for general topological spaces. However, a counter-example would be nice.

Faithful actions of finite groups on topological spaces - MathOverflow

Faithful actions of finite groups on topological spaces