Exercise 16, § 9

Let G be a topological group operating properly on a separated topological space E. Let $x$ be a point in E, H its fixer in G, and S a subset of E containing $x$ and stable under H. The group H operates on the right on $G\times S$ by the formula $(g, s).h= (gh, h^{-1}s)$ for $g\in G, h\in H, s\in S$; the map $(g, s) \rightarrow gs$ induces by passage to the quotient a map from $(G\times S)/H$ to X. Then S is said to be a transversal at $x$ if this map is a homeomorphism onto an open subset of X.

a) Show that, if G is discrete, there exists a transversal at $x$.

b) Let F be a separated topological space on which G operates properly, and let $f: E\rightarrow F$ be a continuous map, compatible with the operations of G, such that the fixer of $f(x)$ in G is equal to H. Show that, if S is a transversal at $f(x)$, then $f^{-1}(S)$ is a transversal at $x$.

c) Let N be a closed normal subgroup of $G,\pi : E\rightarrow E/N$ the canonical projection, T a transversal at $\pi (x)$ for the operation of $G/N$ on $E/N$, and $S\subset \pi^{-1}(T)$ a transversal at $x$ for the operation of HN on $\pi^{-1}(T)$. Show that S is a transversal at $x$ in E (for the operation of G).