# Are the fixed point sets homeomorphic?

Assume $$f:G\rightarrow H$$ is a continuous epimorphism of topological groups and $$K\leq H$$ is a subgroup of $$H$$. One can consider the actions of subgroups $$K$$ and $$f^{-1}(K)$$ on $$H/K$$ and $$G/f^{-1}(K)$$ respectively (here $$H/K$$ and $$G/f^{-1}(K)$$ denote right coset spaces - with quotient topology and the actions are given by left multiplications).

Consider the fixed point sets of these actions, $$Fix(G/f^{-1}(K),f^{-1}(K))$$ and $$Fix(H/K,K)$$ with the induced subset topologies.

Is it true that $$Fix(G/f^{-1}(K),f^{-1}(K))$$ is homeomorphic to $$Fix(H/K,K)$$? If not in egneral, then maybe at least in the case $$K$$ and $$f^{-1}(K)$$ are finite?

• Is $f$ a homomorphism? Then the fixed point sets are $N_H(K)/K$ and $N_G(K)N/KN$, where $N=\ker(f)$., And these are isomorphic groups (via $f$). – Hempelicious Dec 21 '18 at 2:11
• Thinking about this more, $f$ always induces a continuous bijection between the fixed point sets. If $f$ is a quotient map, then you definitely get a homeomorphism. But it's not always true: let $G$ be $\mathbb{R}$ with the discrete topology, $H$ be $\mathbb{R}$ with the usual topology, $f$ the identity map and $K$ the rationals. – Hempelicious Dec 21 '18 at 17:20