The space of conjugacy classes of a topological group

Abstract:

The space ${G^\# }$ of conjugacy classes of a topological group $G$ is the orbit space of the action of $G$ on itself by inner automorphisms. For a class of connected and locally connected groups which includes all analytic $[Z]$-groups, the universal covering space of ${G^\# }$ may be obtained as the space of conjugacy classes of a group which is locally isomorphic with $G$, and the Poincaré group of ${G^\# }$ is found to be isomorphic with that of $G/G’$, the commutator quotient group. In particular, it is shown that the space ${G^\# }$ of a compact analytic group $G$ is simply connected if and only if $G$ is semisimple. The proof of this fact has not appeared in the literature, even though more specialized methods are available for this case.