The space of closed subgroups of $R^n$

Benoit Kloeckner (1)

1 IF - Institut Fourier

The Chabauty space of a topological group is the set of its closed subgroups, endowed with a natural topology. As soon as $n>2$, the Chabauty space of $R^n$ has a rather intricate topology and is not a manifold. By an investigation of its local structure, we fit it into a wider, but too wild, class of topological spaces (namely Goresky-MacPherson stratified spaces). Thanks to a localization theorem, this local study also leads to the main result of this article: the Chabauty space of $R^n$ is simply connected for all $n$. Last, we give an alternative proof of the Hubbard-Pourezza Theorem, which describes the Chabauty space of $R^2$.

Mots clés: Chabauty topology; stratified spaces; closed subgroups; homotopy of pairs

Domaines: Topologie géométrique [math.GT]

Benoit Kloeckner. The space of closed subgroups of $R^n$. Journal of topology, 2009, 2 (3), pp.570-588. ⟨10.1112/jtopol/jtp022⟩. ⟨hal-00346132⟩