The space of closed subgroups of $R^n$

Details The space of closed subgroups of $R^n$ The space of closed subgroups of $R^n$

2008-12-11

arxiv.org/abs/0812.2111v1

Journal of topology 2, 3 (2009) 570-588

Mathematics

Geometric Topology

28 pages

Scientific paper

10.1112/jtopol/jtp022

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 $R^2$.

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