The space of closed subgroups of $R^n$

Benoit Kloeckner

doi:10.1112/jtopol/jtp022

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arXiv · 0812.2111

The space of closed subgroups of $R^n$

Benoit Kloeckner

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Abstract

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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BibTeX RIS

Benoit Kloeckner. 2008-12-11. The space of closed subgroups of $R^n$. https://doi.org/10.1112/jtopol%2Fjtp022

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