When is the group of homeomorphisms of a compact space locally compact? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T08:27:11Z http://mathoverflow.net/feeds/question/47908 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/47908/when-is-the-group-of-homeomorphisms-of-a-compact-space-locally-compact When is the group of homeomorphisms of a compact space locally compact? Spencer 2010-12-01T14:50:38Z 2010-12-02T08:25:17Z <blockquote> <p>When is the group of homeomorphisms of a compact space locally compact?</p> </blockquote> <p>I am interested in finding out when the group of homeomorphisms of a compact topological space $X$ (with appropriate topology e.g. 'weak' or compact-open) is a locally compact space.</p> <p>What extra conditions might we be able to put on $X$ to ensure that it is so?... What if $X$ is, say, a metric space and we ask when the isometry group is locally compact?</p> http://mathoverflow.net/questions/47908/when-is-the-group-of-homeomorphisms-of-a-compact-space-locally-compact/47909#47909 Answer by Theo Buehler for When is the group of homeomorphisms of a compact space locally compact? Theo Buehler 2010-12-01T14:58:28Z 2010-12-02T08:25:17Z <p>I do not know what you mean by automorphism group, I guess you mean homeomorphisms. In that case the answer is no:</p> <p>For instance, the homeomorphisms of the circle are in one-to-one correspondence with continuous strictly monotone functions $[0,1] \to \mathbb{R}$ such that $f(0) \in [0,1)$ and $f(1) = f(0)\pm 1$. Compact-open topology just means uniform convergence, and this obviously is not a locally compact space.</p> <p>As for local compactness of the isometry group, it follows from the Arzelà-Ascoli theorem that that the isometry group of a <em>proper</em> metric space (i.e., closed balls are compact) is locally compact.</p> http://mathoverflow.net/questions/47908/when-is-the-group-of-homeomorphisms-of-a-compact-space-locally-compact/47940#47940 Answer by Keivan Karai for When is the group of homeomorphisms of a compact space locally compact? Keivan Karai 2010-12-01T20:03:01Z 2010-12-01T20:03:01Z <p>For a (connected) smooth Riemannian manifold $M$, it has been shown by Myers and Steenrod that that the group of isometries is a Lie group, hence is locally compact. On the other hand the group of homeomorphisms of a smooth manifold $M$ is never locally compact. When the dimension is at least $2$, this group acts $k$-transitively for any $k$ on $M$ and from here I think it should be easy to show that the groups is not locally compact. </p>