Does a compact semilocally simply connected geodesic space have the homotopy type of a CW complex? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T06:40:47Z http://mathoverflow.net/feeds/question/51842 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/51842/does-a-compact-semilocally-simply-connected-geodesic-space-have-the-homotopy-type Does a compact semilocally simply connected geodesic space have the homotopy type of a CW complex? Jim Conant 2011-01-12T13:12:16Z 2011-01-12T19:56:16Z <p>Does a compact semilocally simply connected geodesic space have the homotopy type of a compact CW complex? Actually what I'd like to know is whether the fundamental group of such a space is finitely presented. </p> <p>Edit: As Bruno Martelli notes, this is obviously false, but the question of whether the fundamental group is finitely presented, which is what I really want to know, is still open.</p> http://mathoverflow.net/questions/51842/does-a-compact-semilocally-simply-connected-geodesic-space-have-the-homotopy-type/51845#51845 Answer by Bruno Martelli for Does a compact semilocally simply connected geodesic space have the homotopy type of a CW complex? Bruno Martelli 2011-01-12T13:33:56Z 2011-01-12T13:33:56Z <p>The bouquet of infinitely many shrinking 2-spheres in $\mathbb R^3$ centered in $(0,0,n)$ and of radius $n$ is a compact simply connected geodesic space, which is not homotopic to a compact CW complex.</p> http://mathoverflow.net/questions/51842/does-a-compact-semilocally-simply-connected-geodesic-space-have-the-homotopy-type/51867#51867 Answer by Theo Buehler for Does a compact semilocally simply connected geodesic space have the homotopy type of a CW complex? Theo Buehler 2011-01-12T19:42:10Z 2011-01-12T19:56:16Z <p>I hope I have not goofed, but I think the answer to your modified question is <strong>yes</strong>:</p> <blockquote> <p>The fundamental group of a semi-locally simply connected, compact and geodesic space is finitely presented.</p> </blockquote> <p>Here are the ingredients - all numbers in parentheses refer to Bridson-Haefliger, <em>Metric spaces of non-positive curvature</em>, Springer Grundlehren, 1999, Part I.</p> <ul> <li>The universal covering space $\widetilde{X}$ equipped with the length metric induced by the covering projection is a length space (3.25).</li> <li>The fundamental group $\pi_{1}(X)$ acts on $\widetilde{X}$ properly and cocompactly by isometries (8.3 (2)), see also (8.5).</li> <li>If a length space $\widetilde{X}$ admits a proper and cocompact action by isometries then it is locally compact (8.4 (1)) and hence proper and geodesic (3.7).</li> <li>A group is finitely presented if and only if it acts properly and cocompactly by isometries on a simply connected geodesic space (8.11).</li> </ul> <p>All this taken together yields that $\widetilde{X}$ is a simply connected geodesic metric space and $\pi_{1}{(X)}$ acts properly and cocompactly by isometries, hence $\pi_{1}{(X)}$ must be finitely presented.</p>