Does a compact semilocally simply connected geodesic space have the homotopy type of a CW complex? - MathOverflow

Does a compact semilocally simply connected geodesic space have the homotopy type of a CW complex?

2011-01-12T13:12:16Z

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.

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.

Answer by Bruno Martelli for Does a compact semilocally simply connected geodesic space have the homotopy type of a CW complex?

2011-01-12T13:33:56Z

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.

Answer by Theo Buehler for Does a compact semilocally simply connected geodesic space have the homotopy type of a CW complex?

2011-01-12T19:42:10Z

I hope I have not goofed, but I think the answer to your modified question is yes:

The fundamental group of a semi-locally simply connected, compact and geodesic space is finitely presented.

Here are the ingredients - all numbers in parentheses refer to Bridson-Haefliger, Metric spaces of non-positive curvature, Springer Grundlehren, 1999, Part I.

The universal covering space $\widetilde{X}$ equipped with the length metric induced by the covering projection is a length space (3.25).
The fundamental group $\pi_{1}(X)$ acts on $\widetilde{X}$ properly and cocompactly by isometries (8.3 (2)), see also (8.5).
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).
A group is finitely presented if and only if it acts properly and cocompactly by isometries on a simply connected geodesic space (8.11).

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.