Example of a compact homogeneous metric space which is not a manifold - MathOverflow

Example of a compact homogeneous metric space which is not a manifold

2011-02-24T14:58:23Z

A metric space $(X,d)$ is isometrically homogeneous if its isometry group acts transitively on points, i.e., for every $x,y \in X$ there is an isometry $\varphi:X\to X$ with $\varphi(x) = y$. I'd like to know an example of a compact isometrically homogeneous metric space which is not a manifold (a space with finitely many points counts as a 0-dimensional manifold).

Googling a bit I've discovered enough recent literature on this general subject to be sure there must be classical examples known to experts, but I haven't managed to find them written down. For example, Theorem 1.2 of this paper implies:

A compact isometrically homogeneous metric space is a finite-dimensional manifold if and only if it is locally contractible.

So equivalently, I'd like an example of a compact isometrically homogeneous metric space which is not locally contractible.

Added: Pete and Neil both gave very nice answers. I'm accepting Neil's since, as Pete points out, it essentially contains Pete's answer as a special case.

Answer by Pete L. Clark for Example of a compact homogeneous metric space which is not a manifold

2011-02-24T15:13:45Z

The ring of $p$-adic integers $\mathbb{Z}_p$ with its standard metric seems to be an example of what you want.

Of course this space is a "$p$-adic analytic manifold", so you may want to see another example. E.g., how about a space which satisfies all of your properties and is also connected?

Answer by Neil Strickland for Example of a compact homogeneous metric space which is not a manifold

2011-02-24T15:28:45Z

Take $X=\prod_{n=0}^\infty S^1$ with $d(x,y)=\sum_n|x_n-y_n|/2^n$. Then the metric topology is the same as the product topology, which is compact by Tychonov. There is an obvious group structure by pointwise multiplication, and multiplication by any fixed element is an isometry, so the space is isometrically homogeneous. It is path-connected but not locally contractible.

More generally, I guess you can take any sequence of compact isometrically homogeneous spaces $X_n$, rescale the metric so that $d(x,y)\leq 2^{-n}$ for all $x,y\in X_n$, and then take $X=\prod_nX_n$ with $d(x,y)=\sum_nd(x_n,y_n)$.