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Let us phrase the question in the title in more detail: I wonder if there exists a metric space $X$ which has at least two points, has finite diameter (in the sense that there is an upper bound for the distance between two points, for example it could be compact), is contractible (there is a homotopy of continuous self-maps of $X$ which starts at the identity and ends a constant map) and such that its isometry group acts transitively.

I suspect that there is no such $X$ (maybe with more assumptions, like asking it to be compact or geodesic; if it is CAT(0) then a solution is obvious), but I have not been able to find a proof of this. So I would be grateful if someone knows a solution to or a reference for this problem.

edit: it turns out (see Swiat Gal's answer) that there are such spaces when one drops the compactness assumption--however, I am more interested in the compact case. I also made an assumption to avoid the dumb example of the one-point space.

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2 Answers 2

up vote 8 down vote accepted

According to this MO question,

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

So a compact isometrically homogeneous contractible metric space is a point, unless I'm missing something.

Edit (after Swiat Gal's comment): I was indeed under the impression that a contractible space should be locally contractible at the contraction point (and hence at any point if the space is also homogeneous), but now I'm not so sure about it. It certainly works if the space can be (strongly) deformation retracted to a point, but that is stronger than mere contractibility.

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That's really quite a remarkable result... –  Todd Trimble May 18 '13 at 1:30
    
Todd: This is a variation on the solution of the Hilbert's 5th problem or, more precisely, its generalization to transitive group actions on various classes of spaces (Gleason-Montgomery-Zippin-Yamabe et al), so not completely unexpected. –  Misha May 18 '13 at 5:05
    
Yes, it has that kind of flavor. That still doesn't prevent me from finding it amazing. –  Todd Trimble May 18 '13 at 7:30
    
Thank you very much! –  Jean Raimbault May 18 '13 at 12:04
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Just to unconfuse me: you do not claim that contractible (isometrically homogeneous metric) space needs to be locally contractible? –  Swiat Gal Jun 3 '13 at 10:38

What about a unit sphere in a Hilbert space?

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I don't think this is contractile (in finite dimension it is most certainly not). –  Jean Raimbault May 17 '13 at 9:28
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The infinite dimensional sphere is contractible. And the one point space appears to be another counterexample. –  Christian Nassau May 17 '13 at 9:35
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It is contractible, see for example math.ucr.edu/home/baez/week151.html –  The User May 17 '13 at 9:36
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{\it Autant pour moi}. I will edit the question to ask for compact examples (which I probably should have done in the first place). –  Jean Raimbault May 17 '13 at 9:41

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