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In the book of Bridson and Haefliger it is said that 'it follows easily' from what they proved before. Does anyone know of a rigorous proof that CAT(0) spaces are contractible?

mathreader
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1 Answers1

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Cat(0) spaces have unique geodesics between points. Fix a base point, and along each geodesic out from that point, pull everything inwards. This is well-defined because geodesics are unique, and is continuous if you use the same map on each geodesic.

i can provide more details as needed.

Brian Rushton
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    This reasoning seems to apply verbatim to a sphere. We don't actually use the uniqueness of geodesics when we pull everything along them to a point. – mathreader Feb 04 '15 at 03:38
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    Spheres do not have unique geodesics. Antipodal points have infinitely many geodesics connecting them. – Brian Rushton Feb 04 '15 at 04:13
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    I understand that. However, the phrase "pull a point towards the origin along a geodesic" doesn't use anything specific to CAT(0). And changing 'a' into 'the unique' doesn't change anything either. What I am searching for is a rigorous proof, without all this handwaving. – mathreader Feb 04 '15 at 04:19
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    @mathreader The fact that the 'pull everything inwards' map is well-defined uses the uniqueness of geodesics. – Steven Stadnicki Feb 04 '15 at 04:29
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    How we define 'pull-everything-inwards' map is irrelevant. We can do that in many ways. The main issue is continuity. For CAT(0) spaces it is satisfied for some reason, for spheres it is not. This is the crucial moment, not the existence/uniqueness of geodesics. – mathreader Feb 04 '15 at 04:41
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    @mathreader Take two points and look at the distance between them. The geodesics from the origin to those points form two edges of a triangle, with the line between them being the third. As both geodesics are retracted evenly, the CAT(0) property should ensure the distance between them decreases. This would make the map Lipschitz, and thus continuous. – Brian Rushton Feb 04 '15 at 16:36