X,Y Top. Spaces, f cont:f: X-- --Y is nullhomotopic if X is contractible
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From: Carsten Schultz on 5 Apr 2010 05:30 Am 05.04.10 05:20, schrieb Bacle: > Hi, everyone: > Let X,Y be topological spaces, f:X-->Y a cont. map. > > I am having trouble seeing that if X is contractible, then for any Y , f is nullhomotopic. The opposite direction, i.e., g:X-->Y, with Y contractible is clear, > but not this direction. > > > Anyway, this seems like a counterexample: > > f(t):I-->S^1 ; f(t)=e^i*2*Pi*t > > I imagine part of the issue is that we allow free homotopies. But even using free homotopies, it seems difficult: we must tear the path f(t) open before deforming it to a point, and I don't see how we can > tear (the image of) f(t) open in a continuous way. X is contractible iff id_X is homotopic to a constant map, say c. Then f = f.id_X is homotopic to f.c, which is constant. If you work out the details, you will also be able to write down an explicit homotopy from your example map to a constant map. -- Carsten Schultz (2:38, 33:47) http://carsten.codimi.de/ PGP/GPG key on the pgp.net key servers, fingerprint on my home page. |