Author 
Topic: Question about Homotopy (Read 1880 times) 

immanuel78
Newbie
Gender:
Posts: 23


Question about Homotopy
« on: Sep 20^{th}, 2006, 6:18am » 
Quote Modify

Let G be a region. Let L : [0,1] > G be a closed rectifiable path. Then there exists a closed, piecewisesmooth path in G which is homotopic to L in G. This question seems to be true to me, but it is not easy for me to prove it.

« Last Edit: Sep 20^{th}, 2006, 9:51am by immanuel78 » 
IP Logged 



Icarus
wu::riddles Moderator Uberpuzzler
Boldly going where even angels fear to tread.
Gender:
Posts: 4863


Re: Question about Homotopy
« Reply #1 on: Sep 20^{th}, 2006, 6:18pm » 
Quote Modify

The image of L is compact and lies in the interior of G. Cover the image with balls that lie within G, then choose a finite number of them that still cover it. You can order the balls B_{1}, B_{2}, ..., B_{n} in the order in which they first occur along L. Choose points x_{i} on L such that x_{i} is in the intersection of B_{i} and B_{i+1} (since L is closed, let x_{n} be in the intersection of B_{n} and B_{1}). The piecewise linear path determined by these points satisfies the conditions.

« Last Edit: Dec 23^{rd}, 2006, 7:18am by Icarus » 
IP Logged 
"Pi goes on and on and on ... And e is just as cursed. I wonder: Which is larger When their digits are reversed? "  Anonymous



immanuel78
Newbie
Gender:
Posts: 23


Re: Question about Homotopy
« Reply #2 on: Sep 21^{st}, 2006, 6:14am » 
Quote Modify

Thank you for solving it.


IP Logged 



Icarus
wu::riddles Moderator Uberpuzzler
Boldly going where even angels fear to tread.
Gender:
Posts: 4863


Re: Question about Homotopy
« Reply #3 on: Sep 21^{st}, 2006, 4:23pm » 
Quote Modify

Compactness and connectedness are the hammer and screwdriver of the analyst's toolbox. They are often the first things you should look to in solving problems.


IP Logged 
"Pi goes on and on and on ... And e is just as cursed. I wonder: Which is larger When their digits are reversed? "  Anonymous



