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Topic: Question about Homotopy (Read 2091 times) 

immanuel78
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Question about Homotopy
« on: Sep 20^{th}, 2006, 6:18am » 
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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 » 
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Icarus
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Re: Question about Homotopy
« Reply #1 on: Sep 20^{th}, 2006, 6:18pm » 
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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 » 
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immanuel78
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Re: Question about Homotopy
« Reply #2 on: Sep 21^{st}, 2006, 6:14am » 
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Thank you for solving it.


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


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



