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        general >> complex analysis >> Question about Homotopy
        
(Message started by: immanuel78 on Sep 20^th, 2006, 6:18am)

Title: Question about Homotopy
Post by immanuel78 on Sep 20^th, 2006, 6:18am

Let G be a region.
Let L : [0,1] -> G be a closed rectifiable path.
Then there exists a closed, piecewise-smooth 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.

Title: Re: Question about Homotopy
Post by Icarus on Sep 20^th, 2006, 6:18pm

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 B₁, B₂, ..., 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 B₁). The piecewise linear path determined by these points satisfies the conditions.

Title: Re: Question about Homotopy
Post by immanuel78 on Sep 21^st, 2006, 6:14am

Thank you for solving it.

Title: Re: Question about Homotopy
Post by Icarus on Sep 21^st, 2006, 4:23pm

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.