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Constructing Explicit Deformation Retractions

  1. Sep 11, 2011 #1
    I don't really know why, but I'm having trouble actually building deformation retractions, although I understand the concepts behind homotopies, etc.


    For example, when constructing a deformation retraction for [tex]\mathbb{R}^n-\{0\}[/tex] to [tex]S^{n-1}[/tex], I found that you could define the mapping [tex]F(x,t) = (\frac{x_1}{t||x||+(1-t)},...,\frac{x_n}{t||x||+(1-t)})[/tex]. However, I still don't see how you can think of such a thing..
    I get the idea of turning the x_ns into unit vectors, but I don't understand the intuition behind the +(1-t), etc.



    Anyone want to give some advice?
     
  2. jcsd
  3. Sep 12, 2011 #2
    I am not sure what you would like to know. Your mapping F(x,t) is such that F(x,0)=x, hence F(.,0)=id on R^n - {0} and F(x,1)=x/|x| is in the (n-1)-sphere. This is the reason the t's and (t-1)'s are put there as they are.
     
  4. Sep 12, 2011 #3
    ^ Never mind; I got my answer somewhere else. But what I meant was how someone comes up with such an idea. It's easy to see that the mapping is a det. ret., but it's harder to actually produce the mapping to begin with (or what I was having trouble on anyways).
     
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