Constructing Explicit Deformation Retractions

[Anonymous217]

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 $$\mathbb{R}^n-\{0\}$$ to $$S^{n-1}$$, I found that you could define the mapping $$F(x,t) = (\frac{x_1}{t||x||+(1-t)},...,\frac{x_n}{t||x||+(1-t)})$$. 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?

 

[holy_toaster]

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.

 

[Anonymous217]

^ 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).