Constructing Explicit Deformation Retractions

In summary, the speaker is having trouble constructing deformation retractions, even though they understand the concept of homotopies. They provide an example of a mapping for a deformation retraction from \mathbb{R}^n-\{0\} to S^{n-1}, but they don't understand the intuition behind it. They ask for advice on how to come up with such a mapping, but later state that they have found the answer elsewhere.
  • #1
Anonymous217
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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?
 
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  • #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.
 
  • #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).
 

Related to Constructing Explicit Deformation Retractions

1. What is a deformation retraction?

A deformation retraction is a continuous transformation of a space onto a subspace, which is also continuous, such that each point in the subspace remains fixed throughout the transformation.

2. Why is constructing explicit deformation retractions useful?

Constructing explicit deformation retractions allows for a better understanding of the topological properties of a given space and can aid in solving problems related to homotopy, which is a fundamental concept in algebraic topology.

3. What are the steps involved in constructing an explicit deformation retraction?

The first step is to identify a subspace of the given space that will serve as the fixed point set. Then, a continuous transformation must be defined from the given space to the subspace. Next, the continuity and fixed point properties must be verified. Finally, if the transformation satisfies these properties, it can be considered an explicit deformation retraction.

4. Can any space have an explicit deformation retraction?

No, not all spaces have an explicit deformation retraction. In order for a space to have an explicit deformation retraction, it must have certain topological properties, such as being contractible or having a homotopy equivalence with a contractible space.

5. How are explicit deformation retractions used in real-world applications?

Explicit deformation retractions have various applications in fields such as computer graphics, robotics, and physics. They can be used to model deformable objects and simulate their behavior, which is crucial for animation and virtual reality. They are also used in motion planning for robots and in studying the behavior of complex systems in physics.

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