Neighborhood Retract of Boundary

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In summary, the conversation discusses finding a retraction from a neighborhood of the boundary of a manifold with boundary. The Collar Neighborhood Theorem is mentioned as a potential solution, but the speaker is using it to prove another theorem. Their approach involves showing local existence and using Zorn's Lemma to construct a neighborhood retraction, but they are unsure about how to prove local extendability. Another person mentions that they did not need to use Zorn's Lemma and it may be because they assumed the boundary was compact. The conversation also mentions that the open sets on the boundary are isomorphic to half spaces.
  • #1
jgens
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Here is the problem: If M is a manifold with boundary, then find a retraction r:U→∂M where U is a neighborhood of ∂M.

I realize the Collar Neighborhood Theorem essentially provides the desired map, but I am actually using this result to prove the aforementioned theorem. My thought on how to prove the theorem is to show local existence, show that you can locally extend a retraction, and then use Zorn's Lemma to construct a neighborhood retraction of the boundary. The only difficulty I run into here is showing local extendability. Can anyone help me with this step?
 
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  • #2
I don't remember needing Zorn's Lemma when I had to do this exercise, but I think we were allowed to assume the boundary was compact.

Can you be more specific about what you mean by locally extending a retraction?
 
  • #3
i think the open sets on the boundary are isomorphic to half spaces, if that helps any.
 

1. What is a neighborhood retract of boundary?

A neighborhood retract of boundary is a topological concept that refers to a subset of a topological space that can be continuously deformed to a lower-dimensional subset of itself, while keeping its boundary fixed. In other words, it is a way to shrink a space down to a smaller space while maintaining its boundary.

2. How is a neighborhood retract of boundary different from a deformation retract?

A neighborhood retract of boundary differs from a deformation retract in that it must preserve the boundary of the subset, while a deformation retract only requires that the subset be continuously deformed to a point. In other words, a neighborhood retract of boundary is a stricter form of a deformation retract.

3. What is the significance of a neighborhood retract of boundary in topology?

A neighborhood retract of boundary is significant in topology because it helps to simplify the study of topological spaces. It allows for the reduction of a complex space to a simpler one while still preserving important topological properties. This makes it easier to analyze and understand the space.

4. Can a neighborhood retract of boundary exist in any topological space?

No, a neighborhood retract of boundary can only exist in certain types of topological spaces, such as manifolds, CW complexes, or homotopy equivalent spaces. It also depends on the specific properties of the space, such as the dimension and compactness.

5. What are some applications of neighborhood retract of boundary in mathematics?

The concept of neighborhood retract of boundary has various applications in mathematics, particularly in algebraic topology and differential geometry. It is used to prove theorems and classify spaces, and also has applications in fields such as robotics and computer graphics where shape deformation is important.

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