Neighborhood Retract of Boundary

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jgens
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Main Question or Discussion Point

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?
 

Answers and Replies

  • #2
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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
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i think the open sets on the boundary are isomorphic to half spaces, if that helps any.
 

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