Neighborhood Retract of Boundary

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SUMMARY

The discussion focuses on finding a retraction map \( r: U \to \partial M \) for a manifold \( M \) with boundary, specifically utilizing the Collar Neighborhood Theorem. The user aims to prove the existence of such a retraction by demonstrating local extendability and employing Zorn's Lemma for constructing the neighborhood retraction. A challenge arises in the local extendability step, particularly when the boundary is not assumed to be compact. Clarification is sought regarding the specifics of locally extending a retraction, with a suggestion that the open sets on the boundary may be isomorphic to half spaces.

PREREQUISITES
  • Understanding of manifold theory, specifically manifolds with boundary.
  • Familiarity with the Collar Neighborhood Theorem.
  • Knowledge of Zorn's Lemma and its applications in topology.
  • Concept of local extendability in the context of topological maps.
NEXT STEPS
  • Research the implications of the Collar Neighborhood Theorem in manifold topology.
  • Study the application of Zorn's Lemma in constructing topological objects.
  • Explore local extendability techniques for retraction maps in topology.
  • Investigate the properties of open sets in manifolds and their isomorphism to half spaces.
USEFUL FOR

Mathematicians, particularly those specializing in topology and manifold theory, as well as graduate students seeking to deepen their understanding of boundary retractions and related theorems.

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

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