Torus Excluding Disc: Boundary of RP^2 X RP^2

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Homework Help Overview

The discussion revolves around the topology of a torus with a disk removed, specifically exploring its homeomorphic properties and boundaries. Participants are examining the implications of excluding a disk from a toroidal surface.

Discussion Character

  • Exploratory, Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants are attempting to determine what the torus excluding a disk is homeomorphic to, with some suggesting it may relate to RP^2 X RP^2. Others question the nature of the remaining surface, debating whether it can be considered simply connected or if it retains a hole.

Discussion Status

The discussion is ongoing, with various interpretations being explored. Some participants have offered insights into the nature of the boundary and the implications of removing a disk, while others are questioning the assumptions regarding homeomorphism and connectivity.

Contextual Notes

There is a lack of consensus on the definitions and properties being discussed, particularly regarding the relationship between the torus and the resulting surface after the disk is removed. Participants are also considering the implications of homotopy in relation to the problem.

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Homework Statement


What is the torus excluding a disc homeomorphic to?

What is the boundary of a torus (excluding a disc)?

The Attempt at a Solution


RP^2 X RP^2?

As a guess.
 
Last edited:
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Excluding a disk? You mean you slice a disk out of the torus? What's left is simply connected and looks to me like it is homeomorhic to a ball.
 
Yes, slice out a disk. A torus is a surface so it hollow? A ball is a solid. The torus still has a hole in it. How can it be homeomorphic to a ball?

I'd say it is homeomorphic to a proper torous which is homeomorphic to what?
 
Last edited:
What is the boundary of a torus excluding a disc?
 
What is it homeorphic to? Infinitely many things, obviously. But I don't immediately see them as being interesting. Now, what is it homotopic to, there is an interesting question.

The boundary of a torus excluding a (closed) disc is obvious, surely. What do you think happens to an object without a boundary if we remove something like a disc?
 

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