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

[pivoxa15]

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.

 

[HallsofIvy]

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.

 

[pivoxa15]

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?

 

[pivoxa15]

What is the boundary of a torus excluding a disc?

 

[matt grime]

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?