Topology: Homeomorphic Mapping

[tylerc1991]

Homework Statement

Let X = [0,2 PI] x [0,2 PI] and let D be the partition of X that contains:
-{(x,y)} when x =/= 0,2 PI and y =/= 0,2 PI
-{(0,y),(2 PI,y)} for 0 <= y <= 2 PI
-{(x,0),(x,2 PI)} for 0 <= x <= 2 PI

Let U be the quotient topology on D induced by the natural map p: X -> D.

Prove that (D,U) is homeomorphic to S^1 x S^1 (let this = Y)

where S^1 = {(x,y) in R^2 : x^2 + y^2 = 1}

The Attempt at a Solution

OUTLINE:
Instead of going from D to Y I was thinking of going from X to Y. Essentially I would be going from a 2 PI by 2 PI square onto the torus of inner radius 0 and outer radius 2. Say this function is f. Then I could show that f is homeomorphic. Then I have a function p relating X and D, as well as a function f relating X and Y. Then I could relate D and Y and be done with the problem.

Is this the right idea for the problem? If so, I have been messing around with a few different functions for f but can't map that square onto the torus for the life of me. Can anyone give me a kind hand of assistance please?

 

[mighty2000]

Your approach seems to be on the right track. Going from X to Y would be a more straightforward way to show the homeomorphism between D and Y. However, instead of trying to map the entire square onto the torus, it might be easier to first map each individual side of the square onto a circle, and then combine them to form the torus.

Here is a possible approach:

1. Map the left and right sides of the square onto the circle S^1 by using the function f(x,y) = (0,y) and g(x,y) = (2 PI,y) respectively. These are the two sides that have been collapsed in D to form the two circles in Y.

2. Map the top and bottom sides of the square onto the circle S^1 by using the function h(x,y) = (x,0) and k(x,y) = (x,2 PI) respectively. These are the two sides that have been collapsed in D to form the two circles in Y.

3. Now, you have four separate circles (two from f and g, and two from h and k) in X that have been mapped onto the four separate circles in Y. You can then combine these four circles in X to form the torus by using a function that wraps the circles around each other. For example, you can use the function m(x,y) = (cosx,cosy,sinx,siny) where (cosx,sinx) and (cosy,siny) are the coordinates on the circle S^1.

4. Finally, you can show that this function m is continuous and bijective, and hence a homeomorphism between X and Y. Since p is also a homeomorphism between X and D, this means that D and Y are homeomorphic.

I hope this helps. Good luck with your proof!