# Thread: Homeomorphism of connected sums

1. ## Homeomorphism of connected sums

Hello.

I'd like to know how to solve the following problem: $\displaystyle \mathbb{R}P^2$#$\displaystyle T^2=\mathbb{R}P^2$#$\displaystyle K^2$ , where $\displaystyle \mathbb{R}P^2$ is the real projective plane, $\displaystyle T^2$ is the torus and $\displaystyle K^2$ is the Klein bottle.

2. ## Re: Homeomorphism of connected sums

I think I did not explain myself well when posting this.

The problem is to prove the equality. But no equations are necessary. For example, in a previous exercise it's asked to prove that $\displaystyle S^2$ is the identity in the operation #. It sufices to say that a sphere without a part that is homeomorphic to an open disk $\displaystyle D^2$ is homeomorphic to a closed disk.

If that's not the problem, could you tell me if this problem is a hard one for a 3rd year student of mathematics?

3. ## Re: Homeomorphism of connected sums

Hi, ModusPonens. I haven't done a problem like this in awhile, but there is essentially a procedure you follow to do stuff like this. It starts with using the polygonal representation for the connected sums and going from there using cuting/pasting techniques. Check out a book by Christine Kinsey for the steps.