# Homeomorphism of connected sums

• Jul 28th 2012, 01:12 PM
ModusPonens
Homeomorphism of connected sums
Hello.

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

(Bow)
• Aug 4th 2012, 07:46 PM
ModusPonens
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 $S^2$ is the identity in the operation #. It sufices to say that a sphere without a part that is homeomorphic to an open disk $D^2$ 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?

(Bow)
• Aug 4th 2012, 08:07 PM
GJA
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.
• Oct 22nd 2012, 02:02 AM