Homeomorphism of connected sums

**ModusPonens** · July 28th 2012, 01:12 PM

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.

**ModusPonens** · August 4th 2012, 07:46 PM

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?

**GJA** · August 4th 2012, 08:07 PM

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.

**AlbertaDiana** · October 22nd 2012, 02:02 AM

Thanks for your sharing.I will concern about it.

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