About Classification of compact 2-manifolds

Question

I have question about classification of compact 2-manifolds. I have read in some sources that every compact 2-manifold is diffeomorphic with sphere with n-holes or sphere with m-mobius strips for some natural n or m (e. g. torus is a sphere with 1 hole and klein bottle is a sphere with 2 mobius strips). But shouldn't there also be a possibility to be a compact 2-manifold with some non-zero number of holes and some non-zero number mobius strips? I don't see a reason why there should be allowed only holes or only mobius strips.

Answer 1

If $\Sigma$ is a closed connected two-dimensional manifold, then

• $\Sigma$ is diffeomorphic to the connected sum of $n \geq 0$ tori if $\Sigma$ is orientable, or
• $\Sigma$ is diffeomorphic to the connected sum of $m \geq 1$ real projective planes if $\Sigma$ is non-orientable.

What about surfaces which are connected sums of tori and real projective planes? The answer is that such a surface is diffeomorphic to the connected sum of real projective planes only. This follows from the fact that $T^2\#\mathbb{RP}^2$ is diffeomorphic to $\mathbb{RP}^2\#\mathbb{RP}^2\#\mathbb{RP}^2$; see here for a proof, and here for a visual illustration. So for $m > 0$, the manifold $nT^2\#m\mathbb{RP}^2$ is diffeomorphic to $(2n + m)\mathbb{RP}^2$.