# About Classification of compact 2-manifolds

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.

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 $$(2n + m)\mathbb{RP}^2$$.
• Could You explain what You mean by $T^{2}$ and $#$ sign respectively - is it cartesian product? Thanks. Commented Dec 31, 2020 at 15:45
• Ah, $T^{2}$ is a tori, sorry. Commented Dec 31, 2020 at 15:46
• $\#$ denotes connected sum. Commented Dec 31, 2020 at 15:48
• @MaciejFicek: Keep in mind, some phrases in your post should be formulated more rigorously in connected sum language, for example: "the sphere with $n$-holes" should be formulated as "the connected sum of $n$ copies of $T^2$". Commented Dec 31, 2020 at 15:52