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.
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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 $(2n + m)\mathbb{RP}^2$.
Michael Albanese
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Could You explain what You mean by $T^{2}$ and $#$ sign respectively - is it cartesian product? Thanks. – robin3210 Dec 31 '20 at 15:45
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Ah, $T^{2}$ is a tori, sorry. – robin3210 Dec 31 '20 at 15:46
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$#$ denotes connected sum. – Michael Albanese Dec 31 '20 at 15:48
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@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 $T^2$". – Lee Mosher Dec 31 '20 at 15:52
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@MichaelAlbanese: Do you know who first realized that $\mathbb{RP}^2#\mathbb{RP}^2#\mathbb{RP}^2\simeq T^2 #\mathbb{RP}^2$? Or it was too obvious in the birth of topology era? – C.F.G Dec 31 '20 at 16:50
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1@C.F.G: According to Wikipedia, this is due to Walther von Dyck. – Michael Albanese Dec 31 '20 at 17:55