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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 $(2n + m)\mathbb{RP}^2$.

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  • $\begingroup$ Could You explain what You mean by $T^{2}$ and $#$ sign respectively - is it cartesian product? Thanks. $\endgroup$
    – robin3210
    Commented Dec 31, 2020 at 15:45
  • $\begingroup$ Ah, $T^{2}$ is a tori, sorry. $\endgroup$
    – robin3210
    Commented Dec 31, 2020 at 15:46
  • $\begingroup$ $\#$ denotes connected sum. $\endgroup$ Commented Dec 31, 2020 at 15:48
  • $\begingroup$ @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 $T^2$". $\endgroup$
    – Lee Mosher
    Commented Dec 31, 2020 at 15:52
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    $\begingroup$ @C.F.G: According to Wikipedia, this is due to Walther von Dyck. $\endgroup$ Commented Dec 31, 2020 at 17:55

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