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Math Help - Proof

  1. #1
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    Proof

    We have to give a 1-page proof showing that the list we have constructed of all closed, oriented 2-manifolds (sphere, torus, 2-hole torus, 3-hole torus, etc....) is a complete list without repetitions of all closed, oriented 2-manifolds.

    So I have to show that the list is complete and I have to show that there are no repetitions.
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  2. #2
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    Quote Originally Posted by Janu42 View Post
    We have to give a 1-page proof showing that the list we have constructed of all closed, oriented 2-manifolds (sphere, torus, 2-hole torus, 3-hole torus, etc....) is a complete list without repetitions of all closed, oriented 2-manifolds.

    So I have to show that the list is complete and I have to show that there are no repetitions.
    According to the classification theorem of closed surfaces (2-manifold), any closed surface is homeomorphic to some member of one of these three families.

    1. sphere
    2. connected sum of g tori, g>=1
    3. connected sum of k real projective planes for k>=1

    1 and 2 are orientable and 3 is non-orientable.

    For 1 and 2, a surface that is the connected sum of n tori is said to be of genus n, while a sphere is of genus 0 (Intuitively speaking, a genus number corresponds a number of handles in a closed orientable surface).

    This genus number is also involved in Euler characteristic, which is topologically invariant. So we can say that we classify closed orientable 2-manifolds up to homeomorphism by using a genus.
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