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Follow the question, how do we characterize a generic unorientable 2d manifolds (whose the first stiefel whitney class of the tangent bundle is nonzero)?

For example, for a generic orientable 2d manifolds, we can use

  • the genus $g$
  • the Euler character $\chi=2-2g$

  1. How about the full characterization and characters of a generic unorientable 2d manifolds?

  2. If you have comments on higher dimensions (3d, 4d, 5d, etc), this will also be welcome too.

annie marie cœur
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    Closed surfaces are fully classified by their Euler characteristic $\chi$. If $\chi_S=2-2g$ is even, $S$ is orientable and homeomorphic to the connected sum of $g$ tori. If $\chi_S = 2-k$ is odd, $S$ is homeomorphc to the connected sum of $k$ projective plane. The classification of closed manifold of heigher dimension is a hard and non-solved problem. For example, in dimension $3$, it took a century to prove that every compact simply connected $3$-manifold is homeomorphic to the 3-sphere: it is the Poincaré conjecture, solved by Perelman in early 2000's. – Didier Nov 22 '20 at 23:45
  • can you remind me $_$ for: 1. Mobius strip, 2. $RP^2$ and 3. Klein bottle? – annie marie cœur Nov 22 '20 at 23:47
  • The mobius strip is not a closed surface (it has a boundary). $\mathbb{R}P^2$ is the projective plane... If you read carefully what I stated, this implies it has Euler characteristic $1$. The Klein bottle is the connecyed sum of two projective planes. – Didier Nov 22 '20 at 23:49
  • But we can still define $_S$ for: 1. Mobius strip $_S=?$ , 2. $^2$ $_S=1$ and 3. Klein bottle $_S=2$ ? – annie marie cœur Nov 22 '20 at 23:50
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    I already answered for the Klein bottle and the projective space. Please read carfully what peaople answer you. The mobius strip is not a closed surface, thus the classification does not concern it, even if there is a notion of Euler characteristic for it: it is 0. – Didier Nov 22 '20 at 23:52
  • The first stiefel whitney class of the tangent bundle. – Connor Malin Nov 22 '20 at 23:56
  • @ConnorMalin I think you misunderstood the question. It is not ask how to know if a surface is orientable or not, but how to classify unoriented surfaces. (The title is not clear) – Didier Nov 22 '20 at 23:59
  • Many thanks -- The point is that I hope to know a set of numbers (like genus, or Euler character ) to have a one to one map to generic unorientable manifolds. So every unorientable manifold has its unique ID. – annie marie cœur Nov 23 '20 at 00:03
  • You should decide if your surfaces are connected, compact, have no boundary, etc. Depending on this, you will have a different set of IDs. For connected compact surfaces with empty boundary, see for instance https://math.stackexchange.com/questions/3203841/classification-of-surfaces – Moishe Kohan Nov 23 '20 at 00:52

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