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Any closed topological surface can be constructed from a fundamental polygon with $2n$ sides, with pairs of edges identified according to a specified orientation.

  1. Which closed surface results from identifying opposite sides of a $2n$-gon, such that all pairs of identified sides have a parallel orientation (with respect to the geometry inherited from the Euclidean plane into which the fundamental polygon is embedded)?

  2. Which closed surface results from identifying opposite sides of a $2n$-gon, such that all pairs of identified sides have an antiparallel orientation? (E.g. if we use arrows to indicate the identification orientation and the arrows form an oriented loop around the fundamental polygon's entire perimeter.)

My naive guess would have been that it's the connected sum of $n-1$ tori in the parallel case and the connected sum of $n-1$ real projective planes in the antiparallel case. But this guess is wrong: in the parallel case it's actually the simple torus for both $n = 2$ (the textbook case) and $n = 3$, while it's the connected sum of two tori for $n = 4$. My revised guess is that in general it might be the connected sum of the floor $\lfloor n/2 \rfloor$ tori in the parallel case, and similarly for real projective planes in the antiparallel case. But that's basically a wild guess based on three data points and the intuition that it should be a simple pattern.

tparker
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  • You should be able to count the number of vertices, edges, and faces (even in the general case), then compute the Euler characteristic, and then work out the surface from the classification theorem. – Gerry Myerson Sep 13 '21 at 04:28
  • @GerryMyerson Unfortunately, I've never formally studied topology (just read the Wikipedia article), so I don't know how to do that. – tparker Sep 13 '21 at 04:31
  • Actually, the way you have set it up, it's just a single projective plane, regardless of the value of $n$, in the second set-up. – Gerry Myerson Sep 13 '21 at 04:34
  • Chapters 2, 3, and 4 of Chris Cooper's notes will show you how to go. http://www.coopersnotes.net/third_topology.html – Gerry Myerson Sep 13 '21 at 04:36
  • I think your guess is exactly right for the tori case. – Gerry Myerson Sep 13 '21 at 04:44
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    See https://math.stackexchange.com/questions/4088573/does-any-polygon-with-side-number-2n-with-n-ge-2-form-a-torus-when-all-pair – Gerry Myerson Sep 13 '21 at 04:46
  • As long as all pairs of opposite sides are identified parallel, you have specified a unique surface. If the sides of a square are named $a,b,c,d$ in order around the square, and $a$ and $c$ are identified parallel, and $b$ and $d$ are identified parallel, you get a torus. I don't see how you're getting a sphere. – Gerry Myerson Sep 13 '21 at 05:16
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    @GerryMyerson Oops, sorry, I misread the diagram for a fundamental polygon for the sphere on Wikipedia - it was identifying adjacent rather than opposite sides together. – tparker Sep 13 '21 at 12:33

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