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I have read a lot of books and a lot of posts in the Internet and I still can't understand few problems. Professor, who presents us a lecture "Algebraic topology", is doing his job like he must to do it. Moreover, he just translates Munkres book "Algebraic topology" without any explanation.

Back to the issue. I have an exam in next week and example exercises.

  • Find a complex with 7 vertices whose underlying space is Torus.
  • Find a complex with 8 vertices whose underlying space is Klein Bottle.

I know Torus and Klein Bottle fundamental polygons and I saw an example triangulations of both of this surfaces. How to solve these exercises with only the fundamental polygons (https://commons.wikimedia.org/wiki/Fundamental_polygon)? How to build this complexes?

Thanks for any hint.

user3725657
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  • I’m not quite sure what you mean by “find a complex.” One “solution” might be to draw a triangulation on the fundamental polygon and make sure everything lines up. This may be complicated if your definition of simplicial complex requires the simplecies to live in some Euclidean space, but one can usually handwave around or omit the part where you prove your triangulation forms simplecies in some Euclidean space. I found this sort of question hard when I was learning Alg top. I think I first went on a massive nearly-correct proof that any simplicial set is isomorphic to some simplicial complex. – Dan Robertson Jan 20 '18 at 01:01
  • Yes, I have to draw a particulary labelled (7/8 labbels) diagram (complex) which is homeomorphic to Torus/Klein Bottle.

    Example diagram for Torus with 9 labbels: https://math.stackexchange.com/questions/1650865/triangulating-torus-using-simplices/1659715.

     – user3725657 Jan 20 '18 at 07:15
  • I can’t quite remember how to do it. But you don’t want something as regular as the linked answer. That answer gives you the rules for such a triangulation to follow – Dan Robertson Jan 20 '18 at 12:43
  • I probably have found a way to solve my exercises. @DanRobertson, can you just check it? link – user3725657 Jan 20 '18 at 21:10
  • Looks right to me – Dan Robertson Jan 20 '18 at 21:12
  • I will give here an answer with my idea soon. Maybe someone will appreciate it. Thank you for help, @DanRobertson! :) – user3725657 Jan 20 '18 at 21:16

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