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Show that the fundamental group of the double-holed torus is given by: $\pi_1=<a, b, c, d | aba^{-1}b^{-1}=cdc^{-1}d^{-1}>$

Double-holed torus

I have ended up with the following identification diagram representing the space:

Diagram

I am stuck with triangulating this space with simplices. Please could you help me with this?

After this I can find the generators $a, b, c$ and $d$ on the remaining 1-simplices that generate the fundamental group

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  • $\begingroup$ Why do you need to triangulate the space? Do you know Seifert -Van Kampen? $\endgroup$ May 19, 2016 at 21:58
  • $\begingroup$ Yes, triangulating the space is an alternative to the Van-Kampen theorem. It involves triangulation, drawing a maximal contractible space and considering remaining 1-simplices as generators for the fundamental group $\endgroup$
    – thinker
    May 19, 2016 at 22:02
  • $\begingroup$ that just seems like a greater hassle. Well, at least for me it would be. An easy way to get the triangulation would be to take two copies of the planar diagram for the torus. For those, we have the standard triangulation and form the connected sum via triangle in one copy, with a triangle in the other. Then you a have a triangulation of the double torus. If you need me to draw that, just let me know. $\endgroup$ May 19, 2016 at 22:06
  • $\begingroup$ if you could draw it that would be great. I think Van Kampen is a better method to use but I would like to be able to do both. $\endgroup$
    – thinker
    May 19, 2016 at 22:16
  • $\begingroup$ I put the image below. $\endgroup$ May 19, 2016 at 22:41

1 Answer 1

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Here is the image. Now just stretch the part where you glued for the connected sum and you get your triangulated octagon.

enter image description here

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  • $\begingroup$ Brilliant , this really clarified it for me.. Finding the generators is pretty fiddly now but not too hard $\endgroup$
    – thinker
    May 20, 2016 at 10:00
  • $\begingroup$ @Faraad Armwood what happen to the triangles in the corner after the cut? $\endgroup$
    – Vajra
    Oct 18, 2021 at 9:53

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