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I want to compute the fundamental group of the double Torus using the Seifert-van Kampen theorem so then I choose $U=\text{double Torus} / \{\text{point} =x_1\}$ and $V=D$ the disc.

The thing is that when I want to compute the fundamental group of $U$ I do a deformation retraction of $x_1$ expanding it to the wire which form is an octagon, I know that a square (no matter which direction we choose in its sides) is homotopic to the figure eight, then I suppose I will have four circles, each pair corresponding to one of the squares that form the octagon wire, but then if I want to know the fundamental group of this I use again Seifert- van Kampen. The thing is that if I choose the open sets $U,V$ as each of the pairs of circles the intersection are only two points which is not path connected.

Then, How can I set four circles that only intersect in a point?, Can someone help me with this?


Using @QuiaochuYuan 's answer how do you compute the fundamental group if you don't have any relations to work with?

Note: I have read this but I think is a little bit involve, I think it should be easier because I think I should only work on the circles not bouquet circles.

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  • $\begingroup$ Do you know polygonal representation of double torus, look at it in Hatcher... $\endgroup$ – Anubhav Mukherjee Apr 29 '16 at 22:42
  • $\begingroup$ Yes the octagon :) $\endgroup$ – user162343 Apr 29 '16 at 22:43
  • $\begingroup$ Do you know how to compute fundamental group of those objects? $\endgroup$ – Anubhav Mukherjee Apr 29 '16 at 22:44
  • $\begingroup$ That's the thing :) jajaj $\endgroup$ – user162343 Apr 29 '16 at 22:44
  • $\begingroup$ I have post what I was trying to do :) $\endgroup$ – user162343 Apr 29 '16 at 22:45
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For using Seifert-van Kampen I think it's easier to think of the surface of genus $2$ as the connected sum of two tori. So take $U, V$ to be tori with a hole cut out so that $U \cap V = S^1$. The fundamental groups of $U$ and $V$ are free groups on $2$ generators (exercise). Can you finish the computation from here?

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  • $\begingroup$ Right :) that is the other way to do it but I am supposing I know nothing about the above :), I will do both of course to check my answer but with out using those things how do you proceed? $\endgroup$ – user162343 Apr 29 '16 at 22:47
  • $\begingroup$ Without using what??? $\endgroup$ – Anubhav Mukherjee Apr 29 '16 at 22:49
  • $\begingroup$ Oh the fact that we can cut the double torus, and compute the fundamental group, I want to do it from scratch just to see how things go, and the only way to do this is to choose U, V as I did :) $\endgroup$ – user162343 Apr 29 '16 at 22:50
  • $\begingroup$ I mean, you can compute the fundamental group of a punctured torus more or less from scratch too. $\endgroup$ – Qiaochu Yuan Apr 29 '16 at 22:50
  • $\begingroup$ yes but I want to imagine the standard way, the base point in one corner of the polygonal representation, then I pick a point such that it has a neighborhood near of the base point , etc $\endgroup$ – user162343 Apr 29 '16 at 22:53

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