Computing the fundamental group

Question

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.

Do you know polygonal representation of double torus, look at it in Hatcher... — Anubhav Mukherjee, Apr 29 '16 at 22:42
Do you know how to compute fundamental group of those objects? — Anubhav Mukherjee, Apr 29 '16 at 22:44

score 0 · Answer 1 · answered Apr 29 '16 at 22:45

0

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?

answered Apr 29 '16 at 22:45

Qiaochu Yuan

419,620

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? – user162343 Apr 29 '16 at 22:47
Without using what??? – Anubhav Mukherjee Apr 29 '16 at 22:49
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 :) – user162343 Apr 29 '16 at 22:50
I mean, you can compute the fundamental group of a punctured torus more or less from scratch too. – Qiaochu Yuan Apr 29 '16 at 22:50
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 – user162343 Apr 29 '16 at 22:53
Should I give up and do it with bouquet circles? – user162343 Apr 29 '16 at 22:58
@Anubhav.K , Quiaochu Yuan what can be done? – user162343 Apr 29 '16 at 23:35
Hello @QiaochuYuan Could you finish the computation please? – user162343 May 02 '16 at 21:04
is because I have no relations to work with – user162343 May 02 '16 at 21:33

Computing the fundamental group

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