# Computing the fundamental group

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
• Yes the octagon :) – user162343 Apr 29 '16 at 22:43
• Do you know how to compute fundamental group of those objects? – Anubhav Mukherjee Apr 29 '16 at 22:44
• That's the thing :) jajaj – user162343 Apr 29 '16 at 22:44
• I have post what I was trying to do :) – user162343 Apr 29 '16 at 22:45

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?