# Triangulation of double-holed torus to calculate fundamental group

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

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

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

• Why do you need to triangulate the space? Do you know Seifert -Van Kampen? May 19, 2016 at 21:58
• 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 May 19, 2016 at 22:02
• 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. May 19, 2016 at 22:06
• 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. May 19, 2016 at 22:16
• I put the image below. May 19, 2016 at 22:41