# Thread: Fundamental group of the connected sum of Tori

1. ## Fundamental group of the connected sum of Tori

How do you compute the Fundamental group of the connected sum of infinitely many tori?

I know the Fund. Group of the connected sum of n tori is given by

$\displaystyle \pi _1 (T\# T\# \cdots \# T)= < \beta _1 , \gamma _1 , \cdots , \beta _n , \gamma _n / \beta _1 \gamma _1 \beta^{-1} _1 \gamma^{-1} _1 \cdots \beta _n \gamma _n \beta^{-1} _n \gamma^{-1} _n =1>$.

But what about the Fundamental group of the surface given by the connected sum of infinitely many tori, and how do you prove this?

2. Have you encountered the Seifert-Van Kampen Theorem yet?

3. yes, but I am not sure how to apply it here in this case. What would be the path connected open sets A and B whose union is the connected sum? Note, the connected sum is infinite, from the left and from the right, when drawing the space,...), and how to show their intersection is path connected?

4. Splitting it up into two (or finitely many) pieces isn't going to be helpful...luckily S-VK holds for decompositions into arbitrarily many subsets. Are you seeing this in Hatcher? If so, there's a suggestive diagram.

5. I can see the diagram in Hatcher, so would you instead split into infinitely many 2-spheres?( I mean, by cutting right through the suggested equators). Then at least they would be path connected...