What's the fundamental group of N in this example?

Question

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In this page, in the first answer with pictures, it offered a computation of the fundamental group of torus by polygon, but what's the fundamental group of N?

Answer 1

Let $U$ be the punctured torus depicted in the link and $V$ be a ball centered at the point removed. Including a loop from $U\cap V$ into $U$ gives a loop homotopic to the boundary loop $A^{-1}B^{-1}AB$. When included into $V$ we get the trivial loop. So $N=<<A^{-1}B^{-1}AB>>$ (normally generated group). This tells you that:

$$\pi_{1}(T^{2})=\mathbb{Z}*\mathbb{Z}/<<A^{-1}B^{-1}AB>>=\mathbb{Z}\times\mathbb{Z}$$