I found this exercise but I can't do it. the text says:
Consider the quotient space $X = T^2/\sim$, where $T^2 = S^1\times S^1$ is the $2$-dimensional torus and this $\sim$ is equivalence relation which identifies two distinct points $p$, $q$ of $T^2$. Prove that fundamental group of $X$ is $(\mathbb Z\times\mathbb Z)*\mathbb Z$.
HINT
Note that $X$ looks like $T^2$ with two points pinched together. This is homotopy equivalent to the space $Y$ which is $T^2$ with a path connecting the points $p$ and $q$ added, which is then homotopy equivalent to $T^2\vee S^1$. Then apply the Seifert-van Kampen theorem, or use a well known corollary if you're familiar with it.
