Exercise 1.2.8

Proof. Let $X$ be this topological space. Recall that the torus has $1$-skeleton $S^1\vee S^1$; by identifying one circle with that of another torus, we get the $1$-skeleton $X^1$ of $X$ to be $S^1\vee S^1\vee S^1$. Thus ${\pi }_1(X^1)=\mathbb{Z}\ast \mathbb{Z}\ast \mathbb{Z}$; let $a,b,c$ be representatives of the generators of each copy of $\mathbb{Z}$. If $a$ is the generator of the circle that is identified from the two tori, then the normal subgroup $N$ on p. 50 is the normalizer of the subgroup generated by all elements of the form $\mathit{ab}{a}^{-1}{b}^{-1}$ and $\mathit{ac}{a}^{-1}{c}^{-1}$, for each $2$-cell is attached along the loop given by these two generators. By Proposition $1.26$, ${\pi }_1(X)\approx {\pi }_1(X_1)∕N=\mathbb{Z}\times (\mathbb{Z}\ast \mathbb{Z})$. □