Lemma 12.5:

If $\gamma_1$ and $\gamma_2$ are two homotopic loops based at $1$. then deg$\gamma_1 =$   deg$\gamma_2$. Thus the map $\phi : \pi_1(S^1, 1) \longrightarrow \mathbb{Z}$ given by $[\gamma] \mapsto$   deg$\;\gamma$ is well-defined.