Definition 12.1:

Let $\gamma : [0,1] \longrightarrow S^1$ be a loop based at $1$. By the lifting lemma there exists unique lift ${\tilde \gamma} : [0,1] \longrightarrow \mathbb{R}$ such that ${\tilde \gamma}(0) = 0$, ex${\tilde \gamma}(1) = 1$. Thus, ${\tilde \gamma}(1) \in \mathbb{Z}$ and we call this integer the degree of the loop $\gamma$.