Corollary 12.7 (Generators for $\pi_1(S^1, 1)$):

(1) The generators for $\pi_1(S^1, 1)$ are given by the loops

$$\eta : t\mapsto \exp(\pm 2\pi it) \eqno(12.4)$$

(2) The loops (12.4) also generate the group $\pi_1(\mathbb{C} - \{0\}, 1)$.