Theorem 36.3:

Suppose that $f : S^1 \longrightarrow S^1$ is a continuous map such that $f(1) = 1$ then the degree of $f$ as defined above agrees with the notion of degree as defined in lectures (12-13). Moreover the generator for the group $H_1(S^1)$ is the homology class of the cycle

$$\eta:t \mapsto \exp(2\pi it),\quad 0\leq t \leq 1. \eqno(36.2)$$

Note that we have tacitly identified the standard one simplex $\Delta_1$ with $[0, 1]$.