The Fundamental Group of the Circle Is ℤ

Let $\varepsilon : \mathbb{R} \to S^1$ be the covering map $\varepsilon(t) = e^{2\pi i t}$, and let $\gamma_n : [0,1] \to S^1$ be the loop $\gamma_n(s) = e^{2\pi i n s}$ based at $1 \in S^1$ (winding $n$ times around the circle).

(a) Without any machinery, explain why $\gamma_1$ and $\gamma_2$ cannot be homotopic as loops based at $1$.

(b) Conclude that $\pi_1(S^1, 1) \cong \mathbb{Z}$, and identify what the integer attached to a loop $\gamma$ represents geometrically.