Theorem 11.8:

(i) The fundamental group of the punctured plane is the additive group $\mathbb{Z}$ and the homotopy class of the loop

$$t\mapsto \exp(2\pi it),\quad 0 \leq t \leq 1 \eqno(11.7)$$

provides a generator for the group.

(ii) The fundamental group of $\mathbb{R}^n - \{0\}$ is the trivial group.