Consider plane $\mathbb{R}^2\setminus \{ x_1, x_2 \}$ without two points, and such closed path on this plane: (points on this picture are deleted points $x_1$ and $x_2$)
Question: how to prove that this path isn't homotopic to zero?
Appendix. As I see, we may fix some point $\alpha\in\mathbb{R}^2\setminus \{ x_1, x_2 \}$ and consider loops $a$ and $b$, which "walk round" point $x_1$ and $x_2$ respectively. Then my path will be homotopic to $b^{-1}a^{-1}ba$ as the element of fundamental group ${\pi}_{1} (\mathbb{R}^2\setminus \{ x_1, x_2 \},~\alpha)$.
Thanks a lot!