Q:

Path on $\mathbb{R}^2\setminus \{x_1, x_2\}$

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!

A:

Here is what happens when you try to draw a pull back of your loop on a rather simple (but still non abelian) covering of your space :

You can pull back an homotopy on your initial space starting from a null path into a loop into on an homotopy on this covering starting from a null path into a pull back of the loop.
So if your loop were homotopic to 0, its pullback should be a loop homotopic to 0 on this covering. But it is not even a loop because the end points are pulled into two different parts of the covering. Therefore it can't be homotopic to the null path in the first place.

