Torsion on $\pi_1(X)$, $X$ connected and open in $\mathbb{R}^n$

Can the fundamental group of an open connected subset $X$ of $\mathbb{R}^n$ have a torsion element?

Solution 1:

The real projective plane $\Bbb R P^2$ embeds into $\Bbb R^4$. Taking a tubular neighborhood $N$ of this embedding, we get for all $n \geq 4$ an open subset $N \subset \Bbb R^n$ such that $\pi_1(N) \cong \Bbb Z/2$, so the fundamental group of this open subset has torsion.

For $n = 2$, there is no such subset; see the following article:

Eda, K. Free $\sigma$-products and fundamental groups of subspaces of the plane. Topology and its Applications Volume 84, Issues 1-3, 24 April 1998, pp. 283-306.

This is actually an open problem for $n = 3$.

Solution 2:

As you can see here, the real projective plane $\mathbb{R}P^2$ can be embedded in to $\mathbb{R}^4$ and so, taking a sufficiently small $\epsilon$ open neighbourhood of the image of our embedding, gives a homotopy equivalent surface with fundamental group $\mathbb{Z}/2\mathbb{Z}$. This is probably the most simple case.