Is there a domain in $\mathbb{R}^3$ with finite non-trivial $\pi_1$ but $H_1=0$?

The exterior of the Alexander Horned Sphere has $H_1=0$ but $\pi_1\neq 0$, in fact, $\pi_1$ is infinite. (See Hatcher p.171-172). Is there an example of a domain (connected open set) in $\mathbb{R}^3$ where $\pi_1$ is non-trivial but finite, and $H_1=0$?

(This question was stimulated by the question Example of a domain in R^3, with trivial first homology but nontrivial fundamental group.)

Solution 1:

Here is a general theorem that does the job:

Theorem. If $\Omega$ is an open connected subset of $R^3$ then $\pi_1(\Omega)$ is torsion-free.

Proof. Suppose not. It is a theorem of D.B.A. Epstein (see theorem 9.8 in the book "3-manifolds" by J.Hempel) that if $M$ is a connected oriented 3-manifold whose fundamental group has nontrivial elements of finite order, then $M$ splits as a connected sum $M=M_1 \# M_2$, where $M_1$ is a compact manifold with $\pi_1(M_1)$ finite and nontrivial. Now, applying this theorem to the domain $\Omega$ we obtain that $S^3$ contains a compact submanifold $N$ with nonempty boundary, such that $\pi_1(N)$ is finite and nontrivial. It follows that the boundary of $N$ is a disjoint union of 2-spheres (see my answer here). However, attaching balls to the spherical boundary components of a manifold of dimension $\ge 3$ does not change its fundamental group. Therefore, $\pi_1(S^3)\cong \pi_1(N)$ is finite and nontrivial. Contradiction. qed