Is there any domain $\Omega\subset\mathbb{R}^3$ with nontrivial homotopy group but with trivial first homology?

Question

I'm using the word domain relatively loosely here. Ideally, I mean by a domain $\Omega\subset\mathbb{R}^3$ the interior of a connected smooth 3-manifold with corners in $\mathbb{R}^3$, though if this question has an answer for any connected open $\Omega\subset\mathbb{R}^3$, that would of course also be good. I know that for general manifolds $M$, it is possible that $\pi_1(M)\ne 0$ but $H_1(M)=0$, but how about if we only consider the case for domains $\Omega\subset\mathbb{R}^3$? Does this remain true?