# Is there a non-simply connected subspace of $\mathbb R^2$ with trivial first homology?

Exactly what the title says. Can you find some topological space $$X\subset\mathbb R^2$$ such that $$\pi_1(X)\neq0$$, but $$\mathrm H_1(X,\mathbb Z)=0$$?

I've been told that this paper shows that the fundamental group of any subspace of $$\mathbb R^2$$ has a torsion-free fundamental group, so at the very least, for such a space to exist there has to be some torsion-free group with trivial abelianization. I do not know if such a group exists.

Edit: It turns out that simple torsion-free groups exist, so there are torsion-free groups with trivial abelianization.

By the references in this answer, fundamental groups of subsets of the plane are residually free, and in particular, if $$\pi_1(X)$$ is nontrivial it surjects onto a nontrivial free group. Because free groups have nontrivial abelianization, we see that $$\pi_1(X)$$ surjects onto an abelian group, and hence $$H_1(X) = \pi_1(X)^{\text{ab}}$$ is nontrivial.