Is there a manifold $X$ such that $H_{1}(X)$ isn't free abelian or trivial and $X$ can be embedded in $\mathbb{R}^3$?

Question

I was hoping to get more intuition for homology, but can't exactly see what would a non-free homology group tell us about a space. The simplest example I know is the real projective plane, but that's still not simple enough to give me any insight. If there is no such space, I would be grateful for a hint towards why is this impossible.

Answer 1

This is a version of several questions which were asked and answered earlier:

1. There are no 2-dimensional manifolds $M$ with torsion in $H_1$ embeddable in $R^3$ (see Mariano's answer here and Kevin's answer here).

2. Also, there are no 3-dimensional manifolds with torsion in $H_1$ homeomorphic to open subsets in $R^3$, see Eric's answer here.