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It seems as though "nice" spaces don't have torsion in their homology groups. What is the underlying characteristic of these nice spaces; that they can be embedded in $\mathbb{R}^3$? So what are some examples of spaces which can be embedded in $\mathbb{R}^3$, but have torsion in their homology groups?

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If by "spaces" you mean "compact surfaces"... – Qiaochu Yuan Jun 8 '12 at 2:18
Are you implying the real projective plane is somewhat unnice?! :) – Mariano Suárez-Alvarez Jun 8 '12 at 2:19
If you want to restrict your attention to subsets of $\mathbb R^3$,… is quite relevant. In particular, for sensible subsets there is no torsion. – Mariano Suárez-Alvarez Jun 8 '12 at 2:20
Dear user, Just as a pedagogical comment: the apparent phenomenon of homology of "nice" spaces being torsion free is just a reflection of the fact that the first examples one sees are low-dimensional, or very simple spaces such as spheres. It's worth noting that already compact three manifolds will very often have torsion in their $H_1$. Regards, – Matt E Jun 8 '12 at 3:00
If $G$ is any group, then there is a $2$-dimensional cell complex $X$ with $\pi_1(X) \cong G$; and then $H_1(X)$ is isomorphic to $G$ made abelian. So, as Mariano asks, are these spaces not nice? – Ronnie Brown Jun 8 '12 at 13:55

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