An orientable manifold can have torsion in its integer homology. But I believe by Poincare duality the manifold must be at least 4-dimensional -- isn't that right? Anyway are there simple examples of such torsion?



Example of torsion in orientable manifolds?
Consider $PSU(2)$, the three-dimensional projective special unitary group (or just $\mathbb{R}\mathbb{P}^3$). It is a Lie group, and therefore orientable. Yet, it is the quotient of the simply-connected group $SU(2)$ by its center $\mathbb{Z}\_2$, so it has 2-torsion in homology. 


