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?
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3$\begingroup$ Three-dimensional lens spaces often have nontrivial torsion in $H_1$ over $\mathbb Z$. $\endgroup$ – John Pardon Jan 28 '15 at 3:05
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5$\begingroup$ $RP^3$ is a simple example as well. $\endgroup$ – Thomas Rot Jan 28 '15 at 5:38
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$\begingroup$ @JohnPardon I learned of lens spaces from Seifert and Threlfall when I first began learning of homology and I did not know what mattered about them so I did not retain much. They include $\mathbb{RP}^3$ and a quick introduction is map.mpim-bonn.mpg.de/Lens_spaces:_a_history $\endgroup$ – Colin McLarty Jan 28 '15 at 14:50
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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.
answered Jan 28 '15 at 3:19
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1$\begingroup$ I think this example would be more recognizable as $\mathbb{RP}^3$. $\endgroup$ – Qiaochu Yuan Jan 28 '15 at 9:23
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$\begingroup$ @QiaochuYuan Yes. I can picture $\mathbb{RP}^3$. But let me check my picture: the torsion 2-cycle is the natural projective plane in $\mathbb{RP}^3$, right? $\endgroup$ – Colin McLarty Jan 28 '15 at 13:56
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2$\begingroup$ The torsion is in the homology group in dimension 1, not in $H_2$. For a closed n-manifold M, Poincaré duality and the Universal Coefficient theorem imply that $H_{n-1}(M)$ is torsion-free. Presumably that's what you were thinking of in your original question. In this case, the torsion 1-cycle is the $RP^1$ in $RP^3$. $\endgroup$ – Danny Ruberman Jan 28 '15 at 17:07
