TITLE: The Heisenberg manifold
QUESTION [18 upvotes]: I am interested in the Heisenberg manifold, which is the quotient of the real Heisenberg group by the discrete Heisenberg (sub)group. It is a $3$-manifold which may be viewed as a circle bundle over the $2$-torus, or as a $2$-torus bundle over the circle. May I know where I can find some introductory materials of this manifold? 
For example: Is it orientable? What is its (co)homology? (Since it is a $K(\pi, 1)$ space, the homotopy groups are clear.) Also, there are many important classes of 3-manifolds; does the Heisenberg manifold belong to some of them?
Any assistance and reference will be greatly appreciated.

REPLY [12 votes]: This is a difficult question to answer completely, partially because classifying $3$-manifolds is difficult. Probably the best answer to the questions "What kind of categories that we can put $3$-manifolds in, can we come up with?" is best answered by Thurston's geometrization conjecture, which roughly says that compact manifolds may be cut into pieces, all of which can look in eight different ways.
As you will find from your list, your particular manifold has nil geometry.
A bit more should be said here: a lot of Thurston's original work involved the study of the geometry of surface bundles or mapping tori in terms of their monodromy. As you mention, your example is a particularly simple case, since it is a torus bundle, i.e. the mapping torus of a torus homeomorphism. As the Wikipedia page states, torus bundles come in three varieties (and that the same is true for higher genus surfaces is known as the Nielsen–Thurston classification), depending on their monodromy. Your manifold is the mapping torus of a Dehn twist and in some sense the simplest non-trivial surface bundle.
Another interesting class of $3$-manifolds tto which your manifold belongs is that consisting of Seifert manifolds. Now, when I was reading your question, the terminology "Heisenberg manifold" didn't really ring a bell, so I had to look it up and came across this survey, where in Example 7.4 it is stated that the Heisenberg manifold is a Seifert manifold with Seifert symbol $\{0,(o_1,1);(1,1)\}$ or, if you like, $\{ 1,(o_1,1)\}$ by the relations mentioned in the Wikipedia article.