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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.

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Let's write $M = H_3(\mathbb{R})/H_3(\mathbb{Z})$. The manifold $H_3(\mathbb{R})$ is a Lie group, so it is certainly orientable. (An $n$-manifold is orientable if the structure group of its tangent bundle can be reduced to $\operatorname{GL}_n^+(\mathbb{R})$. But a Lie group has a trivial tangent bundle: the structure group can be reduced to $\{1\}$!) The elements of $H_3(\mathbb{Z})$ are translations on the Lie group $H_3(\mathbb{R})$, so each one is an orientation-preserving diffeomorphism. It follows that $M$ is orientable. $M$ is also clearly compact and connected, so Poincare Duality applies:

$H_0(M,\mathbb{Z}) \cong H_3(M,\mathbb{Z}) \cong \mathbb{Z}$,

$H_1(M,\mathbb{Z}) \cong H_2(M,\mathbb{Z}) \cong H_3(\mathbb{Z})^{\operatorname{ab}}$.

So it remains to compute the abelianization $H_3(\mathbb{Z})^{\operatorname{ab}}$. If you look at the generators and relations for $H_3(\mathbb{Z})$ given in the wikipedia article, it is almost immediate that the abelianization is $\mathbb{Z}^2$.

Also, there are many important classes of 3-manifold; does the Heisenberg manifold belong to some of them?

Well yes, of course. That's a terribly vague question, and I'm not an expert here, but the Heisenberg group is the simplest example of a compact nilmanifold beyond tori. Nilmanifolds are important in a wide variety of areas of mathematics, e.g. in group cohomology, geometry and algebraic topology. Remarkably, in the last several years they have been studied by Green and Tao for their applications in additive combinatorics! The aforelinked wikipedia article contains some references, enough for you to know much more about this subject area than I do.

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  • $\begingroup$ What point of view makes it clear that M is compact? Or said another way, what's the simplest proof of compactness? If I'm not mistaken, there are somewhat similar situations in which the quotient is non-compact (specifically, there exist discrete subgroups of Lie groups such that for some fixed N, every point of the Lie group is distance at most N from a point in the discrete subgroup, and yet the quotient is non-compact). $\endgroup$ – Dan Ramras May 25 '16 at 21:37
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    $\begingroup$ @Dan: I would suggest writing down an explicit compact fundamental domain. $\endgroup$ – Pete L. Clark May 25 '16 at 21:44
  • $\begingroup$ Thanks; that seems reasonable. Another method is to show that M is a circle bundle over a torus. I was figuring out how to do that yesterday when I came across this answer. $\endgroup$ – Dan Ramras May 26 '16 at 17:10
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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.

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