What is the Classifying Space of the Discrete Heisenberg Group? Which paper/book contains a detailed proof?
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Sign up to join this communityWhat is the Classifying Space of the Discrete Heisenberg Group? Which paper/book contains a detailed proof?
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As was said by Andy, the classifying space of the discrete Heisenberg group $\Gamma$ is $B\Gamma=G/\Gamma$, where $G$ is the 3-dimensional Heisenberg group over the reals. Due to the central extension $$0\rightarrow\mathbb{Z}\rightarrow\Gamma\rightarrow\mathbb{Z}^2\rightarrow 0$$ you may view $B\Gamma$ as a circle bundle over the 2-torus. Alternatively, viewing $\Gamma$ as the semi-direct product
$\Gamma=\mathbb{Z}^2\rtimes\mathbb{Z}$, where $\mathbb{Z}$ acts by (powers of) $\left(\begin{array}{cc} 1 & 1 \\ 0 & 1 \end{array}\right)$, you can view $B\Gamma$ as the mapping torus of this automorphism of $B\mathbb{Z}^2=\mathbb{T}^2$, i.e. as a 2-torus bundle over the circle.
If $\Gamma$ is a finitely generated torsion-free nilpotent group, then Malcev proved that there is a connected nilpotent Lie group $G$ such that $\Gamma$ is a lattice in $G$. The Lie group $G$ is often called the Malcev completion of $\Gamma$. It is an easy exercise to show that a connected simply-connected nilpotent Lie group is homeomorphic to $\mathbb{R}^n$. It follows that $G/\Gamma$ is a classifying space for $\Gamma$.
The baby example of this is $\Gamma = \mathbb{Z}^n$ and $G = \mathbb{R}^n$, so we obtain the usual $n$-torus $\mathbb{R}^n/\mathbb{Z}^n$ for the classifying space of $\mathbb{Z}^n$.
Of course, the Malcev completion of the discrete Heisenberg group is the nondiscrete Heisenberg group.