What is the Classifying Space of the Discrete Heisenberg Group? Which paper/book contains a detailed proof?


Thank you for your time.



The Classifying Space of the Discrete Heisenberg Group
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.


