I'm working on Heisenberg group and I want to understand the suspension viewpoint. Let me be more precise. Let us denote by $\mathbb{H}^3(A)$ the set of matrix
\begin{equation} \begin{pmatrix} 1 & x& z \\ 0 &1 & y\\ 0 &0&1 \end{pmatrix} \end{equation} with $A$ a ring. It is easy to show that is a group for the natural product. Now we put
\begin{equation} G = \mathbb{H}^3(\mathbb{R}) \backslash \mathbb{H}^3(\mathbb{Z}) \end{equation}
A normal subgroup of $G$ is the set of class :
\begin{equation} \begin{pmatrix} 1 & 0& z \\ 0 &1 & y\\ 0 &0&1 \end{pmatrix} \end{equation} (which is isomorphic to the torus $\mathbb{T}^2$)
The quotient is then an other Lie group. Indeed it is a also a suspension of the torus by the following linear map :
\begin{equation} \begin{pmatrix} 1 & 1 \\ 0 &1 \end{pmatrix} \end{equation}
If one have an hint of proof that the both definition give us the same topological object, as a manifold I will be very gratefull (one by suspension and the other by the quotient that I've defined above).