Let $M\_1$ and $M\_2$ be nilmanifolds. We can see them as total spaces of torus bundles $\pi\_i:M\_i \to B\_i\ \ i=1,2$. Suppose that $B\_1=B\_2$ and that the fibers are torus of the same dimension and consider the fiber product $M\_1 \times\_B M\_2$ (which I believe is the same thing as the pullback bundles $\pi\_1^\*M\_2$ and $\pi\_2^\*M\_1$).


Then $M\_1\times\_B M\_2$ is also a torus bundle over $B$. But is it a nilmanifold?



Fiber product of nilmanifolds
The answer is yes.


We can instead view $M\_1\times\_B M\_2$ as a principal torus bundle over $M\_1$, as in the following pullback diagram:
$$
\begin{array}{ccc}
{T}\_2 & = & T\_2 \newline
\downarrow & & \downarrow \newline
M\_1\times\_B M\_2 & \to & M\_2 \newline
\downarrow & & \downarrow \newline
M\_1 & \to & B
\end{array}
$$
Since a (compact) nilmanifold is nothing but an iterated principal torus bundle, and $M\_1$ is a nilmanifold, it follows by induction that $M\_1\times\_B M\_2$ is a nilmanifold.


