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For $u=e^{2\pi iy}$ and $w=e^{2\pi iz}$, defineChanging $y$ or $z$ by an integer changes $v(-t,y,z)$ by left multiplication by an element of $\Gamma$, so $\Phi$ is independent of the chosen lifts. It also respects the mapping-torus seam becausewhich is exactly the identification

Every $\Gamma$-orbit has a representative with its first coordinate in $[-1,0]$, so $\Phi$ is surjective. If two such representatives lie in one orbit, the integer shift of their first coordinates is zero unless they are opposite endpoints; in the first case their torus coordinates agree, and in the endpoint case they are related by the displayed seam. Thus $\Phi$ is injective. It is continuous, $M$ is compact, and $N$ is Hausdorff, so a continuous-bijection theorem givesThis realizes the Heisenberg nilmanifold as a torus mapping torus.