For every maximal dual $\Delta$-complex $\Pi$ there exists a stratified $T^{n}$-fibration $\lambda^{\circ}:{V}^{\circ}\to\Pi$. This fibration satisfies to the following properties

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  the induced map $(\lambda^{\circ})^{*}:H^{n}(\Pi;\mathbb{Z})\to H^{n}({V}^{\circ};\mathbb{Z})$ is injective, where $H^{n}(\Pi;\mathbb{Z})\approx\mathbb{Z}^{p_{g}}$, $p_{g}=h^{n,0}$ is the geometric genus of $V$;

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  for each primitive piece $U_{j}$ of $\Pi$ (see Definition 5) the inverse image $(\lambda^{\circ})^{-1}(U_{j})$ is an open pair-of-pants $\mathcal{P}_{n}$.

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  for each $n$-cell $e$ of $\Pi$ there exists a point $x\in e$ such that the fiber $(\lambda^{\circ})^{-1}(x)$ is a Lagrangian $n$-torus $T^{n}\subset V$;

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  there exist Lagrangian embeddings $\phi_{k}:S^{n}\to{V}^{\circ}$, $k=1,\dots,p_{g}$ such that the cycles $\lambda^{\circ}(\phi_{k}(S^{n}))$ form a basis of $H_{n}(\Pi)$.