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

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

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

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

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

Maximal dual $\Delta_{d}$-complexes exist for every degree $d$ and every dimension $n$.