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Primal and dual complexes

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Given a "primal" complex of dimension $n$, the boundary operator on primal $p$-cells is the transpose of the boundary operator on dual complex's $(n-p+1)$-cells. (That's the Poincare duality.)

We assume that the primal $n$-complex is a combinatorial manifold. This means that each $(n−1)$-cell is the boundary of exactly two $n$-cells and each of them induces an opposite orientation on the shared $(n-1)$-cell. In that case we can define a dual complex by identifying each $p$-cell in the primal complex with a unique $(n−p)$-cell in the dual complex.