We know the following “geometric” version of Poincaré duality:
Let $M$ be a closed $m$-dimensional manifold and let $\mathfrak{X}_*$ be a finite simplicial complex with $|\mathfrak{X}_*|=M$. We can build a dual cell complex $\mathfrak{X}^*$: For each $p$-cell $\Sigma$ we consider as the dual cell $D(\Sigma)$ the convex hull of the barycenters of all $m$-simplices $\Sigma'$ with $\Sigma\preceq\Sigma'$. Then $|\mathfrak{X}^*|\cong M$.
The cells in the dual decomposition are not necessarily simplices, but polyhedra. The master example is the triangulation of $\mathbb{S}^2$ as the surface of the octahedron. The dual decomposition is a cube.
I am looking for a good reference for this. Even more, I want to generalise the statement to other sorts of “simplicial complexes”, where we are allowed to contract faces. (Take e. g. the Fuks complex for the one-point compactification of the unordered configuration space $C^k(\mathbb{C})$, The cells are products of simplices and many boundaries are degenerate and hence contracted to the $\infty$-point.) I would like to have the following statement:
Let $M$ be an open $m$-dimensional manifold and $\mathfrak{X}_*$ a finite “triangulation” of the one-point compactification $M\cup\{\infty\}$ where some faces may be contracted to the $0$-cell $\infty$. Then there is a dual cell complex $\mathfrak{X}^*$ with $\mathfrak{X}^p\cong \mathfrak{X}_{m-p}$ and $$\mathfrak{X}^p\cong\begin{cases}\mathfrak{X}_{m-p} &\text{for } 0\le p\le m-1,\\\mathfrak{X}_0\setminus\{\infty\} &\text{for } p=m.\end{cases}$$ This dual complex has the geometric realisation homotopy equivalent to $M$: $|\mathfrak{X}^*|\simeq M$
Something like this must exist, since it is used often as an argument for the homotopy type of $C^k(\mathbb{C})$: The Fuks complex has only cells in dimension $k+1\le d\le 2k$ and the $0$-cell $\infty$. Thus, the dual is a cellular complex of dimension $k-1$. Furthermore, the usage of the one-point compactification reminds me at the Poincaré–Lefschetz duality $H^p(M)\cong H_{m-p}(M\cup\{\infty\},\infty)$ for orientable $M$.
I would also be happy with some hints to literature about this “old”, geometric approach to duality.
