I'm having trouble understanding Poincaré duality, as it seems unmotivated. Here for instance:
https://math.stackexchange.com/a/14469/454016
Poincaré duality is explained through a duality of simplices and polyhedra. Since these are both two kinds of polytopes, and the general idea of "take the centers of all cells containing a given cell and form the convex closure" makes sense whenever you have polytopes with a clear center, I was thinking aboout whether this duality can be extended to a more complete duality between at least some polytopes. Are there any results in this direction?
