Poincare Duality Reference In Hatcher's "Algebraic Topology" in the Poincaré Duality section he introduces the subject by doing orientable surfaces.  He shows that there is a dual cell structure to each cell structure and it's easy to see that the first structure gives the cellular chain complex, while the other gives the cellular cochain complex.  He goes on to say that this generalizes for manifolds of higher dimension, but that "requires a certain amount of manifold theory".  Is there a good book or paper where I can read about this formulation of Poincaré Duality?

The dual-cell proof of Poincare duality features in my 1992 book Algebraic L-theory and topological manifolds and in my 1999 paper Singularities, double points, controlled topology and chain duality
