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?

See also my 2011 Bochum lectures The Poincare duality theorem and its converse I., II.
