TITLE: Poincare Duality Reference
QUESTION [29 upvotes]: 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?

REPLY [2 votes]: Another German textbook including full details of the geometric proof of Poincaré duality is:
Ralph Stöcker, Heiner Zieschang:  Algebraische Topologie (1988)
Contrary to the claim frequently found in sketches of the argument, the authors stress that the dual ‘cells’ are not in general cells in the topological sense: 

“Examples and sketches in dimensions $\leq$ 3 suggest that for any $q$-simplex $\sigma$ [the dual ‘cell’] is an $(n-q)$-ball and [its ‘boundary’] is an $(n-q-1)$-sphere [...] (if this were the case, then [...] the dual decomposition would be a CW decomposition).  The question whether this is the case was one of the open, difficult and interesting problems in topology for several decades, until it was answered negatively by Edwards in 1975; [...].”
  ${}_{\text{(my translation)}}$

As pointed out by Ryan Budney in his comment, this technical difficulty can be circumvented by restricting attention to PL triangulations.  “Most” manifolds, in particular all differentiable manifolds, admit such triangulations.