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?

As far as I know, the only textbook reference for this approach, which is Poincare's original approach, is Seifert and Threlfall's text "A textbook of topology". It's available in English translation but the original was in German. Moreover, Seifert and Threlfall's proof isn't as efficient as it could be, since they're working entirely with simplicial homology. It's much more efficient to work with both simplicial homology and CW-cohomology (together with the knowledge that simplicial and CW homology / cohomology are canonically isomorphic via the relation to singular homology / cohomology).

This version of the proof only works in the context where your manifold is triangulable, and here it goes:

The idea of the dual cell decomposition in general goes like this. Let $\Delta_n$ be an $n$-simplex, and let $F$ be a facet of $\Delta_n$, meaning the convex hull of some collection of $\Delta_n$'s vertices.

The dual polyhedral bit corresponding to $F$ is the convex hull of the barycentres of all facets $F'$ of $\Delta_n$ which contain $F$ (including $\Delta_n$ itself). So in a tetrahedron $\Delta_3$, if $F$ was an edge, the dual polyhedral bit would be a quadrilateral that intersects $F$ in a single point.

Given a triangulated manifold $M$, if $F$ is a simplex of the triangulation, the dual cell corresponding to $F$ is the union of all the dual polyhedral bits to $F$ in all the top-dimensional simplices containing $F$. If $M$ is $m$-dimensional and $F$ is $k$-dimensional, a little geometry later and you'll see the dual cell is an $(m-k)$-dimensional cell in a genuine CW-decomposition of $M$. Again, in the 3-manifold case, if $F$ were an edge, the dual cell would be a $2$-cell with a single vertex at its centre, decomposed into squares.

That's the basic idea. From there the proof of Poincare duality is very much "follow your nose". It's a fun chase and I encourage you to try to work it out on your own, rather than looking it up.

Moreover, spend as much time as you can thinking about evaluating a homology class $X$ on the dual of a homology class $Y$ (provided $X$ and $Y$ have complementary dimensions). You'll have to be careful about thinking of the simplicial vs. CW-homology when thinking this through, of course.

For those looking for a 3d model, I recently turned this into a 3d print. The data is available at the link.

For compact manifolds (with and without boundary), I would recommend Schubert's Topology. It's these old German books that really go through the detail carefully. You're right that there is an issue here- most books aren't really careful with the dual cell decomposition. The other option, which gives an excellent explanation of the proof for closed manifolds, is Munkres's Elements of Algebraic Topology.

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