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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?

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John Lee's book "Intro to Smooth Manifolds" is a classic with a nice section on DeRham theory (the cohomology of differential forms, dual to submanifolds by the pairing "integration"). Bott and Tu, "Differential Forms in Algebraic Topology" is a little more advanced, but well-written. Certainly take a look at it. This is the classic way to use duality to represent cohomology classes with homology classes (i.e., cycles, submanifolds), and gives you a nice duality between intersecting submanifolds and wedging forms. Probably start with Lee. – Andrew Marshall Dec 16 '10 at 0:36
The problem with this approach is it is restricted to submanifolds -- it doesn't deal with non-representable cycles. – Ryan Budney Dec 16 '10 at 0:48
@AndrewMarshall Well, with these abstract nonsense (say five lemma or even spectral sequence argument), the combinatorial meaning is hidden. I think that original approach is still valuable. – Frank Science Mar 7 '15 at 5:13
up vote 15 down vote accepted

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.

dual polyhedral bits in a tetrahedron

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Is there any reference for the proof that the dual decomposition is really a CW-decomposition? Apparently, it depends on the fact that the original simplicial complex is a manifold. I don't know how to take advantage of this homogeneity explicitly. – Frank Science Mar 7 '15 at 4:53
Yes, there's certainly references for this material. The proofs are quite intuitive though. The facets of the triangulation all have epsilon-neighbourhoods which are trivial vector bundles by the homotopy-classification of vector bundles. This triviality is precisely what you need to prove the dual complex is a CW-decomposition. – Ryan Budney Mar 8 '15 at 7:00
Sorry, I cannot follow your idea. I want to go back to the language introduced in Seifert and Threlfall. The dual cell decomposition is constructed as follows: there's a point, the barycenter, associated to each $n$ dim simplex, then a $1$ dim star complex associated to each $n-1$ dim simplex, whose center is the barycenter of the complex and whose outer boundary is the totality of star complexes associated to each incident $n$ dim simplex, and so forth. Maybe it's easy to prove that such a star complex is a cellular decomposition, i.e. $H_m(X_{n+1},X_n)=0$ for $m\neq n$. – Frank Science Mar 8 '15 at 15:00
The fact that the star complexes constitutes a cellular filtration only depends on the homology of the manifold, just as Serfert & Threlfall shows. I cannot see how to determine the topology, not just homology of that, and how to take advantage of the tubular neighborhood you've mentioned. In addition, sorry for my ignorance, I don't know the tubular neighborhood theorem for triangulated manifolds, but not for differentiable manifolds. – Frank Science Mar 8 '15 at 15:08

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.

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In which section of Schubert's book? I didn't find that. – Frank Science Mar 7 '15 at 4:43

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

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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

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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.

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That article is commenting on an irrellevant side-issue. They're talking about the dual cell decomposition when it is applied to a non-PL triangulation. This means there's no requirement for the triangulation to be compatible with a global PL structure on the manifold. This is a fairly "exotic" idea. Usually when people talk about triangulated manifolds, they're triangulated PL manifolds. So in this article the link of a cell (from the triangulation) need not be a triangulated (standard) sphere. – Ryan Budney Oct 19 '15 at 15:54

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