**Stephen Paul King** (*stephenk1@home.com*)

*Thu, 25 Mar 1999 18:37:44 GMT*

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Stephen Paul King: "[time 55] Re: Euclidean vs Minkowskian"**Previous message:**Stephen P. King: "[time 53] Re: [time 51] SI Units"**In reply to:**ca314159: "[time 51] SI Units"

On Thu, 25 Mar 1999 01:36:18 GMT, John Baez wrote:

*>In article <36F2740B.96314AA3@uiuc.edu>, Eric Forgy <forgy@uiuc.edu> wrote:
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*>
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*>>It looks like maybe I was blurring the notion of duality in my last post. What
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*>>I described (I thought was Poincare duality) appears to be Fourier duality,
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*>>i.e. the functions f in L* map vectors A in L to a real number.
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*>>
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*>>f(A) = f_1 A^1 + f_2 A^2 + ... + f_n A^n
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*>>
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*>>But actually, I thought that notion was defining the space of differential
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*>>forms and that differential forms were related to Poincare duality. Hmm...
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*>
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*>Ah, you're sensing a relationship between Poincare duality and the duality
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*>for vector spaces!
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*>
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*>Like I said, it's good to spend a couple of hours per month trying to
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*>relate different notions of duality. If you spend less than this you'll
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*>never see the grander patterns lurking beneath the surface, but if you
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*>spend much more, you run the danger of going insane.
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*>
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*>In my previous post I described Poincare duality as an operation on cell
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*>decompositions of a fixed n-dimensional manifold M. The Poincare dual of
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*>a p-dimensional cell is an (n-p)-dimensional cell. If M is compact and
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*>oriented, we can use this idea to prove that the pth real homology
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*>of M has as its dual vector space the (n-p)th real homology. In other
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*>words, H_p(M,R)* is isomorphic to H_{n-p}(M,R). So Poincare duality
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*>is related to vector space duality.
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*>
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*>We can also prove a similar thing for cohomology. Here it's convenient
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*>to use differential forms. There's an operation called the "Hodge dual"
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*>that turns a p-form on an oriented Riemannian manifold into an (n-p) form.
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*>Using this we can prove that that if M is compact and oriented, H^p(M,R)*
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*>is isomorphic to H^{n-p}(M,R). Sometimes people call this idea "Hodge
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*>duality".
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*>
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*>>The lattice version of Poincare duality that you mentioned is also called
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*>>barycentric subdivision.
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*>
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*>Actually the Poincare dual is not the same as the barycentric subdivision.
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*>
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*>If you reread my description you'll see the differences. For example, the
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*>barycentric subdivision of a cell decomposition X has one vertex for each
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*>cell of X. The Poincare dual of X only has one vertex for each n-cell of X.
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*>The barycentric subdivision of a triangulation is always a triangulation,
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*>while the Poincare dual is usually not. The Poincare dual of the Poincare
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*>dual of X is X, while the barycentric subdivision of the barycentric
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*>subdivision of X is not the same as X.
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*>
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*>>There is also another way to form a lattice dual via a
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*>>Voronoi subdivision.
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*>
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*>I don't know what this is. Does it really deserve the name "dual"?
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*>I.e., is the Voronoi subdivision of the Voronoi subdivision of a
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*>lattice the original lattice? For some reason I doubt it - probably
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*>because you're calling it a "subdivision".
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*>
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*>
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*>
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