Timeline for What is the Poincare dual of a symplectic form?
Current License: CC BY-SA 2.5
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 23, 2010 at 14:32 | answer | added | user1835 | timeline score: 3 | |
| Oct 21, 2010 at 12:54 | vote | accept | Paul Siegel | ||
| Oct 15, 2010 at 22:30 | answer | added | Tim Perutz | timeline score: 24 | |
| Oct 15, 2010 at 22:13 | comment | added | Tom Goodwillie | @Ryan: Yes, I must have slipped up on the TeX. It got so weird that I had a little trouble finding the X to click to delete the comment. | |
| Oct 15, 2010 at 22:07 | comment | added | Tom Goodwillie | it seems that we're assuming that the real cohomology class of the symplectic form comes from an integral class, in other words that the periods are integers. | |
| Oct 15, 2010 at 22:01 | comment | added | Ryan Budney | I don't think this really sheds much light on symplectic forms. Perhaps the only thing this is saying is that you can interpret a symplectic structure on a manifold $M$ to be Poincare dual precisely to an oriented co-dimension $2$ submanifold $N$ of $M$ such that you can take $n/2$ copies of $N$ in $M$, perturb them so that their total intersection is transverse, and the algebraic intersection number of the total intersection is a non-zero integer. | |
| Oct 15, 2010 at 21:42 | comment | added | Ryan Budney | Some simple general nonsense tells you that $H^2(M) \equiv [M,\mathbb CP^\infty]$. $\mathbb CP^\infty$ contains a co-dimension $2$ sub-$\mathbb CP^\infty$ and the co-dimension $2$ submanifold of $M$ corresponding to a map $M \to \mathbb CP^\infty$ is the pre-image of this co-dimension $2$ sub-$\mathbb CP^\infty$, once the map is made to intersect it transversely. So provided you have this formulation of your symplectic structure you can "readily" get at this manifold. | |
| Oct 15, 2010 at 21:41 | comment | added | Paul Siegel | I would wager a lot of money that the Poincare dual of the standard symplectic form on $\mathbb{C}P^n$ is $\mathbb{C}P^{n−1}$, but haven't yet done the calculation. Even if I can do it, I would still like some nice symplectic machinery that does the calculation for me. | |
| Oct 15, 2010 at 21:35 | history | asked | Paul Siegel | CC BY-SA 2.5 |