Tag: polydisk

 

Symplectic polydisk and ellipsoid embeddings

Some explicit questions about polydisk and ellipsoid embeddings that Olga Buse and I hope to tackle:

  • Consider the map $ f \colon\thinspace B^4(1) \to B^2(1)$ defined in complex coordinates by $ f(w,z) = wz$ . Pick a regular value, say $ z_0 = 1/2$ . Let $ C = f^{-1}(z_0)$ . First of all, what is the area $ A=\int_C \omega_0$ ? Secondly, given any area $ a < A$ , let $ w(a)$ be the supremum over all $ w$ such that there is a polydisk $ B^2(a) \times B^2(w(a))$ embedded in $ B^4(1)$ with $ B^2(a) \times \{0\} \subset C$ . Calculate $ w(a)$ . It should obviously be a decreasing function of $ a$ . Then, for which $ a$ is $ a w(a)$ largest?
  • Same thing for any other Lefschetz fibration, or any complex polynomial for that matter.
  • In principle you might be able to use these same ideas to understand ellipsoid embeddings. Maybe the best way to say it is this: Given the symplectic surface $ C = f^{-1}(z_0)$ , find a smooth positive function $ r \colon\thinspace C \to (0,\infty)$ with a symplectic embedding of $ P(C,r)=\{(p,x,y) \in C \times \mathbb{R}^2 \mid \sqrt{x^2+y^2} < r(p)\}$ into $ B^4(1)$ mapping $ P \times \{(0,0)\}$ to $ P$ by the identity. Hopefully you can find such a function and such an embedding so that it ``fully fills'' $ B^4(1)$ . Then by trimming back $ C$ and trimming back $ r$ just right, we can get embeddings of ellipsoids, and by trimming back $ C$ and trimming $ r$ back to be constant, we can get embeddings of polydisks.
  • My original idea about using Lefschetz fibrations was a little different, but should probably amount to the same thing. Take your favorite topological Lefschetz fibration of $ B^4$ over $ B^2$ . Build it symplectically, with convex boundary, in some standard way, starting with $ F \times B^2$ (where $ F$ is the fiber) and then adding very thin Weinstein handles. You end up with a symplectic structure $ \omega$ on $ B^4$ with convex boundary, with induced contact form $ \alpha$ on $ S^3$ (contact form because you have kept careful track of the form, not just the underlying contact structure, as you attached the handles). And this nonstandard symplectic ball is almost fully filled with a polydisk. Now we know that there exists some function $ f \colon\thinspace S^3 \to [1,\infty)$ so that $ f \alpha = k \alpha_0$ , where $ \alpha_0$ is the standard contact form on $ S^3 = \partial B^4(1)$ , and $ k$ is some constant. I think that then you want to measure $ \int_{S^3} f$ , or something like that - this should be a measure of how much volume you have to add to change $ (B^4,\omega)$ into the standard round ball. Somehow this function is an interesting object attached to a word in Dehn twists.
  • The other idea for polydisks, which ultimately is still really the same basic idea, namely extending a symplectic surface out to a polydisk as much as possible, is to consider ribbon disks for interesting transverse ribbon knots.

 

P.S. I just gave a talk at MSRI on indefinite Morse 2-functions, my main project with Rob Kirby, and the pdf slides can be viewed here.

David Gay 2010-04-28



Towards an NSF proposal

Well, I had plans to dissect those pictures I posted a week ago, but I'm going to postpone that for now. I need to write an NSF proposal, due some time next week. I'm going to start to draft it here, because I started trying to write it and couldn't think of a good title. Also the NSF website is strangely inaccessible at the moment so I can't waste time fiddling with peripheral stuff like budgets and formatting. Here goes:

My main research topics are:

  1. Morse $2$--functions, Cerf theory, and invariants. Here the big dream is to have some unifying picture of Heegaard-Floer-like invariants for smooth manifolds that can be understood as coming from Cerf theory and Morse $2$--function, and to really see that you are counting some obstruction to simplifying some presentation of the manifold. (Just as Morse homology counts an obstruction to simplifying a Morse function or handle decomposition.) There are many subsidiary topics, most not nearly so grand:
    1. Finish the paper with Rob.
    2. Write something about the relationship with open book decompositions. Some interesting related questions are: Can we understand Harer's moves on open book decompositions from the point of view of homotopies of Morse $2$--functions? If two homotopic Morse $2$--functions both have sections, are they homotopic through a homotopy such that at each intermediate time sections exist? How does the homotopy class of the contact structure supported by an open book decomposition relate to the homotopy class of an associated $S^2$--valued Morse $2$--function?
    3. Work on a survey article on Cerf theory with Katrin Wehrheim and Chris Woodward.
    4. What is the proper "combinatorialization" of a Morse $2$--function, in the sense that a handle decomposition (or a handle diagram) is a combinatorialization of a Morse function. In other words, what is the simplest data you can add to a diagram in the base of the singular locus so as to completely determine the total space and the map (up to isotopy)?
    5. Work out something useful to say about the setup in bordered Heegaard-Floer homology; in particular, figure out what the appropriate moves should be on combinatorial descriptions of $4$--dimensional cobordisms between bordered $3$--manifolds.
    6. And then the fanciful stuff, which needs to be said carefully so as not to sound crazy; e.g. might there really be the possibility of finding invariants of homology $4$--spheres? (Homotopy $S^4$'s?!?)
  2. My current delayed project with Tom Mark on relating convex neighborhoods of certain configurations of symplectic surfaces to Lefschetz fibrations that contain those configurations as singular fibers.
  3. My very delayed project with Olga Buse on contact packing problems and symplectic embeddings of polydisks in symplectic balls. The polydisk embedding question is really whether knottedness can help get better embeddings somehow.
  4. The even more delayed collaboration with Margaret Symington on locally toric fibrations on symplectic (and near-symplectic) $4$--manifolds. I'm very interested in whether there is a reasonable notion of locally toric fibrations over non-manifold $2$--complexes with integral affine structures, and whether, allowing for singularities on the boundaries, as we did in our near-symplectic paper, you can get something that is in anyway dual to broken Lefshetz fibrations.
OK, enough for now, more tomorrow I hope. One trick will be how to thread all those projects into one coherent story.





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