Convexity, an explicit example:

2010-10-11

OK, I think some of what I wrote yesterday is wrong, but now I'm going to try to carry out an explicit calculation to point out some subtleties. I'm interested in $X$ being a neighborhood of a symplectic sphere $S$ of square $-1$, and I want to look at three things:

  1. A Lefschetz fibration $f$ of $X$ over $B^2$ with $S$ contained in the single singular fiber over $0$.
  2. A symplectic structure on $X$ making $S$ symplectic and all fibers of the $f$ symplectic.
  3. A Liouville vector field transverse to $\partial X$ and, ideally, defined on all of $X \setminus S$ and behaving well with respect to $f$, whatever that means.

We'll see how far we get - the hard part is number 3, of course.

So we write $X$ as a $0$-handle $H$ and a $2$-handle $H'$, but $H$ is not your ordinary round $0$-handle. Both are subsets of $\R^2 \times \R^2$ with polar coordinates $(r_1,\theta_1,r_2,\theta_2)$ on $H$ and $(r_1',\theta_1',r_2',\theta_2')$ on $H'$. The precise definitions are: \[ H = \{ r_1 r_2 \leq 1, r_1 \leq a, r_1 \leq b \}, \; a,b>1 \] and \[ H' = \{ r_1'  \leq a', r_2' \leq 1/a \} \]

Also use polar coordinates $(r,\theta)$ on the base $B^2$ of the Lefschetz fibration.

  1. The Lefschetz fibration $f$ is defined on $H$ by $f(r_1,\theta_1,r_2,\theta_2) = (r_1 r_2, \theta_1 + \theta_2)$. The $2$-handle $H'$ is glued to the $0$-handle $H$ along $\{r_1' = a'\}$ via the glueing map \[ (\theta_1' = -\theta_1, r_2' = r_2, \theta_2' = \theta_1 + \theta_2 \] Then $f$ transforms appropriately and, on $H'$, is defined by $f(r_1',\theta_1',r_2',\theta_2') = (a r_2,\theta_2)$. This is because, at the boundary where the gluing is happening, $r_1 = a$. But notice that I am doing something fishy here, and I think this is actually important: I'm just describing how to glue along the boundary, not actually giving a gluing of overlapping collar neighborhoods. So in some sense this is just a piecewise smooth manifold, which, I guess, has a canonical smoothing? To properly specify things, I should say that $r_1' = a+a'-r_1$ and $r_2' = r_1 r_2/a$, I think, but this makes the following calculations much messier, if not intractable. 
  2. The symplectic structure $\omega$ on $H$ is the standard $\omega = r_1 dr_1 d\theta_1 + r_2 dr_2 d\theta_2$. This restricts to the region where $H'$ is attached, $\{r_1 = a\}$, as the form $a dr_1 d\theta_1 + r_2 dr_2 d\theta_2$, which then transforms to the attaching region $\{r_1' = a'\}$ as $r_2' dr_2' d\theta_1' + r_2' dr_2' d\theta_2'$, which looks problematic for extending across $H'$. Hmmm... I'm already feeling stuck here... this is surprisingly different than the construction in my paper with Andras Stipsicz. But it's supposed to extend... grrr.
  3. Well, I may already be out of luck with the Liouville vector field since I can't even get the symplectic structure right here. But here are the calculations I have. I will rather work with the contact forms. On $H$ we have the form $\alpha = \frac{1}{2}(r_1^2 d\theta_1 + r_2^2 d\theta_2)$, with $d\alpha = \omega$. Note also that $dr_1 \wedge \alpha \wedge \omega >0$, which means that $\alpha$ restricts to the region $\{r_1 = a\}$ of $\partial H$ as a positive contact form. Thus, when we get the Liouville vector field $V_\alpha$ such that $\imath_{V_\alpha} \omega = \alpha$, we know it points out along that part of the boundary. Now, pull $\alpha$ back to the attaching region of $\partial $H$ (and use here that $r_1' = a+a'-r_1$ and we get $\alpha' = \ldots$.

Gotta go, will finish this later...

Symplectic convexity and Lefschetz fibrations

2010-11-10

This is what I'm working on right now with Tom Mark:

A convex Lefschetz fibration on a compact $4$-manifold $X^4$ with nonempty boundary is a smooth map $f \co X \to D^2$ which restricts to $\partial X$ as an open book decomposition, and such that the local model near the singularities is the complex map $(z_1,z_2) \mapsto z_1 z_2$. (Here an open book decomposition is a smooth map $f \co M^3 \to D^2$ such that $M^3$ decomposes as a union of solid tori, on which $f$ is the projection $S^1 \times D^2 \to D^2$, and $f^{-1}(S^1=\partial D^2)$, on which $f$ is a fibration over $S^1$.) Notice that, in particular, regular fibers are compact surfaces with boundary. Here there's no mention of anything symplectic.

Now, we say that the Lefschetz fibration is symplectic if there is a symplectic form $\omega$ on $X$ with respect to which all fibers are symplectic, and probably also at all singularities the transverse intersections should be $\omega$-orthogonal.

Consider the special case where all the singularities lie in the central fiber $F_0 = f^{-1}(0)$, such that $F_0$ is a union of closed surfaces and disks meeting transversely. We can associate to this an intersection graph, with one vertex for each closed surface and one edge for each intersection (and so the disks contribute half-edges), and we assume that every vertex has at least one half-edge coming from it. Then the self-intersection number of each of the closed surfaces is precisely the negative of its corresponding valence (degree) in this graph. Let $C$ be the union of the closed surfaces. Then it is known in this case that, for any arbitrarily small neighborhood $\nu$ of $C$, there is a smaller neighborhood $\nu'$ of $C$ and a Liouville vector field $V$ on $\nu' \setminus C$, such that $\nu'$ is symplectically convex with respect to $V$, i.e. $V$ points out along $\partial \nu'$.

The question is this: Can we take $\nu'$ to be a neighborhood on which $f$ restricts as a convex Lefschetz fibration over $D^2_\epsilon$ (the disk of radius $\epsilon$ for some small $\epsilon>0$) and such that, for all $\delta$ with $0 < \delta < \epsilon$, $V$ is transverse to $\partial(f^{-1}(D^2_\delta) \cap \nu')$?

At the moment my calculations seem to lead me to think that, well, we can't quite do that. Instead, we can take $\nu'$ to be $f^{-1}(D^2_\epsilon)$ for some $\epsilon > 0$, but there will be some other $\epsilon'>0$, with $\epsilon' < \epsilon$, and we will only be able to say that, for all $\delta$ with $\epsilon' < \delta < \epsilon$, $V$ is transverse to $\partial(f^{-1}(D^2_\delta) \cap \nu')$. Further in (i.e. for $\delta < \epsilon'$, $V$ will still be Liouville and still transverse to the boundary of some neighborhoods of $C$, but those neighborhoods won't be of the form $f^{-1}(D^2_\delta)$; somehow $V$ will have twisted too much.

Here's what I'm thinking: We start with a convex Lefschetz fibration $f \co X_0 \to D^2$ in which all the singularities are on the central fiber $F_0 = f^{-1}(0)$, such that $F_0$ is a configuration of punctured surfaces $\Sigma_1, \ldots, \Sigma_n$ and disks $D_1, \ldots, D_m$, with each disk $D_j$ intersecting exactly one $\Sigma_i$ and each $\Sigma_i$ intersecting at least one $D_j$. We know how to make this symplectic with symplectically convex boundary, by building it with Weinstein handles. Now on $X_0$ there is also another Lefschetz fibration $f' \co X \to D^2$ in which the central fiber is $F_0' = (f')^{-1}(0) = \Sigma_1 \cup \ldots \cup \Sigma_n$ and the disks $D_j$ are sections. Now we have a Liouville vector field $V$ on $X_0$ giving the symplectic convexity there, and an associated 1-form $\alpha = \imath_V \omega$, which is contact when restricted to $\partial f^{-1}(D^2_\epsilon)$ for any $\epsilon>0$. We will attach $2$-handles along all components of each $\partial \Sigma_i$, with framings coming from the $f$ open book. The symplectic form $\omega$ will extend across the new $2$-handles, but $\alpha$ will not. Rather, we need to let $\alpha_0 = d(\theta \circ f')$, where $\theta \co D^2 \to S^1$ is the angular coordinate, and then, for some suitably small $k$, $\alpha + k \alpha_0$ will be contact and extend across the new boundary. Actually, it will always extend, but to be contact we need $k$ suitably small, I think. But now, if we trim the neighborhood back, $\alpha$ gets smaller, and so, eventually, we need a smaller $k$; this is the point at which I believe that you can't keep $V$ transverse to the boundary of arbitrarily small neighborhoods.

More later... But, to summarize, I now think that you have to pick your small neighborhood of the configuration in advance, and then you get some system of convex neighborhoods which behave well with respect to the Lefschetz fibration going some ways in but not all the way in. If you wanted to go further in, you could do it, you'd just have to start all over again, and you'd get a different Liouville vector field which goes further in, but still not all the way in, etc. Presumably you will be getting isotopic but not identical contact structures on the boundary, depending on how small of a neighborhood you choose.

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.

 

Draft of paper with Rob Kirby ready, and some cool drawings

This is a short post to get back in the flow of posting blog entries. First off, Rob Kirby and I finally have a publicly accessible draft of our paper on indefinite Morse 2-functions, which you can access here as a PDF file. We welcome any comments you may have, and will probably post it to the arXiv in a week or so, after Rob visits and we hash out some remaining technical issues. Incidentally, we are not entirely settled on terminology - finding the right names for things in math is worse than finding a name for a rock band, it seems. So you're welcome to weigh in on terminology, if you care.

The second thing I want to share is a sequence of drawings I made and scanned, here as a PDF file for now. This is a drawing of a 1-parameter family of Morse 2-functions from a solid torus to a disk. In my next post I plan to break up the PDF file into lots of little images and explain - for now, just enjoy and see if it makes any sense.