A biased survey on symplectic fillings, part 1 (definitions)

Earlier this week, I gave a talk in my working group seminar entitled “A biased survey on symplectic fillings”. The talk was meant in large part as an excuse to state and discuss certain open questions in my research area that I find especially interesting, and I included the word “biased” in the title so as to absolve myself of any obligation to be balanced or comprehensive, or to mention all sorts of things that, while possibly very important, are not so relevant to the things I usually think about. With this understood, I am going to inaugurate this blog by attempting over the next few posts to reproduce most of what I said in that talk, including several things that I had intended to say but skipped due to lack of time. This can also be interpreted as a sketch of the essential motivations and goals of my current EPSRC-funded project on symplectic cobordisms.

This first post will deal only with the basic definitions.

For a proper discussion of symplectic fillings, it helps to be a bit more general and discuss symplectic cobordisms. To that end, assume $(W,\omega)$ is a $2n$ -dimensional compact symplectic manifold whose boundary can be written as a disjoint union of two (each possibly empty) components $M_-$ and $M_+$ . We say that $M_+$ is a convex boundary component if a neighborhood of $M_+$ admits a so-called Liouville vector field $V$ that points transversely outward at the boundary; here the “Liouville” condition means that $V$ satisfies

${\mathcal L}_V \omega = \omega$ ,

so its flow dilates the symplectic form. Similarly, we call $M_-$ a concave boundary component if the same condition holds near $M_-$ , except that the Liouville vector field points transversely inward instead of outward.

There are a few reasons why these are natural conditions to consider at the boundary of a symplectic manifold. One reason is that a convex or concave boundary component naturally inherits a contact structure, and this contact structure determines the symplectic form in a collar neighborhood of the boundary. To see this, suppose $V$ is the Liouville vector field near $M_\pm \subset \partial W$ , and consider the 1-form

$\lambda := \omega(V,\cdot)$ .

By Cartan’s formula for the Lie derivative, the Liouville condition now translates into the statement that $d\lambda = \omega$ , and outward/inward transversality to the boundary means that the 1-forms $\alpha_\pm := \lambda|_{TM_\pm}$ on $M_\pm$ satisfy

$\pm \alpha_\pm \wedge d\alpha_\pm^{n-1} > 0$ ,

in other words, $\alpha_\pm$ is a positive/negative contact form on $M_\pm$ . (Note that we are assuming $M_+$ and $M_-$ are each assigned the natural boundary orientation, where $W$ carries the orientation determined by its symplectic form.) Denote the resulting (positive or negative) contact structure by $\xi_\pm = \ker\alpha_\pm$ . Now we observe two important facts:

By flowing from the boundary along $V$ for a sufficiently small time, we can identify a collar neighborhood of $M_+$ with $(-\epsilon,0] \times M_+$ for sufficiently small $\epsilon > 0$ . Under this identification, if $t$ denotes the coordinate on $(-\epsilon,0]$ , we find that $\lambda$ takes the form $e^t \alpha_+$ , hence $\omega = d(e^t \alpha_+)$ in this collar. In the same manner, a collar neighborhood of $M_-$ in $(W,\omega)$ can be identified symplectically with $([0,\epsilon) \times M_- , d(e^t \alpha_-))$ for sufficiently small $\epsilon > 0$ .
The choice of Liouville vector field $V$ is far from unique: indeed, whenever such a vector field exists, its sum with any sufficiently small Hamiltonian (or more generally, locally symplectic) vector field will be another Liouville vector field that is still transverse to $\partial W$ . Thus there is an infinite-dimensional space of choices involved in defining the contact structures $\xi_\pm$ … but importantly, it is easy to check that that space of choices is convex, and thus contractible. By Gray’s stability theorem, it follows that while $\xi_+$ and $\xi_-$ are not uniquely determined by $\omega$ , they are unique up to isotopy.

The second observation above provides motivation for the following:

Definition. $(W,\omega)$ as described above is a (strong) symplectic cobordism from $(M_-,\xi_-)$ to $(M_+,\xi_+)$ .

Caution: some authors prefer to call what I’ve just defined a symplectic cobordism “from $(M_+,\xi_+)$ to $(M_-,\xi_-)$ “, i.e. from the convex to the concave side instead of the other way around. To some extent this is a matter of taste, though I would say there are good topological reasons for following the convention as I have written it; these reasons have to do with the Morse theoretic description of Stein manifolds via contact surgery, which we’ll touch upon later.

Whenever a strong cobordism exists from $(M_-,\xi_-)$ to $(M_+,\xi_+)$ , we shall write $(M_-,\xi_-) \preceq (M_+,\xi_+)$ . This notation is meant to evoke a partial order, and indeed, the cobordism relation is almost a partial order:

It is reflexive, meaning that every contact manifold $(M,\xi = \ker \alpha)$ is cobordant to itself via the trivial cobordism $([0,1] \times M, d(e^t \alpha))$ . This cobordism is simply a compact subset of the symplectization of $(M,\xi)$ , in fact the latter can be interpreted as the symplectic completion of the trivial cobordism, obtained by gluing cylindrical ends to both boundary components.
It is transitive, meaning that if $W_{01}$ and $W_{12}$ are two symplectic cobordisms such that the convex boundary of $W_{01}$ is contactomorphic to the concave boundary of $W_{12}$ , then they can be glued together to form a symplectic cobordism from the concave boundary of $W_{01}$ to the convex boundary of $W_{12}$ . This observation follows easily from the collar neighborhoods described above, after possibly rescaling the symplectic forms by positive factors.

Unlike the situation in topology, the symplectic cobordism relation is not an equivalence, i.e. it is not symmetric. The difference is one of orientations: in topology, if $X$ is an oriented cobordism from $Y_-$ to $Y_+$ , then the same manifold with reversed orientation gives an oriented cobordism from $Y_+$ to $Y_-$ . But you can’t just reverse the orientation of a symplectic manifold; the symplectic structure determines the orientation, and there are plenty of examples of symplectic manifolds that do not admit any symplectic structure compatible with the opposite orientation! (Exercise for the reader: this is true on ${\mathbb C} P^2$ . Do you see why?)

Less obviously, but just as important, the symplectic cobordism relation is not antisymmetric, i.e. it is not true that if $(M_-,\xi_-) \preceq (M_+,\xi_+)$ and $(M_+,\xi_+) \preceq (M_-,\xi_-)$ , then $(M_-,\xi_-)$ and $(M_+,\xi_+)$ must be contactomorphic. We will see counterexamples to this later in the discussion.

To conclude this first post, here is the main definition that we’ve been leading up to:

Definition. A (strong) symplectic filling of a contact manifold $(M,\xi)$ is a strong symplectic cobordism from the empty manifold to $(M,\xi)$ . Similarly, a symplectic cobordism from $(M,\xi)$ to the empty manifold is called a symplectic cap.

For a first example of a symplectic filling, take the unit ball in the standard symplectic ${\mathbb R}^{2n}$ . It is not hard to write down an example of a Liouville vector field on ${\mathbb R}^{2n}$ that points radially outward, thus the unit ball is a symplectic filling of the unit sphere; the induced contact structure on $S^{2n-1}$ is the one we usually call the standard contact structure $\xi_{\text{std}}$ on the sphere. (Exercise: $\xi_{\text{std}}$ can also be defined as the field of complex tangencies, i.e. the maximal complex subbundle of $T S^{2n-1}$ when $S^{2n-1}$ is regarded as the unit sphere in ${\mathbb C}^n$ .)

It is similarly easy to find symplectic caps for $(S^{2n-1},\xi_{\text{std}})$ , and in fact there is an unfathomably large variety of them: take any closed symplectic manifold $(X,\omega)$ and define $W \subset X$ to be the complement of an open Darboux ball. The arbitrariness of this construction should tip you off to the fact that symplectic caps are in general much less interesting objects than symplectic fillings, i.e. they exist in abundance and do not seem to carry information about the contact structure on the boundary. We will see more evidence of this when we discuss the known results on fillings and caps.

I will further expand this list of definitions in the next post.

2 Responses to A biased survey on symplectic fillings, part 1 (definitions)

aaa says:

2015-01-11 at 19:46

Everything typed was very reasonable. However,
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I ain’t suggesting your content isn’t solid., however what if you added a title to possibly get folk’s
attention? I mean A biased survey on symplectic fillings, part 1 (definitions) | Symplectic Field
Theorist is a little boring. You might glance at Yahoo’s front page
and note how they create article headlines to get people
to click. You might try adding a video or a pic or two
to get people excited about what you’ve got to say. Just my opinion,
it might make your posts a little bit more interesting.

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- Chris Wendl says:
  
  2015-01-11 at 20:05
  
  Ladies and gentlemen, this blog has achieved a milestone today: we have our very first troll!
  
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