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Contact and Symplectic Topology: Mastering the Art of Front Cooking

Roger Casals
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A Bird’s Eye View

Classification results are relevant in many areas of mathematics. The initial data is often a set of objects and an isomorphism relation—we invite the reader to think of their favorite classification problem. A few examples are:

(i)

knots , classified up to smooth isotopy, or smooth manifolds up to diffeomorphism,

(ii)

metric spaces, e.g., Riemannian manifolds, understood up to isometries or quasi-isometries,

(iii)

discrete groups, up to (virtual) isomorphism, or Lie groups and their Lie algebras,

(iv)

the classification of integer quadratic forms, classified up to classes or genera—or the solutions to a Diophantine equation up to descent,

(v)

the (space of) functions solving a partial differential equation, solutions possibly considered up to compactly supported smooth functions or gauge equivalence.

Classification problems aim to either prove two given objects and are isomorphic, , or that they are not isomorphic. In the former case, an isomorphism must be constructed, either explicitly or by abstract means. In the latter, one must argue such an equality cannot exist; this is frequently achieved by constructing and computing an invariant , associated to each object .

Figure 1.

Legendrian wavefront for a (2,4)-torus link (left). Legendrian wavefront for a 2-sphere (right).

Graphic for Figure 1.  without alt text

This invariant could be anything: a number, a polynomial, an Abelian group, a category,⁠Footnote1 a topological space, or a kind of potato—as long as it satisfies that if , it can be used to argue that by showing that .

1

E.g., “computing a category” might mean finding a set of generators, or presenting it as the differential graded derived (dg-derived) category of modules over an algebra we understand.

Examples.

Instances of invariants include the dimension of a vector space, the index of Fredholm operator, the class number of a number field, the genus of a surface, curvatures in Riemannian manifolds, the Jones polynomial of a knot, the Poincaré polynomial of a space, the homology groups of a space, or the category of coherent sheaves on an algebraic variety.

In general, one still has to show , which is also a classification problem—the trick is to choose invariants which are easily distinguished: most mathematicians would be comfortable telling numbers, polynomials, and Abelian groups apart, and such invariants are most frequently used (categories, spaces, and potatoes might have a more niche crowd).

The field of contact and symplectic topology 16 has seen developments on both sides of the aisle: new invariants distinguishing objects, and new techniques to construct isomorphisms.

Developments: Two New Directions

This article will introduce some of the objects studied in contact and symplectic topology and discuss some of the results that have been proved.⁠Footnote2 We focus on two particular developments:

2

The four main protagonists (contact, symplectic, Legendrian, and Lagrangian) will be defined momentarily and the reader is welcome to skip to the next section if needed.

(1)

The discovery of a flexible class of contact structures and symplectic structures.

(2)

The study of Legendrian and Lagrangian submanifolds using microlocal sheaf theory.

In these results, Legendrian wavefronts, which are certain singular hypersurfaces , have played a crucial role; see Figure 1. These will be discussed in depth in the upcoming section. The title of this article is a reference to Mastering the Art of French Cooking. For us, it is the study of (wave)front diagrams, with all its flavors and different recipes, that allows us to prove new results and enjoy some of the wonders of our mathematical kitchen.

Example.

For a Legendrian knot , the front is akin to a planar knot diagram, except it allows for cusps and all planar crossings are overcrossings;⁠Footnote3 see Figure 1 (left) and Figure 4. For a Legendrian surface , the front is a singular surface in , as in Figure 1 (right) or the blue flying saucer in Figure 2. Note that we are drawing surfaces in 5-dimensions by drawing fronts in 3-space.

3

Northwest-Southeast strand above Southwest-Northeast.

Given a Legendrian wavefront , we will build a symplectic manifold and two contact manifolds , and also construct invariants. Figure 2 schematically shows these possibilities.

Figure 2.

A Legendrian wavefront , from which we can build geometric manifolds (left) or compute algebraic invariants (right). is the smooth manifold where the wavefront lives, typically .

Graphic for Figure 2.  without alt text

Part (1) above belongs to the realm of -principles 12. The theory of -principles is constructive, a tenet being that “if certain algebraic obstructions vanish, then a geometric construction is possible.” (In broad terms, -principles are theorems where “algebra implies geometry.”) In our context, -principles are results allowing one to conclude that, for certain classes of objects and (actually computable) invariants , the equality implies that . By definition, are flexible with respect to an invariant when implies . Thus, should these flexible objects exist, the invariant gives a complete classification! For a given invariant , it is interesting to study whether a nonempty class of flexible objects exists and, if so, decide whether a given object is flexible.

In contact and symplectic topology, these flexible classes are defined by the existence of a local zig-zag in Legendrian wavefronts;⁠Footnote4 such a zig-zag is shown in Figure 1 (right). A wavefront with a zig-zag is said to be stabilized. The following result 256 summarizes⁠Footnote5 three flexible developments.

5

For notational simplicity, we write smooth isotopy for a smooth isotopy which includes the corresponding formal data required by the -principle; the same for diffeomorphisms.

4

The importance of zig-zags in -principles, and in general wrinkled maps, goes back to Eliashberg-Mishachev (1997).

Theorem 1.

In higher dimensions, i.e., for Legendrians and for symplectic manifolds, the following hold:

(i)

Let be smoothly isotopic Legendrian submanifolds, each admitting a stabilized wavefront. Then is Legendrian isotopic to .

(ii)

Let be diffeomorphic symplectic manifolds, each admitting a stabilized front handlebody. Then is symplectic isomorphic to .

(iii)

Let be diffeomorphic contact manifolds, each obtained by -surgery on a stabilized wavefront. Then is contact isomorphic to .

In fact, we also know several equivalent characterizations 5 of the hypothesis in Theorem 1(iii), e.g., in terms of contact open books and the existence of overtwisted disks.

Each of the three statements in Theorem 1 is of the following form: we have two objects which we want to compare. Then if both verify a certain topological property (being diffeomorphic and “stabilized”), then . The notions featured in Theorem 1 will be explained shortly. The message is that, in contact and symplectic topology, we now have verifiable local properties (of a wavefront) which globally characterize certain isomorphism classes of geometric objects, i.e., if you see a zig-zag somewhere in your wavefront diagram , this characterizes completely the symplectic and contact topology of , , and given the underlying smooth data. (This is rather exclusive to contact and symplectic topology: e.g., for a knot diagram or a Riemannian manifold, no local property near one point will typically be able to globally characterize the knot or the Riemannian metric.)

In contrast, part (2) above has provided new invariants to help the classification and study of contact and symplectic structures.⁠Footnote6 This relies on new and fruitful connections to cluster algebras 317, differential equations, and homological mirror symmetry. A class of invariants that are shaping the field is given by studying categories of constructible sheaves, with appropriate constraints 131415.

6

Floer-theoretic invariants date back to the 1980s, the dg-algebras being developed in the 2000s. The resurgence of microlocal sheaf theory belongs to the last decade 2010–20.

Let be a smooth manifold. These sheaves are intuitively given by assigning a finite-dimensional vector space at each point of . The dimensions of the vector spaces might potentially jump as we vary the point. Legendrian wavefronts serve as constraint: we only allow these dimensional jumps to happen when we cross the wavefront. This is morally what being constructible with respect to a wavefront means; the set of such sheaves is denoted by . In precise terms, let be a unit cotangent bundle. Then, we have the following.

Theorem 2 (131418).

The category of constructible sheaves microlocally supported at a Legendrian is a Legendrian invariant . Furthermore, it can often be computed combinatorially by using a wavefront for .

The computability is a strong virtue of such categories . The result above is remarkably efficient at distinguishing Legendrian submanifolds which are smoothly isotopic but not Legendrian isotopic. Many interesting results in the field have been discovered or reproved thanks to the use of microlocal sheaf theory 131415. Two new examples 317 are given in the following theorem.

Theorem 3.
(i)

The moduli space of Lagrangian fillings of positive Legendrian links admits a (partial) cluster structure.

(ii)

The Legendrian (max-tb) representatives of torus links, , admit infinitely many distinct Lagrangian fillings. In fact, most max-tb Legendrian representatives of a positive braid admit infinitely many distinct Lagrangian fillings.

The Hamiltonian isotopy class of an embedded exact Lagrangian filling specifies a cluster chart in the moduli space of Theorem 3(i), and it is shown that these cluster charts can be used to distinguish exact Lagrangian fillings as in Theorem 3(ii).⁠Footnote7

7

This relates to the wall-crossing phenomena present in Morse theory and mirror symmetry.

Contact and Symplectic Topology

Let us first introduce the four main characters in the field: symplectic and contact structures, and Legendrian and Lagrangian submanifolds. In this article, the symplectic and Lagrangian dimensions will be and , and the contact and Legendrian dimensions will be and , respectively.

Definition 1.

A symplectic structure on a real -dimensional smooth manifold is the choice of a nondegenerate closed 2-form on .

In practice, the integral over a 2-dimensional surface generalizes the notion of 2-dimensional area to any surface . In contrast to a Riemannian metric, the 2-form is antisymmetric and perceives orientations. Thus, nonpositive (symplectic) areas are allowed. Note that the three infinite families of simple Lie algebras are , corresponding to volume-preserving geometry, , giving (pseudo)Riemannian geometry, and , which yields symplectic geometry.

Example.

Consider the -dimensional ball

Then is a symplectic form. Given an embedded surface , the real number is obtained as follows: for each , project to the 2-plane and compute the signed area of this image. Then is the sum . See Figure 3 and note that some, even all, of the might be zero, e.g., the 2-torus has all .

Figure 3.

A surface and the number given by the symplectic structure.

Graphic for Figure 3.  without alt text

Symplectic structures are abundant: complex projective manifolds, or any manifold cut out by complex polynomials, are symplectic. The phase space of any physical system on a configuration space , i.e., the cotangent bundle and its reductions, also admits symplectic structures. These come from the 2-from , where is the canonical Liouville 1-form on . Most interesting aspects of symplectic topology are global: near a point , the symplectic structure is always given by the above example 1.

Definition 2.

A contact structure on a real -dimensional smooth manifold is a (locally) generic⁠Footnote8 and smooth choice of hyperplane at each tangent space.

8

The precise condition is “maximally nonintegrable”; by the Frobenius Integrability Theorem, this is equivalent to the algebraic condition and , where is a -form in .

Example.

Consider the sphere . The kernel is a contact structure, where is the restriction of the 1-form to the boundary . Removing a point from the sphere yields a contact structure .

Contact structures are also bountiful. In many cases, the intersection of a manifold with the unit sphere has a canonical contact structure . Similarly, the energy level sets of phase spaces and their reductions generically have contact structures coming from . Contact topology is also global in nature: near a point , the contact structure is given by the example above 9.

Remark.

É. Cartan classified all distributions that locally have a unique normal form. There are four classes: (i) nonvanishing vector fields, which lead to smooth dynamics, (ii) Engel structures, which are 4-dimensional, (iii) even–contact structures, and (iv) contact structures.

A salient relation between a symplectic manifold and a contact manifold arises when is the boundary of and is the differential of a 1-form. Then the hyperplanes are the linear subspaces , and these often form a contact structure if is appropriately chosen. Note that the two examples above fit into this framework as . In general, it is fruitful to think of contact manifolds as boundaries of (exact) symplectic manifolds .⁠Footnote9

9

This apparently innocent pairing is strengthened by a tenet in the field: “the contact boundary knows about the symplectic interior .” For instance, suppose that symplectically coincides with away from a compact set. Then must be diffeomorphic to , and even symplectic isomorphic to for .

Figure 4.

A Legendrian trefoil knot and its wavefront . Legendrian means that the tangent vectors to belong to the contact 2-plane .

Graphic for Figure 4.  without alt text

A Lagrangian submanifold is any submanifold such that the restriction is zero and . On the contact side, an isotropic submanifold is a submanifold of such that at each point , the tangent space is contained in the hyperplane . An isotropic submanifold is said to be Legendrian if .

In fact, implies the dimensional inequality and, similarly, implies . Thus, Lagrangian and Legendrian submanifolds have the maximal possible dimension given their defining (isotropic) constraints. Figure 4 (left) depicts a Legendrian knot .

Example.

Lagrangian submanifolds generalize the graphs of the derivative of an -valued function , which are indeed Lagrangian for . In fact, any Lagrangian is locally of this form. Similarly, Legendrian submanifolds , with , generalize the 1-jet graphs of , containing the information of the function and its derivatives . Any such is Legendrian for and, conversely, any Legendrian is locally of this form.

The relation between and parallels that of and : in many cases, the boundary of a properly embedded Lagrangian submanifold is a Legendrian submanifold .⁠Footnote10

10

The previous tenet also holds: “the Legendrian boundary knows about the Lagrangian interior .” E.g., if an embedded (exact) Lagrangian surface bounds a Legendrian knot , then must have the minimal -genus of the knot .

Figure 5.

A symplectic manifold with contact boundary , and a Lagrangian with Legendrian boundary .

Graphic for Figure 5.  without alt text

Pictorial conclusion. Figure 5 depicts a rather general situation where the four main characters appear at once. Researchers in the field look at many different questions; including the classification of all symplectic (or contact) structures on a smooth manifold, as well as the classification of Lagrangian (or Legendrian) submanifolds. Theorems 1, 2, and 3 above help us classify some such structures.

Studying Legendrians: Wavefronts

Symplectic and contact structures are rather hard to visualize: one is a 2-form and the other a hyperplane distribution. We will be able to translate problems on contact and symplectic structures into Legendrians, which are often more easily visualized and manipulated. This leads to some key questions: e.g., given two Legendrian submanifolds , how do we show or prove that ?

At its core, the answer to this question is by drawing them: by picturing a Legendrian we will be able to both pin down properties of the drawing that might characterize the Legendrian isotopy class of and compute invariants that distinguish Legendrians. Let us focus on , as this is the local model. The projection , , has the following property: the image of any Legendrian recovers the Legendrian by setting the -coordinates to be the -slope of the tangent plane of , i.e., . Indeed, this is just the analytic incarnation of the condition .

The image is known as the wavefront of . Figures 1, 2, and 4 depict examples of wavefronts. The wavefront is typically a singular hypersurface, despite being embedded: the singularities acquired by come from the singularities of the restricted projection . Hence, we can study Legendrians in by studying certain singular hypersurfaces in . Building on the theory of singularities 1, there is a diagrammatic calculus for wavefronts 1411. In the same way that one may tackle knots through their diagrams, and manipulate smooth 4-manifolds with Kirby calculus, contact and symplectic topologists can manipulate Legendrians and Lagrangians using wavefront diagrams.

Two Applications

Sample problems that we can now address with contact and symplectic techniques include the following two problems.

Problem 1 (Affine varieties).

Let us consider the two symplectic manifolds and

where inherits the symplectic structure from . It can be shown that as smooth manifolds , hence smoothly. In contrast, as affine algebraic varieties. (This is an algebraically exotic structure on , known as the Koras-Russell cubic. Being algebraic isomorphic implies being symplectic isomorphic, but the converse may fail.) Now we can study the question: are and isomorphic as symplectic manifolds? That is, does there exist a diffeomorphism such that ?

Here is a variation on Problem 1: consider . As smooth manifolds but, for instance, are algebraically distinct. Are they symplectically isomorphic? Both versions of this problem will be solved using Legendrian handlebodies, Legendrian wavefronts, and Theorems 1 and 2.

Problem 2 (Propagation of singularities).

Consider a planar wavefront in the shape of an ellipse moving inwards. Imagine an elliptical source of light, or a elliptical water wave⁠Footnote11 propagating inwards, as depicted in Figure 6 (upper-left). As wavefronts evolve in time, they develop singularities and it is interesting to understand which singularities are created and how they propagate.⁠Footnote12 For instance, the Four Cusps Theorem states that, generically, any sequence of wavefronts that starts and ends as in Figure 6 must have a wavefront in the middle with at least four cusps.

12

These problems already appear in classical geometric optics; this is the theory of caustics and wavefronts, key to understanding oscillatory integrals.

11

These are rare in the ocean, but common in round kiddie pools, fish farms, or a cup of tea.

Figure 6.

An eversion of an elliptic wavefront. Fixing the first and last fronts, we can ask which and how many singularities must occur for any sequence of intermediate fronts.

Graphic for Figure 6.  without alt text

In general, given two (possibly singular) wavefronts in a smooth manifold, one can ask which (and how many) singularities must appear or persist for any (generic) sequence of fronts starting at and ending at . The invariant Sh, which studies sheaves constrained by a wavefront , will allow us to answer some such questions.

Legendrian Handlebodies

Consider the situation in Figure 5, focusing on a Legendrian sphere in the contact boundary.

From Legendrians to contact and symplectic. We can use this Legendrian to construct a new symplectic manifold , where denotes the (unit disk) cotangent bundle of . This is obtained by attaching the symplectic piece , called a handle, to (a neighborhood of) the Legendrian along its boundary .⁠Footnote13 For the appropriate choice of framing (cf. 620), A. Weinstein showed that admits a symplectic structure 20 and its boundary is a contact manifold 9.

13

The second factor of is written in a smaller font to emphasize that its only purpose is to thicken to the necessary dimension , and make the total space symplectic.

Figure 7.

The handle attachment construction of symplectic manifolds: the cores of the handles are attached along (open neighborhoods of) the isotropic spheres . Differing spheres may lead to different contact and symplectic structures.

Graphic for Figure 7.  without alt text

Given a -dimensional isotropic sphere , we can similarly construct a new symplectic manifold by attaching the handle along its boundary to . The contact boundary of is denoted . In either case, the pieces we attach are simple, as they are standard symplectic disks . The data that most enriches this construction is the choice of where the piece is attached, which we specify by choosing an isotropic submanifold. Intuitively, the wealth of Legendrian submanifolds which are smoothly isomorphic (i.e., smoothly isotopic), but distinct as Legendrians, accounts for the additional richness of contact and symplectic topology, in comparison to differential topology.

Remark.

The above construction is a symplectic incarnation of handlebody decompositions for smooth manifolds; see Figure 8. The advantage of this technique—exemplified by 4-dimensional Kirby calculus—is that it translates problems about smooth manifolds into (a generalization of) knot theory, where pictorial and combinatorial techniques can be successfully used to manipulate diagrams.

Figure 8.

The space constructed by attaching a 3-dimensional 2-handle (an igloo) to a yellow solid 3-disk. The attachment is specified by the red circle in .

Graphic for Figure 8.  without alt text

A key distinction is that isotropic submanifolds of are significantly different than Legendrian submanifolds, i.e., those with .

Theorem 4.

Let be isotropic, , and smoothly isotopic⁠Footnote14 to . Then is isotopic to , i.e., there exists a -parameter family of isotropic submanifolds, , starting at and ending at . In contrast, there exist pairs of Legendrian submanifolds which are smoothly isotopic but not Legendrian isotopic.

14

For experts, it should read “formal isotropic isotopic” 6, as we use “smoothly isotopic” for “formally isotopic.”

The message of Theorem 4 is that when we encode contact and symplectic structures in terms of isotropic submanifolds, only those of largest dimension, i.e., Legendrian submanifolds, contain any information beyond smooth topology. In the statement, are Legendrian isotopic if there exists a 1-parameter family of Legendrian submanifolds, , starting at and ending at . Let us formalize the above construction.

Definition 3.
(i)

Let be a symplectic manifold. A Legendrian handlebody is a decomposition of the form where and is a Legendrian (or isotropic) sphere 620.

(ii)

Let be a contact manifold. A surgery diagram is a sequence