Eliashberg proved that overtwisted contact structures up to isotopy are classified by their homotopy class in the space of 2-plane distributions in a 1989 paper. The higher dimensional case shares some similar structural aspects to the proof in 3 dimensions, so it seems worth going through the original result. I will try to mention the relations with the higher dimensional proof throughout. My sources here are Eliashberg’s original paper (Inventiones 1989), and the explanation of the proof in Geiges’ Introduction to Contact Topology book in section 4.7.

Starting at the beginning: an overtwisted disk in a contact 3-manifold is an embedding of the disk where the contact structure on is the kernel of . In dimension 3, we can alternatively define an overtwisted disk as an embedded disk whose boundary is Legendrian (tangent to the contact planes), such that the framing given by the contact planes agrees with the framing given by the surface. The existence of an overtwisted disk by one definition implies the existence of an overtwisted disk by the other definition, so we say a contact structure is overtwisted if it contains an overtwisted disk (using either definition).

The main idea is to start with a distribution, and homotope it piece by piece until it becomes a contact structure on the entire manifold. In order to *extend* the contact structure over the entire manifold, the existence of overtwisted disks is needed. This shows that every homotopy class contains a contact structure. In dimension 3 this followed from the work of Martinet who constructed a contact structure on each 3-manifold with surgery techniques and that of Lutz who showed that you can use Lutz twisting to modify the homotopy class of the contact structure however you want to without changing the 3-manifold. To show that any two overtwisted contact structures and which are homotopic are isotopic uses a *parametric* version of the extension construction. The homotopy between them gives an interpolating interval family of 2-plane distributions , and using the same ideas, we can homotope the intermediate distributions piece by piece until they become a smooth family of actual contact structures interpolating between and . Then by Gray’s theorem and are contact isotopic.

Note: everything here can be done relative to a closed subset; namely, if the 2-plane distribution is already contact on an open neighborhood of a closed subset, the homotopies can be chosen to fix the 2-plane distribution on that closed subset. This is in fact necessary to preserve the existence of the overtwisted disk throughout the modifications of the distribution, and to preserve the contact structures which are the end points of a 1-parameter family of distributions.

In fact the parametric version of the proof can be done when the parameter space is any compact set, so this can be used to show a more general statement. Let denote the space of overtwisted contact structures on M with a fixed overtwisted disk, and let denote the space of 2-plane distributions which also contain the fixed overtwisted disk. Then the inclusion is a homotopy equivalence. (Technically, the parametric version shows that this is a weak homotopy equivalence, but the spaces are CW complexes so the Whitehead theorem implies it is a full homotopy equivalence.) The idea to reprove the extension theorem in a parametric version is also used in the higher dimensional proof.

Now for the argument that we can homotope the distributions to genuine contact structures piece by piece (while fixing the pieces that we like already). We start with a triangulation of the manifold. We will make the distribution contact first in a neighborhood of the vertices, and then in a neighborhood of the 2-skeleton in a controlled manner, so that it will extend over the 3-cells at the end. The overtwisted disks needs to show up on the boundary of the ball that needs to be filled in at the end, so the 3-cells are all connected together with tubes and then connected to a neighborhood of the overtwisted disk. The contact condition in dimension 3 geometrically indicates whether the planes are twisting at the correct speed in the correct direction determined by the orientations. Over neighborhoods of vertices, we can easily homotope the planes to twist as much as necessary. Then we need to ensure that the planes form a contact structure over a neighborhood of the 2-skeleton, and moreover, we need to control what the contact structure looks like on the remaining boundary spheres of the 3-cells. The way to keep track of this is through the characteristic foliations.

** Characteristic foliations in dimension 3 and higher **

Given a surface in a contact 3-manifold, the intersection of the contact planes with the tangent planes to the surface produces a 1-dimensional singular foliation of the surface called the characteristic foliation. Equivalently, we can look at the restrictions and to the surface, and define a foliation by the vector field defined by . Notice that such a vector field is necessarily in the kernel of the contact form, and is identically zero exactly when the contact planes are tangent to the surface (since then and is an area form). In higher dimensions, the intersection of the contact planes with the tangent planes no longer forms an integrable distribution, but there is still a 1-dimensional singular characteristic foliation defined via the contact form in the same way. However, the characteristic foliation in higher dimensions can look considerably more complicated, and controlling its behavior takes a few additional steps than what is needed for the 3-dimensional proof.

The characteristic foliations that we will aim for on the 2-spheres are as follows. First we want the foliation to be * simple * meaning it has exactly two singular points, one a source (the “north pole”) and the other a sink (the “south pole”), and all the limit cycles (closed orbits) are isolated. These limit cycles necessarily form parallels between the two poles. If additionally there is a curve running from the south pole to the north pole which is positively transverse to all of the (oriented) leaves of the foliation, then the foliation is called * almost horizontal*. This condition is met when all of the limit cycles are oriented from east to west when viewing the sphere as a globe and the limit cycles as lines of longitude. The benefit of the almost horizontal condition, is that the foliation is determined up to homeomorphism by a * monodromy map * from the interval to itself where the interval is the transverse curve, and the holonomy map is determined by flowing around the characteristic foliation once. The limit cycles correspond to fixed points of the holonomy. In the 2n+1 dimensional case, the monodromy will be a map from the 2n-1 disk to itself. In the 3-dimensional case, the homeomorphism type of the foliation is determined completely by whether the points are moved up or down the interval by the holonomy between each pair of fixed points. This is important because of the following lemma:

** Lemma:** Let be a contact structure defined near the boundary of a 3-ball. The question of whether extends over the ball depends only on the topological type of the characteristic foliation induced on the boundary sphere.

To prove the lemma, given two characteristic foliations on the sphere that are topologically equivalent, first identify the poles and limit cycles. Then show that in a small neighborhood of the boundary sphere which one of the characteristic foliations, there is another sphere which realizes the other characteristic foliation. Therefore, if the contact structure extends over the ball for the first foliation, it extends over the ball for the second.

Now we want to prove two things:

1. We can homotope the 2-plane field over a neighborhood of the 2-skeleton so that it is contact there and it induces almost horizontal characteristic foliations on the spheres which bound the remaining finitely many holes where the distribution may not yet be contact (inside the 3-cells).

2. Given a contact structure on the manifold in the complement of a collection of finitely many balls, such that the characteristic foliations are almost horizontal, and such that the contact structure contains an overtwisted disk, the balls can be connected together to each other and to the overtwisted disk so that the contact structure in a neighborhood of the resulting boundary sphere can be extended inside the resulting ball to a contact structure.

**Part 1:** Over the 2-skeleton, the failure of the 2-plane distribution to be contact amounts to the planes not twisting positively enough. However, we need to identify which direction we need to twist along. For this we find an auxiliary 2-dimensional foliation defined near the 2-skeleton (maybe not necessarily defined near the 0-skeleton) which is everywhere transverse to , and is parallel to the 1-simplices (each 1 simplex is contained in a leaf), and is perpendicular to the 2-simplices. Now consider the characteristic foliation determined by on each leaf of this auxiliary foliation. The characteristic foliation is nonsingular by the transversality condition, so we can cover this neighborhood of the 2-skeleton by pieces which can be identified with a subset of with 1-dimensional foliation given by curves parallel to the y-axis. By ensuring that the 2-planes twist enough along the leaves of this 1-dimensional foliation, we can identify each of these pieces of manifold together with the 2-plane distribution with a piece of . The idea is to twist along these “Legendrian curves” in a neighborhood of each simplex one by one, in a relative way so that we fix the parts that we have already made contact. The thing we want to avoid is at some point, one of the Legendrian curves may have two ends in the relative piece that we have already made contact and do not want to mess up. In order to avoid this, keep very close track of the angles between and the simplices and the angles between different adjacent simplices.

A question: why is the 2-dimensional foliation chosen parallel to the 1-simplices and perpendicular to the 2-simplices? Some thoughts: this might be needed to make the Legendrian foliation consistent from the neigbhorhood of the 1-simplices to the neighborhood of the 2-simplices, or it might be needed for the Legendrian foliation to accurately capture the angle the contact planes make with the simplices.

In the parametric case where we want to keep track of the angles of a family of distributions , we simply subdivide the compact parameter space into sufficiently small pieces so that the angles between the contact planes at a fixed point but at different times in the parameter space remain sufficiently small relative to the chosen simplicial complex. Then one can make a homotopy through contact structures for t in each subinterval, and by doing everything relative to the end points, this will gradually extend across the entire parametrizing interval.

The precise details of the argument are in Geiges’ exposition, but without getting bogged down in notation, here is the idea of the angle tracking argument. First choose a very fine simplicial complex where the maximal diameter, d, of the simplices becomes very small, but the angles between simplices remains bounded above by , and the minimum distance between disjoint simplices is less than some fixed constant multiplied by d. On each simplex, the amount that the angles of the contact planes change relative to each other is measured by a Gauss map from the simplex to . This change can be captured by a norm, which by choosing a good enough simplicial subdivision, can be assumed to be small relative to , so that across each simplex, the angle of the contact planes only changes by a very small fraction of .

Eliashberg defines “special simplices” as 1- or 2-simplices which contain some point p at which makes an angle less than with the simplex. The other 1- and 2-simplices are considered non-special. The idea of the special simplices is that the Legendrian curves which tell you which direction to twist in, make a small angle with the simplices, whereas with the non-special simplices, the Legendrian curves are “sufficiently transverse” to the simplices that the curves will have at least one end in the 3-simplices, away from the neighborhoods of the 0-, 1-, and 2-simplices where we may have already modified to be contact. By carefully keeping track of the angles between simplices and , and between at one point versus another (using the small norm assumption from the previous paragraph), one can show with triangle inequalities that if two special simplices were adjacent, the angle between them would be less than , which is not possible. Therefore the special simplices are isolated from each other, so we will perturb the distribution to become contact along the special simplices first and not worry about whether this changes the plane field near the other adjacent simplices as we will fix them later. Once this is done, we assume that we have modified to be contact in a neighborhood of all special simplices, and in a neighborhood of any 0-simplices (vertices) which are disjoint from the special simplices. The sizes of these neighborhoods are chosen relative to the constants and . Next modify the contact structure in small neighborhoods of the non-special 1-simplices (which are not the boundary of a special 2-simplex) rel boundary (where the structure is already contact and we don’t want to modify it anymore). The angle between the non-special simplices and the Legendrian foliation curves defined by is at least at each point. By having chosen the neighborhoods of the special simplices and 0-simplices small in terms of and the minimal distance between disjoint simplices, , we can ensure that none of the Legendrian curves through the non-special simplex hit the already contact neighborhoods in more than one end so we can twist the planes towards the end where we do not need to fix the planes.

In this picture the blue curves represent the “Legendrian foliation.” The black is the non-special 1-simplex, and the grey regions are where the 2-planes have already been perturbed to be contact.

Finally, we homotope the planes in a neighborhood of the non-special 2-simplices rel boundary (since the planes are contact over the entire 1-skeleton now). Again having chosen the previous neighborhoods sufficiently small in terms of , we can ensure that the Legendrian curves through these 2-simplices only intersect the neighborhood of the special and 1-simplices at one end so we can twist towards the free end.

A note about homotoping relative to a fixed closed subset: During this process, we modify the planes to be contact in certain areas and then freeze the planes there as we homotope the planes in other regions. In a similar way, if our planes were already contact in a certain region that we wanted to keep fixed from the beginning (e.g. a neighborhood of an overtwisted disk), we can do this. We just need to refine the simplicial subdivision enough near this relative set, so that we never get Legendrian curves with two ends inside the relative set.

Now we have reached a distribution which is contact in a neighborhood of the 2-skeleton, and thus in the complement of finitely many balls. We need to slightly enlargen these balls into the neighborhood of the 2-skeleton but without hitting the 2-skeleton, so that the balls are sufficiently round (they have normal curvatures bounded below by a positive constant). We can also assume some genericity of the contact structure on the boundary sphere within the roundness constraint. This together with the restrictions on the norm of the contact structure (it the planes can only twist a little bit over any simplex), ensures that the characteristic foliations on the boundaries of these enlargened balls are almost horizontal. The idea is to compare the Gauss maps along the boundary sphere of the tangent planes and the contact planes. If the characteristic foliation were not almost horizontal, the two Gauss maps, which agree at the positive singular point, would at some point become far apart (separated by angle ), which cannot happen in these tiny simplices where does not change its angle much.

Before going on to part 2, I want to briefly mention that in the higher dimensional version there is a part of the argument which involves keeping track of the angles of the hyperplanes relative to a foliation. There is again a distinction between pieces where the angles are sufficiently large and those that are relatively small. However, controlling the angles gets to a contact structure in the complement of balls (with a “saucer” type almost contact structure) but they do not have a sufficiently controlled characteristic foliation to fill in directly without a number of additional steps.

**Part 2:** At this point we have a finite set of balls in our manifold. The planes of have been homotoped to be contact in the complement of the interiors of these balls, and we have almost horizontal characteristic foliations on the boundaries. Furthermore, based on our original assumption, somewhere in the contact region is a neighborhood of an overtwisted disk, which we have fixed throughout, homotoping everything else relative to this piece. We will need to connect the almost horizontal balls to the overtwisted ball in order to fill them in. We first connect the finitely many almost horizontal balls to each other, by ordering them and then choosing a path from the north pole of one almost horizontal sphere to the south pole of the next. The connect sum of these spheres appears as the boundary of the balls together with a tiny neighborhood of the connecting path.

Note: if there is a closed set where we want to keep the contact planes fixed, we should choose our connect sum paths to avoid this set since we will modify the planes on the interior of the connected up ball.

The resulting characteristic foliation on the connect sum of all the almost horizontal spheres and the boundary of the neighborhood of the overtwisted disk is a simple foliation where there are two limit cycles oriented east to west (coming from the neighborhood of the overtwisted disk), and the rest of the limit cycles are oriented west to east (coming from the almost horizontal foliations). The standard overtwisted ball of radius in can be isotoped to a ball with boundary whose characteristic foliation is topologically equivalent to this connect sum. By the lemma mentioned before part 1, this is enough to fill in the holes. To see this isotopy consider the surfaces of revolution of smooth curves around the z-axis in in . Every time the curve intersects the line , we obtain an extra limit cycle. The orientation of the limit cycle (east to west or west to east) is determined as follows. The characteristic foliation is generated by a vector field defined by $laetx \iota_X\Omega=\alpha$ (where is restricted to the surface and is a positive area form on the surface). Therefore if we pair with a vector in the tangent plane to the surface which has a positive Reeb component, we obtain a positively oriented basis for the surface. The sphere surface is oriented as the boundary of the ball, so the outward normal to the sphere, followed by $X$ followed by a vector with a positive Reeb component forms a positive basis for . The Reeb vector field coorients the tangent planes, and points in the negative z direction at , so vectors with a negative z component have a positive Reeb component.

In this picture, the red vectors lie in the plane and are outward normals to the surface of revolution. The vector field generating the characteristic foliation on the limit cycles are pointing either directly in or out of the plane as indicated by the blue words. The green vector fields each have a negative z component and thus have a positive Reeb component. Observe that red, blue, green gives a positive basis for at each of these points. Furthermore notice that each limit cycle oriented by blue going out is moving east to west (there are two of these), and each limit cycle oriented by glue going in is moving west to east so we can create any number of these by creating intersections of the curve with on the “inside” of the curve. To make an odd number of limit cycles, have the curve intersect at a tangency on the inside.

This completes the 3-dimensional argument to homotope 2-plane fields containing an overtwisted disk to a contact structure. By doing this for an interval family of contact structures rel end points, we get that a family of overtwisted 2-plane fields can be homotoped to an isotopy of overtwisted contact structures.

Step 2 in the higher dimensional proof of Borman, Eliashberg and Murphy is somewhat different. First of all, one needs to establish more clearly what the necessary boundary structure of the balls is through an explicit model. The amount of flexibility in this model is not quite as much as the topological equivalence of characteristic foliations on spheres. Once the boundary model is established, each of the spheres is connect-summed to a neighborhood of an overtwisted disk (though they are not connected summed to each other, instead a bunch of copies of overtwisted disks are used, one for each hole). Then, using some contactomorphisms and various lemmas, it is possible to show that these connect sums of holes which have model contact structures on their boundaries with neighborhoods of overtwisted disks can be filled in with a contact ball. The filling of the holes is not as explicit as the surface of rotation model above in the 3-dimensional case. More on the higher dimensional case coming in later posts.