## Overtwisted disks and filling holes

This post is on the end of the proof by Borman, Eliashberg, and Murphy that there is an overtwisted contact structure in every homotopy class of almost contact structures in higher dimensions and via the parametric version, any two overtwisted contact structures which are homotopic as almost contact structures, are isotopic as contact structures. There are a number of other posts preceding this one that are meant to be read first, and there are a few pieces of the proof that we skipped, but I think this will be my last post on this topic.

Overtwisted disks in higher dimensions and filling the holes

In dimension three, an overtwisted disk is a certain model germ of a contact structure on a two dimensional disk. The key property of this overtwisted disk which generalizes in higher dimensions, is its role in the proof of the h-principle: after connecting the codimension zero “holes” where the almost contact structure resists becoming genuinely contact, with a neighborhood of the overtwisted disk, one is able to extend the contact structure. One useful feature of overtwisted disks in dimension three, is that they can be recognized simply by finding an embedded unknotted circle with Thurston-Bennequin number 0 (the contact planes along the unknot do not twist relative to the Seifert framing determined by the disk that is bounded by the unknot). This is not true in higher dimensions: there are quantitative properties of the contact structure on the interior of the disk which are needed for the h-principle proof to work.

Recall, from Roger’s post, that in the presence of an overtwisted disk, we can reduce the problem of extending the contact structure over the hole, to extending the contact structure over an annulus (interval times sphere) whose germ on one boundary component is modelled by the contact Hamiltonian obtained by concatenating the Hamiltonian modelling the hole with the overtwisted model Hamiltonian, and whose germ on the other boundary component is given by the overtwisted Hamiltonian. (Remember this picture?)

This is because we can connect each hole to an overtwisted annulus by a tunnel, and then forget that we already had a genuine contact structure on the tunnel and the overtwisted annulus and just look at the contact germs on the two boundary components of the boundary sum of the ball with the annulus, like in this schematic picture.

This is the key point where we use the overtwistedness of the contact structure. The arguments to get to this point are made in a relative way that just fixes the contact structure in the overtwisted regions. At this point, we need to change the contact structure on the overtwisted annulus. In order to fill in the larger annulus (the overtwisted annulus connected to the hole) with a genuine contact structure, we need to show that, up to conjugation, the overtwisted Hamiltonian is less than the connect sum of the Hamiltonian for the hole with the overtwisted Hamiltonian. We are assuming at this point, that we know how to homotope the almost contact structure so that it is genuinely contact in the complement of holes, and each of the holes has its almost contact structures given by a circle model. Moreover, by doing this extra carefully (using equivariant coverings), we can assume that there are finitely many different types of contact Hamiltonians defining the circle models for the holes. The number of types of contact Hamiltonians needed a priori depends on the dimension. An easier reduction is to assume that the Hamiltonian $K: \Delta\times S^1 \to \mathbb{R}$ is independent of the $S^1$ (time) direction since the circle is compact so $\overline{K}(x)=\min_{\theta\in S^1} K(x,\theta)$ is well-defined and satisfies $\overline{K}\leq K$ so there is a genuine contact annulus extending the contact structure from the boundary of the circle model for $K$ inward to the boundary of the circle model for $\overline{K}$.

In order to prove the key lemma that we can fill in the appropriate annuli, we need a more concrete family of contact Hamiltonians. Consider a contact Hamiltonian $K_{\varepsilon}$ on the cylinder $\Delta_{cyl}=\{(z,u_i,\theta_i): |z|\leq 1, u=\sum u_i\leq 1\}\subset \mathbb{R}^{2n-1}$ which is negative on the region where $|z|$ and $u$ are both less than $1-\varepsilon$, and which increases linearly from 0 in $z$ and $u$ towards the boundary with slope 1. These are called special Hamiltonians . The main thing which is special about such a Hamiltonian $K_{\varepsilon}$ is that there is a contact embedding $\Theta$ of $\Delta_{cyl}$ with the standard contact form, into the boundary sum of $\Delta_{cyl}$ with itself, such that $\Theta_*K_{\varepsilon}$ is less than the connected sum of $K_{\varepsilon}$ with itself. Given this, if the hole and the overtwisted annulus are both modelled by such Hamiltonians with the same $\varepsilon$, we can fill in the holes by genuine contact structures.

Notice that any contact Hamiltonian which is positive on $\partial \Delta_{cyl}$ must dominate (is greater than) some special Hamiltonian for sufficiently small $\varepsilon$. It is important that it is possible to reduce to assuming that the holes are modelled by finitely many types of contact Hamiltonian circle models, therefore in a given dimension, there is a certain universal $\varepsilon_{univ}$, such that for any $\varepsilon<\varepsilon_{univ}$, every hole dominates a circle model for a special $K_{\varepsilon}$. Therefore, the key overtwisted annuli are given by circle models for special Hamiltonians corresponding to such an $\varepsilon$.

To get from overtwisted annuli to overtwisted disks, we use the fact that the main lemma embedding $\Theta$ fixes the end where $z\in[1-\varepsilon,1]$. Therefore we do not need the full annulus (neighborhood the boundary of the cylinder), only the topological disk obtained but cutting off the end of the cylinder.

The overtwisted disk is thus defined to be the disk with the contact germ on the boundary of a circle model over a cylinder (excluding one end) defined by a special contact Hamiltonian $K_{\varepsilon}$ for some $\varepsilon<\varepsilon_{univ}$ where $\varepsilon_{univ}$ depends only on the dimension. I think that dependence on the dimension is not really understood at this point, but the idea is that $\varepsilon_{univ}$ probably gets smaller as the dimension increases, so the region where the contact Hamiltonian is negative would be larger.

Proving the main lemma

We want to show that there is a contact embedding $\Theta:\Delta_{cyl}\to \Delta_{cyl}\# \Delta_{cyl}$ such that for a special Hamiltonian $K_{\varepsilon}$, $\Theta_*K_{\varepsilon} (where here $\#$ denotes the boundary sum obtained by tubing the two cylinders together so that the contact Hamiltonian is positive on the tube). For the parametric version, the main lemma shows there is a family $\Theta_s$ interpolating between the identity and $\Theta$.

Recall the things we know how to do with contactomorphisms from the previous post:
(1) We can reorder contact Hamiltonians however we want in regions where they are negative by the disorder lemma.
(2) We have transverse scaling contact embeddings which shrinks/expands $\Delta_{cyl}$ in the $z$ direction by a diffeomorphism $h:\mathbb{R}\to \mathbb{R}$ at the cost of correspondingly shrinking/expanding $\Delta_{cyl}$ in the $u$ direction by rescaling by $h'(z)$. The effect on the contact Hamiltonian is $(\Phi_h)_*K(h(z),h'(z)u,\theta)=h'(z)K(z,u,\theta)$.
(3) We have twist embeddings which shrink/expand $\Delta_{cyl}$ in the radial $u$ direction by rescaling by $\frac{1}{1+g(z)u}$ if you allow the angular $\theta$ directions to be twisted. The effect on the contact Hamiltonian if we ignore the angular coordinate is $(\Psi_g)_*K(z,\frac{u}{1+g(z)u})=(1-g(z)u)K(z,u)$.

To prove the main lemma, we want to stretch out the $z$ direction of $\Delta_{cyl}$ so that it spreads the length of the connected sum. We can do this with a transverse scaling contactomorphism, but the $u$ directions will expand: $(z,u)\mapsto (h(z),h'(z)u)$. Since we don’t want to mess with the contact structure on the $z$ ends, we choose $h$ to look like a translation so $h'(z)=1$ when $z$ is within $\varepsilon$ of the ends. We can compensate for the expansion in the $u$ directions away from the ends with a twist embedding which rescales the expanded $u$ directions to fit back inside a (longer) cylinder where $u\leq 1$, by choosing $g(z)=1-\frac{1}{h'(h^{-1}(z))}$. The total effect of composing these two maps is an embedding $\Gamma$ mapping $(z,u)\mapsto (h(z),\frac{h'(z)u}{1+(h'(z)-1)u})$ (the angular directions get twisted some amount but we don’t care). $\Gamma$ sends a short cylinder $\Delta_{cyl}$ to a longer cylinder $\Delta_{cyl}\# \Delta_{cyl}$, so that the points where $u=1$ are sent to points where $u=1$, but points where $u<1$ are sent to points with $u$-coordinate $\frac{h'(z)u}{1+(h'(z)-1)u}> u$. So this contactomorphism inflates the cylinder in the $u$ directions towards the boundary. By choosing a family of diffeomorphisms $h_s$ starting with a basic translation we get a family of embeddings $\Gamma_s$ which look like this:

Now, we want to see the effect on the contactomorphisms on a special Hamiltonian $K_{\varepsilon}$. We find that

$(\Gamma_s)_*K_{\varepsilon}(h_s(z),\frac{h_s'(z)u}{1+(h_s'(z)-1)u})=(h_s'(z)-(h_s'(z)-1)u)K(z,u)$

which can be rewritten as

$(\Gamma_s)_*K_{\varepsilon}(h_s(z),u)=(h_s'(z)-(h_s'(z)-1)u)K\left(z,\frac{u}{h_s'(z)-(h_s'(z)-1)u}\right)$.

When $z$ is within $\varepsilon$ of the ends, we have chosen $h$ to be a translation, so $(\Gamma_s)_*K_{\varepsilon}(h_s(z),u)=K_{\varepsilon}(z,u)$, i.e. the Hamiltonian is basically fixed to be standard on these ends. When we reach $s=1$, the ends of $\Gamma_1(\Delta_{cyl})$ coincide with the ends of $\Delta_{cyl}\#\Delta_{cyl}$ so in these regions $(\Gamma_1)_*(K_\varepsilon)=K_{\varepsilon}\#K_{\varepsilon}$.

The rescaling factor for the Hamiltonian, $(h_s'(z)-(h_s'(z)-1)u)$ is always greater than or equal to 1, so the region where $(\Gamma_s)_*K \leq 0$ is the image under $\Gamma_s$ of the region where $K\leq 0$ and similarly $\{(\Gamma_s)_*K\geq 0\}=\Gamma_s(\{K\geq 0\})$. Since we can use the disorder lemma, we don’t care much about the exact negative values of $(\Gamma_s)_*K$, but we do need $(\Gamma_1)_*K(z,u)\leq K\#K(z,u)$ wherever $(\Gamma_1)_*K\geq 0$. Therefore we need to check this inequality on points $\Gamma_1(z,u)$ where $u\in [1-\varepsilon,1]$ and $z$ is more than $\varepsilon$ away from the ends (since we already understand the behavior when $z$ is within $\varepsilon$ of the boundary). On this region, the special Hamiltonian $K_{\varepsilon}$ is just a linear function of $u$ with slope 1. Therefore

$(\Gamma_s)_*K_{\varepsilon}(h_s(z),u)=(h_s'(z)-(h_s'(z)-1)u)\left(\frac{u}{h_s'(z)-(h_s'(z)-1)u}-(1-\varepsilon) \right)$

which as a function of $u$ is linear, has the value $0$ when $\frac{u}{h_s'(z)-(h_s'(z)-1)u}=1-\varepsilon$, and the value $\varepsilon$ at $u=1$. Notice that $\frac{u}{h_s'(z)-(h_s'(z)-1)u}=1-\varepsilon$ when $u=\frac{h_s'(z)(1-\varepsilon)}{1+(h_s'(z)-1)(1-\varepsilon)}>1-\varepsilon$ so in this region $(\Gamma_s)_*K_{\varepsilon}$ compares to $K\#K$ like this:

Therefore $(\Gamma_1)_*K_{\varepsilon}(z,u)\leq K\#K(z,u)$ wherever $(\Gamma_1)_*K_{\varepsilon}\geq 0$. Then we can use the disorder lemma to produce a contactomorphism which fixes everything on this positive region but makes the Hamiltonian sufficiently negative in the region where $K_\varepsilon\#K_\varepsilon\leq 0$ so that after composing $\Gamma_s$ with this disorder contactomorphism we get the embedding $\Theta_s$ such that $(\Theta_1)_*K_{\varepsilon}\leq K_{\varepsilon}\#K_{\varepsilon}$ as required. Notice that $\Theta_s$ fixes the end where $z\in[1-\varepsilon,1]$ so we do not actually need to use that end of the overtwisted annulus to fill in the hole.

It is worth noting that an overtwisted disk could be modelled using any Hamiltonian for which the main lemma could be proven, not just the ones that increase linearly near the boundary. The tricky part to check for a more general function is the inequality near the $u$-boundary. When the contact Hamiltonian was linear, the contactomorphism transformation and the rescaling factor cancelled in just the right way so that the pushed forward contact Hamiltonian was still linear in $u$ so the inequality could be determined simply by understanding the values near end points. For more general contact Hamiltonians you would probably need to do more work to get the required estimates.

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## Contact Hamiltonians II

This post is a continuation from Roger’s last post on Contact Hamiltonians about Borman, Eliashberg, and Murphy’s h-principal result on higher dimensional overtwisted contact structures. Here we will start to get into some of the main pieces of the proof.

First lets recall what we are trying to prove: given an almost contact structure that contains a particular model “overtwisted disk”, this almost contact structure can be homotoped through almost contact structures to a genuine contact structure. A parametric version of this theorem implies that homotopic overtwisted contact structures are isotopic through contact structures. So far, we still have not actually defined an overtwisted disk in higher dimensions (but will soon); for now just keep in mind that there is a model piece of contact manifold that we assume is embedded in the almost contact manifold from the start. The broad idea of the proof is to modify the almost contact structure to be genuinely contact on larger and larger pieces of the manifold until all the “holes” (pieces where the almost contact structure has not been made contact yet) are filled in. Gromov’s (relative) h-principal for open contact manifolds implies that the almost contact structure can be homotoped to be contact in the complement of a compact codimension zero piece (while fixing the structure near the overtwisted disk). A technical argument which keeps track of the angles between the contact planes and the boundary of the hole reduces the argument to extending the contact structure over holes which near the boundary agree with a certain circular model. We put off this technical argument for now, but mention that it is analogous to the argument in the 3-dimensional case called part 1 in this earlier post.

Refer to section 6 of the BEM paper for more details on the first half of this post, and to section 8 for the second half.

The circular model

The goal here is to define a model almost contact structure on a ball, which near the boundary is a genuine contact structure encoded by a contact Hamiltonian. View the 2n+1 dimensional ball as the product of a 2n-1 dimensional ball $\Delta$ with a 2-dimensional disk $D^2$, viewed as a subset of $\mathbb{C}$. The contact Hamiltonian is a function

$K: \Delta \times S^1 \to \mathbb{R}$

Using the standard contact structure $\lambda_{st}=dz+\sum_i r_i^2d\theta_i$ on $\Delta\subset \mathbb{R}^{2n-1}$, recall that an extension of this function $\widetilde{K}: \Delta \times D^2 \to \mathbb{R}$ defines an almost contact structure $\alpha = \lambda_{st}+\widetilde{K}d\theta$ on $\Delta\times D^2$ which is genuinely contact wherever $\partial_r\widetilde{K}>0$ (compute $\alpha\wedge d\alpha>0$). Using the conventions from the BEM paper, we will use the coordinate $v=r^2$. If $K$ is everwhere positive, we can realize this contact structure near the boundary of the following embedded subset of the standard contact $(\mathbb{R}^{2n+1},\ker(\lambda_{st}+vd\theta)$

$B^{S^1}_{K}:=\{(x,v,\theta)\in \Delta\times \mathbb{C} : v\leq K(x,\theta)\}$

If $K$ is negative anywhere, then we need to look at a modified version. We can still encode the shape of $K$ by shifting everything up by a sufficiently large constant $C$ so that $K+C$ is positive. Then define

$B^{S^1}_{K,C}:=\{(x,v,\theta)\in \Delta \times \mathbb{C} : v\leq K(x,\theta)+C \}$.

In order to have the contact form encodes the contact Hamiltonian $K$ near the boundary, we want to shift the contact form from $\lambda_{st}+vd\theta$ to $\lambda_{st}+(v-C)d\theta$ near the boundary. However, because the polar coordinates degenerate near $v=0$, in a neighborhood of $v=0$, we need to keep the form standard: $\lambda_{st}+vd\theta$. Define a family of functions $\rho_{(x,\theta)}(v)$ to interpolate between these two, and then define the almost contact structure on $B^{S^1}_{K,C}$ by the form $\eta_{\rho}=\lambda_{st}+\rho d\theta$. We want this almost contact form to be genuinely contact near the boundary since we are looking for a model for the holes. You can compute $\eta_{\rho}\wedge d\eta_{\rho}$ to see that $\eta_{\rho}$ defines a genuine contact form exactly when $\partial_v\rho_{(x,t)}(v)>0$. The boundary of the ball $B^{S^1}_{K,C}$ has two pieces: the piece where $v=K(x,\theta)+C$ and the piece where $x\in \partial \Delta$. In a neighborhood of the former piece, $\rho(v)=v-C$ so it has positive derivative, but on the latter piece we have to impose the condition directly that $\partial_v\rho_{(x,t)}>0$ in an open neighborhood of points where $x\in \partial\Delta$.

One can show that different choices for $C,\rho$ which satisfy these conditions do not yield genuinely different almost contact forms $\eta_\rho$ because up to diffeomorphism, different choices do not change the contact structure near the boundary or the relative homotopy type of the almost contact structure on the interior.

The key point is that this almost contact structure on $B^{S^1}_{K,C}$ can be chosen to be a genuine contact structure only along $x$ slices where $K$ is positive. Remember that $\rho_{(x,\theta)}$ says how much the almost contact planes are rotating in the radial direction, and if $\partial_r\rho_{(x,\theta)}=0$ this means the twisting has stopped. If $K(x,\theta)$ is negative then since $\rho_{(x,\theta)}(K(x,\theta)+C)=K(x,\theta)<0$ and $\rho_{(x,\theta)}(v)=v$ near 0, $\rho_{(x,\theta)}$ must have a critical point and the almost contact planes must stop twisting and thus fail to be genuinely contact. In particular, to define the circle model for a contact Hamiltonian $K$ we need $K(x,\theta)>0$ near points where $x\in \partial \Delta$, so we only consider such Hamiltonians.

Here is a 3-dimensional example. The arrows indicate the twisting of the almost contact planes defined by $\rho$. Note that where K fails to be positive the planes start twisting counterclockwise as you move radially outward, but then have to switch to turning clockwise at some point. The functions $\rho_{(x,\theta)}$ are indicated by the graphs above–they start having critical points when K fails to be positive.

If we have two contact Hamiltonians $K_0$ on $\Delta_0$ and $K_1$ on $\Delta_1$ such that $\Delta_0\subset \Delta_1$ and $K_0 \leq K_1$, then it is not hard to see that we can choose circle models for each such that $(B^{S^1}_{K_0,C},\eta_{\rho_0})$ embeds into $(B^{S^1}_{K_1,C},\eta_{\rho_1})$ and so that $(\rho_1)_{(x,\theta)}(v)=v-C$ in a neighborhood of the entire region where $K_0(x,\theta)\leq v \leq K_1(x,\theta)$. In other words, the almost contact structure is contact and twisting in the standard way along the radial direction on the region between $K_0$ and $K_1$. In the terminology of the BEM paper, $K_1$ directly dominates $K_0$. View of the extendability a contact structure from one contact germ defined by a contact Hamiltonian $K_1$ to another germ defined by $K_0$, as an ordering. The thing that makes this ordering interesting is that using contactomorphisms to change coordinates, a contact germ can be modelled by a different contact Hamiltonian. Therefore if $K_0$ and $K_1$ cannot be directly compared (i.e. at some points $K_0\leq K_1$ but at others $K_0>K_1$), then there may be a different contact Hamiltonian $\widetilde{K}_0$ which corresponds to the same contact germ in different coordinates such that $\widetilde{K}_0$ can be compared to $K_1$. This will be the subject of the rest of this post.

Contactomorphisms and conjugating the Hamiltonian

Given a contactomorphism on the domain $(\Delta,\lambda)$, we want to construct an induced contactomorphism on $(\Delta\times \mathbb{C},\lambda+\rho d\theta$. Because contactomorphisms only preserve the contact planes, and not the contact form, a contactomorphism $\Phi: (\Delta,\ker(\lambda))\to (\Delta, \ker(\lambda))$ satisfies $\Phi^*(\lambda)=c_{\Phi}\lambda$ where $c_{\Phi}$ is a positive real valued function on $\Delta$. Because the pull-back rescales $\lambda$, we need to rescale the Hamiltonian on the image as well so that it fits together with $\Phi^*\lambda$ to give a contact form for the same contact structure. Therefore define $\Phi_*K$ by $(\Phi_*K)(\Phi(x),\theta)=c_{\Phi}(x)K(x,\theta)$.

$\Phi$ natural induces an extension on $\Delta\times \mathbb{C}$ defined by $\widehat{\Phi}(x,v,\theta)=(\Phi(x),\phi_{(x,\theta)}(v),\theta)$ for any family of functions $\phi_{(x,\theta)}$. If $\widetilde{\rho}$ defines the contact structure on the image $\lambda+\widetilde{\rho}d\theta$ then

$\widehat{\Phi}^*(\lambda+\widetilde{\rho}d\theta)=\Phi^*\lambda+\widetilde{\rho}\circ\phi d\theta=c_{\Phi}\lambda+\widetilde{\rho}\circ \phi d\theta$

Therefore the function defining the contact Hamiltonian on the image must satisfy $\widetilde{\rho}_{(x,\theta)}\circ \phi_{(x,\theta)}(v)=c_{\Phi}(x)\rho_{(x,\theta)}$.

Why did we include the function $\phi_{(x,\theta)}$ in the above definition of $\widehat{\Phi}$? This is to allow us to reparameterize $\widetilde{\rho}_{(x,\theta)}$ so that it satisfies the required conditions to define the circular model (should look like the identity near $v=0$, and should look like the identity shifted by the constant near $v=K+C$). Before the contactomorphism, to define the circular model, you choose a constant $C$ and then $\rho_{(x,\theta)}$ is considered on the domain $[0, K(x,\theta)+C]$ and is required to have certain behavior near the endpoints of this interval. After rescaling, we have a new Hamiltonian $(\Phi_*K)(\Phi(x),\theta)=c_{\Phi}(x)K(x,\theta)$, so we pick a new constant $\widetilde{C}$ so that $\Phi_*K+\widetilde{C}>0$. Then we consider $\widetilde{\rho}_{\Phi(x),\theta}$ on the interval $[0,c_{\Phi}(x)K(x,\theta)+\widetilde{C}]$ and require it to have particular behavior near $0$ and $c_{\Phi}(x)K(x,\theta)+\widetilde{C}$. Because $\widetilde{\rho}_{(x,\theta)}=c_{\Phi}(x)\rho_{(x,\theta)}\circ \phi_{(x,\theta)}^{-1}$, modifying the functions $\phi_{(x,\theta)}$ allows us to make $\widetilde{\rho}_{\Phi(x),\theta}$ have the desired behavior near the end points of the interval $[0,c_{\Phi}(x)K(x,\theta)]$ so that $\widetilde{\rho}$ can be used to define a circular model for $\Phi_*K$.

The action of the contactomorphism $\Phi$ on the contact Hamiltonian $K$ is referred to as conjugating the Hamiltonian for the following reason. If the contact Hamiltonian is generated by a contact isotopy $\phi^t_K$ in the sense that $\lambda(\partial_t\phi^t_K)=K(\phi^t_K,t)$, then you can compute that $\Phi\phi^t_K\Phi^{-1}=\phi^t_{\Phi_*K}$.

Important types of contactomorphisms and their effects on the Hamiltonian

What kinds of changes can we make in the contact Hamiltonian through contactomorphisms? A key lemma is that in a (star-shaped) region where the contact Hamiltonian is negative, contactomorphisms can be used to make the values arbitrarily close to zero. This basically means that the exact negative values of a contact Hamiltonian do not matter in the ordering, since a contactomorphism can make any given negative values larger than any other given negative values. This indicates that the key difficulty in filling in the contact structure on holes whose boundary looks like a contact Hamiltonian circular model, is where and how large are the regions where the contact Hamiltonian is positive.

The idea of the proof of this “disorder lemma” (Lemma 6.8 in the BEM paper) is as follows. Let $\Delta$ be the region where the contact Hamiltonian $K$ is defined and let $\widetilde{\Delta}$ be a subset containing the piece where $K$ is negative. Construct a contactomorphism $\Phi$ which shrinks $\widetilde{\Delta}$ into itself a lot, but fixes the points of $\Delta$ sufficiently away from $\widetilde{\Delta}$. (You can do this by looking at the flow of an inward pointing contact vector field–this is where the star-shaped condition comes in–cut off to zero sufficiently away from $\widetilde{\Delta}$.) Because $\widetilde{\Delta}$ is being shrunk, the rescaling function $c_{\Phi}(x)$ for the contact form defined by $\Phi^*\lambda=c_{\Phi}\lambda$ is a positive function with very tiny values close to 0, for $x\in \widetilde{\Delta}$. The more $\widetilde{\Delta}$ is shrunk, the tinier the values of $c_{\Phi}$. The corresponding Hamiltonian $(\Phi_*K)(\Phi(x),\theta)=c_{\Phi}(x)K(x,\theta)$ has values rescaled by $c_{\Phi}$. Therefore, by choosing a contactomorphism which shrinks $\widetilde{\Delta}$ enough, the values of $c_{\Phi}$ can be made sufficiently small so that $c_\Phi K(x,\theta)>-\varepsilon$ for $x\in \widetilde{\Delta}$.

In addition to the disorder lemma, we need two types of contactomorphisms of $(\mathbb{R}^{2n-1},\xi_{st})$ which rescale in certain directions. Choose cylindrical coordinates $(z,r_i,\theta_i)$ on $\mathbb{R}^{2n-1}$ and let $u_i=r_i^2$ so $\xi_{st}=\ker(dz+\sum_i u_i\theta_i)$.

A transverse scaling contactomorphism $\Phi_h$ is defined by a diffeomorphism $h:\mathbb{R}\to \mathbb{R}$ by $\Phi_h(z,u_i,\theta_i)=(h(z),h'(z)u_i,\theta_i)$. You can check directly that this diffeomorphism is a contactomorphism which rescales the standard contact form by $h'(z)$. Therefore this contactomorphism modifies a contact Hamiltonian by

$(\Phi_h)_*K(z,u_i,\theta_i)=h'(h^{-1}(z))K\circ\Phi_h^{-1}(z,u_i,\theta_i)$

The tagline for this type of contactomorphism is you can “trade long for thin”. By choosing a shrinking $h$, you can shrink a domain which is long in the $z$ direction at the cost of shrinking the radial $u_i$ directions.

A twist embedding contactomorphism $\Psi_g$ allows you to rescale the radial directions $u_i$ by $\frac{1}{1+g(z)u}$ at the cost of twisting in the angular directions by an amount that depends on $g$ (see section 8.2 of the BEM paper for the exact formulas). The points at radii where $g(z)u>-1$ get sent to points where $g(z)u<1$ since $g(z)\frac{u}{1+g(z)u}-1$. The rescaling factor for the contact form is $(1-g(z)u)$, so the contact Hamiltonian is rescaled accordingly. For positive functions $f_1,f_2$, setting $g=\frac{1}{f_1}-\frac{1}{f_2}$ gives $\Psi_g$ taking the region where $u\leq f_2(z)$ to the region where $u\leq f_1(z)$. Therefore twist embeddings allow you to modify the radial directions however you want to, with basically no cost (just twisting the angular directions).

By composing these two types of contactomorphisms we can use transverse scaling to stretch or shrink in the $z$ direction at the cost of stretching or shrinking radially. Then we can use a twist embedding to counteract the stretching or shrinking in the radial directions, with only the cost of twisting in the angular direction, which does not significantly change the shape of the region.

These contactomorphisms are the key ingredients towards filling in circular model holes connected summed with neighborhoods overtwisted disks, as will be discussed in the next post.

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## Contact Hamiltonians (Part I)

This entry follows the post Contact Hamiltonians (Introduction), where we discussed normal forms for contact forms and the appearance of contact Hamiltonians. In this entry we will focus on the 3–dimensional situation and hence we will be able to write formulas and draw (realistic) pictures.

Consider a 2–sphere of radius 1 in the standard tight contact Euclidean space $(\mathbb{R}^3,\lambda_{st}=dz+r^2d\theta)$. Its characteristic foliation (defined by the intersection of the tangent space and the contact distribution) has two elliptic singular points in the north and south poles and all the leaves are open intervals connecting the north and the south pole. Take a transversal segment I=[0,1] connecting the poles (a vertical segment will do). Given a point in the segment we can consider the unique leaf through that point and move around the leaf until we hit the interval I=[0,1] again. This defines a diffeomorphism of the interval [0,1] fixed at the endpoints. We will call this diffeomorphism the monodromy of the foliation (and note that conversely any diffeomorphism will give a foliation on the 2–sphere via a mapping torus construction and collapsing the boundary). This is drawn in the following figure:

In the figure the monodromy map is represented by the orange arrow. This monodromy does not have fixed points (this is crucial). Let us look at the monodromy in the sphere of radius $\pi+c$ , where c is a small positive constant, in the overtwisted contact manifold $(\mathbb{R}^3,dz+rtg(r)d\theta)$. The overtwisted monodromy is drawn in the next figure:

There are 3 types of points in the vertical transverse interval I=[0,1]. The Type 1 points belong to a leaf, Leaf I in the figure, such that the points move down in the segment. The Type 2 points are the points between the unique pair of closed leaves, these belong to Leaf II and move up. The Type 3 points are fixed points, there are two leaves of this type (Leaf III). The monodromy is represented by the blue arrows.

Hence, we can encode the tight and the overtwisted foliations on the 2–sphere in terms of their monodromies in the following figure:

In the last entry we explained a relation between monodromies and contact Hamiltonians. Consider a contact form $dz-H(x,y,z)dx$ in $\mathbb{R}^3$, this is a quite general normal form (which we can obtain by trivializing along the y–lines of $\mathbb{D}^2(x,y)$). If we restrict to the sphere $x^2+y^2+z^2=R^2$ we can write H in terms of $H=H(x,z)$ at points where the implicit function theorem works. Then the characteristic foliation is nothing else than the solution of the time–dependent (x is the time) differential equation $dz-Hdx=0$ on the interval I=[-1,1] given by the coordinate z. Hence the contact Hamiltonian yields the ODE  to which the monodromy is a solution.

Tool: How do we obtain a piece of a disk in standard contact $(\mathbb{R}^3,dz-ydx)$ with a given characteristic foliation ?

Answer: Consider a disk in the (z,x)–plane and a function H(z,x). The standard contact structure $dz-ydx$ restricts to the graph of H in $\mathbb{R}^2(z,x)\times\mathbb{R}(y)$ as $dz-ydx|_{\{y=H\}}=dz-Hdx$.

For instance, let us consider the following function H(z) for z=[-1,1]:

This function H can be considered as a function on the polydisk (x,z) which is represented by the lower square in the third figure (the whole figure is PL immersed in the standard contact 3–space). Its image is the bumped square drawn above it, and we may consider the PL sphere obtained by adding the vertical annulus connecting the domain and the graph. The characteristic foliation on the bottom piece is by the horizontal z–lines, on the annulus the foliation is vertical and on the top piece the foliation is drawn on the left. Note that the characteristic foliation in this immersed PL sphere has a closed leaf (in red) coming from the fixed point (or zero, if we look at it horizontally) of H.

Let us briefly focus on the existence of a contact structure in a region bounded by a domain and a graph as in the previous paragraph.

Exercise: Does there exist a contact structure filling the following pink region ?

(The contact structure should restrict to the germs (in purple) already defined on the boundary.)

Answer: Yes. This is already embedded in $\mathbb{R}^3$, hence we just need to restrict the ambient contact structure. (This should be compared with the previous post where this question was also formulated and answered in terms of the positivity of the function H).

The second exercise we need to solve is as simple as the previous one, let us however draw the figures in order to keep them in mind.

Annulus Problem (weak): Does there exist a contact structure in the (yellow) annulus ?

The contact structure should also restrict to the germs (in purple and green) already defined on the boundary.

Answer: Yes, again this is already embedded in standard contact Euclidean space. This is yet another instance of the relevance of order. If one Hamiltonian is less than another one, then we can obtain a contact structure on the annulus.

This will be formalized in subsequent posts using the notion of domination of Hamiltonians and their corresponding contact shells. We shall not use this language right now.

We are now going to prove Eliashberg’s existence theorem in dimension 3 from the contact Hamiltonian perspective (i.e. from the monodromy viewpoint). The fundamental fact is that we only need to extend contact structures up to contactomorphism and this is translated to the fact the Hamiltonians can be conjugated.

Annulus Problem (strong): Does there exist a contact structure on the following region ?

Answer: If we are able to conjugate the bottom Hamiltonian (in green) strictly below to the upper one (in purple), then we can use the contact structure of the embedded annulus (weak version of the annulus problem). Hence, it all reduces to the order (or rather, the lack thereof).

Fundamental Fact: There exists a conjugation of the bottom Hamiltonian such that it is strictly less than the upper one. In general, given two Hamiltonian with fixed points which are positive at the endpoints of the interval, there exists a conjugation bringing one of them below the other.

(This is an exercise with functions in one variable, in higher dimensions this is no longer simple and this is precisely the main point that M.S. Borman, Y. Eliashberg and E. Murphy have understood).

Let us prove Eliashberg’s 3–dimensional existence theorem, we focus on the extension part (part 2 according to the post three entries ago).

Extension Problem (Version I): Suppose that there exists a contact structure on the complement of a ball $B^3$ in a 3–fold (which is given by Gromov’s h–principle, see previous posts) and that the characteristic foliation on the boundary $S_h^2$ has monodromy with fixed points (h stands for hole). Can we extend the contact structure ?

Suppose that there exists a sphere $S_{ot}^2$ somewhere inside the manifold with an overtwisted monodromy (in blue, see above) in its characteristic foliation. Consider the annulus $A_{ot}=S_{ot}^2\times(-\tau,\tau)$. Use the south poles of $S_{ot}^2\times\tau$ and $S_h^2$ to connect both and obtain an annulus $A$ such that the monodromy in the exterior boundary sphere is the concatenation of the contactomorphisms of the intervals (green#pink). Hopefully this figure helps:

The monodromies of the foliations in the two spheres bounding the annulus $A_{ot}$ are drawn in pink (exterior boundary) and blue (interior boundary). The monodromy in green is that of $S_h^2$. Connecting the spheres $S_h^2$ and $S_{ot}^2\times\{\tau\}$ yields a sphere with the monodromy green#pink (the transition area is purple, this has some relevance but it is not essential). Consider the annulus A bounded by $S_h^2\#(S_{ot}^2\times\{\tau\})$ and $S_{ot}^2\times\{-\tau\}$. We have reduced the problem of extending the contact structure to the interior of $S_h^2$ to the problem of extending the contact structure in the annulus A. In the exterior boundary of A the characteristic foliation is green#pink and on the interior is red (which comes from moving blue).

Extension Problem (Version II): Does there exists a conjugation such that (the graph of) any contactomorphism can be conjugated to lie beneath any other (graph) ?

Answer:  No. Fixed Points are an obstruction. However, if we restrict ourselves to the same question in the class of contactomorphisms with fixed points the answer is yes. This is exactly the Fundamental Fact stated above.

How do we conclude the proof ? Conjugate the red Hamiltonian to lie beneath the green#pink Hamiltonian and use the contact structure in the resulting annulus (as embedded in standard contact space). Assuming Gromov’s h–principle and the technical work in order for the foliation to be controlled, this argument concludes the theorem.

(We have disregarded some details, but the idea of the argument is the one described above. Observe that the parametric version of the existence problem in dimension 3 is quite immediate from the Hamiltonian perspective.)

Note also that we do not need the whole sphere $S^2_{ot}$: in order to use the argument with the Hamiltonians we can cut the North pole of $S^2_{ot}$ and retain just the remaining disk, which is an overtwisted disk.

There is a substantial advantage in this proof of the 3–dimenisonal case: we can define an overtwisted disk $\mathbb{D}^{2n}$ in higher dimensions 2n+1 to be the object that appears when using the contact Hamiltonian on a simplex $\Delta^{2n-1}$ given by

(We will give precise definitions in the subsequent entries.)

The strategy of the argument works in higher dimensions if we can prove the Fundamental Fact stating that there is enough disorder for contact Hamiltonians. In the next entries we will focus on this crucial step in higher dimensions and conclude existence.

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## Contact Hamiltonians (Introduction)

This entry is part of the series of posts on the recent work of  M. S. Borman, Y. Eliashberg and E. Murphy on the existence and classification of overtwisted contact structures in all dimensions. In the previous two entries the construction in the 3–dimensional case and Gromov’s h–principle for the open case have been explained.

The essential fact in Eliashberg’s 3–dimensional argument (this is part 2, two entries ago) is the control on the characteristic foliation: the extension problem is reduced to being able to fill a 2–sphere with a contact ball given a particular characteristic foliation on the boundary. This can be done explicitly by deforming the local model provided by the standard overtwisted contact ball in order to have the given characteristic foliation on the boundary. The construction in higher dimensions is not quite the same and it relies on the use of contact Hamiltonians, hence this and subsequent entries. This first introductory entry should help the reader to follow the next entries, each entry should however be readable on its own.

Consider a contact manifold of dimension 2n+1. The information of a contact structure is contained in a 1–form (locally this is the data of 2n+1 functions, plus another one as a conformal factor, satisfying 1 non–degeneracy equation). This data can be drastically reduced when restricted to simple topological subsets if we choose appropriate coordinates: for instance, the Darboux theorem tells us that the normal form of a contact 1–form around a point is $\alpha=dz-\sum_{n=1}^{2n}y_idx_i$. It will be helpful for the reader to understand the geometric proof of the Darboux theorem, see Theorem 2 in Page 5 of Topological methods in 3-dimensional contact geometry. The strategy is finding a suitable flow to obtain the desired coordinates.

Suppose that we have a contact manifold $(M,\alpha_0)$ and a codimension–1 distribution $\xi=\ker(\alpha)$ on the manifold $M\times\mathbb{D}^2(r,\theta)$ such that it restricts to $\ker(\alpha_0)$ on each slice $M\times\{pt.\}$. In these hypotheses:

Lemma: There exists the following normal form for the 1–form $\alpha$, we can write $\alpha=\alpha_0+H(p,r,\theta)d\theta$ for some smooth function $H:M\times\mathbb{D}^2\longrightarrow\mathbb{R}$.

Proof: Consider the product manifold $M\times\mathbb{D}^2$ as a trivial fibre bundle over the disk $\mathbb{D}^2$. The data in the hypothesis gives a connection in this bundle whose parallel transport is by contactomorphisms, it is defined as the skew–orthogonal complement of the symplectic subspace $\ker(\alpha_0)$ in the bundle $(\xi,d\alpha)$ with respect to the 2–form $d\alpha$ (which is not necessarily symplectic). We can then consider the radial vector field in the base $\mathbb{D}^2$ and lift it to the total space $M\times\mathbb{D}^2$. The pull–back of the contact form by this flow is (conformally) of the form $\alpha_0+H(p,r,\theta)d\theta$ for some function $H:M\times\mathbb{D}^2\longrightarrow\mathbb{R}$. The reason being that the radial factor $dr$ cannot appear because in the trivializing coordinates (provided by the flow of the lift), the lift of the radial vector field belongs to the distribution. $\hfill\Box$

There are a couple of technical details regarding the existence of the flow, which can be translated into the size of the base disk. Let us not focus on that. Thanks to the Lemma we have the following reduction of the extension problem.

Suppose that on a given almost contact (2n+1)–fold V we have a contact structure on all of V except on a neighbourhood $Op(M)\cong M\times\mathbb{D}^2$ of a codimension–2 submanifold M with trivial normal bundle. If the almost contact structure $\xi$ satisfies the hypothesis for the Lemma in $Op(M)$, then the extension problem for the contact structure is reduced to:

Problem: Given a germ of a contact structure on $M\times S^1$ described by a function $H:M\times S^1\times(1-\varepsilon,1+\varepsilon)\longrightarrow\mathbb{R}$, does there exist a contact structure on $M\times\mathbb{D}^2$ such that it restricts to the given germ on $M\times S^1$ ?

There are two remarks at this point. First, the meaning of the function H is really geometric. It describes the angle of rotation of the contact structure in the radial direction, in particular the condition for $\alpha_0+Hd\theta$ to be a contact structure on $M\times S^1\times(1-\varepsilon,1+\varepsilon)$ reads $\partial_r H>0$ (this is often stated as the contact structure has to rotate). Second, the extension does not need to be of the form $\alpha_0+Hd\theta$, we just need a contact structure on $M\times\mathbb{D}^2$.

Example 1 (Tight): Consider $(M,\alpha_0)=((-1,1),dz)$ and the function $H(p,r,\theta)=r^2$. The contact form is $\alpha=dz+r^2d\theta$ and since the function H verifies the contact condition on $B^3=(-1,1)\times\mathbb{D}^2$ this defines a contact structure on $B^3$. This is the standard contact structure on the ball.

Example 2 (Overtwisted): Consider $(M,\alpha_0)=((-1,1),dz)$ and $H(p,r,\theta)=r\cdot tg(r)$. The contact form is then $\alpha=dz+rtg(r)d\theta$, which should be read as $\alpha=cos(r)dz+rsin(r)d\theta$. This is the standard overtwisted contact structure on the ball $B^3=(-1,1)\times\mathbb{D}^2$ if the radius of the disk is larger than $\pi$.

This second example has the following very nice feature: the function $H(r)=rtg(r)$ is negative at r=2. This provides a solution to the problem of extending a germ in $(-1,1)\times S^1\times\{2\}$ to the interior $(-1,1)\times\mathbb{D}^2$ if this germ is everywhere negative. Although a priori it seems non–sense to go from 0 to a negative value growing (in order to preserve the contact condition) this can be done by inserting a pole, i.e. going to infinity (and then continuing from minus infinity). This phenomenon underlies many h–principles, try to solve for instance Section 4.1.1 from Chapter 4 in Eliashberg–Mishachev book.

The functions H appearing in the above constructions are called contact Hamiltonians.

Problem (Easy Case): Given a germ of a contact structure on $M\times S^1$ described by a positive function $H:M\times S^1\times(1-\varepsilon,1+\varepsilon)\longrightarrow\mathbb{R}$, does there exist a contact structure on $M\times\mathbb{D}^2$ such that it restricts to the given germ on $M\times S^1$ ?

Answer: Yes. In this case the extension can be a contact structure of the form $\alpha_0+\widetilde{H}d\theta$ where $\widetilde{H}$ extends H and is such that $\partial_r\widetilde{H}>0$. Certainly, we just need to construct a function which at the origin looks like $\widetilde{H}=r^2$ and then it grows in the radial direction until we reach the value given by H on the boundary $M\times S^1$. The existence of such a function is immediate. $\hfill\Box$

The difficult case is that of a germ of a contact structure defined by a Hamiltonian which is negative in some points and positive in others (the presence of such negativity requires overtwistedness). The situation described above is quite hard because we may not even understand the (contact) topology of M. The first step is to focus on $M=\Delta^{2n-1}$ a (2n-1)-ball, or star–shaped domain, in $\mathbb{R}^{2n-1}$.

In the next entry, Contact Hamiltonians (Part I) we will continue to use contact Hamiltonians and relate them to Eliashberg’s 3–dimensional argument using the characteristic foliation. The essential word will be monodromy.

In the context above, monodromy arises as follows: consider the contact germ on $M\times S^1(\theta)$ and lift the vector field $\partial_\theta$ to the connection defined before. Its flow at time equal to the length of the circle (say 1) defines a contactomorphism of the fibre $M\times\{0\}$. This is the monodromy contactomorphism.

There is however another way to obtain a contactomorphism of $(M,\alpha_0)$ if we have a  function $H:M\times S^1\longrightarrow\mathbb{R}$ (referred to as a time–dependent contact Hamiltonian). Indeed, compute the Hamiltonian contact vector field X associated to H, which is the unique solution of

$\alpha_0(X_\theta)=H_\theta$ and $d\alpha_0(X_\theta,\cdot)=-dH_\theta+dH(R_{\alpha_0})\cdot\alpha_0$

where $R_{\alpha_0}$ is the Reeb vector field. Then the time–1 flow of the Hamiltonian vector field is a contactomorphism of M. This contactomorphism is said to be generated by the contact Hamiltonian H.

Lemma: Given the contact germ $\alpha_0+Hd\theta$ on $M\times S^1$, the monodromy contactomorphism coincides with the contactomorphism generated by H.

The proof of this lemma is a nice exercise on linear algebra using the defining equation of the connection. This setup can be explicitly studied in 3–dimensions where the monodromies (and the functions H) can be drawn and they correspond to ODEs in the plane. In the next post we will proof Eliashberg’s theorem in dimension 3 from the contact Hamiltonian perspective.

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## Gromov’s h-principle for open contact manifolds

Continuing towards a discussion of the proof of existence and classification of overtwisted contact structures in higher dimensions, here I want to talk about h-principles, and contact structures on open manifolds.

h-principles

Given a partial differential equation or partial differential relation (like the contact condition $\alpha \wedge d\alpha > 0$), one can formally replace the derivatives of the variables with independent formal variables (i.e. $\alpha \wedge \eta >0$ for a 2-form $\eta$). Solving this new problem where the derivatives are replaced by independent formal variables is purely an algebraic topology problem. If the algebraic topology problem has no solution then certainly the partial differential relation has no solution. However, it is generally surprising when the converse holds: namely, the existence of a solution to the algebraic problem implies the existence of a solution of the partial differential relation. A theorem that proves this type of statement is referred to as an h-principle.

The language of jet bundles and holonomic sections will help make this more precise below.

The main results of Borman, Eliashberg, and Murphy that we are heading towards are an h-principle for contact and almost contact structures on higher dimensional closed manifolds which says that any almost contact structure is homotopic through almost contact structures to an actual (overtwisted) contact structure, and a parametric version of this h-principle which says that any family of almost contact structures connecting two genuine overtwisted contact structures can be homotoped to a family of genuine contact structures connecting the fixed overtwisted contact structures on the ends.

While contact structures on closed manifolds can have incredibly complicated classifications (because of the rigidity of tight contact structures), it is a result of Gromov that on open manifolds the geometric subtlety disappears and the classification of contact structures is reduced to algebraic topology by an h-principle. This post is based on a talk given by Kyler as part of the discussion of the proof of flexibility of overtwisted contact structures in higher dimensions, though the original source for the content is Gromov’s Partial Differential Relations book.

Define a (cooriented) almost contact structure on an odd dimensional manifold to be a cooriented hyperplane distribution, together with a non-degenerate 2-form on the distribution. In dimension 3, this is homotopy equivalent to the space of co-oriented 2-plane distributions. Gromov’s theorem is:

Let V be an open manifold. Then the inclusion of cooriented contact structures on V into cooriented almost contact structures on V is a homotopy equivalence.

The proof is based on two main ideas: the holonomic approximation theorem on neighborhoods of codimension one polyhedra, and the fact that all open smooth manifolds smoothly retract onto a neighborhood of a complex of codimension at least one. I’ll start with the former.

The 1-jet space of a fiber bundle $X\to V$, is a bundle $J^1(X)\to V$ where the fiber over $p \in V$ consists of sections of X defined over a neighborhood of p up to an equivalence which equates sections that agree up to 1st order near p. (The r-jet bundle is defined similarly where you equate sections which agree up to rth order, but here we will only need the 1-jet bundle.) A section of $J^1(X)\to V$ chooses an equivalence class of sections over each point in $V$: for each $p\in V$, $s(p)=(f(p),\alpha(p))$ where $f(p)$ is a point in the fiber $X_p$, and $\alpha(p)$ specifies the first partial derivatives of a function at that point. However, even though the section is smooth, $\alpha(p)$ need not specify the actual derivative of $f(p)$ since $\alpha(p)$ is encoded as an independent direction in the fibers of $J^1(X)$. A holonomic section of a 1-jet space is one where this linear variation specified by $\alpha(p)$ agrees with the actual partial derivatives of the differentiable section of $X\to V$ given by the 0th order information of the section. The holonomic approximation theorem aims to approximate an arbitrary section of the 1-jet bundle by a holonomic section as well as possible.

Here the blue curve represents a section of $J^1(X)$. The grey curve represents its projection to the 0th order information, and the 1st order information is encoded in the dimension coming out of the page. Representing the value the blue curve takes in this dimension by a green line of the appropriate slope centered at each point on the grey curve, we see that this is not a holonomic section because the 1st order information is not tangent to the curve.

The important relevant example for Gromov’s theorem is when $X=\Lambda^1(V)$, so sections of the bundle are 1-forms. Sections of the 1-jet space keep track of two coordinates: the pointwise values of the underlying 1-form and its formal linear variation. Locally, $\Lambda^1(V)$ is a trivial bundle, and a section is just the graph of a function on $U\subset V$. Modding out by the equivalence relation, we get that for a section $s:V\to J^1(\Lambda^1(V))$, $s(p)$ keeps track of the point $p$, a point in $T^*_p(V)=\Lambda^1_p(V)$ and an n by n matrix at that point which specifies the formal first partial derivatives of a graph in that equivalence class (where n is the dimension of V). Symmetrizing this matrix ($A-A^T$), gives the coefficients for a 2-form. When the section of $J^1(\Lambda^1(V))$ is a holonomic section, this 2-form built from the 1st order information of the section, is the exterior derivative of the 1-form which gives the 0th order information of the section. Given any pair $(\alpha, \beta)$ of a 1-form and a 2-form, there is a section of $J^1(\Lambda^1(V))$ such that the 0th order information gives $\alpha$ and after symmetrizing the 1st order information we get $\beta$. For holonomic sections, this process gives a pair $(\alpha, d\alpha)$ where $\alpha$ is a 1-form.

There are certain limitations on the extent to which we can approximate an arbitrary section by a holonomic one. For example if we consider the 1-jet space of the bundle $\pi_1: \mathbb{R}\times \mathbb{R}\to \mathbb{R}$. A section of the 1-jet space is given by specifying the pointwise data and a formal 1st derivative. An example of a section has pointwise data given by the graph of $f(x)=x$, and formal derivative specified as 0 (horizontal lines at each point). To approximate this by a holonomic section, we would need to find a function g whose pointwise values only differ from those of $f(x)=x$ by $\varepsilon$, and whose derivative only differs from zero by $\varepsilon$. Such a function would contradict the mean value theorem. So we cannot hope to approximate an arbitrary section by a holonomic one at every point. On the other hand, we can approximate the section in a small neighborhood of a point.

This motivates the idea to look at codimension 1 subspaces. Taking the previous example and just taking the product with a trivial extra dimension with coordinate y, we run into the same problem: that if we can only move up with a tiny slope in the x-direction, we cannot get up far enough by just moving along a path that has slope 1 in the x direction and does not move in the y-direction. However if we are allowed to perturb the path to lengthen it in the extra y-dimension that we have by adding many zig-zags, then we can do this approximation.

Moving along the black curve, there is no holonomic approximation which stays close to the horizontal planes. However, moving along the perturbed red curve, we can find a closer approximation which is holonomic.

This leads us to the precise theorem:

Holonomic approximation theorem: Let $A \subset V$ be a polyhedrong of codimension at least 1 and suppose we have a section of the jet bundle defined over a neighborhood of A. Then for any $\varepsilon>0$, there is a $\varepsilon$ small isotopy $h_t$ of A (measured in the $C^0$ topology), and a holonomic section defined in a smaller neighborhood of $h_1(A)$ which is $\varepsilon$ close to the chosen section.

Suppose we have an almost contact structure $(\alpha, \eta)$ on the open manifold V of dimension n. In order to use this theorem to prove Gromov’s theorem, we must identify a good codimension 1 subset of our open manifold V, where we can use holonomic approximation to find a genuine contact structure on a neighborhood of this subset which is $\varepsilon$ close to the almost contact structure we are considering. Choose a triangulation of V, and for each top dimensional simplex, choose a path from the barycenter of that simplex out to infinity which avoids the barycenters of other simplices. The parts of the 2-skeleton which do not intersect these paths form a codimension 1 subcomplex S. The entire manifold smoothly deformation retracts onto arbitrarily small neighborhoods of S.

Now apply the holonomic approximation theorem to the pair $(\alpha, \eta)$ (which corresponds to a section of $J^1(\Lambda^1(V))$) along S. Then on a tiny perturbation of S, there is an actual holonomic section corresponding to $(\widetilde{\alpha},d\widetilde{\alpha})$ which is very close to $(\alpha,\beta)$. By choosing our $\varepsilon$ sufficiently small so that $(\alpha, \beta)$ and $(\widetilde{\alpha},d\widetilde{\alpha})$ are sufficiently close, we can ensure that the straight line homotopy between them stays in the space of almost contact structures (since the almost contact condition is an open condition ($\alpha\wedge \eta>0$). Therefore the holonomic approximation theorem implies we can homotope our almost contact structure to be contact on a neighborhood of the perturbed S.

Observe that if $g_1:V\to V$ is the end of the deformation retraction which sends V into the neighborhood of S where the almost contact structure is now genuinely contact, then $g_1$ pulls back the almost contact structure on V to a genuine contact structure on V. The deformation retraction provides a homotopy between the almost contact structure which is contact on the neighborhood of S to the genuine contact structure coming from this pullback. Therefore concatenating the homotopy provided by the holonomic approximation theorem with the homotopy provided by the deformation retract, gives a homotopy from our original almost contact structure to an actual contact structure on V.

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## Classifying Overtwisted contact 3-manifolds

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 $\{z=0, r\leq \pi\}\subset \mathbb{R}^3$ where the contact structure on $\mathbb{R}^3$ is the kernel of $\cos(r)dz+r\sin(r)d\theta$. 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 $\xi_0$ and $\xi_1$ 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 $\xi_t$, and using the same ideas, we can homotope the intermediate distributions piece by piece until they become a smooth family $\xi_t'$ of actual contact structures interpolating between $\xi_0$ and $\xi_1$. Then by Gray’s theorem $(M,\xi_0)$ and $(M,\xi_1)$ 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 $Cont^{ot}(M)$ denote the space of overtwisted contact structures on M with a fixed overtwisted disk, and let $Dist(M)$ denote the space of 2-plane distributions which also contain the fixed overtwisted disk. Then the inclusion $Cont^{ot}(M)\to Dist(M)$ 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 $\beta =i^*\alpha$ and $\Omega=i^*d\alpha$ to the surface, and define a foliation by the vector field defined by $\iota_X\Omega=\beta$. 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 $\beta =0$ and $\Omega$ 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 $\xi$ be a contact structure defined near the boundary of a 3-ball. The question of whether $\xi$ 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 $\xi (\xi_t)$, 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 $\xi$ 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 $\mathbb{R}^3$ 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 $(\mathbb{R}^3,\xi_{std}=\ker(dz-ydx))$. 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 $\xi$ 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 $\xi_t$, 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 $\xi_t$ 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 $\alpha$, and the minimum distance $\delta$ 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 $S^2$. This change can be captured by a norm, which by choosing a good enough simplicial subdivision, can be assumed to be small relative to $\alpha/d$, so that across each simplex, the angle of the contact planes only changes by a very small fraction of $\alpha$.

Eliashberg defines “special simplices” as 1- or 2-simplices which contain some point p at which $\xi_p$ makes an angle less than $\alpha/4$ 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 $\xi$ to be contact. By carefully keeping track of the angles between simplices and $\xi$, and between $\xi$ 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 $\alpha$, 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 $\xi (\xi_t)$ 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 $\alpha$ and $\delta$. 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 $\xi$ is at least $\alpha/8$ at each point. By having chosen the neighborhoods of the special simplices and 0-simplices small in terms of $\alpha$ and the minimal distance between disjoint simplices, $\delta$, 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 $\alpha, \delta$, 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 $\pi$), which cannot happen in these tiny simplices where $\xi$ 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 $\xi$ 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 $\pi+\delta$ in $(\mathbb{R}^3,\xi_{ot}=\ker(\cos(r)dz+r\sin(r)d\theta))$ 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 $\pi+\delta$ in $(\mathbb{R}^3,\xi_{ot}=\ker(\cos(r)dz+r\sin(r)d\theta))$. Every time the curve intersects the line $r=\pi$, 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 $X$ defined by $laetx \iota_X\Omega=\alpha$ (where $\alpha$ is restricted to the surface and $\Omega$ is a positive area form on the surface). Therefore if we pair $X$ 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 $\mathbb{R}^3$. The Reeb vector field coorients the tangent planes, and points in the negative z direction at $r=\pi$, 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 $\mathbb{R}^3$ 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 $r=\pi$ on the “inside” of the curve. To make an odd number of limit cycles, have the curve intersect $r=\pi$ 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.

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## Overtwisted contact structures and h-principles

Twenty-five years ago Eliashberg established a dichotomy of contact structures on 3-manifolds as either tight or overtwisted. The overtwisted manifolds were motivated by the property that they satisfy certain flexibility principals. 1. Every co-oriented 2-plane field on a 3-manifold is homotopic to an overtwisted contact structure. 2. Any two overtwisted contact structures which are homotopic as 2-plane fields are isotopic as contact structures. (More generally, there is a homotopy equivalence between the space of contact structures and the space of 2-plane fields when you restrict each space to only include those with a particular overtwisted disk.) Recently, this theorem has been generalized to higher dimensions by Borman, Eliashberg, and Murphy , who explained their work at a recent workshop. The next few posts will be an attempt to share the intuition they gave, to work through some of the steps of their paper, to provide some historical background, and to discuss some related constructions and interesting open questions.

I also want to mention, that we are happy to give author permissions to young mathematicians who want to contribute blog posts. Thanks to Roger for agreeing to contribute on this topic, and welcome to the blog.

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