Author Archives: lstarkston

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}<K\# K (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


which can be rewritten as


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


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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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.


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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Graduate Student Topology and Geometry Conference

The 12th annual graduate student topology and geometry conference will be at UT Austin this April. The registration deadline is coming up on February 1. The website for registration and more information is Many of the talks are given by graduate student participants, and we have lots of interesting faculty speakers in a range of topology/geometry topics. I encourage all grad students to register and submit a talk proposal (expository or original research).

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Seiberg Witten 4: Moduli spaces and invariants

This is my last post on defining the Seiberg-Witten equations and invariants for closed 4-manifolds based on a learning seminar at UT. Maybe I’ll post about some applications later on.

The Seiberg-Witten Configuration Space

We start with a Riemannian 4-manifold (M,g) and a Spinc structure \sigma on M. As we have seen, this data gives rise to the associated bundle S_\sigma=S_\sigma^+\oplus S_\sigma^- and the determinant line bundle det(\sigma).

Let \mathfrak{A}_\sigma(M) be the set of all Hermitian connections on det(\sigma). We have seen that such a connection gives rise to a connection on S_\sigma which is compatible with the Clifford multiplication.

The Seiberg-Witten configuration space is defined as
\mathcal{C}_\sigma(M)=\mathfrak{A}_\sigma(M)\times \Gamma(S_\sigma^+)

The Seiberg-Witten equations

The Seiberg-Witten equations take an element (A,\psi)\in \mathcal{C}_\sigma(M) as their input. We are now prepared to define these equations.

As discussed in an earlier post, a connection A on det(\sigma) gives rise to a connection on S_\sigma^+. Note that for \xi\in T^*M, c(\xi)\in c(Cl^-(4)) so c(\xi):S_\sigma^+\to S_\sigma^-. Therefore we have a Clifford structure
c: \Gamma(T^*M\otimes S_\sigma^+)\to \Gamma(S_\sigma^-)
which composes with the connection
\nabla^A: \Gamma(S_\sigma^+)\to \Gamma(T^*M\otimes S_\sigma^+)
to get a Dirac operator
D_A: \Gamma(S_\sigma^+)\to \Gamma(S_\sigma^-).

Denote the curvature of the connection A by F_A. Then the curvature is a matrix of 2-forms on M, so we can consider its self-dual and anti-self dual parts F_A^+ and F_A^-.

Let (\psi\otimes \psi^*)_0 denote the traceless part of the endomorphism \psi\otimes \psi^*:S_\sigma^+\to S_\sigma^+.

Now we can define the (perturbed) Seiberg-Witten equations. Fix a closed 2-form \eta\in \Omega^2(M) (the pertubation parameter). Then the Seiberg-Witten equations are:
SW_{(\sigma,\eta)}=\begin{cases}  \frac{1}{2}c(F_A^++i\eta^+)-(\psi\otimes \psi^*)_0=0\\ {D}_A\psi = 0\end{cases}

The input to these equations is an element (A,\psi)\in \mathcal{C}_\sigma(M). The elements of \mathcal{C}_\sigma(M) which are solutions to these equations are called ((\sigma,\eta)-)monopoles.

The Gauge Action

The gauge group is \mathfrak{G}_\sigma(M)=\{\gamma: M\to U(1)| \text{ smooth}\}. It acts on \mathcal{C}_\sigma(M) by
\gamma\cdot (A,\psi) = (A-2d\gamma/\gamma, \gamma\psi)

While it seems natural enough to act on the section \psi by multiplication, why do we define the action \gamma\cdot A=A-2d\gamma/\gamma? Specifically where is the 2 coming from?

A is the connection of the determinant line bundle L of S_{\sigma}^+. We would really like to think of the gauge group as acting on S_{\sigma}^+. If g\in \mathfrak{G} acts on s\in S^+ by multiplication s \mapsto gs, then the induced action on \sigma\in L=\wedge^2 S_{\sigma}^+ is multiplication by g^2. (This goes back to the fact that in coordinate charts, the spinc structure is obtained by tensoring the spin structure with the square root of the determinant line bundle L.) Now we can look at how this acts on the covariant differentiation \nabla_A induced by the connection A on L. Here the natural action is conjugation

g^2\nabla_A(g^{-2}s)=g^2d(g^{-2})\otimes s +\nabla_As=-2g^{-1}dg\otimes s +\nabla_As

For C=(A,\psi)\in \mathcal{C}_\sigma(M) we can consider its stabilizer in \mathfrak{G}_\sigma(M). If the stabilizer of C is trivial, we say C is irreducible, otherwise we say C is reducible. It is easy to show that the reducible elements are exactly those with \psi\equiv 0, and that their stabilizers are the constant maps into S^1.

The Seiberg-Witten moduli space

The Seiberg-Witten solution space is the space of elements (A,\phi) for which the Seiberg-Witten equations are satisfied. To obtain the moduli space from this, we want to mod out by the gauge action. In order for this to be well defined, we first need to check that the space is invariant under the gauge action.

For the first equation, we can prove that F_A^+=F_{A-2g^{-1}dg}^+ because F_{A-2g^{-1}dg}^+=(F_A-d(2g^{-1}dg))^+ and d(g^{-1}dg)=0 because we can think locally that g^{-1}dg=d(log(g)) so taking its exterior derivative gives 0. Furthermore (g\phi)\otimes (g\phi)^*=gg^{-1}\phi\otimes \phi^*=\phi\otimes \phi^*, so the first equation is invariant under the gauge action.

For the second equation, D_{A-2g^{-1}dg(g\phi)} can be understood by breaking up the dirac operator into the composition of the Clifford multiplication and the connection \nabla_A on S_\sigma^+.

The discussion above about why the gauge group acts as it does on A is related to the fact that \nabla_(A-2g^{-1}dg)=\nabla_A-g^{-1}dg\otimes I_{S^+}. Applying the Clifford multiplication to this connection acting on g\phi and using the Leibniz rule for connections eventually simplifies to show that D_{A-2g^{-1}dg}(g\phi)=g D_{A}\phi so the solutions to D_A\phi=0 are invariant under the gauge action.

Therefore we can mod out the Seiberg Witten solution space by the gauge action to get a well-defined space.

Properties of the Seiberg Witten moduli space

The reason the Seiberg-Witten equations are so useful is that the moduli space is actually a compact smooth manifold in many cases. When there are no reducible solutions to the equations, the moduli space defined by a generic perturbation is a smooth manifold (one needs to show that the linearization of a map defined by the Seiberg Witten equations and the gauge action is Fredholm and then use Sard-Smale to show that generic perturbations correspond to regular values).

Compactness of the manifold requires some analytic estimates. The Weitzenbock forumla is the main tool in obtaining bounds on solutions to the Seiberg-Witten equations.

After going through hard work to show these properties, which I am avoiding here, one just needs to worry about reducible solutions. Notice that if there are reducible solutions (A,0) then they satisfy F_A^+=\eta for our chosen perturbation. Since both of these forms are closed, they represent cohomology classes. The cohomology class of the curvature [F_A^+]=-2\pi ic_1(L)^+ is independent of A, so we only have reducible solutions when [\eta]=-2\pi ic_1(L)^+. When the dimension of the positive second homology is at least 1, then a generic perturbation will avoid this phenomenon.

The Seiberg-Witten invariant of a 4-manifold is given by the homology class of the moduli space of solutions in the configuration space. This configuration space is homotopy equivalent to \mathbb{CP}^\infty so its cohomology has a canonical generator in even degrees. By evaluating this generator against the homology class of the Seiberg-Witten moduli space we obtain an integer SW_{M,g,\eta}(L)\in \mathbb{Z}.

A priori this integer depends on the metric and perturbation, but when b_2^+>1, the subspace of perturbations which allows for reducible solutions (bad perturbations) is codimension 2. Since the space of metrics on a manifold is convex, we can find a path through the space of metrics and good perturbations connecting any two pairs (g_1,\eta_1), (g_2,\eta_2) which lifts to a cobordism between the moduli space at (g_1,\eta_1) and the moduli space at (g_2,\eta_2). Therefore SW gives a diffeomorphism invariant of the 4-manifold, and it has been used very effectively to distinguish many homeomorphic but not diffeomorphic 4-manifolds (exotic pairs).

When b_2^+=1, there is a codimension 1 space of bad perturbations which forms a wall between two chambers. Within each chamber SW_{M,g,\eta}(L) stays constant, and there is a well-understood wall-crossing formula describing the difference of SW in the two different chambers. By keeping track of a little more information, it is still possible to use information from the Seiberg-Witten invariants to distinguish exotic pairs (this has been used a lot for finding exotic \mathbb{CP}^2\#N\overline{\mathbb{CP}^2}).


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Seiberg Witten 3: Dirac operators, Spin and Spinc connections

This is my third post on the set-up for the Seiberg-Witten invariants of 4-manifolds. The next post will finally define the Seiberg-Witten equations and invariants, so this is the last bit of background.

Symbols, generalized Laplacians, and Dirac operators

In order to define the Seiberg-Witten equations, we need to understand certain partial differential operators called Dirac operators. If you don’t know the formal definition of partial differential operators and their symbols, here is a link with some definitions and examples.

The class of all second order partial differential operator with the same symbol as the usual Laplacian: \sigma_L(\xi)=-|\xi|^2I\in End(E,E) are called generalized Laplacians. Note that the symbol \sigma_L(\xi): E_x\to E_x of a generalized Laplacian is an isomorphism on each fiber for \xi\neq 0, which means generalized Laplacians are elliptic operators. An elliptic operator L is good because there are estimates on the norms of solutions to equations of the form Lu=v. This allows us to use Fredholm theory to describe the space of solutions to equations using elliptic operators. (In particular the linearization of an elliptic operator is Fredholm, i.e. has finite dimensional kernel and cokernel).

Dirac operators are 1st order partial differential operators which square to a generalized Laplacian. Dirac operators inherit many of the nice properties of Laplacians, specifically they are also elliptic (though in a weaker sense than the Laplacian–my vague understanding is that the bounds we get from ellipticity of the Laplacian are uniform, whereas the bounds we get from ellipticity of a Dirac operator depend on the point in the manifold; in the case of compact manifolds these coincide).

Dirac Operators and Clifford multiplication

We mentioned above that the symbol of a generalized Laplacian, (which is the square of a Dirac operator) is \sigma_L(\xi)=-|\xi|^2I, for \xi\in \Gamma(T^*M). Additionally, one can show that the symbol of a Dirac operator (which squares to a generalized Laplacian), is the square root of the symbol of the generalized Laplacian. Therefore (\sigma_D(\xi))^2=-|\xi|^2I so \sigma_D gives us a Clifford multiplication. In conclusion, a Dirac operator give rise to a Clifford structures by taking its symbol.

Conversely, given a Clifford structure, c: \Gamma(T^*M)\to \Gamma(End(E)) (equivalently c: \Gamma(T^*M\otimes E)\to \Gamma(E)) and a connection \nabla: \Gamma(E)\to \Gamma(T^*M\otimes E) we can compose them

D:\Gamma(E)\xrightarrow{\nabla}\Gamma(T^*M\otimes E)\xrightarrow{c}\Gamma(E)

and the resulting operator is a Dirac operator.

Spin connections

A Riemannian manifold M has a distinguished connection, the Levi-Civita connection \nabla^M, which has nice properties namely it preserves the metric g (this can be phrased either as \nabla^Mg=0 or \nabla(g(X,Y))=g(\nabla X,Y)+g(X,\nabla(Y))), and it is torsion free meaning \nabla_XY-\nabla_YX-[X,Y]=0. Basically, this is a natural connection on TM when a Riemannian metric g is given.

Using the metric and orientation on M, the structure bundle of TM reduces to an SO(n)-bundle. Namely, we can find gluing maps defining the tangent bundle that map into SO(n): \{g_{\alpha\beta}: U_\alpha\cap U_\beta \to SO(n)\} which define a principal SO(n)-bundle P_{SO(n)}\to M. The Levi-Civita connection on TM induces a principal SO(n)-connection on P_{SO(n)} specified locally by

\omega_{\alpha}\in \Omega^1(U_{\alpha})\otimes \mathfrak{so}(n).

We have the double cover map \tau: Spin(n)\to SO(n), which induces, by differentiating at 1, an isomorphism \tau_*: \mathfrak{spin}(n)\to \mathfrak{so}(n).

If we have a Spin structure on M, this means there are lifts \widetilde{g}_{\alpha\beta}: U_\alpha\cap U_\beta\to Spin(n) such that \tau\circ \widetilde{g}_{\alpha\beta}=g_{\alpha\beta}. These define a principal Spin(n) bundle P_{Spin(n)}. In this case, the Levi-Civita connection on P_{SO(n)} induces a connection \widetilde{\nabla}^M on P_{Spin(n)} which is locally defined by

\tau_*^{-1}\omega_{\alpha}\in \Omega^1(U_{\alpha})\otimes \mathfrak{spin}(n).

So Riemannian manifolds with spin structures have a distinguished connection on the Spin(n) bundle.

The representations \rho_\pm: Spin(4)\to Aut(\mathbb{S}^\pm), and \rho=(\rho_+,\rho_-) give rise to an associated bundle S_0=P_{Spin}\times_\rho \mathbb{S}. The spin connection on M induces a connection \nabla^{S_0} on S_0 whose local matrix valued 1-forms are defined by

\rho_*\tau_*^{-1}\omega_{\alpha}\in \Omega^1(U_\alpha)\otimes End(\mathbb{S}).

Recall that T^*M acts on S_0 by the Clifford multiplication c: Cl(TM)\otimes \mathbb{C}\to End(S_0). The composition of the Clifford multiplication with the induced connection on S_0 yields a Dirac operators D_0.

\mathbf{Spin^c} connections

Remember, a Spin^c(n)-bundle is specified by gluing data

\{(h_{\alpha\beta}, z_{\alpha\beta}): U_{\alpha}\cap U_{\beta} \to Spin(n)\times U(1)\}
satisfying the cocycle condition

(h_{\alpha\beta}h_{\beta\gamma}h_{\gamma\alpha}, z_{\alpha\beta}z_{\beta\gamma}z_{\gamma\alpha})=\pm (1,1).
We want to understand Spin^c structures for M and their connections. Let \sigma be a Spin^c structure on M given by the Spin^c(4) bundle P_{Spin^c}.

Letting \rho^c=(\rho^c_+,\rho^c_-), the associated spinor bundle to \sigma is S_\sigma=P_{Spin(4)}\times_{\rho^c} \mathbb{S}, which splits into S^\pm_\sigma = P_{Spin(4)}\times_{\rho^c_\pm}\mathbb{S}^{\pm}. A connection on the Spin^c bundle will induce a connection on S_\sigma, S^+_\sigma,S^-_\sigma. Also note that S_\sigma has a Clifford structure, inherited from the map c: Cl(V)\otimes \mathbb{C}\to End(\mathbb{S}).

In the case that M has a spin structure, P_{Spin^c}=P_{Spin}\otimes (det\sigma)^{1/2} and S_\sigma = S_0\otimes (det\sigma)^{1/2}.

In the general case, we will construct connections on the associated bundles using the Levi-Civita connection on M, and a choice of connection on the determinant line bundle of \sigma.

In the case that TM is the trivial bundle, the determinant line bundle has a square root, and P_{Spin^c}=P_{Spin}\otimes (det\sigma) and S_\sigma=S_0\otimes (det\sigma)^{1/2}. We have the natural lift \widetilde{\nabla}^M of the Levi-Civita connection to P_{Spin}. This induces a natural connection \nabla^{S_0} on the associated bundle S_0, which we can tensor with any connection on the line bundle (det\sigma)^{1/2} to get a connection on S_{\sigma}=S_0\otimes (det\sigma)^{1/2}.

Remember that S_0 had a Clifford structure c as well as a natural connection S_0 which together give rise to a Dirac operator. We obtain a similar structure on S_\sigma by twisting the triple (S_0,\nabla^{S_0}, c) with a line bundle with connection (L,\nabla^L) to obtain a triple (S_0\otimes L, \nabla, c_L) where

\nabla(s\otimes x) = \nabla^{S_0}s\otimes x +s\otimes \nabla^Lx


c_L: \Omega^*M \xrightarrow{c}End(S_0)\xrightarrow{\cdot \otimes I_L} End(S_0\otimes L)

Therefore over trivial charts, a choice of connection A on (det\sigma)^{1/2} gives rise to a Dirac triple (S_\sigma,\nabla_A, c_\sigma).

In general the determinant line bundle does not have a global square root, though over any trivial chart it does. When the determinant line bundle has a square root, the connections on det(\sigma) are related to the connection on (det(\sigma))^{1/2} as follows. If the connection on det(\sigma) is defined by

\{\omega_\alpha \in \Omega^1(U_\alpha)\otimes \mathfrak{u}(1)\}

then the induced connection on (det(\sigma))^{1/2} is defined by

\{\frac{1}{2}\omega_\alpha \in \Omega^1(U_\alpha)\otimes \mathfrak{u}(1)\}.

We can always choose a connection on det(\sigma). This induces a connection over each trivial chart on (det(\sigma))^{1/2}. Then we can twist this in to the locally defined Dirac triples (S_0,\nabla^{S_0},c), to obtain (S_\sigma, \nabla, c) on each trivial chart U_\alpha. Finally, one can use a partition of unity to glue all these pieces back together to a global Dirac triple (S_\sigma,\nabla, c).

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Seiberg-Witten Theory 2: Clifford Structures and Spinors

Here is the second post on setting up the Seiberg-Witten equations on a 4-manifold, based on our learning seminar at UT Austin. The first post is here.

Clifford Algebras and Structures

For a vector space V with inner product g, its Clifford Algebra is defined as the tensor algebra of V modded out by all relations generated by setting v\otimes v=-g(v,v)1,

Cl(V)= \otimes V/\langle v\otimes v +g(v,v)1: v\in V\rangle.

For a vector bundle E\to M, any map c: T^*M\to End(E) satisfying c(u)c(v)+c(v)c(u)=-2g(u,v)I_E for all u,v\in \Gamma(T^*M) (equivalently satisfying c(v)^2=-|v|I_E for all v) extends to a representation
c:Cl(T^*M)\to End(E)
Such a map is called a Clifford structure.

There are two reasons we are interested in Clifford algebras and Clifford structures for Seiberg-Witten theory. The first is their relation to Spin and Spinc structures. The second is their relation to Dirac operators. In this post we will focus on their relation to Spin and Spinc structures, and discuss Dirac operators next.

Clifford algebras and Spin

Let Cl(n) denote the Clifford algebra of \mathbb{R}^n with its standard inner product. Let (e_1,\cdots, e_n) denote the standard orthonormal basis for \mathbb{R}^n. Consider the multiplicative subgroup of Cl(n) generated by unit vectors of \mathbb{R}^n. This is called Pin(n).

There is a natural \mathbb{Z}/2 grading on Cl(n) induced by a bijection Cl(n)\leftrightarrow \bigwedge^* \mathbb{R}^n identifying e_{i_1}\cdots e_{i_k} \leftrightarrow e_{i_1}\wedge \cdots \wedge e_{i_k}. The integer grading on the exterior power reduces to a \mathbb{Z}/2 grading (even/odd) on the Clifford algebra. This yields a splitting Cl(n)= Cl^+(n)\oplus Cl^-(n). Define Spin(n) to be the intersection of Pin(n) with the even summand Cl^+(n).

Before, we defined Spin(n) to be the universal double cover of SO(n). We can show this new definition of Spin agrees with the old definition, by explicitly constructing a double cover map from this subset of Cl(n) to SO(n).

There is an action of Cl(n) on \mathbb{R}^n given by signed conjugation (using the multiplicative structure of the Clifford algebra). If v\in \mathbb{R}^n is a unit vector (i.e. a generator of Pin(n)) then for any x\in \mathbb{R}^n
-vxv^{-1} = vxv = x-2\langle x,v \rangle v
Here we have used the fact that for unit vectors -vv=1 so v^{-1}=-v, and the relation vx+xv=-2g(v,x) in the Clifford algebra. This can be interpreted geometrically: the action (v,x)\mapsto -vxv^{-1} is the reflection of x over the hyperplane orthogonal to v.

The group of orthogonal transformations is generated by reflections over hyperplanes, so we have a representation called the twisted adjoint representation:
\rho: Pin(k)\to O(k)
defined by \rho(y)x = yx\varepsilon(y^{-1}) where \varepsilon(Cl^\pm(n))=\pm 1 (extend linearly). Restricting this to Spin(k) this is just usual conjugation, which corresponds to an even number of reflections so the image lies in SO(k):
\rho: Spin(k)\to SO(k)
This map is a surjective group homomorphism, and by studying the elements of Spin(k) which lie in the center of Cl(k), we see that the kernel of \rho: Spin(k)\to SO(k) is two elements \{\pm 1\}. Because these are nice smooth compact Lie groups, this implies that \rho is a covering map. To check it is not the trivial double cover, we can find a path in Spin(k) between -1 and 1 given by
for t\in [-\pi,\pi] [observe this path is a product of two unit vectors at each t and is thus in Spin(n)].
Therefore this definition of Spin(n) agrees with the previous one.

The spinor representation

We have already seen that Spin(4)\cong SU(2)\times SU(2) so Spin(4) and Spin^c(4) naturally admit two complex rank two representations coming from the projections onto the two factors of SU(2). However, it is useful to understand these representations from the Clifford algebra perspective so that the representations carry the additional information of a Clifford structure. In fact, there is a complex representation of the entire (complexified) Clifford algebra Cl(4) which splits into a direct sum of two complex rank two representations, which behave nicely with respect to the \mathbb{Z}/2 grading on the Clifford algebra. More specifically:

Theorem: There is a complex vector space \mathbb{S}=\mathbb{S}^+\oplus \mathbb{S}^- with \dim_{\mathbb{C}}\mathbb{S}^+=\dim_{\mathbb{C}}\mathbb{S}^-=2, and an \mathbb{C}-linear isomorphism
c: Cl(4)\otimes \mathbb{C}\to End(\mathbb{S})
such that c(Cl^+(4))\cong End(\mathbb{S}^+)\oplus End(\mathbb{S}^-) and c(Cl^-(4)\cong Hom(\mathbb{S}^+,\mathbb{S}^-)\oplus Hom(\mathbb{S}^-,\mathbb{S}^+).

To prove this, we have to define \mathbb{S}, \mathbb{S}^\pm, and the map c, and then verify that c is an algebra isomorphism satisfying the specified properties. There are a lot of things to check so I will define everything, and say a few things about how the map c works which hopefully make it more believable that c is an algebra isomorphism.

Let V=\mathbb{R}^4 with standard coordinates and standard almost complex structure J. This almost complex structure gives rise to a splitting of V\otimes \mathbb{C} = V^{1,0}\oplus V^{0,1}, where V^{1,0} is the i-eigenspace of J and V^{0,1} is the -i-eigenspace of J. We have orthonormal bases for these pieces given by:
V^{1,0}=span\left(\varepsilon_1 := \frac{1}{\sqrt{2}}(e_1-if_1), \varepsilon_2 := \frac{1}{\sqrt{2}}(e_2-if_2)\right)
V^{0,1}=span\left(\overline{\varepsilon}_1 := \frac{1}{\sqrt{2}}(e_1+if_1), \overline{\varepsilon}_2 := \frac{1}{\sqrt{2}}(e_2+if_2)\right)

Define \mathbb{S}:= \bigwedge^* V^{1,0}, and its splitting by \mathbb{S}^+ := \bigwedge^{even}V^{1,0} and \mathbb{S}^- := \bigwedge^{odd}V^{1,0}.

Now we need to define c: Cl(V)\otimes \mathbb{C} \to End (\mathbb{S}) with the properties specified in the theorem. We will define c on elements of V\otimes \mathbb{C} and then extend this to a map on the Clifford algebra by setting c(e_{i_1}\cdots e_{i_k})=c(e_{i_1})\cdot \cdots \cdot c(e_{i_k}) and extending complex linearly. To specify c on V\otimes \mathbb{C}, it suffices to say what c does to vectors in V^{1,0} and V^{0,1}.

For v\in V^{1,0}, c(v) is the endomorphism of \mathbb{S} obtained by wedging with v:
c(v)(u_1\wedge \cdots u_k)=\sqrt{2}v\wedge u_1\wedge \cdots \wedge u_k

For \overline{v}\in V^{0,1} c(\overline{v}) is contraction with \overline{v}:
c(\overline{v})(u_1\wedge \cdots u_k) = \sqrt{2}\sum_{j=1}^k (-1)^j g(v,u_j)u_1\wedge \cdots \wedge \widehat{u_j} \wedge \cdots u_k

One needs to check that this respects the Clifford algebra structure, and is an isomorphism. Initially, this may look wrong because for example when v\in V^{1,0}
c(v)^2(u_1\wedge \cdots \wedge u_k) = v\wedge v\wedge u_1\wedge \cdots \wedge u_k=0
and it seems like we should have c(v)^2=-|v|^2I. However, the algebra structure we want to preserve is complex linear on Cl(V)\otimes \mathbb{C} and has the Clifford structure only on the Cl(V) piece. Therefore, for example when v=e_j-if_j\in V^{1,0},
0=c(e_j-if_j)^2 = (c(e_j)-ic(f_j))^2 = (c(e_j))^2-ic(e_j)c(f_j)-ic(f_j)c(e_j)-(c(f_j))^2 = |e_j|^2-i2g(e_j,f_j)-|f_j|^2

For basis elements, the map c is a sum of the exterior and interior products. To compute for example, c(e_j) we split this into the V^{1,0} and V^{0,1} parts, so
c(e_j)=c\left(\frac{1}{2}(e_j-if_j)+\frac{1}{2}(e_j+if_j)\right)=\frac{\sqrt{2}}{2}\left((e_j-if_j)\wedge\cdot +\iota_{e_j-if_j} \right)
If you want to be slightly more convinced without completing the proof that c(v)^2=-|v|^2I for real elements of Cl(V) it is fairly easy at this point to check that c(e_j)^2=-I at least on the \bigwedge^0V^{1,0} part of \mathbb{S}=\bigwedge V^{1,0} (since any map that starts with contraction vanishes and \iota_x(y\wedge f)=-fg(x,y) for f\in \bigwedge^0V^{1,0} a complex number, and x,y\in V\subset Cl(V)\otimes 1).

We get the last property in the theorem easily from the definition of c. For v\in V\otimes \mathbb{C}, c(v) either raises or lowers by 1, wedge power of an element of \mathbb{S}=\bigwedge V^{1,0}. Therefore c(v) sends \mathbb{S}^+ to \mathbb{S}^- and vice versa. Extending this over the entire Clifford algebra, we see that the endomorphisms in c(Cl^+(4)) preserve \mathbb{S}^+ and \mathbb{S}^- (since they switch between \mathbb{S}^\pm an even number of times) and c(Cl^-(4)) sends \mathbb{S}^\pm to \mathbb{S}^\mp.

Note: We can rewrite the isomorphism c: Cl(4)\otimes \mathbb{C}\to End(\mathbb{S}) as a map
c: Cl(4)\otimes \mathbb{C}\otimes \mathbb{S}\to \mathbb{S}.
This will be useful when we use this representation to form associated bundles and consider sections of those bundles and maps between the spaces of sections.

This theorem generalizes for Cl(2n), producing a complex vector space \bigwedge V^{1,0} which splits where dim(V)=2n, whose endomorphisms are isomorphic to Cl(2n)\otimes \mathbb{C}, where Cl^+ preserves the splitting and Cl^- switches the components. In the odd dimensional case, the situation is slightly different, but reduces to the even case by showing that Cl(2n-1)\cong Cl^+(2n). For the purposes of Seiberg-Witten Floer homology, it will be useful to know Cl(3)\cong Cl^+(4) which implies Cl(3)\otimes \mathbb{C}\cong End(\mathbb{S}^+)\oplus End(\mathbb{S}^-).

Spinor bundles

Now that we have this representation of the complexification of the Clifford algebra, we can restrict to get a representation of Spin. Because Spin(4)\subset Cl^+(4), and c(Cl^+(4)) preserves the splitting \mathbb{S}=\mathbb{S}^+\oplus \mathbb{S}^-, we get two representations
\rho_{\pm}: Spin(4)\to Aut(\mathbb{S}^\pm)
Note the image of Spin(4) lands in automorphisms instead of only endomorphisms because elements of Spin(4) are invertible in Cl(4). These two representations correspond to the same ones we obtain by identifying Spin(4)\cong SU(2)\times SU(2) and projecting onto one component.

We can extend these maps to Spin^c by defining
\rho^c_{\pm}: Spin^c(4)\to Aut(\mathbb{S}^\pm)
by \rho^c_\pm((g,z))=z\rho_{\pm}(g) for g\in Spin(4), z\in U(1).

Note this is well defined since \rho^c_\pm((-g,-z))=\rho^c_\pm((g,z)).

Given a Spin or Spinc structure on a manifold, these representations give rise to associated bundles S^\pm \to M. These bundles show up in the set-up for the Seiberg-Witten configuration space, which I will get to in another post.

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Seiberg Witten Theory 1: Spin, Spinc structures

Seiberg-Witten theory has been one of the most useful tools for understanding exotic 4-manifolds. It has been around for awhile now, but it involves a lot of geometric definitions and analytic proofs so it is difficult to approach as a grad student. Ciprian Manolescu’s recent disproof of the triangulation conjecture has brought Seiberg-Witten Floer homology into the spotlight again, which has convinced a group of us at UT Austin to go through and learn this stuff carefully from the beginning. Since this seems potentially useful to a wider audience, I’m posting some of what we have gone through. The first 2 or 3 posts will be loosely based on talks in our learning seminar given by Cagri, Richard, and me, and the written reference we have relied most on is a book by Nicolaescu called Notes on Seiberg-Witten Theory.

We started from the beginning defining curvature and connections on vector bundles and principal bundles. That part is a bit definitional/computational and not too blog friendly, so I’ll just include a link: ConnectionsCurvature. Here I’ll start with spin and spinc structures from a fairly topological perspective. In the next post I’ll talk about their relations to Clifford structures and Dirac operators, and eventually get to connections on these bundles and various associated bundles, so we can define the Seiberg Witten invariants.

Spin and Spinc structures on 3 and 4-manifolds

We can specify any vector bundle , (specifically we will be concerned with the tangent bundle) over a manifold M with fiber V^n uniquely up to isomorphism by cocycle data (\{U_{\alpha}\}, \{g_{\beta \alpha}\}), where \{U_\alpha\} is an open cover of M, and g_{\beta \alpha}: U_\alpha \cap U_\beta \to GL(V^n) satisfy the “cocycle condition”
g_{\alpha \beta}(x)g_{\beta \gamma}(x)g_{\gamma \alpha}(x)=I_V

The vector bundle is formed by taking the disjoint union over all \alpha of the trivial bundles U_\alpha \times V and quotienting out by identifications of the fibers x\times V\subset U_\alpha\times V and x\times V \subset U_\beta \times V above a point x\in U_\alpha \cap U_\beta by the isomorphism g_{\beta \alpha}(x). The cocycle condition ensures that everything glues up coherently to a global vector bundle. Similarly a principal G-bundle can be specified by cocycle data (\{U_\alpha\}, \{g_{\beta \alpha}\}) where now g_{\beta \alpha}: U_\alpha\cap U_\beta \to G and g_{\alpha \beta}(x)g_{\beta \gamma}(x)g_{\gamma \alpha}(x)=1_G for x\in U_\alpha\cap U_\beta. It is frequently useful for computations to think about vector bundles and principal bundles through these local trivializations, keeping track of the gluing maps.

Given an orientable n-manifold, we consider its tangent bundle described by gluing maps \{g_{\alpha\beta}: U_\alpha\cap U_\beta\to GL(n,\mathbb{R})\}. By choosing a metric and orientation, we can reduce its structure group to SO(n), meaning we can assume the maps g_{\alpha\beta} have image in SO(n). We can use these gluing maps to construct a principal SO(n) bundle (the bundle of orthonormal frames) whose associated bundle is the tangent bundle. Spin and Spinc structures are types of lifts of this principal bundle.

For all n\geq 3, \pi_1(SO(n))=\mathbb{Z}/2 so SO(n) has a double cover, which conveniently, is also a Lie group called Spin(n).

One can show explicitly that
Spin(4)=SU(2)\times SU(2)
by constructing 2-fold covers SU(2)\to SO(3) and SU(2)\times SU(2)\to SO(4) using the following idea. Identify \mathbb{R}^4 with the quaternions. Observe that SU(2)\cong S^3 can be identified with the unit quaternions, and that the unit quaternions act by conjugation on the imaginary quaternions in a norm-preserving way. This action by conjugation induces a map from SU(2) to SO(3) whose kernel is \pm 1. Similarly an action of SU(2)\times SU(2) on the quaternions can be defined by (P,Q)\cdot X = PXQ^{-1}. Again one can check this action is orthogonal so there is an induced map SU(2)\times SU(2)\to SO(4) whose kernel is two points.

A spin structure on an n-manifold M is a Spin(n) bundle over M which lifts the principal SO(n) bundle associated to TM.

We can also define Spin^c(n)=(U(1)\times Spin(n)/\{\pm 1\}\to SO(n), and similarly define Spin^c structures on a manifold.

On a 4-manifold, a spin structure gives rise to two rank 2 complex associated bundles as follows. Spin(4)=SU(2)\times SU(2) has two natural projection maps onto SU(2), pr_1,pr_2. These can be viewed as representations, so if \widetilde{P} is a Spin structure on M^4, we obtain two complex rank 2 associated bundles
S^\pm = \widetilde{P}\times_{pr_{1,2}}\mathbb{C}^2
We will see these representations again in the context of Clifford structures, when we discuss how Spin(n) sits inside a Clifford algebra.

If M^4 admits a spin^c structure, we also have two projections:
pr_{1,2}: Spin^c(4)=U(1)\times SU(2)\times SU(2)/\{\pm 1\}\to U(2)
These similarly admit two complex rank two associated bundles W^{\pm}=\widetilde{P}\times_{pr_{1,2}}\mathbb{C}^2. In this case \bigwedge^2W^+=\bigwedge^2W^-.

Obstructions to Spin and Spinc structures:

The obstruction to a Spin structure is the Stiefel-Whitney class w_2(M), which can be viewed as a Cech cohomology class as follows. If \{g_{\alpha\beta}: U_\alpha\cap U_\beta\to SO(n)\} are the gluing maps for the SO(n) bundle defining the structure group for TM, each map g_{\alpha\beta} has exactly two lifts to maps h_{\alpha\beta}: U_{\alpha}\cap U_\beta\to Spin(n). Then
w_{\alpha\beta\gamma}:=h_{\alpha\beta}h_{\beta\gamma}h_{\gamma\alpha}\in \mathbb{Z}/2=\ker(Spin(n)\to SO(n))
This collection \{w_{\alpha\beta\gamma}\} is a Cech 2-cycle and so it represents a Cech cohomology class, called the second Stiefel-Whitney class w_2(M). When h_{\alpha\beta} satisfy the cocycle condition: w_{\alpha\beta\gamma}=1, this cohomology class vanishes and the h_{\alpha\beta} define a spin structure.

Because Spin^c(n)=(Spin(n)\times S^1)/\{\pm (1,1)\}, a Spin^c structure can be specified by cocycle data relating to the gluing maps \{g_{\alpha\beta}\} for the SO(n) structure bundle of the tangent bundle. This cocycle data is given by a collection of maps
h_{\alpha\beta}: U_\alpha\cap U_\beta \to Spin(n)
z_{\alpha\beta}: U_\alpha\cap U_\beta \to S^1
satisfying two requirements

(1) \rho(h_{\alpha\beta})=g_{\alpha\beta} (the Spin^c structure is a cover of the SO(n) bundle)
(2) (h_{\alpha\beta}h_{\beta\gamma}h_{\gamma\alpha}, z_{\alpha\beta}z_{\beta\gamma}z_{\gamma\alpha})\in\{(1,1),(-1,-1)\} (cocycle condition)

Focusing on the maps z_{\alpha\beta}, we almost get a U(1) bundle except that instead of the cocycle condition we have that z_{\alpha\beta}z_{\beta\gamma}z_{\gamma\alpha}=\pm 1. Because U(1)=S^1 is abelian, by looking instead at \lambda_{\alpha\beta}=z_{\alpha\beta}^2, we obtain gluing maps satisfying the cocycle condition, so they form a genuine U(1) bundle, or equivalently a complex line bundle L. We can calculate c_1(L) as follows. Write \lambda_{\alpha\beta}=e^{i\theta_{\alpha\beta}}. Then
is an integer (since e^{i(\theta_{\alpha\beta}+\theta_{\beta\gamma}+\theta_{\gamma\alpha})})=1), and the n_{\alpha\beta\gamma} define the cocycle representing c_1(L)\in H^2(M;\mathbb{Z}).

There is a relationship between w_{\alpha\beta\gamma}:= h_{\alpha\beta}h_{\beta\gamma}h_{\gamma\alpha} given by requirement (2) above. Namely,
w_{\alpha\beta\gamma} = sign(z_{\alpha\beta}z_{\beta\gamma}z_{\gamma\alpha}) = n_{\alpha\beta\gamma} \mod 2
(where w_{\alpha\beta\gamma}\in \{-1,1\}=\mathbb{Z}/2 where the group structure is multiplication, and n_{\alpha\beta\gamma}\in \{0,1\}=\mathbb{Z}/2 where the group structure is addition.) Since \{w_{\alpha\beta\gamma}\} represents w_2(M), and \{n_{\alpha\beta\gamma}\} represents c_1(L), the existence of a Spin^c structure implies w_2(M)\equiv c_1(L) \mod 2.

For any Spin^c structure \sigma the associated line bundle L constructed above is called det(\sigma). It is not obvious from this definition that this is the determinant of any vector bundle, but in fact it will be the determinant of the spinor bundle associated to the Spinc structure via the spinor representation we will discuss in the next post.

Note that the set of line bundles over M acts on the set of Spin^c structures as follows. For a line bundle L defined by gluing maps \zeta_{\alpha\beta}: U_{\alpha}\cap U_\beta \to S^1, and a Spin^c structure \sigma defined by gluing maps [h_{\alpha\beta}, z_{\alpha\beta}], \sigma\otimes L is defined by [h_{\alpha\beta}, z_{\alpha\beta}\zeta_{\alpha\beta}]. Observe that det(\sigma\otimes L)=det(\sigma)\otimes L^2 so c_1(\sigma\otimes L)=c_1(det(\sigma))+2c_1(L). One can prove that the action of the line bundles on the set of Spin^c structures is free and transitive.

For any manifold with a spin structure, there is a canonical Spin^c structure \sigma_0, obtained by composing the maps h_{\alpha\beta}:U_{\alpha}\cap U_\beta \to Spin(n) with the obvious map Spin(n)\to Spin(n)\times S^1/\{\pm 1\} sending g\in Spin(n) to [(g,1)] where [\cdot] denotes the equivalence class by modding out by \pm 1. Given this canonical Spin^c structure, any other is represented by gluing cocycles \{(h_{\alpha\beta},z_{\alpha\beta})\} where \{z_{\alpha\beta}\} satisfy the cocycle condition (since \{h_{\alpha\beta}\} do). In other words the \{z_{\alpha\beta}\} define a complex line bundle L. Any Spin^c structure is given by \sigma_0\otimes L, and the associated line bundle is det(\sigma_0\otimes L)=L^{\otimes 2}. Therefore any Spin structure canonically determines a square root of det(\sigma) for any Spin^c structure \sigma.


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