## Kylerec – Seiberg-Witten and Fillings

The first post on Kylerec is here, so if you haven’t been keeping up with these posts, you might want to start there. This is also the last post on the Kylerec 2017 workshop, which has been fun and rewarding to write about (with some much appreciated help from Agustín Moreno)!

If you’re following along with the lecture notes from Kylerec, then this post corresponds to Day 4 (Talks 11-14), consisting of the following four talks:

### A pre-introduction to Seiberg-Witten theory

The Seiberg-Witten equations have been discussed in this blog by Laura Starkston in a sequence of four posts. For more information, details, and clarification, the interested reader should go there, or to the notes of Hutchings and Taubes mentioned in the introduction to this post. I call this a pre-introduction because the details will be rather sketchy. I will not even write the Seiberg-Witten equations down. The reader interested in skipping to fillings may wish to jump ahead to the two-sentence summary at the end of this section.

I should mention that the Seiberg-Witten equations arise naturally in physics, although I’ve not yet personally taken the time to understand Witten’s motivation for first writing down these equations, which he called the monopole equations. If you are interested in that sort of thing, maybe check out this MathOverflow post.

Consider a closed oriented smooth 4-manifold $X$, together with the following data:

• a Riemannian metric $g$
• a self-dual 2-form $\mu$ (meaning $*\mu = \mu$ where $*$ is the Hodge star with respect to $g$)
• a $\textbf{spin}^c$-structure $\mathfrak{s} \in \mathcal{S}_X$

You might be asking – what’s a $\text{spin}^c$-structure? Recall that for $n \geq 3$, one has $\pi_1(\text{SO}(n)) = \mathbb{Z}/2\mathbb{Z}$, so one can form the connected double cover $\text{Spin}(n)$. Then one defines the Lie group

$\text{Spin}^c(n) = (\text{Spin}(n) \times U(1))/ \pm 1.$

This comes with a map to $\text{SO}(n)$ with fiber $U(1)$. The metric $g$ yields a principal $\text{SO}(n)$-bundle called the frame bundle, which topologically doesn’t depend on the metric, and a $\text{spin}^c$-structure is just a principal $\text{Spin}^c(n)$-bundle such that quotienting by the $U(1)$-action (sitting inside the $\text{Spin}^c(n)$-action) recovers the frame bundle.

For $X^4$ oriented, which is the case of interest to us, the space of $\text{spin}^c$-structures, $\mathcal{S}_X$, is an affine space modelled on $H^2(X;\mathbb{Z})$ (this is not obvious). Also in the 4-dimensional case, representation theory of the Lie group $\text{Spin}^c(4)$ yields two complex 2-dimensional spinor bundles $S_{\pm}$ and a complex line bundle $L = \det S_{\pm}$. The Seiberg-Witten equations are then equations on pairs $(A,\psi)$ consisting of a $U(1)$-connection on $L$ and a positive spinor (a section of $S_+$). We write this simply as

$\mathcal{F}_{g,\mu,\mathfrak{s}}(A,\psi) = 0$.

Let $\mathfrak{m}_{g,\mu,\mathfrak{s}}$ be the solutions to this eqution. There is an action of the gauge group $\mathcal{G} = C^{\infty}(X,S^1)$ (given by $h \cdot (A,\psi) = (A-2h^{-1}dh,h\psi)$). This action is free except for reducible solutions where $\psi = 0$, in which case the stabilizer is $S^1$. The quotient yields the moduli space (where we suppress $g,\mu,\mathfrak{s}$ from the notation):

$\mathcal{M} = \mathfrak{m}/\mathcal{G}$

Theorem (key, nontrivial): The space $\mathcal{M}$ is always compact.

Let $b_2^+$ be the rank of the positive-definite part of $H^2(X;\mathbb{R})$ with respect to the intersection product. We will assume $b_2^+ > 1$. This implies that generically paths of choices $(g,\mu)$ for a fixed $\mathfrak{s}$ will avoid reducible solutions, yielding the following.

Theorem (standard Fredholm theory): Consider $X$ with $b_2^+ > 1$ and some fixed $\mathfrak{s}$. Then generically (with respect to $(g,\mu)$),

• $\mathcal{M}$ is a smooth finite-dimensional manifold of dimension given by topological data (only depending on $X$ and $\mathfrak{s}$)
• $\mathcal{M}$ can be given an orientation with some auxiliary topological choice (not depending on $\mathfrak{s}$)
• $\mathcal{M}$ is a cobordism invariant

Definition: For $b_2^+ > 1$, and for $\mathfrak{s}$ with $\dim \mathcal{M} = 0$, we define the Seiberg-Witten invariant $\text{SW}_X(\mathfrak{s}) = \#\mathcal{M} \in \mathbb{Z}$, where we count $\mathcal{M}$ with signs according to the auxiliary topological choice.

One can also define the Seiberg-Witten invariant when the dimension is positive, but there is the simple type conjecture that in such cases, this invariant is zero. In the case of symplectic manifolds, which is the case we care about, this is known to be true (by Taubes). By construction, the Seiberg-Witten invariants are diffeomorphism invariants (once we have fixed our auxiliary data for determining an orientation of $\mathcal{M}$).

We are interested in the case of symplectic manifolds. In this case, there is a canonical choice for the data which orients the moduli spaces of solutions to the Seiberg-Witten equations. There is a natural morphism $\mathcal{S}_X \rightarrow H^2(M;\mathbb{Z})$ given by the $c_1(L)$ where $L$ is the line bundle mentioned before. (This is not an isomorphism if $H^2(X;\mathbb{Z})$ has 2-torsion.)

Definition: A class $c \in H^2(M;\mathbb{Z})$ is basic if there is a $\text{spin}^c$-structure $\mathfrak{s}$ with $c_1(L_{\mathfrak{s}}) = c$ such that $\text{SW}_{X}(\mathfrak{s}) \neq 0$.

We finish by stating the following facts without proof (although we did discuss the proofs at Kylerec).

Theorem [Taubes]: For a symplectic manifold, $\pm c_1(X,\omega)$ are basic classes (with Seiberg-Witten invariants $\pm 1$).

Theorem [Corollary of the same Taubes paper]: When $(X,\omega)$ is minimal, Kähler, and of general type (the last condition meaning $c_1(X,\omega)[\omega] < 0$ and $c_1^2(X,\omega) > 0$), then $\pm c_1(X,\omega)$ are the only basic classes.

Theorem [Corollary of Morgan-Szabó]: If $(X^4,\omega)$ has $c_1(X,\omega) = 0$, $b_1 = 0$, and $b_2^+ > 1$, then it is a rational homology K3 surface.

SUMMARY OF THIS SECTION: The Seiberg-Witten invariants form a diffeomorphism invariant, hence so do basic cohomology classes. This fact, plus the previous three theorems, are all we need.

### Filling unit cotangent bundles

Unit cotangent bundles, which we shall notate as $S^*M$, have canonical Weinstein fillings given by the unit disk cotangent bundles. It is a natural question to ask if this natural filling is in fact the only one up to some notion of equivalence. We shall restrict ourselves in this discussion to when the base space is a closed orientable surface $\Sigma_g$ of genus $g$. We mostly focus on the $g \geq 2$ case, but we quickly review the case of $g = 0,1$.

Let us begin with $g=0$. In the first post on J-holomorphic curves, when discussing McDuff’s classification result, I mentioned that $L(2,1) = S^*S^2$ has a unique minimal strong filling up to diffeomorphism. Further, Hind proved that Stein fillings are unique up to Stein homotopy.

For $g=1$, in the second post on J-holomorphic curves, when discussing Wendl’s J-holomorphic foliations, I mentioned that every minimal strong filling of $S^*T^2$ to the standard one. In fact, he proves further that every minimal strong filling is symplectically deformation equivalent, which is a little stronger. Also, Stipsicz proved that all Stein fillings are homeomorphic (to $D^*T^2 = D^2 \times T^2$).

To summarize roughly (though we know a little more), for $g=0,1$, exact fillings (which are automatically minimal) are unique up to symplectic deformation equivalence.

So now we move on to $g \geq 2$. We focus on exact fillings because strong fillings (even minimal ones) are too weak to get a handle on. One can build strong fillings with arbitrarily large positive second Betti number $b_2^+$. This involves cutting out a cap (with concave boundary) from one particular strong filling (McDuff) and gluing in other caps with higher $b_2^+$ (Etnyre and Honda).

The idea, in this paper of Li, Mak, and Yasui, is similar to the idea we encountered in McDuff’s approach to the $g=0$ (and more generally $L(p,1)$) case – attach a cap, and then use classification results to figure out what you had in the first place. The following definition is the correct version of cap that we need.

Definition:Calabi-Yau cap for a contact 3-manifold $(M,\xi)$ is a strong cap (like a filling, but with a concave end instead) $(P,\omega)$ with $c_1(P,\omega)$ torsion.

Theorem 1 [LMY]: If a Calabi-Yau cap exists for $(M,\xi)$, then the set of triples of Betti numbers $(b_1(X), b_2(X), b_3(X))$ is finite as $X$ ranges over all exact fillings.

Remark: This theorem is not true if instead we let $X$ range over all strong fillings. This was noted above when we remarked that $b_2^+$ could be arbitrarily large for a strong filling.

Theorem 2 [LMY]: In the case of the unit cotangent bundle, then for any exact filling $(X,\omega)$, its homology $H^*(X;\mathbb{R})$ and intersection form $H^2(X;\mathbb{R}) \otimes H^2(X, \partial X;\mathbb{R}) \rightarrow \mathbb{R}$ are the same as that of the standard filling.

Sketch Proof of Theorem 1: Some messing around with Chern classes tells us that if we have an exact filling $(X,\omega_X)$ and a Calabi-Yau cap $(P,\omega_P)$ for $(M,\xi)$, then the glued manifold $(Z,\omega)$ satisfies $c_1(X) \cdot [\omega] = 0$. Then one can plug this into classification theorems by an invariant called the symplectic Kodaira dimension $\kappa^s(Z,\omega)$. In the case when $X$ is minimal with $c_1(X) \cdot [\omega] = 0$, we must have $\kappa^s = 0$. In this case, when $b_1 = 0$, then the Morgan-Szabó result mentioned in the Seiberg-Witten section implies that we have a rational K3 surface, hence we know its Betti numbers. Tian-Jun Li extended this result to a classification for $\kappa^s = 0$ and $Z$ minimal but with arbitrary $b_1$. Otherwise, if $X$ is not minimal, it must have $\kappa^s = - \infty$, and one needs to be a little more careful, working with a symplectic surface in $P$ to which an adjunction inequality ends up bounded the Betti numbers.

Sketch Proof of Theorem 2: The key lemma is to construct a symplectic K3 surface $(X,\omega)$ with $g$ non-intersecting Lagrangian tori in the same homology class which all intersect a Lagrangian sphere transversely in one point. Then we can perform Lagrangian surgery to give an embedded Lagrangian genus $g$ surface $L$. Then $X \setminus \text{Op}(L)$ is a Calabi-Yau cap for $S^*\Sigma_g$. Playing around with intersection forms, we see that attaching this cap must yield a rational K3 surface (one can rule out all other possibilities given by the classification theorems mentioned in the proof of Theorem 1), from which playing around more with exact sequences of homology and intersection forms gives the result.

Remark: The classification-type results with respect to symplectic Kodaira dimension are the only place in this section where Seiberg-Witten equations enter the picture, and are really the meat of the argument, in some sense. The rest just comes from exact sequences and understanding intersection forms, which is comparatively simple, staying far away from gauge theory.

The main theorem of Sivek and Van Horn-Morris is the following:

Theorem [SV]: Weinstein fillings of $S^*\Sigma_g$ are unique up to s-cobordism rel boundary.

If you’re worried about the word “s-cobordism,” just think of this as a beefed up version of homotopy equivalence that comes relatively easily in this case once we prove the homotopy type of the filling is unique (is a $K(\pi_1(\Sigma_g),1)$). There are some beautiful group-theoretic arguments which go into this argument, but we have essentially already seen how the Seiberg-Witten invariants come into play, so I won’t include a sketch of the proof.

Finally, I mention a little bit of history with regards to these two papers, because I was confused looking at the most recent versions as I was writing this, not for lack of improper attributions, just by my own confusions about reading them concurrently. The theorems stated are quite similar, as are aspects of the proofs, despite them being stumbled upon independently. To clarify, Theorem 2 of LMY did not exist in version 1 of their paper. About a year later, within a month of each other, SV posted their paper and LMY posted version 2 of their paper. Independently, SV had proved some subset of Theorem 2 (with some small fudge factor in $H_1$ and the intersection form) while LMY had proved the full version. SV’s result was good enough for them to prove the s-cobordism statement, and as far as I can tell, version 2 of SV is just version 1 but where they mention that they have learned that LMY proved the strong version of Theorem 2.

### Homotopic tight contact structures which are different

The main theorem, due to Lisca and Matić, is the following:

Theorem [LM]: For any $n \geq 0$, there exists a rational homology 3-sphere with at least $n$ distinct contact structures up to contactomorphism which are homotopic as plane fields.

In this short section, we simply sketch the proof.

Sketch of proof: One must begin by simply writing the Gompf surgery diagrams (described in the post on Weinstein fillings) for the contact structures in question. One has that a rational homology sphere can be obtained by 0-surgery on a trefoil and $-n$-surgery on an unknot which links with the trefoil once, and so suggests the following surgery diagrams so that the canonical framing on the Legendrians drawn below gives exactly what we want.

We denote these contact structures by $\xi_n^k$ for $1 \leq k \leq n-1$. We will show that for a fixed $n$, all of these are homotopic but not contactomorphic. One computes via results of Eliashberg that for the corresponding Weinstein fillings $W_n^k$ (which are diffeomorphic) that $c_1(W_n^k) = (2k-n)\text{PD}(T)$, where $T$ is the class in $H_2(W_n^k)$ given by the handle coming from the trefoil in the surgery. We shall call the smooth underlying manifold $N_n$.

The homotopic part is rather simple. By classical results (clutching functions, and computing Pontrjagin classes to plug into the Hirzebruch signature theorem) following an argument attributed to Gompf, one can show that the homotopic result can be reduced to proving that $c_1(W_n^k)^2 = c_1(W_n^{k'})^2$, which is itself clear since $\text{PD}(T)^2 = 0$.

As for the contactomorphism part, one embeds $W_n^k$ into a minimal compact Kähler surface $S$ of general type and $b_2^+ > 0$. This is a nontrivial statement, but is nonetheless true. In fact, because $W_n^k$ and $W_n^{k'}$ have isomorphic collars, one can attach the same cap to produce Kähler surfaces $S_n^k$ and $S_n^{k'}$. One can extend the identity on these caps to an orientation preserving diffeomorphism $\phi \colon S_n^k \cong S_n^{k'}$ acting by $\pm 1$ on $H^2(N_n) \subset H^2(S_n)$ (by work of Gompf). But also, since we have a minimal compact Kähler manifold, by the theorem mentioned in the first section as a corollary of Taubes’ work, one has that $\{\pm c_1(S_n^k)\}$ is a diffeomorphism invariant, and so we see that $c_1(S_n^k) = \pm \phi^*c_1(S_n^{k'})$. So these must restrict to the same thing on $H_2(N_n)$, where we showed $c_1(S_n^k)|_{H_2(N_n)} = (2k-n) \text{PD}(T)$. Hence, $2k-n = \pm (2k'-n)$, so either $k = k'$ or $k = n-k'$. Thus, increasing $n$, we can find arbitrarily many homotopic but non-contactomorphic contact structures.

Filed under Uncategorized

## SH & SH^+ [Momchil Konstantinov’s talk]

Let us begin with a rather informal and sketchy overview of the basics behind symplectic homology (this is by no means the most general version, and we refer the reader to the vast and growing literature, of which we give some references below).

Consider $(V,\lambda)$ a Liouville domain with contact boundary $(M=\partial V, \alpha= \lambda\vert _{\partial V})$ and its completion $(\widehat{V},\widehat{\lambda})$, obtained from $(V,\lambda)$ by attaching cylindrical ends. Given a nondegenerate Hamiltonian $H:S^1\times \widehat{V}\rightarrow \mathbb{R}$,  we have an associated action functional $\mathcal{A}^H: C^\infty(\mathbb{R}/\mathbb{Z}, \widehat{V})\rightarrow \mathbb{R}$, defined by

$\mathcal{A}^H(x)=\int_{S^1}x^*\widehat{\lambda}-\int_{S^1}H_t(x(t))dt$

Its differential is given by $d_x\mathcal{A}^H(\xi)=\int_{S^1}d\lambda(\xi(t),\dot{x}(t)-X_{H_t}(x(t)))dt$, and it follows that its critical points correspond to closed Hamiltonian orbits. Given a $d\lambda$-compatible almost complex structure $J$ which is cylindrical on the ends, this induces a metric on the loop space, for which the gradient of $\mathcal{A}^H$ can be written as $\nabla_x\mathcal{A}^H=-J(\dot{x}-X_H(x))$, so that the gradient flow equation becomes the Floer equation. We define the symplectic homology chain complex (with mod 2 coefficients) as

$CF_*(H)=\bigoplus_{x \in \mbox{crit}(\mathcal{A}^H)}\mathbb{Z}_2.x$

By simplicity, assume that $x \in \mbox{crit}(\mathcal{A}^H)$ is contractible (so that we don’t have to worry about homology classes and whatnot), and also assume that $c_1(V)=0$ (this condition can be relaxed to $c_1\vert_{\pi_2(V)}=0$, and is needed for the grading). Then we can define the Conley-Zehnder index of $x$ by choosing spanning disks for $x$ and trivializing $TV$ along this disk, and we choose the grading $|x|=\mu_{CZ}(x)-n$, which is independent on the trivialization by the assumption on $c_1(V)$. The differential is now $d_H: CF_k(H)\rightarrow CF_{k-1}(H)$, given by

$d_H(x)=\sum_{\substack{y\in \mbox{crit}(\mathcal{A}_H)\\|y|=|x|-1}}\#_{\mathbb{Z}_2}\mathcal{M}(y,x)y$

where $\mathcal{M}(y,x)$ is the moduli space of Floer trajectories joining $x$ to $y$ divided by the natural $\mathbb{R}$-translation action. This moduli space is a zero dimensional manifold when $|y|=|x|-1$ (for generic $J$). Recall that Gromov compactness requires uniform $C^0$-bounds (which in our situation are not for free, since $\widehat{V}$ is non-compact) and uniform energy bounds (which we have for $u \in \mathcal{M}(y,x)$, since $E(u)=\mathcal{A}^H(x)-\mathcal{A}^H(y)$).

Def.  The spectrum of $(M,\alpha)$ is

$spec(M,\alpha)=\{ T \in \mathbb{R}: \mbox{ there exists a }\alpha-\mbox{Reeb orbit of period }T\}$

Def. The space of admissible Hamiltonians $Ad(V,\lambda)$ is the set of Hamiltonians $H: S^1 \times \widehat{V}\rightarrow \mathbb{R}$ satisfying

$H_t(r,y)=Ae^r+B$ on $r>R>>0$,  for some $R$,  where $A>0, A \notin spec(M,\alpha)$.

Denote by $h(s)=As+B$, so that $H_t(r,y)=h(e^r)$ on the ends.

If one chooses an admissible $H$ and a $J$ which is cylindrical on the ends, one gets $C^0$-bounds, as follows from the maximun principle: indeed, consider $\Omega \subseteq \mathbb{R}\times S^1$ an open subset, and $u: \Omega \rightarrow \widehat{V}$ a holomorphic map, which has a portion lying on the cylindrical ends. This portion can be parametrized by $u(s,t)=(a(s,t),v(s,t))\in \mathbb{R} \times M$, and a computation gives

$\Delta a + \partial_s(h^\prime(e^a))=\Delta a + h^{\prime\prime}(e^a)e^a\partial_sa=||\partial_s v ||^2\geq 0$

The maximum principle then implies that a sequence of Floer cylinders with fixed asymptotics cannot escape to infinity, since we would get a maximum of $a$, which implies $\Delta a\geq 0$, and this cannot happen if one assumes that the maximum is non-degenerate (a clever trick then gets rid of this assumption). So we get the $C^0$-bounds, which leads to compactness by Gromov, which implies $d_H$ well-defined and $d_H^2=0$ (as follows by studying the boundary of 1-dimensional moduli spaces of Floer trajectories). From this, one gets the Floer homology group

$HF_k(H):=H_k(CF_*(H),d_H)$

The first thing one asks is: is it independent of $H$? And the answer is…well… nope. BUT…

Consider two different $H_+, H_- \in Ad(V,\lambda)$, and choose a smooth path of Hamiltonians $H^s:\widehat{V}\rightarrow \mathbb{R}$ for $s \in \mathbb{R}$, such that $H^s=H_-$ for $s<<0$, $H^s=H_+$, for $s>>0$, and $H^s(r,y)=h_s(e^r)=A_se^r+B_s$ for $A_s,B_s \in \mathbb{R}$, on the cylindrical ends. This gives the parametrized Floer equation $\partial_su + J(\partial_tu - X_{H^s}(u))=0$ and a corresponding moduli space $\mathcal{M}_{\{H^s\}}(x_-,x_+)$ joining the orbits $x_-$ and $x_+$, which is zero dimensional when $|x_-|=|x_+|$ (now we don’t have a translation action). This ideally would allow us to define a map

$\Phi: CF_*(H_+)\rightarrow CF_*(H_-)$

given by

$\Phi(x_+)=\sum_{\substack{x_- \in \mbox{crit}(\mathcal{A}_H)\\ |x_-|=|x_+|}} \#_{\mathbb{Z}_2} \mathcal{M}_{\{H_s\}}(x_-,x_+)x_-$

satisfying $d_{H_-}\circ \Phi = \Phi \circ d_{H_+}$, as follows by studying how trajectories in 1-dimensional moduli spaces can break. But this, again, requires Gromov compactness. A similar computation gives

$\Delta a + \partial_s(h_s^\prime (e^a))=\Delta a + h_s^{\prime\prime}(e^a)e^a\partial_sa + (\partial_s h_s^\prime)(e^a)=||\partial_s v||^2$

So, to have $\Delta a + h_s^{\prime\prime}(e^a)e^a\partial_sa\geq 0$ it suffices with

$\partial_sh_s^\prime=\partial_s A_s<0$

In other words, the slope of $H_-$ is necessarily steeper than that of $H_+$. This means that we only get compactness in “one direction”, and we do not get a homotopically inverse map.

If we define a partial order $\prec$ on $Ad(V,\lambda)$ by $H_1\prec H_2$ if $H_1 outside of a compact set, the previous discussion gives us a map $HF_*(H_1)\rightarrow HF_*(H_2)$. Moreover, we get commutative diagrams for any $H_1 \prec H_2 \prec H_3$, giving a direct system, so that we may define the symplectic homology of $(V,\lambda)$ as

$SH_k(V,\lambda)=\varinjlim_{H \in Ad(V,\lambda)} HF_k(H)$

Observe that, as with any direct limit, one can compute it by taking cofinal sequences. Now we identify the generators of this homology. Let us recall the following fact from Floer theory:

Fact. If $H$ is sufficiently $C^2$-small then all the 1-periodic orbits of $X_H$ are critical points of $H$, and every Floer trajectory between them is a Morse flow-line.

This means that if $H$ is sufficiently $C^2$-small and positive on $V$, then the generators on this region of $SH_k$ will correspond to critical points (graded by $|x|=\mu_{CZ}(x)-n=n-ind_x(H)-n=-ind_x(H)$), and observe that $\mathcal{A}^H(x)=-H(x)<0$. On the cylindrical ends, we have $X_H=h^\prime(e^r)e^{-r} R_\alpha$, where $R_\alpha$ is the Reeb vector field of $\alpha$ on $r=0$, so that closed Hamiltonian orbits lie in the contact slices $\{r=r_0\}$ and are reparametrizations of closed Reeb orbits of period $T:=h^\prime(e^{r_0})$, and these have action

$\mathcal{A}^H(x)=T-h(e^{r_0})>0$

Since we assume that the slope of $H$ does not lie in the spectrum, there are no closed orbits for $r>R>>0$, and between $0$ and $R$ we see potential closed Hamiltonian orbits of bounded action. Since the differential decreases action, we have a subcomplex $CF_*^{-}(H)$ of $CF_*(H)$ generated by orbits of negative action (critical points), and an exact sequence of chain complexes

$0\rightarrow CF_*^{-}(H)\rightarrow CF_*(H)\rightarrow CF_*^+(H)\rightarrow 0$

where $CF_*^+(H)=\frac{CF_*(H)}{CF_*^{-}(H)}$. If we define

$SH_*^+(H)=\varinjlim_{H \in Ad(V,\lambda)}H_*(CF_*^+(H),d_H)$

and we take direct limit in the resulting long exact sequence (which preserves exactness), we get an induced exact triangle

Here we have used the Floer theory fact, and the maximum principle, to say that $CF_*^-(H)$ computes $H^{-k}(V)$ for every $H$ ($C^2$-small on $V$). Observe that we get cohomology of $V$ rather than homology, since we get a minus in the grading ($-ind_x(H)$ goes to $-ind_x(H)-1=-(ind_x(H)+1)$ under the differential). Yes, it’s confusing.

We can now state a few theorems.

Thm. [Bourgeois-Oancea] If all Reeb orbits of $(M,\alpha)$ satisfy

$\mu_{CZ}(x)+n-3>0$

that is, if $(M,\alpha)$ is dynamically convex, and $V,W$ are two Liouville fillings of $M$ with $c_1(V)=c_1(W)=0$, then $SH_*^+(V)\simeq SH_*^+(W)$.

In other words, $SH_*^+$ is an invariant of $M$, rather than the fillings (with $c_1=0$). The idea is to show that no critical points can be connected to a non-constant orbit by a Floer trajectory, and that no cylinder connecting two of the latter ventures into the filling $V$ (there is a stretching the neck argument here).

Thm. [ML Yau] If $(M,\xi)$ is subcritically Stein fillable (for a filling with $c_1=0$), then $M$ admits a dynamically convex contact form.

Thm. [Cieliebak] If $V$ is subcritically Stein (with $c_1=0$), then it has vanishing symplectic homology.

Cieliebak proves that $V$ is isomorphic to a split Stein manifold $W \times \mathbb{C}$, for $W$ Stein, and using a version of the Künneth formula for $SH_*$, the result follows from the fact that $SH_*(\mathbb{C})=0$, which one can compute by hand.

Cor. If $V,W$ are subcritical Stein fillings of $(M,\xi)$ with $c_1(V)=c_1(W)=0$, then $H^*(V)\simeq H^*(W)$.

This follows from the exact triangle, and all theorems stated above, since $H^{-*}(V)\simeq SH^+_*(V)$ for a subcritical Stein manifold with $c_1(V)=0$.

References

A few references on symplectic homology (by all means very much non-exhaustive):

A begginer’s overview: https://www.mathematik.hu-berlin.de/~wendl/pub/SH.pdf

A nice survey: https://arxiv.org/abs/math/0403377

A Morse-Bott version (relevant for Cédric’s talk below): https://arxiv.org/abs/0704.1039

A related theory (Rabinowitz Floer homology): https://arxiv.org/abs/0903.0768

## Contact manifolds with flexible fillings [Scott Zhang’s talk]

The main reference for this post is this paper: https://arxiv.org/pdf/1610.04837.pdf.

Let us recall the following result, which appeared in Momchil’s talk:

Thm. [M.L Yau] If $W_1, W_2$ are two subcritical fillings of a contact manifold $(M^{2n-1},\xi)$, (with $c_1(W_1)=c_1(W_2)=0$) then $H^*(W_1)\simeq H^*(W_2)$.

The goal for this talk was to discuss the following generalization to the $\emph{flexible}$ case:

Thm 1. [O. Lazarev] If $W_1,W_2$ are two flexible fillings of $(M,\xi)$, then $H^*(W_1)\simeq H^*(W_2)$.

Remark: The same conclusion is true if we consider fillings with vanishing symplectic homology.

The idea is to replace the dynamical convexity condition in Bourgeois-Oancea’s result by an asymptotic version. In the following, given $\alpha_1,\alpha_2$ contact forms for the same contact structure, we will denote $\alpha_1\geq \alpha_2$ if $\alpha_1=f \alpha_2$ for some smooth function $f\geq 1$, and by $\mathcal{P}^{ the set of $\alpha$-Reeb orbits $\gamma$ with action $\int_\gamma \alpha . The degree of a Reeb orbit $\gamma$ is $|\gamma|=\mu_{CZ}(\gamma)+n-3$.

Def.  $(M^{2n-1},\xi)$ is asymptotically dynamically convex (ADC) if there exists a sequence of contact forms $\alpha_1\geq \alpha_2\geq \dots$ for $\xi$ and a sequence $0 with $\lim_{i}D_i=\infty$ such that every element in $\mathcal{P}^{ has positive degree.

We have the following:

Thm 2. [O. Lazarev] If $(M,\xi)$ is ADC, then $SH^+$ is independent of the Stein filling with $c_1=0$.

Recall that  flexible Weinstein manifolds have vanishing symplectic homology. This follows by the Bourgeois-Ekholm=Eliashberg surgery formula (https://arxiv.org/pdf/0911.0026.pdf), but there are alternative arguments not using the SFT machinery, based on an h-principle for exact codimension zero embeddings, and the Künneth formula for symplectic homology, which even works for twisted coefficients (see e.g. Murphy-Siegel https://arxiv.org/abs/1510.01867). From the exact triangle for $SH_+$, we know that $SH_*^+(W)\simeq H^{-*}(W)$ for flexible $W$, so to get thm. 1 it suffices to show that flexible fillings induce ADC contact structures on their boundaries.

Thm 3. [O. Lazarev] If  $(M^\prime,\xi^\prime)$ is obtained from $(M,\xi)$ by flexible surgery and $(M,\xi)$ is ADC, then so is $(M^\prime,\xi^\prime)$.

Remark. The subcritical case where the ADC condition is replaced by DC (dynamical convexity) is already due to Yau.

Since the standard sphere is ADC, thm. 1 follows.

Here are a few ingredients in the argument. Let us recall first the following:

Prop. [Bourgeois-Ekholm-Eliashberg] After surgery along a Legendrian sphere $\Lambda^{n-1} \;(n\geq 3)$, we have a 1-1 correspondence between the newly created Reeb orbits with action bounded by $D>0$, and words of Reeb chords on $\Lambda$ with action bounded by $D$ (up to cyclic permutation). Moreover, we have $|\gamma_{c_1\dots c_n}|=\left(\sum_i |c_i|\right)+n-3$, where $\gamma_{c_1\dots c_n}$ denotes the Reeb orbit corresponding to the word $c_1\dots c_n$.

The idea is to slightly perturb the data so that given a collection of ordered chords, there is a closed Reeb orbit which enters the handle and is close to the original chords in the complement of the handle (the fact that all closed orbits that enter the handle have to leave it boils down to the fact that the geodesics on the flat disk leave the disk).

Key lemma. If $\Lambda$ is loose, there exists a Legendrian isotopy such that (action bounded) Reeb chords have positive degree.

The point is that stabilizing a loose Legendrian, which in general does not change the formal homotopy type, actually does not change the genuine isotopy type, by Murphy’s h-principle, and one can explicitly see that the degree of the resulting Reeb chords is greater or equal than 1 after the stabilization. The fact that we get decreasing contact forms comes form this stabilization process.

## Computations on Brieskorn manifolds [Cédric De Groote’s talk]

The goal for this talk, much more computational in spirit, was to discuss how invariants like contact and symplectic homology can be used to distinguish contact structures on Brieskorn manifolds, specially when the underlying manifolds are diffeomorphic, and in certain cases even when the contact structures are homotopic as almost contact structures.  A useful tool is a Morse-Bott version of symplectic homology, which applies in many cases where a lot of symmetry in present in the setup.

Brieskorn manifolds and Ustilovsky exotic contact spheres

The Brieskorn manifold associated to $a=(a_0,\dots,a_n)$, where $a_i\geq 2$ is an integer, is defined by $\Sigma(a)^{2n-1}=\{z_0^{a_0}+\dots + z_n^{a_n}=0\}\cap S^{2n+1}\subseteq \mathbb{C}^{n+1}$. In other words, it is the link of the (isolated) singularity associated to the complex polynomial $f(z)=z_0^{a_0}+\dots + z_n^{a_n}$. It is the binding of an open book on $S^{2n+1}$, with pages which are diffeomorphic to $\{f(z)=\epsilon\}\cap \mathbb{D}^{2n+2}$, for small $\epsilon>0$ (the Milnor fiber of $f$, see Milnor’s classic book: “Singular points of complex hypersurfaces”).

Brieskorn manifolds come with a contact form $\alpha_a=\frac{i}{8}\sum_{j=0}^na_j(z_jd\overline{z_j}-\overline{z_j}dz_j)$, which is induced by the “weighted” exact symplectic form $\omega_a=\frac{i}{4}\sum_{j=0}^n a_j dz_j\wedge d\overline{z_j}$ on $\mathbb{C}^{n+1}$, with associated Liouville vector field $V(z)=z/2$, which is transverse to $\Sigma(a)$. The corresponding Reeb vector field is $R_a=(\frac{4i}{a_0}z_0,\dots,\frac{4i}{a_n}z_n)$, which has flow $\phi_a^t(z)=(e^{\frac{4it}{a_0}}z_0,\dots,e^{\frac{4it}{a_n}}z_n)$. We also have a filling for $\Sigma(a)$, given by $W_a=\{f(z)=\epsilon \varphi(|z|)\}$, where $\varphi: [0,+\infty)\rightarrow \mathbb{R}$ satisfies $\varphi\equiv 1$ close to $0$, and vanishes close to $1$ (so that $W_a$ is a non-singular interpolation between the Milnor fiber and the singular hypersurface $\{f=0\}$). It comes endowed with the restriction of $\omega_a$, and is therefore an exact filling (it is actually Stein). By thm. 5.1 in Milnor’s book, it is parallelizable, and hence $c_1(W_a)=0$.

Some interesting facts:

1. $\pi_1(\Sigma(a))=\dots=\pi_{n-1}(\Sigma(a))=0$, i.e $\Sigma(a)$ is $(n-1)$-connected (lemma 6.4 in Milnor, which works for any Milnor fiber).
2. If $n\neq 2$, $\Sigma(a)$ is homeomorphic to a sphere if and only if it is a homology sphere (For $n \geq 3$ it follows by 1. above -which implies simply connectedness-, and the generalized Poincaré hypothesis, and is trivial for $n=1$). By 1., Poincaré duality and Hurewicz’ theorem, this is equivalent to the reduced homology $\widetilde{H}_{n-1}(\Sigma(n))=0$.
3. There exist conditions on $a$ which are equivalent to $\Sigma(a)$ being homeomorphic to the sphere $S^{2n-1}$. Namely, If there exist $a_i,a_j$ which are relatively prime to all other exponents, OR there exist $a_i$ which is relatively prime to all others and a set $\{a_{j_1},\dots,a_{j_r}\} (r\geq 3 \mbox{ odd })$ such that every $a_{j_k}$ is relatively prime to every exponent not in the set, and $gcd(a_{j_k},a_{j_l})=2$ for $k\neq l$.
4. $\Sigma(2,2,2,3,6k-1)$ for $k=1,\dots,28$ gives all smooth structures in $S^7$ (it is homeomorphic to the sphere by the previous criterion).
5. Any simply connected spin 5-manifold is a connect sum of Brieskorn 5-manifolds.

Thm.[Brieskorn] If $p \equiv \pm 1 (mod \;8)$ then $\Sigma(p,2,\dots,2)$, where the number of 2’s is $2m+1$, is diffeomorphic to $S^{4m+1}$.

Denote by $\xi_p$ the contact structure on $\Sigma(p,2,\dots,2)$ that we obtain by the weighted symplectic form, as above. Observe that by the above criterion these manifolds are all homeomorphic to spheres.

Thm.[Ustilovsky] If $p_1 \neq p_2$, then $\xi_{p_1}$ is not contactomorphic to $\xi_{p_2}$.

The proof uses contact homology. One can take an explicit perturbation making the contact form non-degenerate, and compute the degrees of the resulting non-degenerate Reeb orbits, which are all even. This implies that the differential vanishes, so that contact homology is isomorphic to the underlying chain complex. For different values of $p$, the degrees of the generators differ, and hence contact homology does also (and this is an invariant of the contact structure).

Def.  An almost contact structure on $Y^{2n+1}$ is  a pair $(\alpha,\beta)$ of a 1-form $\alpha$ and a 2-form $\beta$ such that $\beta\vert_{\ker \alpha}$ is non-degenerate. This is equivalent to having a reduction of the structure group of $TY$  to $U(n)\times 1$.

Def. A contact sphere $(S^{2n+1},\xi)$ is called exotic if it is not contactomorphic to  $(S^{2n+1},\xi_{std})$, the standard contact structure on $S^{2n+1}$. It is homotopically trivial if it is homotopic to $(S^{2n+1},\xi_{std})$ as almost contact structures.

An almost contact structure on $S^{4m+1}$ is equivalent to a lift of the classifying map $S^{4m+1}\rightarrow BSO(4m+1)$ to a map $S^{4m+1}\rightarrow B(U(2m)\times 1)$, under the natural map $B(U(2m)\times 1) \rightarrow BSO(4m+1)$ induced by inclusion. This map has fibers $S0(4m+1)/ (U(2m)\times 1)$, and therefore almost contact structures are classified by the group $G:= \pi_{4m+1}( S0(4m+1)/ (U(2m)\times 1))$.

Thm.[Massey] $G$ is cyclic of order $d=(2m)!$ if $m$ even, and $d=(2m)!/2$ if $m$ odd.

Thm.[Morita] The contact structure $\xi_p$ on $\Sigma(p,2,\dots,2)$ represents $\frac{p-1}{2} (mod \; d)$ in $G$ when viewed as an almost contact structure.

It follows that if $p\equiv 1 (mod \; 2(2m)!)$ and $p\equiv \pm 1 (mod \; 8)$ then $\xi_p$ is homotopically trivial.  Since there are infinitely many $p$‘s satisfying these conditions, we obtain:

Thm.[Ustilovsky] There exist infinitely many exotic but homotopically trivial contact structures on $S^{4m+1}$.

Morse-Bott techniques

The Morse-Bott condition is morally the next best thing to having non-degeneracy (in fact, one can argue that it is the best thing when one wishes to do computations), and it can be thought of as a manifestation of symmetry.

Recall that a function $f:M \rightarrow \mathbb{R}$ is Morse-Bott if its critical set $\mbox{crit}(f)=\bigsqcup_i C_i$ is a disjoint union of connected submanifolds $C_i$, such that, if we denote by $\nu(C_i)$ the normal bundle of $C_i$ inside $M$, then $Hess_p(f)\vert_{\nu(C_i)}$ is non-degenerate.

Loosely speaking, the degeneracies are “well-controlled”, and come in “families”. In general, in the Morse-Bott situation, one hopes for a perturbation scheme which recovers the non-degenerate/Morse case, by a small perturbation of the data, in such a way that one gets a 1-1 correspondence between the symmetric (i.e Morse-Bott) data, and the generic (i.e Morse) one, and so that compuations can be carried out in the Morse-Bott setting in the first place. For instance, if one wishes to compute Morse homology from a Morse-Bott function $f$, one can choose a Morse function $h$ on $\mbox{crit}(f)$, and consider $f_\epsilon:=f+\epsilon \rho h$, for $\epsilon>0$ small, and $\rho$ is a bump function with support near $\mbox{crit}(f)$. The critical points of $f_\epsilon$ are exactly those of $h$, and there is a well-defined notion of convergence of flow-lines of $f_\epsilon$ to “cascades” (when the perturbation parameter $\epsilon$ is taken to go to zero). The latter consist of a flow-line of $f$ hitting a critical manifold, followed by a flow-line segment of $h$ along this manifold, followed by another flow-line of $f$ hitting another critical manifold, and so on, finishing in a critical point of $f$ (see the figure below). One can define the index of a cascade in such a way that the index is preserved under this convergence, and there is a 1-1 correspondence between index $I$ cascades and index $I$ Morse flow-lines of $f_\epsilon$. Hence, one can define a Morse-Bott differential which counts cascades, and the resulting Morse-Bott (co)homology coincides with the usual Morse (co)homology.

In the setting of symplectic homology, if $W$ is a Liouville filling of a contact manifold $(M,\xi)$ and $H$ is an admissible autonomous Hamiltonian, then we have closed Hamiltonian orbits in the contact slices $\{r\}\times M$ corresponding to closed Reeb orbits, which come in $S^1$-families obtained by reparametrizations (since $H$ is time-independent). This is then a Morse-Bott situation.

[Bourgeois-Oancea] In the Morse-Bott situation described above, if we assume that the orbits come in $S^1$-families (and there are no further directions of degeneracy), then there is a Morse-Bott version of symplectic homology of $W$, $SH_{MB}(W)$.

More generally, one can ask the following Morse-Bott conditions: $\mathcal{N}_T:=\{m|\varphi^T(m)=m\}$ is closed submanifold (where $\varphi^T$ is the time $T$ Reeb flow), such that $rank(d\alpha\vert_{\mathcal{N}_T})$ is locally constant and $T\mathcal{N}_T=\ker(d\varphi^T-id)$. Informally, one can think of this as an infinite-dimensional version of the Morse-Bott conditions, applied to the action functional defined on the loop space, whose critical points are closed Hamiltonian orbits. Assuming that $c_1(W)=0$ and the closed orbits are contractible (so we get an integer grading), fix a choice of Morse functions $f_T$ on $\mathcal{N}_T$ for every $T$. The generators will correspond to pairs $(\gamma,T)$  where $\gamma \in \mbox{crit}(f_T)$, and the differential counts “Floer cascades”, consisting of a Floer cylinder, followed by a flow-line segment of a $f_T$, followed by another Floer cylinder…(finitely many times). The grading is defined by $|(\gamma,T)|=\mu_{RS}(\mathcal{N}_T)+ind_{\gamma}(f_T)-\frac{1}{2}(\dim(\mathcal{N}_T)-1)$, where $\mu_{RS}$ is the Robin-Salamon index, and with this definition the differential has degree -1. Under these conditions, we have a Morse-Bott version of symplectic homology $SH_{MB}$.

Uebele’s computation

We focus now on the Brieskorn manifolds $\Sigma_l^n:=\Sigma(2l,2,\dots,2)$, where there are $n$ 2’s, for odd $n$, endowed with the contact structure discussed in the first part of this talk. Randell’s algorithm gives $H_{n-1}(\Sigma_l^n)=\mathbb{Z}$, and it follows from Wall’s classification of highly-connected manifolds that $\Sigma_l^n \simeq S^{n-1}\times S^n$ if $l\equiv 0 (mod 4)$, $\Sigma_l^n \simeq S^*S^n$ if $l\equiv 1 (mod 4)$, $\Sigma_l^n \simeq S^{n-1} \times S^n \# K$ if $l\equiv 2 (mod 4)$, $\Sigma_l^n \simeq S^*S^n \#K$ if $l\equiv 3 (mod 4)$. Here, $K=\Sigma(2,\dots,2,3)$ is Kervaire’s sphere. If $n=3$, $K$ is diffeomorphic to $S^5$, and hence $\Sigma_l^5$ is always $S^2 \times S^3$.

These contact manifolds manifolds are actually not distinguishable by contact homology. However, we have:

Thm. [Uebele] The manifolds $\Sigma_l^n$ are pairwise non-contactomorphic.

This uses the following lemma:

Lemma. For $\Sigma_l^n$, $SH_{MB}^+$ is independent of the filling, as long as $c_1(W)\vert_{\pi_2(W)}=0$.

This is proved by showing that these manifolds are dynamically convex, and using an analogous version of Bourgeois-Oancea result. Therefore one can regard $SH_{MB}^+$ as a contact invariant.

The idea now is to compute $SH_{MB}^+$ of the natural filling of these Brieskorn manifolds, using the Morse-Bott techniques, and showing that they are pairwise different. One can choose perfect Morse functions along the critical manifolds (or “formally pretend” that one can, by a spectral sequence argument due to Fauck), making the Morse differential trivial, and between different critical manifolds, one sees that for each consecutive degrees $N, N+1$ there exists a unique pair of generators having these degrees, the one with bigger degree $N+1$ having lower action than the one with smaller degree $N$. Since the differential has degree -1 and lowers the action, it has to vanish (this works for $n\geq 5$, and a different argument is needed for $n=3$). The upshot is that the Morse-Bott symplectic homology coincides with its chain complex, and the degrees differ for different values of $l$.

References

A nice reference for a survey of Brieskorn manifolds in contact topology can be found here: https://arxiv.org/abs/1310.0343

Ustilovsky’s exotic spheres: 1999-14-781

Uebele’s computations: https://arxiv.org/abs/1502.04547

Fauck’s thesis (related, and uses RFH): https://arxiv.org/abs/1605.07892

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## Kylerec – Weinstein fillings

Continuing on with the Kylerec posts… (see the first one here as well as notes to follow along with here).

This post is a synthesis of the following talks:

• Day 1 Talk 2 – François-Simon Fauteux-Chapleau’s talk on Weinstein handles and contact surgery
• Day 1 Talk 3 – Orsola Capovilla-Searle’s talk on Kirby calculus for Stein manifolds
• Day 1 Talk 4 – Alvin Jin’s talk on Lefschetz fibrations and open books
• Day 2 Talk 1 – Bahar Acu’s talk on mapping class factorizations and Lefschetz fibration fillings
• Day 3 Talk 2 – Sarah McConnell’s talk on applications of Wendl’s theorem to fillings
• Day 5 Talk 1 – Ziva Myer’s talk on flexible and loose Legendrians

### Weinstein surgery theory

I assume the reader is familiar with smooth surgery theory. Recall the following definition.

Definition: A Weinstein cobordism consists of a quadruple $(W,\omega,V,\phi)$, where

• $(W,\omega)$ is a compact symplectic manifold with boundary
• $V$ is a Liouville vector field for $(W,\omega)$, meaning $\mathcal{L}_V\omega = \omega$, which is also transverse to the boundary $\partial W$
• $\phi \colon W \rightarrow \mathbb{R}$ is a Morse function
• $V$ is gradient-like for $\phi$, meaning there is some constant $\delta$ with $d\phi(V) \geq \delta(|d\phi|^2 + |V|^2)$ with respect to a given Riemannian metric.

In this case, the boundary decomposes as $\partial W = \partial^+ W \sqcup \partial^-W$, where $V$ points out of $\partial^+ W$ and into $\partial^- W$. Note that the 1-form $\lambda = \iota_V \omega$ satisfies $d\lambda = \omega$, and is sometimes called the Liouville 1-form, since it encodes the same data as $V$. Also note that a Weinstein cobordism with $\partial^- W = \emptyset$ is what we called a Weinstein filling.

The gradient-like condition is meant to give $V$ some directionality (since $d\phi(V) > 0$) and ensure that the critical points of $V$ are non-degenerate. One typically doesn’t think of the precise choice of pair $(V,\phi)$ as very important, but rather the data up to some notion of homotopy. For example, one can always perturb the Morse function so that each of $\partial^- W$ and $\partial^+ W$ is a regular $\phi$-level set, regardless of the number of components, and so we might as well assume this from the start. The equivalence hinted at here is called Weinstein homotopy, by which we perturb the pair $(V,\phi)$, possibly through birth-death type singularities.

Lemma: The descending manifolds in a Weinstein cobordism, i.e. the set of points which flow along $V$ to a given critical point in infinite time, are isotropic submanifolds.

Proof: Standard Morse theory implies these submanifolds are smooth. Let $\phi_V^t$ be the flow along $V$ at time $t$, and suppose we choose some $q \in D_p^-$ where $D_p^-$ is some descending manifold for a given critical point $p$. Suppose $v \in T_qD_p^-$ is a vector in the tangent space. Then since $\mathcal{L}_V\lambda = d\iota_V\lambda + \iota_V d\lambda = d\iota_V^2\omega + \iota_V\omega = 0 + \lambda = \lambda$, we have that

$e^t\lambda_q(v) = ((\phi_V^t)^*\lambda)(v) = \lambda(d\phi_V^t(v))$

As $t \rightarrow \infty$, the right hand side goes to zero since $\phi_V^t(q') \rightarrow p$ for all $q'$ in a curve $\gamma$ along $D_p^-$ with tangent vector $v$ at $q$. Hence, $\lim_{t\rightarrow \infty} e^t \lambda_q(v) = 0$, from which it follows that $\lambda_q(v) = 0$. Hence, $\lambda|_{D_p^-} = 0$, and so also $\omega|_{D_p^-} = d\lambda_{D_p^-} = 0$.

Corollary: All critical points in a Weinstein cobordism $(W^{2n},\omega,V,\phi)$ are of index at most $n$. Smoothly, any such manifold can be built up by surgery starting from a neighborhood of $\partial^-W$ and attaching handles of index at most $n$.

One would like to be a bit more precise about how the surgery interacts with the symplectic geometry. As a first step, along a regular level set $W_c := \phi^{-1}(c)$, the symplectic condition on $\omega$ implies that $\lambda|_{W_c}$ is a contact form. The proof of the lemma above further implies that $D_p^- \cap W_c$ gives an isotropic submanifold of $W_c$ with respect to $\lambda|_{W_c}$.

So we can think, at least smoothly, that our Weinstein cobordism is built up, starting from $\partial^- W$, by attaching handles with isotropic cores and attaching spheres along isotropics in level sets of $\phi$ (which are contact submanifolds). But there’s a little more that we know about neighborhoods of isotropics. In a symplectic manifold, the neighborhood of an isotropic $M \subset (W,\omega)$ is completely determined up to symplectomorphism by its symplectic normal bundle, $(TM)^{\omega}/TM$, as a symplectic vector bundle (with symplectic structure induced by $\omega$ on the fibers).  A similar statement holds for isotropic submanifolds in contact manifolds, but now with their neighborhoods determined up to contactomorphism by the conformal symplectic normal bundle $(TM)^{d\alpha}/TM$, where $\alpha$ is a contact form so that $d\alpha$ is symplectic on $\xi$. Furthermore, if we fix $\alpha$, then the symplectic vector bundle structure determined by $d\alpha$ on the nose determines the neighborhood up to exact contactomorphism. Patching these two things together, one finds:

Theorem [Weinstein, before the term “Weinstein handle” was coined]: Weinstein handle attachment is completely specified (up to Weinstein homotopy) by matching the symplectic framing data determined by $\lambda$ along the isotropic attaching spheres.

One therefore thinks of $\partial^+ W$ as being built up from $\partial^- W$ by contact surgery along isotropic submanifolds with given framing information compatible with the underlying symplectic topology.

Consider a Weinstein cobordism of dimension $2n$. Then the handles of index $k \in \{0,1,\ldots,n-1\}$ are called subcritical handles, whereas the handles of index $k = n$ are called critical handles. When $k = n$, the aformenetioned symplectic normal bundles are trivial automatically, and so one specifies critical handle attachment simply by drawing a Legendrian sphere on $\partial^- W$.

Recall that the proof of the h-cobordism theorem requires some ability to cancel (and create) pairs of handles with index differing by 1 whose ascending and descending manifolds intersect in a 1-dimensional manifold, to move around attaching spheres, and to move critical values around. The last of these we can always do, so we can attach the handles in order of their index. It turns out that when $2n > 4$, we can recreate all parts of the proof of the h-cobordism theorem for subcritical Weinstein cobordisms. In some sense, subcritical Weinstein domains have no symplectic geometry in them – they are encoded by algebro-topological information, and so this gives some flexibility phenomena.

It turns out that some critical handles behave the same way. The key obstruction to the aforementioned flexibility is that sometimes the data of an attaching Legendrian does not boil down to purely toplogical information. However, Emmy Murphy defined a class of Legendrians, called loose Legendrians, for which there is such a so-called h-principle. The Weinstein h-cobordism theorem works for Weinstein cobordisms which can be built (up to Weinstein homotopy) out of subcritical and loose critical handle attachments. We call such Weinstein cobordisms flexible.

We often care about the case when $2n = 4$. In this case, it is pretty easy to describe a connected Weinstein domain (or its contact boundary). One can first order the handles by index, and then cancel 0-handles with 1-handles until we are in the situation where there is precisely one 0-handle and possibly many 1- and 2-handles. The boundary of the 0-handle is just a standard contact $S^3$, and 1-handle attachment is trivially described by picking pairs of points in $S^3$ (the bundle data boils down to showing $\pi_0(\text{Sp}(2,\mathbb{R})) = 0$). So it suffices to draw Legendrians on $S^3$ with $k$ pairs of points identified, which is just $\#^k (S^1 \times S^2)$. Any Legendrian $L$ has a canonical framing of its normal bundle given by the twisting of the Reeb chord around the Legendrian. Eliashberg showed that adding a left twist to this framing gives the smooth framing which determines the corresponding smooth surgery data.

Gompf showed that in this case $2n = 4$, one can draw standard Kirby calculus type surgery diagrams. We think of all of these 1-handle attachments and Legendrians as missing a point in $S^3$, so that we can draw our diagrams in $(\mathbb{R}^3, \ker dz - ydx)$. The front projection is the projection to the coordinates $(x,z)$, so that $y$ is determined by $dz/dx$. It might not be obvious how to draw a smooth knot in this projection since the curve can’t have infinite slope, but we are allowed semi-cubical cusps, corresponding to $(x,y,z) = (t^2,3t/2,t^3)$. Note that transverse crossings are also allowed, since the $y$-coordinates are distinct. One usually draws the front projection of a Legendrian without showing which strand lies over the other, but we include this extra information in the next figure, where we imagine the $y$-axis as pointing into the page.

A Legendrian trefoil knot

Gompf’s standard form for these Legendrians looks like the following, where the pairs of balls in each row corresponds to where the 1-handles are attached, and the Legendrian strands simply go through the handles as though they were wormholes.

An example of a Gompf surgery diagram. There are three 1-handles (in blue, red, and green) and two 2-handles with attaching spheres given by the Legendrian tangle above. All of the information can be made to live inside of the purple rectangle (i.e. without going horizontally or vertically outside of where the 1-handles are attached).

### Weinstein fillings, Lefschetz fibrations, and open book decompositions

Definition:Lefschetz fibration is a smooth map $\pi \colon W^4 \rightarrow \Sigma^2$ with finitely many critical points with distinct critical values such that locally around the critical points, $\pi$ looks like a complex Morse function (i.e. $(z_1,z_2) \mapsto z_1^2+ z_2^2$ in local coordinates). When $\Sigma$ has boundary, we assume the critical values of $\pi$ are all in the interior of $\Sigma$.

We shall typically be concerned with the case where $\Sigma = \mathbb{D}$ (although see this post by Laura Starkston which slightly generalizes some of what is discussed here).

A schematic for a Lefschetz fibration over the disk

In the case where $\Sigma = \mathbb{D}$, we see that the boundary decomposes as $\partial W = \partial^v W \cup \partial^h W$, where the superscripts are meant to indicate vertical and horizontal. That is, $\partial^v W = \pi^{-1}(\partial \mathbb{D})$, while $\partial^hW = \sqcup_{p \in \mathbb{D}} \partial \pi^{-1}(p)$. If we write $F$ for a regular fiber of $\pi$, then $\partial^h W = \partial F \times \mathbb{D}$. Meanwhile, we see that $\partial^v W$ is just a fibration over $S^1$ with fiber $F$, and hence can be described by some monodromy map $\phi \colon F \rightarrow F$ fixing the boundary, so that $\partial^v W = F \times [0,1]/{\sim}$ where $(\phi(x),0) \sim (x,1)$ (the mapping torus of $\phi$).

The structure on the boundary, in which we have a fibration over $S^1$ with fiber $F$ glued together with $\partial F \times \mathbb{D}$ in the natural way, is called an open book decomposition. It is given completely by the pair $(F,\phi)$. We think of each fiber over $S^1$ as a page, and the subset $F \times \{0\}$ as the binding, analogous to what one would get if one took their favorite book and matched the covers so that the pages radiate outwards. So Lefschetz fibrations yield open books on the boundary. To be a little more precise, one should extend each page so that the boundary of each page is actually the binding.

Some pages near the binding of an open book. I guess the name “Rolodex” wasn’t as catchy as “open book.” (Image from Wikipedia)

Now suppose $0 \in \mathbb{D}$ is a regular value (which can always be arranged up to small perturbation of $\pi$). Then $\pi^{-1}(\epsilon \mathbb{D}) \cong F \times \mathbb{D}$. One can ask what happens when we extend to $\pi^{-1}(U)$, where $\epsilon \mathbb{D} \subset U$ and there is exactly one critical value $p$ on $U \setminus \epsilon\mathbb{D}$.

Since we have a nice fibration away from critical points, we see that paths in $\mathbb{D}$ yield monodromy maps (up to isotopy preserving boundary) on the fibers. We can choose a connection on the fibration if we wish to make this a map on fibers, not just a map up to isotopy. If we take a path $\gamma$ from 0 to $p$ which intersects $\partial \epsilon \mathbb{D}$ once and otherwise avoids critical values then for whatever connection we chose, we can see what points flow to the critical point over $p$. Over each regular fiber, this is just a circle, and the union of all of them together with the critical point yields a disk. The path $\gamma$ is called a vanishing path, and each circle on the regular fiber is called a vanishing cycle (one really should think of it as a homology cycle, but for concreteness, one can think of it as a curve). The disk consisting of the union of vanishing cycles above a path is called a thimble.

The green circles in the regular fibers above the purple vanishing path are the vanishing cycles. Their union is the thimble.

It is then not hard to see that $\pi^{-1}(U)$ is obtained from $\pi^{-1}(\epsilon \mathbb{D})$ by 2-handle attachment, where the attaching curve is just the vanishing cycle above $\gamma \cap \partial \epsilon \mathbb{D}$ and the core of the handle is the thimble. Furthermore, one can check by a local computation that the monodromy map in a loop around $p$ is just given by a Dehn twist (positive or negative, depending on orientations) around the vanishing cycle. Hence, one can write out the open book determined by the Lefschetz fibration explicitly – it is just the product of the Dehn twists on the vanishing cycles, performed in an order determined by a sequence of vanishing paths.

Notice that for a given regular value on $\partial \mathbb{D}$, one can choose a different basis of vanishing paths, and this yields a possibly different factorization for the monodromy. Such changing of the basis is generated by so-called Hurwitz moves, as drawn below.

A Hurwitz move swapping the $i$th and $(i+1)$st critical points. Note that the corresponding vanishing cycles for the critical point corresponding to $\gamma_i$ and $\gamma_{i+1}'$ are actually different, but the overall monodromy on the open book at the boundary is the same.

Hence, understanding Lefschetz fibrations over the disk essentially corresponds to understanding factorizations of mapping class group elements into Dehn twists.

Now, this whole story can be repeated in the symplectic context, as follows.

Definition:symplectic Lefschetz fibration is a Lefschetz fibration with $(W,\omega)$ a symplectic manifold such that each fiber is symplectic submanifold away from the critical points, while at the critical points the coordinates in which $\pi$ locally looks like a complex Morse function can be taken to be holomorphic for some compatible almost complex structure $J$.

In this case, one can take the connection to be the symplectic connection given the symplectic orthogonal complement to the vertical directions. In this way, the thimbles produced will actually be Lagrangian disks, which suggests one can think of these as the descending disks for a Weinstein domain filling the boundary. In addition, the monodromy maps are now compositions of positive Dehn twists only, since the symplectic condition gives the proper orientations. In other words, our Lefschetz fibration is itself positive. If the vanishing cycles of a Lefschetz fibration are homologically nontrivial, we shall call it allowable.

With a little more work, we can obtain the following theorem of Loi and Piergallini (although an alternative proof by Akbulut and Özbağci is more in line with the exposition presented here):

Theorem: Any positive allowable Lefschetz fibration (PALF) yields a Weinstein domain, and any Weinstein domain comes from a PALF in this way.

Furthermore, one obtains a little bit more compatibility at the boundary.

Definition: An open book decomposition on a manifold $M$ is said to support a cooriented contact structure $\xi$ if there is some contact form $\alpha$ for $\xi$ such that the binding is a contact submanifold, $d\alpha$ is a symplectic form on the pages, and the boundary orientation of the page (with respect to $d\alpha$) matches the orientation of the binding with respect to $\alpha$.

One checks that the open book on the boundary of a PALF does indeed support the contact structure determined by being the boundary of a Weinstein domain.

Our surgery theory for these Lefschetz fibration builds the fiber up by subcritical surgery, and the 2-handle attachments correspond to the critical points of the fibration. One can always produce, for any Weinstein manifold, a cancelling pair consisting of a 1-handle and a 2-handle. The way that this affects the open book is by positive stabilization, meaning that one adds a 1-handle to the page, but kills it by adding an extra Dehn twist to the monodromy through a circle which passes through the handle.

The following theorem implies that all 3-dimensional contact geometry can actually be encoded (somewhat non-trivially) in the study of open books up to positive stabilization, and hence the study of Weinstein fillings reduces to studying positive factorizations of given elements of the mapping class group of a surface with boundary (up to this not-so-easy-to-work-with notion of positive stabilization).

Theorem [Giroux correspondence]: There is a one-to-one correspondence between contact structures on a closed 3-manifold up to isotopy with open books up to positive stabilization.

### Applications to Weinstein fillings

To summarize the previous section, an explicit surgery decomposition of a Weinstein filling yields a PALF which in turn gives an open book structure supporting the contact boundary of the Weinstein filling with monodromy factored into positive Dehn twists. Conversely, given a supporting open book for a contact structure with monodromy factored into positive Dehn twists, one obtains a Weinstein filling.

One common question we ask is whether a single contact manifold has multiple Weinstein fillings. From the above construction, one possible way to attack this problem is to look for distinct positive factorizations of a given element in a mapping class group.

Theorem [Auroux]: There is an element in the mapping class group of the surface $\Sigma_{1,1}$ (of genus 1 and with one boundary component) with two distinct factorizations into positive Dehn twists such that the Weinstein fillings are distinguished by their first homology.

Remark: In this setting, the first homology is just given by $H_1(F)/V$ where $V$ is the span of the vanishing cycles. The only real trick of Auroux is therefore to find a good candidate for the above theorem to hold, and just compute.

Generalizing a bit more:

Theorem [Baykur – Van Horn-Morris]: There exists an element in the mapping class group of $\Sigma_{1,3}$ (of genus 1 with three boundary components) which admits infinitely many positive factorizations such that the corresponding Weinstein fillings are all distinguished from each other by their first homology.

Finally, as one last application, I want to consider a result of Plamenvskaya and Van Horn-Morris, but I need to define the contact structures in question to begin. Honda’s classification of tight contact structures on the lens spaces $L(p,1)$ can be formulated in Gompf’s surgery diagrams by the following diagrams, coming from a single 2-handle attachment to standard $S^3$. We denote the corresponding contact structures by $\xi_1,\xi_2,\ldots, \xi_{p-2}$.

The surgery diagram for the contact structure $\xi_k$.

Of these, the universal covers of $\xi_1$ and $\xi_{p-2}$ are also tight, where as the others’ universal covers are overtwisted. We say $\xi_2, \ldots, \xi_{p-3}$ are virtually overtwisted.

Theorem [PV]: Each virtually overtwisted $(L(p,1), \xi_k)$ has a unique Weinstein filling (up to symplectic deformation) and a unique minimal weak filling.

Proof sketch: Let us first discuss the Weinstein part. There are a few nontrivial theorems which go into this, which we won’t discuss, but essentially we have the following sequence of results. The open book given by the surgery diagrams above induce open books with genus 0 pages. When we discussed Wendl’s theorem in part 2 of the J-holomorphic curve posts, one thing we mentioned was that one can apply his techniques when there is a planar open book (meaning pages have genus 0). He proves that if a contact manifold has a given supporting planar open book, then every Weinstein filling is diffeomorphic to one compatible with that specified planar open book. Hence, it suffices to study Lefschetz fibrations compatible with the one just described, which in turn becomes studying factorizations of an element in the mapping class group of $\mathbb{D}_n$, the disk with $n$ holes. A nontrivial result of Margalit and McCammond gives that every such presentation must be in a certain form, from which one can use smooth Kirby calculus to conclude that the surgery diagram must come from $-p$-surgery on some knot. Finally, an appeal to work of Kronheimer, Mrowka, Ozsváth, and Szabó using Seiberg-Witten Floer homology (also called monopole Floer homology) yields that this knot must have been an unknot, and since the framing is $-p$, this determines the canonical framing of the knot, which in turn implies we could only have had one of our original surgery diagrams.

Finally, to obtain the weak part, one can use work of Ohta and Ono to boost a weak filling up to a strong filling, from which Wendl’s theorem implies that any minimal weak filling is symplectic deformation equivalent to a Weinstein filling.

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## Kylerec – On J-holomorphic curves, part 2

This is a continuation post following part 1. Hope the delay wasn’t too long – more coming soon.

### A preparatory comment on last time

Let us quickly recall the proof sketched in part 1 that fillable implies tight for contact $(M^3,\xi)$. The idea was that if we had a filling $W$, then the presence of an overtwisted disk locally gave a Bishop family of holomorphic disks as part of a 1-dimensional moduli space, but the compactified moduli space was seen to have only one boundary point. This was because continuing the family away from the center of the overtwisted disk could not lead to a possible boundary point – in our version, such a point would require a bubble, but we considered exact fillings.

It was mentioned last time that Eliashberg’s paper on fillings by holomorphic disks actually covers the weak case instead. The difference is that one now instead shows that any closed surface in a contact boundary which is weakly filled indeed bounds a 3-manifold which can be foliated by holomorphic disks. With this result, we never actually need to consider bubbles on the interior, so we can remove the exactness assumption.

I want to point this out because although this isn’t perfectly analagous to the discussion in the next section on Wendl’s paper, it is related in basic idea. The set-up is different, sure, and also one cannot work directly with the contact manifold with its filling (the necessity of a strong filling appears in the need to attach a positive cylindrical end), and finally also the completed foliation does have (isolated) nodal curves, but in the end we end up with a nice Lefschetz fibration with holomorphic fibers, and that’s pretty powerful, just like Eliashberg’s disk fillings. We explain this… now!

### On Wendl’s Strongly fillable contact manifolds and J-holomorphic foliations

I should say, first of all, that one can generalize the results I am discussing here. For a somewhat more general discussion, see this post by Laura Starkston from 2013.

In this section, all contact manifolds are 3-dimensional and all symplectic manifolds are 4-dimensional.

We begin by recalling that the symplectization of a cooriented contact manifold $(M,\xi = \ker \alpha)$ is the symplectic manifold $(\mathbb{R} \times M, d(e^t\alpha))$, where $t$ is the $\mathbb{R}$-coordinate. This symplectic manifold does not depend upon the choice of $\alpha$, since if we chose $\alpha' = e^f \alpha$, then $d(e^t\alpha') = d(e^{t+f}\alpha) = \phi^* d(e^t\alpha)$ where $\phi$ is the diffeomorphism of $\mathbb{R} \times M$ sending $(t,m) \mapsto (t+f(m),m)$.

Remark: One can define this in a more invariant way. The symplectization of $(M,\xi)$ is the set of covectors vanishing on $\xi$. This is an $\mathbb{R}^*$-bundle over $M$ and is symplectic with respect to the standard symplectic form on $T^*M$. Fixing a local section $\alpha$ over $M$ gives a coordinate $w$ for the fiber such that the symplectic form is just $d(w\alpha)$. We simply take the component where $e^t := w > 0$ and $\alpha$ coorients $\xi$.

Given a symplectization, explicitly determined by a chosen contact form $\alpha$, one typically studies J-holomorphic curves only for choices of $J$ which are admissible, meaning:

• $J$ is $\mathbb{R}$-invariant
• $J \partial_t = R_{\alpha}$
• $J|_{\xi}$ is a compatible almost complex structure for $d\alpha|_{\xi}$

Under these conditions, finite energy J-holomorphic curves from punctured Riemann surfaces are analytically easy to understand – the punctures are asymptotic to Reeb orbits at the positive and negative ends, and Gromov compactness extends to this setting in that one needs to include holomorphic buildings. One can imagine, for example a sequence of $J$-holomorphic curves which look like some union of cylinders over Reeb orbits for all but a union of two intervals $(-A-C,-A) \cup (A,A+C)$ on which there is nontrivial behavior, where $C$ remains fixed but $A \rightarrow \infty$. In the limit, as these two intervals get farther apart, we break into two holomorphic curves in the symplectization. This forms what is sometimes called a holomorphic building. In general, there may be multiple levels in the limit, as in the figure below.

Here, a J-holomorphic curve gets stretched out in two places, shown in green, until eventually these green almost cylindrical parts get infinitely long. In the limit, we obtain a three story holomorphic building.

For more details in much more generality, one should consult this paper of Bourgeois, Eliashberg, Hofer, Wysocki, and Zehnder.

Now suppose that we have a strong filling $(W,\omega)$ of a contact manifold $(M,\xi)$. Then by definition, we have a Liouville vector field $V$ whose flow allows me to identify a neighborhood of $M$ with a subset of the symplectization of the form $((-\epsilon,0] \times M, d(e^t\alpha))$ where $\alpha$ is a contact form for $\xi$ on $M$. One can append the rest of the positive end of the symplectization, $([0,\infty) \times M, d(e^t\alpha))$, to form a completed symplectic manifold $(\widehat{W},\widehat{\omega})$. I can choose some compatible almost complex structure $J$ which far enough into the positive end is a restriction of some admissible $J_+$. In this case, one can study J-holomorphic curves, and we have a similar Gromov compactness statement. In this case, our curves can either bubble, or form holomorphic buildings where the lowest level is just $(\widehat{W},J)$ and whose higher levels are all $(\mathbb{R} \times M, J_+)$.

Theorem (vaguely stated): Under some technical analytical conditions, an $\mathbb{R}$-invariant foliation of $\mathbb{R} \times M$ by $J$-holomorphic curves of uniformly bounded energy will extend, with isolated nodal singularities, to the interior of $W$ (and hence to all of $\widehat{W}$).

Proof (sketch): We study the compactification of the moduli space $\mathcal{M}$ of finite energy J-holomorphic curves in $\widehat{W}$, and in particular, the closure of the component $\mathcal{M}_0$ of the moduli space containing a special leaf in the symplectization end. This component is 2-dimensional, and hence is precisely given by the foliating leaves around it (recall $\widehat{W}$ is 4-dimensional). The closure of this component yields the full J-holomorphic foliation, where some isolated finite subset of the leaves are actually nodal curves.

In the end, by considering on which curve in the foliation a point is located, this yields a map $\pi \colon \widehat{W} \rightarrow \overline{\mathcal{M}_0}$, where the fibers are symplectic (since they are J-holomorphic and $J$ is compatible with $\widehat{\omega}$) and generically smooth except with finitely many nodal singular fibers, forming what is called a symplectic Lefschetz fibration.

We will discuss this notion more in a future post, where we will also see that Stein fillings correspond in some sense to certain (“allowable”) symplectic Lefschetz fibrations over a disk. Hence, one can ask – are there some examples of contact manfiolds $(M,\xi)$ on which we can find a finite energy foliation on the symplectization $\mathbb{R} \times M$ satisfying the correct analytical assumptions and such that $\overline{\mathcal{M}_0} = \mathbb{D}$? The answer is yes in the case when $(M,\xi)$ is supported by a so-called planar open book, as was proved in this paper by Wendl. We will define this in a future post, but this discussion implies (up to how to tackle the word “allowable”) that:

Corollary: For contact 3-manifolds supported by a planar open book, strong and Stein fillability are equivalent.

Along similar lines, one can find finite energy foliations for the standard 3-torus $(\mathbb{T}^3,\xi_0)$ (with contact structure induced by the restriction of the Liouville form on $T^*\mathbb{T}^2$ to the unit cotangent bundle). In this case, any strong filling, not just the standard one, would have $\overline{\mathcal{M}_0} = [0,1] \times S^1$, and so any strong filling of $(\mathbb{T}^3,\xi_0)$ arises as the boundary of a Lefschetz fibration to $[0,1] \times S^1$. Wendl then beefs this up to prove, for example, that every minimal strong filling of $(\mathbb{T}^3,\xi_0)$ is diffeomorphic to $\mathbb{T}^2 \times \mathbb{D}$.

Finally, one can use these results to obstruct strong fillability in a manner analogous to the Bishop family argument. That is, if $(M,\xi)$ has a finite energy foliation satisfying the technical analytic assumptions, then one should be able to extend that foliation to a strong filling. Recall that the foliation extended by considering the component of the moduli space containing some specified leaf $u_0$ satisfying some conditions. If there is some other leaf $u_1$ which is not diffeomorphic to $u_0$, then they cannot both be fibers of the same Lefschetz fibration, and so there couldn’t have been a strong filling in the first place. There are also other more technical versions of this argument, which for example allow one to reprove that positive Giroux torsion, i.e. that there is a contact embedding of $([0,1] \times T^2, \cos(2\pi t)d\theta_1 + \sin(2\pi t)d\theta_2)$, obstructs fillability, originally proved by David Gay using gauge-theoretic methods which are completely avoided in this approach.

### On Barth-Geiges-Zehmisch’ The diffeomorphism type of fillings

As we have now seen twice, the technique of comparing moduli spaces of J-holomorphic curves to the topology of the situation in question is very powerful. We saw this both in our discussion last time of McDuff’s rational ruled classification, and we also just saw in our discussion of Wendl’s paper that the breaking which occurs in the compactification of a certain moduli space of curves in a strong filling of the positive end of a symplectization actually cooks up a Lefschetz fibration. One can view this paper as another instance of this way of thinking – here evaluation maps end up directly producing strong restrictions on the topology of a filling.

As we will see in a future post, Weinstein fillings of contact manifolds $(M^{2n+1},\xi)$ have a surgery theory consisting of handles of index at most $n$, and so they have the homotopy type of a CW complex of at most this dimension. A subcritical Weinstein filling is then one where all the handles have index at most $n-1$. The main theorem states that the existence of just one subcritical Weinstein filling places restrictions on the topology of any strong symplectically aspherical filling $(W,\omega)$. By symplectically aspherical, we mean that $\omega|_{\pi_2(W)} = 0$.

Theorem [BGZ]: If $(M,\xi)$ is a contact manifold of dimension $\geq 3$ admitting a subcritical Stein filling with the homotopy type of a CW complex of dimension $\ell_0$, then any strong symplectically aspherical filling $(W,\omega)$ satisfies

• $H_k(W) = H_k(M)$ for $k = 0,\ldots, \ell_0$ via the isomorphism induced by inclusion
• $H_k(W) = 0$ otherwise
• If $\pi_1(M) = 0$, then all strong aspherical fillings of $M$ are diffeomorphic.

Corollary [Eliashberg-Floer-McDuff ’91]: Every symplectically aspherical filling of the standard contact sphere is diffeomorphic to a ball.

Remark: For $S^3$, which is just the lens space $L(1,1)$, McDuff’s theorem from last time about fillings of lens spaces implies that there is a unique minimal filling up to diffeomorphism. By positivity of intersection, symplectically aspherical fillings are minimal, which implies the above result. But also, since $\omega$ is automatically a trivial cohomology class on the ball, McDuff’s result implies that the filling is in fact unique up to symplectomorphism. This result goes back to Gromov’s ’85 paper.

We won’t quite make it to a proof of the full theorem, but we will see some of the inner workings in the statement of the theorem stated below. We proceed by making an extra definition (not in Barth-Geiges-Zehmisch) to clarify the exposition.

Definition: Let $(M^{2n+1},\xi)$ be a connected contact manifold and $(V^{2n},\omega = d\lambda)$ a Liouville manifold of finite type (meaning it is modelled after a positive symplectization outside of some compact region). Let $\mathcal{L}$ be the corresponding Liouville vector field (satisfying $i_{\mathcal{L}}\omega = \lambda$). The $M$ is called $V$spliffable (yes, this is what we called it at Kylerec) if $M$ is contactomorphic to a hypersurface $\widetilde{M}$ in $V \times \mathbb{C}$ such that:

• $\widetilde{M}$ is convex, meaning it is transverse to the vector field $\mathcal{L} \oplus \partial_r$ where $\partial_r$ is the standard radial Liouville vector field on $\mathbb{C}$
• the infinite component of $V \times \mathbb{C} \setminus \widetilde{M}$ is modelled after the positive symplectization of $M$ meaning this component is the union of the positive flow of $\widetilde{M}$ along $\mathcal{L} \oplus \partial_r$

Remark: A contact manifold which is fillable by a subcritical Weinstein manifold is spliffable. This follows from a result of Cieliebak that subcritical Stein manifolds are split.

Theorem: Let $(W,\omega)$ be an aspherical strong filling of a $V$-spliffable contact manifold $M$. Then there exists a commutative diagram of the form

(and similarly with $H_*$ replaced with $\pi_1$).

Remark: The Eliashberg-Floer-McDuff theorem is already a corollary of this weaker statement, using that $S^{2n-1}$ is $\mathbb{C}^{n-1}$-spliffable, and using smooth topology.

Corollary: The unit cotangent bundle $M = S^*\Sigma$ of a closed manifold $\Sigma^n$ admits no subcritical Weinstein fillings.

Proof: We need $H_n(V)$ surjects onto $H_n(W) \neq 0$ with $W$ the standard unit disk filling and $M$ is $V$-spliffable. But if $M$ admits a subcritical filling, then $V$ can be chosen to be subcritical so that $H_n(V) = 0$. This is a contradiction.

Proof of the main theorem:

We begin by embedding $M$ into $V \times \mathbb{D}$ so that it is convex (which we can do by the spliffability condition). The interior component determine by the splitting through $M$ can then be replaced by $W$ by gluing in (since strong gluings are set up to be Liouville near the boundary). Call the interior of this manifold $Z$. We can then choose a map $\mathbb{D} \rightarrow \mathbb{C}P^1$ such that the interior embeds diffeomorphically onto $\mathbb{C}P^1 \setminus \{\infty\}$. This embedding then gives us a smooth manifold $\widetilde{Z}$ which looks like $V \times \mathbb{C}P^1$ but with the interior component replaced by $W$. That is, $\widetilde{Z} = Z \cup (V \times \{\infty\})$.

We then wish to study some $J$-holomorphic curves on this manifold. We pick a compatible $J$ which away from $W$ is of the form $J_V \oplus i_{\text{std}}$, where $J_V$ is admissible (as discussed in the previous section) for $V$. We study the moduli space $\mathcal{M}$ of $J$-holomorphic spheres $u \colon \mathbb{C}P^1 \rightarrow \widetilde{Z}$ such that $[u] = [\{v\} \times \mathbb{C}P^1]$ (for some $v$ large enough so that this slice misses $W$). We really want this up to reparametrization, so we fix slice conditions to define this moduli space: that $u(-1) \in V\times \{a\}$, $u(+1) \in V \times \{b\}$, and $u(\infty) \in V \times \{\infty\}$, for some choice of $a,b \in \mathbb{C}P^1$ distinct and not $\infty$.

The key about positive symplectization ends is that admissibility of the almost complex structure $J$ implies a maximum principle for these curves. This implies the following two items.

• Since $V \times \mathbb{C}P^1$ looks like a symplectization, in $\widetilde{Z}$, any curve in our moduli space must have actually just been $\{v\} \times \mathbb{C}P^1$.
• Any curve in our moduli space intersecting $W$ must intersect $V \times \{\infty\}$. First of all, $W$ is symplectically aspherical, so any holomorphic sphere must leave $W$. But then, if it didn’t intersect $V \times \{\infty\}$, it would be contained completely in $V \times \mathbb{C}$, which contradicts this maximum principle.

Now, this moduli space is an oriented manifold of dimension $2n$, and it comes with an evaluation map of the form $\mathcal{M} \times \mathbb{C}P^1 \rightarrow \widetilde{Z}$. This map is actually proper and degree 1, which follows from the maximum principles just described, plus a little boost from positivity of intersection which implies that there is no need to worry about stable maps in the compactification of $\mathcal{M}$. This then restricts to a proper degree 1 evaluation map $\mathcal{M} \times \mathbb{C} \rightarrow Z$.

Hence, we obtain the following commutative diagram.

In homology, the right triangle becomes the desired triangle from the theorem.

As for the surjectivity part of the theorem, note that the leftmost vertical arrow is an isomorphism. Meanwhile, the bottom horizontal arrow is surjective for standard topological reasons (because one can cook up an explicit shriek map $\text{ev}_!$ which is right inverse to $\text{ev}_*$ on the level of homology).

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## Kylerec – On J-holomorphic curves, part 1

This to-be-2-part-because-this-got-long post is a continuation of the series on Kylerec 2017 starting with the previous post, and covers most of the talks from Days 2-3 of Kylerec, focusing on the use of J-holomorphic curves in the study of fillings. I should mention that two more sets of notes, by Orsola Capovilla-Searle and Cédric de Groote, have been uploaded to the website on this page. So if you wish to follow along, feel free to follow the notes there, and in particular, the relevant talks I’ll be discussing in this post are:

Part 1

• Day 1 Talk 1 – The introductory talk by (mostly) Roger Casals (with some words by Laura Starkston)
• Day 2 Talk 2 – Roberta Gaudagni’s talk introducing J-holomorphic curves
• Day 2 Talk 3 – Emily Maw’s talk on McDuff’s rational ruled classification

Part 2

It should be obvious in what follows which parts of the exposition correspond to which talks, although what follows is perhaps a pretty biased account, with some parts amplified or added, and others skimmed or skipped.

### J-holomorphic curves – basics

Gromov introduced the study of J-holomorphic curves into symplectic geometry in his famous 1985 paper, immediately revolutionizing the field. One might wonder why we care about these objects, and the rest of this post (along with part 2) should be a testament to some (but certainly not all) aspects of the power of the theory.

The “J” in “J-holomorphic” refers to some choice $J$ of almost complex structure on a manifold $M^{2n}$. Given an almost complex manifold, a J-holomorphic curve is a map $u : (\Sigma,j) \rightarrow (M^{2n},J)$ such that $(\Sigma,j)$ is a Riemann surface and $J \circ du = du \circ j$. In the case where $(M,J)$ is a complex manifold, we see this is precisely what it means to be holomorphic.

We are mostly concerned a choice of $J$ which is compatible with a symplectic manifold $(M,\omega)$. By this, we mean that the (0,2)-tensor $g(\cdot, \cdot) = \omega(\cdot,J\cdot)$ is a Riemannian metric. We say $J$ is tame if $\omega(v,Jv) > 0$ for each nonzero vector $v$ (note that $g$ as defined above is not necessarily symmetric in this case).

Proposition: The space of compatible almost complex structures on a symplectic manifold $(M,\omega)$ is non-empty and contractible. So is the space of tame almost complex structures.

This suggests either:

• Studying the space of J-holomorphic curves into $M$ for some particular choice of $J$.
• Study some invariant of spaces of J-holomorphic curves which does not depend on the choice of $J$ compatible (or tame) with respect to a given symplectic form $(M,\omega)$.

In walking down either of these paths, there are a large number of properties at our disposal. What is presented in this section is far from a conclusive list, and I have completely abandoned including proofs and motivation, so beware that there is a lot of subtlety involved in the analytic details. For many many many more details, consult this book of McDuff and Salamon.

Firstly, there is a dichotomy between somewhere injective curves and multiple covers. Some J-holomorphic curves will factor through branched covers, meaning that $u : \Sigma \rightarrow (M,J)$ factors as $(\Sigma,j) \rightarrow (\Sigma',j') \rightarrow (M,J)$ such that the first map is a branched cover of Riemann surfaces. J-holomorphic curves which are not multiply covered are called simple, and it turns out that simple curves are characterized by being somewhere injective, meaning there is some $z$ for which $u^{-1}u(z) = \{z\}$ and $du_z \neq 0$. Even better, somewhere injective means that $u$ is almost everywhere injective.

The main tool in the theory is the study of certain moduli spaces of J-holomorphic curves. There are many flavors of this, but we discuss a specific example to highlight the relevant aspects of the theory. The analytical details are typically easier for simple curves, so we denote by $\mathcal{M}^*(M,J)$ the moduli space of all simple $J$-holomorphic curves. It turns out to be fruitful to focus in on a specific piece of this space, so we often restrict to a given domain of definition, say some $\Sigma_g$, and also restrict the homology class $u_*[\Sigma]$ of the map $u : \Sigma_g \rightarrow (M,J)$ to some $A \in H_2(M)$. The main question is:

When is such a moduli space $\mathcal{M}_g^*(M,A,J)$ actually a smooth manifold?

This is certainly a subtle question, and it turns out that not every $J$ works. However, it is a theorem that for generic $J$, this moduli space is a smooth manifold of dimension $d = n(2-2g) + 2c_1(A)$, where $\dim M = 2n$.

Given our nice moduli space, we also might be interested in what happens as we change our choice of $J$, so that we go from one regular choice to another. A generic path of such almost complex structures will give a smooth cobordism between the moduli spaces, a property which allows us to cook up invariants which do not depend, for example, on choices of $J$ compatible with a given symplectic structure.

To note a few variants of the discussion so far, sometimes we will study J-holomorphic disks with certain boundary conditions, or J-holomorphic curves with punctures sent to a certain asymptotic limit. In all cases, the same analytic machinery already swept under the rug (Fredholm theory) will give that the moduli spaces in question are smooth for generic choices of almost complex structure, and the dimension of this moduli space is given by some purely topological quantity (by, for example, the Atiyah-Singer index theorem).

One common thing to do is to quotient out by the group action given by reparametrizing the domain of a given J-holomorphic curve. That is, we consider the equivalence relation $u \sim u \circ \phi$ where $\phi : (\Sigma,j) \rightarrow (\Sigma,j)$ is a biholomorphism. A more careful author would probably distinguish between the map $u$ as opposed to the corresponding equivalence class, which is really what one should mean when they say curve. Hence, one can quotient our moduli spaces $\mathcal{M}^*$ by reparametrization to obtain moduli spaces of curves. Usually, these are the main objects of interest.

So now we have our nice moduli space, in whatever situation we desire, and we can ask about studying limits of J-holomorphic curves in that moduli space. In general, no such curve might exist. The first reason for this is that any such curve $u : (\Sigma,j) \rightarrow (M,\omega,J)$ has an energy $E = \int_{\Sigma}u^*\omega$ attached to it (when $J$ is compatible with $\omega$). If this quantity diverges to $\infty$, then there can be no limiting curve. One can ask instead about what happens when the energy is bounded.

Consider the following sequence of holomorphic curves $u_n \colon \mathbb{C}P^1 \rightarrow \mathbb{C}P^1 \times \mathbb{C}P^1$ given by $z \mapsto (z, 1/(nz))$. We see that away from $z=0$, this is just converging to the curve $\mathbb{C}P^1 \times \{0\}$. But near $z = 0$, if we reparametrize the domain by $1/(nz)$, we see this converges to the sphere $\{0\} \times \mathbb{C}P^1$. In this case, our curve formed what is often called a bubble. More generally, a curve can split off many bubbles at a time. For an example of this, consider instead $u_n \colon \mathbb{C}P^1 \rightarrow \mathbb{C}P^1 \times \mathbb{C}P^1 \times \mathbb{C}P^1$ given by $z \mapsto (z,1/(nz),1/(n^2z))$, in which a new bubble forms at $\{0\} \times \{\infty\} \times \mathbb{C}P^1$ in addition to the one discussed above. More generally, a sequence of curves can limit to a curve with trees of bubbles sticking out.

Such bubble trees are called stable or nodal or cusp curves (or probably a lot of other things), depending upon how old your reference is and to whom you talk. The incredible theorem, which goes under the name of Gromov compactness, is that this is the only phenomenon which precludes a limit from existing. We state this vaguely as follows:

Theorem [Gromov ’85]: The moduli space of curves of energy bounded by some constant $E$ (modulo reparametrization of domain) can be compactified by adding in stable curves of total energy bounded by $E$.

Another generally important tool is that of the evaluation map. Suppose that we wish to study the moduli space $\mathcal{M}_g^*(M,A,J)$ of simple J-holomorphic maps $u : (\Sigma_g,j) \rightarrow (M,J)$ in the homology class $A \in H_2(M)$. Suppose $G = \text{Aut}(\Sigma_g,j)$ is the group of biholomorphisms of $(\Sigma_g,j)$. Then the group $G$ acts on $\mathcal{M}_g^*(M,A,J) \times \Sigma_g$ by $\phi \cdot (u,z) = (u \circ \phi^{-1},\phi(z))$. Notice then that the evaluation map $(u,z) \mapsto u(z)$ only depends on the orbit, and hence descends to a map $\text{ev} : \mathcal{M}_g^*(M,A,J) \times_G \Sigma_g \rightarrow M$. Proving enough properties of such an evaluation map sometimes allows us to compare the smooth topology of $\mathcal{M}_g^*(M,A,J)$ to that of $M$. There are other variants of this – sometimes we wish to evaluate at multiple points, or sometimes we consider J-holomorphic discs and want to evaluate along boundary points. And often the evaluation map extends to the compactified moduli spaces considered above.

Finally, we come to dimension 4, where curves might actually generically intersect each other. With respect to these intersections, there are two key results to highlight. The first is positivity of intersection (due to Gromov and McDuff), which states that if any two J-holomorphic curves intersect, then the algebraic intersection number at each intersection point is positive (and precisely equal to 1 at transverse intersections). This can be thought of as some sort of rudimentary version of a so-called adjunction inequality (due to McDuff), which states that if $u \colon (\Sigma,j) \rightarrow (M,J)$ is a simple J-holomorphic curve representing the class $A$ with geometric self-intersection number $\delta(u)$, then

$c_1(TV,J) \cdot A + 2\delta(u) \leq \chi(\Sigma) + A \cdot A$.

Further, when $u$ is immersed and with transverse self-intersections, this is an equality, yielding an adjunction formula.

### A first example – Fillable implies tight (in 3 dimensions)

On a first pass, I want to expand upon the example of fillability implying tightness in three dimensions which Roger Casals discussed in his introductory talk. Really, we prove the contrapositive – that an overtwisted contact manifold cannot be filled. For simplicity, we will consider exact fillings. This result is typically attributed to Gromov and Eliashberg, referencing Gromov’s ’85 paper as well as Eliashberg’s paper on filling by holomorphic discs from ’89. This is essentially the same proof in spirit, although we take a little bit of a cheat by considering exact fillings.

Firstly, recall that an overtwisted contact manifold $(M^3,\xi)$ is one such that there exists an embedding of a disk $\phi : D^2 \hookrightarrow M$, such that the so-called characteristic foliation $(d\phi)^{-1}\xi$ on $D^2$, which is actually a singular foliation, looks like the following image, with one singular point in the center and a closed leaf as boundary.

So now suppose $(M^3,\xi)$ has an exact filling $(W^4,\omega = d\lambda)$. We study the space of certain J-holomorphic disks with boundary on the overtwisted disk. The key is that a neighborhood of the overtwisted disk $D$ actually has a canonical neighborhood in $W$ up to symplectomorphism, and one can pick an almost complex structure $J$ to be in a standard form in this neighborhood. It turns out that with this standard choice, in a close enough neighborhood of the singular point $p$ in the interior of $D$, all somewhere injective J-holomorphic curves are precisely those living in a 1-parameter family, called the Bishop family, which radiate outwards from the singular point $p$.

Let us be a bit more precise, so that we can see this Bishop family explictly. Consider the standard 3-sphere $S^3 \subset \mathbb{C}^2$, with its standard contact structure given by the complex tangencies, i.e. $\xi = TS^3 \cap iTS^3$, with $i$ the standard complex structure on $\mathbb{C}^2$. Then consider the disk given by $z \mapsto (z, \sqrt{1-|z|^2})$. The characteristic foliation on this disk looks like the characteristic foliation near the center of the overtwisted disk, so a neighborhood of this disk in $D^4 \subset \mathbb{C}^2$ yields a model for a neighborhood of the center of the overtwisted disk. We may assume the almost complex structure in this neighborhood is just given by the standard one, $i$. Then the Bishop family is just the sequence of holomorphic disks given by $z \mapsto (sz,\sqrt{1-s^2})$ for $s$ a real constant near 0. That these are all of the somewhere injective disks is a relatively easy exercise in analysis. Namely, suppose we had such a disk of the form $z \mapsto (v_1(z),v_2(z))$. Then since boundary points are mapped to the overtwisted disk, $v_2(\partial D^2) \subset \mathbb{R}$. But each component of $v_2$ is harmonic, hence satisfies a maximum principle. Therefore, $v_2(D^2) \subset \mathbb{R}$. But by holomorphicity, $v_2$ cannot have real rank 1 and so must be constant. Hence, any disk in consideration must have $v_2$ is a real constant.

All of these disks live in $D^4 \subset \mathbb{C}^2$, but in particular in the slice where the second component $z_2$ is real, so we can draw this situation in $\mathbb{R}^3$ by forgetting the imaginary part of $z_2$. This is depicted in the following figure.

This Bishop family lives in some component of the moduli space of somewhere injective J-holomorphic disks with boundary on $D$. Perturbing $J$, one can assume this component is actually a smooth 1-dimensional manifold. We can compactify this moduli space by including stable maps, i.e. disks with bubbles, via Gromov compactness. On the Bishop family end, we see explicitly that the limit is just the constant disk at the point $p$. So there must be another stable curve at the other boundary of this moduli space. We prove no such other stable curve can exist.

Similar to how we proved that the only disks completely contained in a neighborhood of the singular point on the overtwisted disk must have been part of the Bishop family, one can use a maximum principle argument to conclude that every holomorphic disk entering this neighborhood must have been in the Bishop family. Alternatively, one can use a modified version of positivity of intersections to conclude that continuing the moduli space away from the Bishop family, these boundaries have to continue radiating outward. Either way, the moduli space has to stay away from the central singularity of the overtwisted disk $D$. But also, the boundary of a J-holomorphic disk cannot be tangent to $\xi$, and in particular cannot be tangent to $\partial D$. This is by a maximum principle which comes from analytic convexity properties of a filled contact manifold.

The only possible explanation is that this is a stable curve with some sphere bubble having formed in the interior of $(W,\lambda)$. But one checks that the relation $g(\cdot,\cdot) = \omega(\cdot,J\cdot)$ implies that for a $J$-holomorphic sphere $u : (S^2,j) \rightarrow (W,J)$, we have $\text{Area}_g(u) = \int_{S^2}u^*\omega$. This vanishes by Stokes’ Theorem since $\omega = d\lambda$ is exact, and so $u$ must be constant, and so there is no bubble. In other words, this cannot explain the other boundary point of the component of the moduli space containing the Bishop family, so this yields a contradiction.

### On McDuff’s The structure of ruled and rational symplectic 4-manifolds

Emily Maw’s talk from the workshop followed this paper by Dusa McDuff. In what follows, we shall consider triples $(V,C,\omega)$ such that $(V,\omega)$ is a smooth closed symplectic 4-manifold and $C$ is a rational curve, by which we mean a symplectically embedded $S^2$. We call a rational curve $C$ exceptional if $C \cdot C = -1$ with respect to the intersection product on $H_2(V)$ (with respect to its orientation coming from $\omega$). We say $(V,C,\omega)$ is minimal if $V \setminus C$ contains no exceptional curves. The main theorem is as follows:

Theorem [McDuff ’90]: If $(V,C,\omega)$ is minimal and $C \cdot C \geq 0$, then $(V,\omega)$ is symplectomorphic to either:

• $(\mathbb{C} P^2, \omega_{FS})$, in which case $C$ is either a complex line or a quadric (up to symplectomorphism).
• A symplectic $S^2$-bundle over a compact manifold $M$, in which case $C$ is either a fiber or a section (up to symplectomorphism).

Before describing the proof, which is the part involving J-holomorphic curve techniques, we apply this to strong fillings. We shall concern ourselves with fillings of the lens spaces $L(p,1)$ with their standard contact structures, where $p > 0$ is an integer. Let us first define this contact structure. Recall that the standard contact structure on $S^3$ is the one coming from complex tangencies by viewing $S^3 \subset \mathbb{C}^2$. Then the standard contact structure on $L(p,1)$ is the one given by the quotient $L(p,1) = S^3/(\mathbb{Z}/p\mathbb{Z})$ where the action of $1 \in \mathbb{Z}/p\mathbb{Z}$ given by $(z_1,z_2) \mapsto e^{2\pi i/p}(z_1,z_2)$ preserves the contact structure, so that it descends.

Theorem [McDuff ’90]: The lens spaces $L(p,1)$ all have minimal symplectic fillings $(Z,\omega)$, and when $p \neq 4$, these fillings are unique up to diffeomorphism, and further up to symplectomorphism upon fixing the cohomology class $[\omega]$. The space $L(4,1)$ has two nondiffeomorphic minimal fillings.

Proof (sketch): The complex line bundle $\mathcal{O}(p)$ over $S^2$ comes with a natural symplectic structure, and this forms a cap for $L(p,1)$. The zero section of $\mathcal{O}(p)$ is a rational curve of self intersection $p > 0$. McDuff’s explicit classification includes examples $(V,C)$ for any such given $p$, and $V \setminus C$ thus gives a minimal filling for $L(p,1)$. The remaining statements come from a more detailed analysis of the classification result.

Now, I will not go through all of the details of McDuff’s proof of the main theorem, but I will highlight where various J-holomorphic tools appear in the proof. Let me break up the proof into two big pieces.

Step 1: “Mega-Lemma” Consider $(V,C,\omega)$ minimal as above. There is a tame almost complex structure $J$ such that $[C]$ can be represented by a $J$-holomorphic stable curve of the form $S = S_1 \cup \cdots \cup S_m$, where:

• Each $A_i := [S_i]$ is $J$-indecomposable (meaning any stable curve representing $A_i$ must actually be a legitimate curve of one component)
• The almost complex structure $J$ is regular for all curves in the class $A_i$.
• The $S_i$ are distinct and embedded curves of self-intersection -1, 0, or 1, with at least one index for which $A_i \cdot A_i \geq 0$.

We didn’t prove this at the workshop, so I won’t discuss it in detail here. But this is a major reduction into cases. For example, if $m = 1$ and $S \cdot S = 1$, then it had already been shown that this implies that $V = \mathbb{C}P^2$. This bleeds into…

Step 2: Using the evaluation maps constructively

Let us discuss the proof of this last fact briefly. The idea is as follows. We consider the moduli space $\mathcal{M}^*(A,J)$ consisting of simple holomorphic spheres representing the class $A = [S]$. This comes with an evaluation map of the form

$\mathcal{M}^*(A,J) \times_{G} (S^2 \times S^2) \rightarrow V \times V$

where $G$ is the group of automorphisms of $S^2$. Both sides have dimension 8 and this evaluation map is injective away from the diagonal since $A \cdot A = 1$ and we have positivity of intersection. Therefore, this map has degree 1, and so any pair of distinct points on $V$ has a unique curve passing through it. This is enough to show $V = \mathbb{C}P^2$.

Let us do another case, but show that the adjunction formula also comes into play.

Proposition: Suppose $B$ is a simple homology class in $(V,\omega)$ (i.e. is not a multiple of another homology class) with $B \cdot B = 0$, and suppose $F$ is a rational embedded sphere representing $B$. Then there is a fibration $\pi \colon V \rightarrow M$ with symplectic fibers and such that $F$ is one of the fibers.

Proof (sketch): The idea is to consider the moduli space $\mathcal{M}^*_{0,1}(V,J,B)$ of rational embedded $J$-holomorphic curves with 1 marked point $p \in S^2$, and where $J$ is chosen to tame $\omega$ and such that $F$ is itself a $J$-holomorphic curve, and where we have quotiented by reparametrization of the domain. Then one can compute the dimension of this moduli space at a given curve $C$ in the appropriate way as

$d = \dim V + 2c_1(TV) \cdot [C] - 4$,

where the last -4 comes from quotienting by the subgroup of $PSL_2(\mathbb{C})$ fixing the marked point. Applying adjunction for the curve represented by $F$, so that $[C] \cdot [C] = 0$, yields $d = 4$. We also have an evaluation map

$\text{ev} : \mathcal{M}^*_{0,1}(V,J,B) \rightarrow V$

Since $B \cdot B = 0$, there is at most one $B$-curve through each point in $V$, so it follows that this evaluation map has degree at most 1, and hence equal to 1 by regularity. This yields the structure of a fibration $\pi : V \rightarrow M$ where the fibers are precisely the curves in our moduli space. Since the fibers are holomorphic, they are symplectic by the taming condition.

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## Kylerec Overview

Updates (June 11, 2017): Added link to other notes from Kylerec workshop, and fixed an error caught by Chris Wendl in the comments.

I’m very excited to be joining this blog!

This is the first of a series of posts about the content of the Kylerec workshop, held May 19-25 near Lake Tahoe, which focused on fillings of contact manifolds. Under the guidance of our mentors, Roger Casals, Steven Sivek, and Laura Starkston, we worked from the basic theory of fillings through some state-of-the-art results. Many of the basics have been discussed on this blog already in Laura Starkston’s posts from January 2013: Part 1 and Part 2 on Fillings of Contact Manifolds. For a more thorough introduction to types of filling and the differences between them, I suggest reading those posts (and the accompanying comments by Paolo Ghiginni and Chris Wendl). This post will remain self-contained anyway.

One can find notes that I took (except for three lectures, due to technical difficulties) at the Kylerec 2017 tab at this link. Other notes (with shorter load times, and including the ones I’m missing) will be posted on the Kylerec website soon are now posted on the Kylerec website here.

Comments and corrections are very welcome!

### Definitions

We quickly review the various notions of fillings of a contact manifold. We shall always assume that our manifolds are oriented and contact structures cooriented. As a starting point, one might be interested in smooth fillings of contact manifolds. It turns out that this problem is rather uninteresting. Every contact manifold of dimension $2n+1$ has a structure group which can be reduced to $U(n) \times 1$, but the complex bordism group is well known to satisfy $\Omega^U_{2n+1} = 0$. As a consequence, every contact manifold is smoothly fillable. We must therefore consider fillability questions which extend beyond the realm of complex bordism in order to discover interesting phenomena.

These notions are as follows, in (strictly!) increasing order of strength.

• We say a contact 3-manifold $(M^3,\xi)$ is weakly fillable if it is the smooth boundary of a symplectic manifold $(W^4,\omega)$ such that $\omega|_{\xi} > 0$. There is a generalization in higher dimensions due to Massot, Niederkrüger, and Wendl, but we omit it here. (Simply requiring that $\omega|_{\xi}$ is a positive symplectic form in the same conformal symplectic class as the natural one on $\xi$, i.e. is $d\alpha|_{\xi}$ up to scaling where $\alpha$ is a contact form for $\xi$, implies strong fillability in higher dimensions, by McDuff.)
• We say a contact manifold $(M^{2n-1},\xi)$ is strongly fillable if there is a weak filling $(W^{2n},\omega)$ such that one can find a Liouville vector field $V$ in a neighborhood of $M$, i.e. one such that $\mathcal{L}_V\omega = \omega$, such that $(\iota_V\omega)|_M$ gives a (properly cooriented) contact form for $\xi$.
• We say a contact manifold $(M^{2n-1},\xi)$ is exactly fillable if there is a strong filling such that the Liouville vector field $V$ can be extended to all of $(W,\omega)$. In other words, $M$ is the contact boundary of a Liouville domain $(W,\omega = d\alpha)$ where $\alpha = \iota_V\omega$.
• We say a contact manifold $(M^{2n-1},\xi)$ is Weinstein (or Stein) fillable if it is exactly fillable by some $(W,\omega = d\alpha)$, where $\alpha = \iota_V\omega$, such that there is also a Morse function $f$ on $W$ such that $V$ is gradient-like for $f$ and $M$ is a maximal regular level set. In other words, $M$ is the contact boundary of a Weinstein domain.

As a final remark, there is a notion of overtwistedness in contact manifolds. In 3-dimensions, this is characterized by the existence of an overtwisted disk. This was known to obstruct all types of fillings, due to Eliashberg and Gromov. In higher dimensions, overtwistedness was defined in a paper of Borman, Eliashberg, and Murphy, which was discussed on this blog by Laura Starkston and Roger Casals, starting with this post and concluding with this one. This definition implies the existence of a plastikstufe as defined by Niederkrüger, which had been already shown to obstruct fillings (strongly in the same paper, weakly in the paper by Massot, Niederkrüger, and Wendl). In other words, in any dimension, overtwistedness implies not fillable. A contact manifold which is not overtwisted is called tight, so equivalently, fillable implies tight, in all dimensions.

To summarize this section:

Tight < Weakly fillable < Strongly fillable < Exactly fillable < Weinstein fillable

where all of the inclusions turn out to be strict.

### Two Motivating Questions

Question 1: What tools do we have at each level of fillability?

The easiest type of filling to understand is that of the Weinstein filling, since Weinstein domains have an explicit surgery theory, which lends themselves to concrete geometric descriptions. Most notably, a Weinstein domain can be thought of as a symplectic Lefschetz fibration, which naturally has an open book decomposition on its boundary whose monodromy is a product of positive Dehn twists. Hence, Weinstein fillings and fillability can be studied through studying supporting open book decompositions for a contact manifold $(M,\xi)$.

Another rather powerful tool is the study of J-holomorphic curves. Let us provide a quick example: the proof that fillability of a contact 3-manifold implies tightness. One assumes by way of contradiction that an overtwisted contact 3-manifold has a filling. Then one considers a certain compact 1-dimensional moduli space of J-holomorphic curves with boundary on the overtwisted disk. One finds an explicit component of this moduli space which has one endpoint (a constant disk) but cannot have another endpoint, which contradicts the compactness of the moduli space. In higher dimensions, studying similar moduli spaces of J-holomorphic curves yields obstructions to fillings.

There are some other miscellaneous techniques. For example, Liouville domains have attached to them a symplectic homology, which provides another tool for the case of exact fillings. And in the case of 3-dimensional contact manifolds, one can also study the Seiberg-Witten invariants of a given filling.

Question 2: How can we study the topology of different fillings? Or tell when fillings are distinct even if they have the same homology?

J-holomorphic curves come with extra evaluation maps which allow one to study how the moduli space of curves compares to some underlying topology, e.g. of the filling or of the contact manifold. This is a technique which comes up many times in different contexts, and it sometimes allows us to produce maps between the filling or the contact manifold in question which do not exist for any other obvious reason.

Similarly, symplectic homology in its two flavors $SH$ and $SH^{+}$ fits into an exact triangle with Morse homology, and so one can understand the topology of a filling from its symplectic homology. One might be interested, for example, in studying fillings with $SH = 0$, in which case the homology of the filling is completely determined by $SH^{+}$. Alternatively, $SH$ can be used directly to distinguish fillings.

### Overview of Kylerec

More detailed posts about the contents of Kylerec will appear in future blog posts, but I will outline here precisely what was covered.

Day 1: After an overview talk, we spent the rest of the day studying the surgery theory of Weinstein manifolds, and began our study of the correspondence between Weinstein fillings, Lefschetz fibrations, and open book decompositions.

Day 2: We highlighted some results from this correspondence, and then turned towards an introduction to the theory of J-holomorphic curves, including applications of this theory to fillings via McDuff’s classification result as well as Wendl’s J-holomorphic foliations.

Day 3: On our short day, we first discussed some applications of J-holomorphic curves to high-dimensional fillings due to Barth, Geiges, and Zehmisch (for example reproving the result of Eliashberg, Floer, and McDuff that the standard sphere has a unique aspherical filling), and applied Wendl’s theorem (as discussed in Day 2) following a paper of Plamenevskaya and Van Horn-Morris to show that many contact structures on the lens spaces $L(p,1)$ have unique Weinstein fillings up to deformation equivalence.

Day 4: We discussed the Seiberg-Witten equations, how they appear in symplectic geometry, and how they are used by Lisca and Matic to distinguish contact structures on homology 3-spheres which are homotopic (through plane fields) but not isotopic (through contact structures). We also discussed how Calabi-Yau caps, as defined by Li, Mak, and Yasui, can be used to prove certain uniqueness results on fillings of unit cotangent bundles of surfaces, as in this paper by Sivek and Van Horn-Morris.

Day 5: On our last day, we focused mainly on symplectic homology (and its variants). In one talk, we performed computations which allowed us to distinguish contact structures on standard spheres (see Ustilovsky’s paper) and to compute the symplectic homology of fillings of certain Brieskorn spheres (see Uebele’s paper). We also discussed Lazarev’s generalization of M.-L.Yau’s theorem (that subcritical Weinstein fillings have isomorphic integral cohomology) to the flexible case.

Filed under Uncategorized

## Some open computational problems in link homology and contact geometry

I’m thrilled to join everyone at the best-named math blog.

I am just home from Combinatorial Link Homology Theories, Braids, and Contact Geometry at ICERM in Providence, Rhode Island.  The conference was aimed at students and non-experts with a focus on introducing open problems and computational techniques.  Videos of many of the talks are available at ICERM’s site.  (Look under “Programs and Workshops,” then “Summer 2014”.)

One of the highlights of the workshop was the ‘Computational Problem Session’ MC’d by John Baldwin with contributions from Rachel Roberts, Nathan Dunfield, Joanna Mangahas, John Etnyre, Sucharit Sarkar, and András Stipsicz.  Each spoke for a few minutes about open problems with a computational bent.

I’ve done my best to relate all the problems in order with references and some background.  Any errors are mine.  Corrections and additions are welcome!

### Rachel Roberts

Contact structures and foliations

Eliashberg and Thurston showed that a $C^2$ one-dimensional foliation of a three-manifold can be $C^0$-approximated by a contact structure (as long as it is not the product foliation on $S^1 \times S^2$).  Vogel showed that, with a few other restrictions, any two approximating contact structures lie in the same isotopy class.  In other words, there is a map $\Phi$ from $C^2$, taut, oriented foliations to contact structures modulo isotopy for any closed, oriented three-manifold.

Geography: What is the image of $\Phi$?

Botany: What do the fibers of $\Phi$ look like?

The image of $\Phi$ is known to be contained within the space of weakly symplectically fillable and universally tight contact structures.  Etnyre showed that if one removes “taut”, then $\Phi$ is surjective.  Etnyre and Baldwin showed that $\Phi$ doesn’t “see” universal tightness.

L-spaces and foliations

A priori the rank of the Heegaard Floer homology groups associated to a rational homology three-sphere Y are bounded by the first ordinary homology group: $\text{rank}(\hat{HF}(Y)) \geq |H_1(Y; \mathbb{Z})|$. An L-space is a rational homology three-sphere for which equality holds.

Conjecture: Y is an L-space if and only if it does not contain a taut, oriented, $C^2$ foliation.

Ozsváth and Szabó showed that L-spaces do not contain such foliations.  Kazez and Roberts proved that the theorem applies to a class of $C^0$ foliations and perhaps all $C^0$ foliations.  The classification of L-spaces is incomplete and we are led to the following:

Question: How can one prove the (non-)existence of such a foliation?

Existing methods are either ad hoc or difficult (e.g. show that the manifold does not act non-trivially on a simply-connected (but not necessarily Hausdorff!) one-manifold). Roberts suggested that Agol and Li’s algorithm for detecting “Reebless” foliations via laminar branched surfaces may be useful here, although the algorithm is currently impractical.

### Nathan Dunfield

What do random three-manifolds look like?

First of all, how does one pick a random three-manifold?  There are countably many compact three-manifolds (because there are countably many finite simplicial complexes, or because there are countably many rational surgeries on the countably many links in $S^3$, or because…) so there is no uniform probability distribution on the set of compact orientable three-manifolds.

To dodge this issue, we first consider random objects of bounded complexity, then study what happens as we relax the bound.  (A cute, more modest example: the probability that two random integers are relatively prime is $6/\pi^2$.1).  Fix a genus $g$ and write $G$ for the mapping class group of the oriented surface of genus $g$.  Pick some generators of $G$. Let $\phi$ be a random word of length $N$ in the chosen generators.   We can associate a unique closed, orientable three-manifold to $\phi$ by identifying the boundaries of two genus $g$ handlebodies via $\phi$.

Metaquestion: How is your favorite invariant distributed for random 3-manifolds of genus $g$?  How does it behave as $g \to \infty$?  Experiment! (Ditto for knots, links, and their invariants.)

Challenge: Show that your favorite conjecture about some class of three-manifolds or links holds with positive probability. For example:

Conjecture: a random three-manifold is not an $L$-space, has left-orderable fundamental group, admit a taut foliation, and admit a tight contact structure.

These methods can also be used to prove more traditional-sounding existence theorems. Perhaps you’d like to show that there is a three-manifold of every genus satisfying some condition. It suffices to show that a random three-manifold of fixed genus satisfies the condition with non-negative probability! For example,

Theorem: (Lubotzky-Maher-Wu, 2014): For any integers $k$ and $g$ with $g \geq 2$, there exist infinitely many closed hyperbolic three-manifolds which are integral homology spheres with Casson invariant $k$ and Heegaard genus $g$.

### Johanna Mangahas

What do generic mapping classes look like?

Here are two sensible ways to study random elements of bounded complexity in a finitely-generated group.

• Fix a generating set. Look at all words of length N or less in those generators and their inverses. (word ball)
• Fix a generating set and the associated Cayley graph. Look at all vertices within distance N of the identity. (Cayley ball)

A property of elements in a group is generic if a random element has the property with probability, so the meaning of “generic” differs with the meaning of “random.” For example, consider the group $G = \langle a, b \rangle \oplus \mathbb{Z}$ with generating set $\{(a,0), (b,0), (id,1)\}$.  The property “is zero in the second coordinate” is generic for the first notion but not the second.  So we are stuck/blessed with two different notions of genericity.

Recall that the mapping class group of a surface is the group of orientation-preserving homeomorphisms modulo isotopy. Thurston and Nielsen showed that a mapping class $\phi$ falls into one of three categories:

• Finite order: $\phi^n = id$ for some $n$.
• Reducible: $\phi$ fixes some finite set of simple closed curves.
• Pseudo-Anosov: there exists a transverse pair of measured foliations which $\phi$ stretches by $\lambda$ and $1/\lambda$.

The first two classes are easier to define, but the third is generic.

Theorem: (Rivin and Maher, 2006) Pseudo-Anosov mapping classes are generic in the first sense.

Question: Are pseudo-Anosov mapping classes generic in the second sense?

The braid group on n strands can be understood as the mapping class group of the disk with n punctures. But the braid group is not just a mapping class group; it admits an invariant left-order and a Garside structure. Tetsuya Ito gave a great minicourse on both of these structures!

Fast algorithms for the Nielsen-Thurston classification

Question: Is there a polynomial-time algorithm for computing the Thurston-Nielsen classification of a mapping class?

Matthieu Calvez has described an algorithm to classify braids in $O(\ell^2)$ where $\ell$ is the length of the candidate braid. The algorithm is not yet implementable because it relies on knowledge of a function $c(n)$ where $n$ is the index of the braid. These numbers come from a theorem of Masur and Minsky and are thus difficult to compute. These difficulties, as well as the power of the Garside structure and other algorithmic approaches, are described in Calvez’s linked paper.

Challenge: Implement Calvez’s algorithm, perhaps partially, without knowing $c(n)$.

Mark Bell is developing Flipper which implements a classification algorithm for mapping class groups of surfaces.

Question: How fast are such algorithms in practice?2

### John Etnyre

Contactomorphism and isotopy of unit cotangent bundles

For background on all matters symplectic and contact see Etnyre’s notes.

Let $M$ be a manifold of any (!) dimension.  The total space of the cotangent bundle $E = T^*M$ is naturally symplectic:  the cotangent bundle of $E$ supports the Liouville one-form $\lambda$ characterized by $\alpha^*(\lambda) = \alpha$ for any one-form $\alpha \in T^*M$; the pullback is along the canonical projection $T^* T^* \to T^*M$.  The form $d\lambda$ is symplectic on $T^*M$.

Inside the cotangent bundle is the unit cotangent bundle $S^*M = \{(p,v) \in T^*M : |v| = 1\}$. (This is not a vector bundle!) The form $d\lambda$ restricts to a contact structure on the $S^*M$.

Fact: If the manifolds $M$ and $N$ are diffeomorphic, then their unit cotangent bundles $S^*M$ and $S^*N$ are contactomorphic

Hard question: In which dimensions greater than two is the converse true?

This question is attributed to Arnol’d, perhaps incorrectly.  The converse is known to be true in dimensions one and to and also in the case that $M$ is the three-sphere (exercise!).

Tractable (?) question: Does contactomorphism type of unit cotangent bundles distinguish lens spaces from each other?

Also intriguing is the relative version of this construction. Let $K$ be an Legendrian embedded (or immersed with transverse self-intersections) submanifold of $M$. Define the unit cosphere bundle of $K$ to be $L_K = \{w \in T^*M : w(v) = 0, \forall v \in TK\}$. You can think of it as the boundary of the normal bundle to $K$. It is a Legendrian submanifold of the unit cotangent bundle $T^*M$.

Fact: If $K_1$ is Legendrian isotopic to $K_2$ then $L_{K_1}$ is Legendrian isotopic to $L_{K_2}$.

Relative question: Under what conditions is the converse true?

Etnyre noted that contact homology may be a useful tool here.  Lenny Ng’s “A Topological Introduction to Knot Contact Homology” has a nice introduction to this problem and the tools to potentially solve it.

### Sucharit Sarkar

How many Szabó spectral sequences are there, really?

Ozsváth and Szabó constructed a spectral sequence from the Khovanov homology of a link to the Heegaard Floer homology of the branched double cover of $S^3$ over that link. (There are more adjectives in the proper statement.) This relates two homology theories which are defined very differently.

Challenge: Construct an algorithm to compute the Ozsváth-Szabó spectral sequence.

Sarkar suggested that bordered Heegaard Floer homology may be useful here. Alternatively, one could study another spectral sequence, combinatorially defined by Szabó, which also seems to converge to the Heegaard Floer homology of the branched double cover.

Question: Is Szabó’s spectral sequence isomorphic to the Ozsváth-Szabó spectral sequence?

Again, the bordered theory may be useful here. Lipshitz, Ozsváth, and D. Thurston have constructed a bordered version of the Ozsváth-Szabó spectral sequence which agrees with the original under a pairing theorem.

If the answer is “yes” then Szabó’s spectral sequence should have more structure. This was the part of Sarkar’s research talk which was unfortunately scheduled after the problem session. I hope to return to it in a future post (!).

Question: Can Szabó’s spectral sequence be defined over a two-variable polynomial ring? Is there an action of the dihedral group $D_4$ on the spectral sequence?

### András Stipsicz

Knot Floer Smörgåsbord

Link Floer homology was spawned from Heegaard Floer homology but can also be defined combinatorially via grid diagrams. Lenny Ng explained this in the second part of his minicourse. However you define it, the theory assigns to a link $L$ a bigraded $\mathbb{Z}[U]$-module $HFK^-(L)$. From this group one can extract the numerical concordance invariant $\tau(L)$. Defining $HFK^-$ over $\mathbb{Q}[U]$ or $\mathbb{Z}/p\mathbb{Z}[U]$ one can define invariants $\tau_0$ and $\tau_p$.

Question: Are these invariants distinct from $\tau$?

Harder question: Does $HFK^-$ have $p$-torsion for some $p \in \mathbb{Z}$? (From a purely algebraic perspective, a “no” to the first question suggests a “no” to this one.)

Stipsicz noted that there are complexes of $\mathbb{Z}[U]$-modules for which the answer is yes, but those complexes are not known to be $HFK^-(L)$ of any link. Speaking of which,

“A shot in the dark:” Characterize those modules which appear as $HFK^-$.

In another direction, Stipsicz spoke earlier about a family of smooth concordance invariants $\Upsilon_t$. These were constructed from link Floer homology by Ozsváth, Stipsicz, and Szabó. Earlier, Hom constructed the smooth concordance invariant $\epsilon$. Both invariants can be used to show that the smooth concordance group contains a $\mathbb{Z}^\infty$ summand, but their fibers are not the same: Hom produced a knot which has $\Upsilon_t = 0$ for all t and $\epsilon \neq 0$.

Conversely: Is there a knot with $\epsilon = 0$ by $\Upsilon_t \neq 0$?

Stipsicz closed the session by waxing philosophical: “When I was a child we would get these problems like ‘Jane has 6 pigs and Joe has 4 pigs’ and I used to think these were stupid. But now I don’t think so. Sit down, ask, do calculations, answer. That’s somehow the method I advise. Do some calculations, or whatever.”

1. An analogous result holds for arbitrary number fields — I make no claims about the cuteness of such generalizations.
2. An old example: the simplex algorithm from linear programming runs in exponential time in the worst-case, but in

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## Overtwisted disks and filling holes

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

Overtwisted disks in higher dimensions and filling the holes

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

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

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

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

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

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

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

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

Proving the main lemma

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

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

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

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

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

which can be rewritten as

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

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

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

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

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

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

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

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## Contact Hamiltonians II

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

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

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

The circular model

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

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

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

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

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

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

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

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

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

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

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

Contactomorphisms and conjugating the Hamiltonian

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

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

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

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

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

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

Important types of contactomorphisms and their effects on the Hamiltonian

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

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

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

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

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

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

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

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

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

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## Contact Hamiltonians (Part I)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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