# Monthly Archives: January 2013

## Fillings of Contact Manifolds Part 2

Here is the overdue part 2 post on symplectic fillability. In part 1 , we gave several equivalent definitions of weak symplectic fillings, strong symplectic fillings, and Stein fillings. From these definitions we had the following inclusions:
$\{\text{Stein fillable}\} \subseteq \{\text{Strongly sympl. fill.}\} \subseteq \{\text{Weakly sympl. fill.}\}$
Now I want to summarize the results that show that each of these notions are distinct from each other, namely that each of these inclusions is strict, and also mention some conditions that can ensure some of these notions do coincide.

Historically, it was first shown that there are weakly symplectically fillable contact manifolds which are not strongly symplectically fillable by Eliashberg (the paper is called “Unique holomorphically fillable contact structure on the 3-torus”). The examples were contact structures on $T^3$. Identify $T^3=T^2\times S^1$, with coordinates $(x,y,\theta)\in (\mathbb{R}/\mathbb{Z})^3$. Then one can verify that for $n=1,2,3,\cdots$, $\alpha_n=\cos(n\theta)dx+\sin(n\theta)dy$ is a contact form ($\alpha_n\wedge d\alpha_n>0$). Then $\xi_n=\ker(\alpha_n)$ is a contact structure on $T^3$ for each $n=1,2,3\cdots$, and it was shown by Giroux and independently Kanda that these contact structures are not contactomorphic for different values of $n$, and all tight contact structures on $T^3$ are contactomorphic to one of these. The claim is that if $n>1$, $(T^3,\xi_n)$ is not strongly symplectically fillable, but all $(T^3,\xi_n)$ are weakly symplectically fillable.

First we will show that $(T^3,\xi_n)$ is weakly symplectically fillable. View $T^3$ as the boundary of $T^2\times D^2$. Put coordinates $(x,y)$ on the $T^2$ factor, and let $\theta$ parametrize the boundary of $D^2$. We can put a symplectic structure on $T^2\times D^2$ by taking the product of area forms on $T^2$ and $D^2$. We want to show that this is a weak symplectic filling of $(T^3,\xi_n)$. This is not immediately obvious, so instead we look at a family of contact structures, $\xi_n^{\varepsilon}$ which are contactomorphic to $\xi_n$, and show that $(T^2\times D^2,\omega =\omega_{T^2}\oplus \omega_{D^2})$ is a weak symplectic filling of $(T^3,\xi_n^{\varepsilon_0})$ for some $\varepsilon_0$. It is clear that the submanifolds $T^2\times \{p\}$ for $p\in \partial D^2$ are symplectic submanifolds of $T^2\times D^2$, so $\omega$ is positive on the tangent planes to these submanifolds. These tangent planes are described by $\{d\theta =0\}$ on $T^3$. Clearly these planes do not form a contact structure on $T^3$ since they are integrable, but $\omega$ will evaluate positively on a nearby contact structure. With this in mind, set
$\alpha_n^{\varepsilon} = (1-\varepsilon)d\theta +\varepsilon\alpha_n)$
Note that
$\alpha_n^{\varepsilon} = (1-\varepsilon)\varepsilon d\theta \wedge d\alpha_n +\varepsilon^2 \alpha_n\wedge d\alpha_n = \varepsilon^2\alpha_n\wedge d\alpha_n$
so $\xi_n^{\varepsilon}=\ker(\alpha_n^{\varepsilon})$ is a contact structure for any $\varepsilon>0$. When $\varepsilon=1$, $\xi_n^1=\xi_n$ so we have a homotopy through contact structures from $\xi_n$ to any $\xi_n^{\varepsilon}$ for $\varepsilon>0$. By Gray’s theorem there are diffeomorphisms $\phi^{\varepsilon}$ such that $(\phi^{\varepsilon})_*\xi_n^{\varepsilon}=\xi_n$, therefore each $\xi_n^{\varepsilon}$ is contactomorphic to $\xi_n$ for $\varepsilon>0$. When $\varepsilon$ is close to zero, $\xi_n^{\varepsilon}$ is close to $\ker(d\theta)$. Therefore, since $\omega|_{\xi}>0$ is an open condition, there exists some $\varepsilon_0>0$ such that $\omega|_{\xi_n^{\varepsilon_0}}>0$. Therefore $(T^2\times D^2, \omega)$ is a weak symplectic filling of $(T^3,\xi_n^{\varepsilon_0})$ which is contactomorphic to $(T^3, \xi_n)$. This shows that all $\xi_n$ are weakly symplectically fillable.

Eliashberg shows that for $n>1$, $(T^3, \xi_n)$ is not strongly symplectically fillable by contradiction. Recall that if it did have a strong symplectic filling, this filling could be glued symplectically to any symplectic manifold whose concave boundary is $(T^3,\xi_n)$. The idea is to find a symplectic manifold with concave boundary, such that if we could close off this boundary symplectically, we would obtain a symplectic manifold that cannot exist.

Consider $(\mathbb{R}^4, \omega_{std}=dx_1\wedge dy_1+dx_2\wedge dy_2)$. Take the product of circles $x_1^2+y_1^2=1, x_2^2+y_2^2=1$. This is a Lagrangian 2-torus inside $(\mathbb{R}^4,\omega_{std})$. Any Lagrangian submanifold, has a neighborhood symplectomorphic to a neighborhood of the zero section of its cotangent bundle. One can explicitly show that a disk bundle inside $T^*(T^2)$ has convex boundary inducing the contact structure $\xi_1$, using the radially outward pointing Liouville vector field. Therefore the complement of a neighborhood of this Lagrangian torus in $\mathbb{R}^4$ has concave boundary inducing the contact structure $\xi_1$. We can take an n-fold cover of the complement of this neighborhood in $\mathbb{R}^4$ so that on the $T^3$ boundary, the $S^1$ factor parameterized by $\theta$ is covered n times. This cover has a symplectic form given by the pull-back for which the $T^3$ boundary is still concave and the induced contact structure on the resulting boundary is $\xi_n$. This n-fold cover has $n$ ends, each symplectomorphic to the complement of a large compact set in $(\mathbb{R}^4,\omega_{std})$. If it were possible to cap off the concave boundary component with a convex symplectic filling of $(T^3,\xi_n)$, we would obtain a symplectic manifold without boundary, but with n standard ends. A theorem of Gromov states that this is impossible (if someone has a good explanation of the idea of this proof, that might make a good new post that I would appreciate). Thus we have reached a contradiction to the assumption that $(T^3,\xi_n)$ is strongly symplectically fillable for $n>1$.

While this example establishes the inequivalence of weak and strong symplectic fillability, the proof by contradiction and the reliance on a difficult theorem of Gromov which requires holomorphic curve techniques makes it difficult to see what the difference between weak and strong fillability would be in general. A generalization of this example was established by Ding and Geiges. They proved that a more general class of 2-torus bundles over the circle have contact structures which are weakly but not strongly symplectically fillable. Their proof that there are no strong fillings reduces the more general case to the original examples on $T^3$ using contact surgery. While this doesn’t make the underlying cause of non-fillability more clear, it does illuminate the fact that certain surgery operations on contact manifolds preserve different types of fillability and non-fillability.

In particular, Legendrian surgery (surgery along a Legendrian knot, with framing given by the contact framing with one additional negative twist) preserves weak symplectic fillability, strong symplectic fillability, and Stein fillability. The proofs that it preserves strong symplectic fillability and Stein fillability are due to Weinstein and Eliashberg respectively, who show that the convex symplectic or Stein structures can be extended over the corresponding handle attachments which provide cobordisms between the original manifold and the result of Legendrian surgery. The fact that Legendrian surgery preserves weak symplectic fillability is proven here in Theorem 2.3 by Etnyre and Honda, by showing that in a neighborhood of the Legendrian knot, the contact structure can be slightly perturbed so that the weak symplectic filling is a convex filling in a neighborhood of the knot. Then the surgery can be performed to preserve the strong fillability in a neighborhood of the knot, thus preserving the weak fillability of the entire contact manifold.

While we still do not seem to have a full understanding of when strong and weak fillability coincide and when they differ, there are certain situations where weak fillability implies strong fillability. Eliashberg proved the following proposition:

Proposition 4.1 in A few remarks about symplectic filling: Suppose that a symplectic manifold $(W,\omega)$ weakly fills a contact manifold $(V,\xi)$. Then if the form $\omega$ is exact near $\partial W=V$ then it can be modified into a symplectic form $\widetilde{\omega}$ such that $(W,\widetilde{\omega})$ is a strong symplectic filling of $(V,\xi)$.

The idea is essentially to consider the primitive $\eta$ for $\omega$ near the boundary and the contact form $\alpha$, and to interpolate between the primitives $t\alpha$ and $\eta$, so that $\widetilde{\omega} =d(t\alpha)$ near the boundary and $\widetilde{\omega} = d\eta$ a little further in from the boundary so it glues up to the original $\omega$ on the interior of the weak filling. The condition that $\omega$ be a weak filling ensures one can do this interpolation while maintaining the symplectic (non-degeneracy) condition.

A consequence of this proposition is that in order to find a 3-manifold which supports weakly fillable contact structures that are not strongly fillable, the 3-manifold must carry some nonzero second homology. In particular, an integer homology sphere which is weakly fillable is also strongly fillable. This fact comes into play in the next example.

While Eliashberg’s examples distinguishing weak and strong fillings were published in 1996, it took much longer to find examples of contact 3-manifolds which were strongly fillable but not Stein fillable. The technology needed to obstruct Stein fillability was the Heegaard Floer contact invariant. Ghiggini found the first examples in this paper, which were the Brieskorn homology spheres (with reversed orientation) $-\Sigma(2,3,6n+5)$ ($n$ even and $\geq 2$). These manifolds can be understood as 0-surgery on the positive trefoil knot in $S^3$, together with $-n-1$-surgery on its meridian (or alternatively as Seifert fibered spaces over $S^2$ with three singular fibers with coefficients 2,-3,$\frac{6n+5}{-n-1}$).

The 3-manifold obtained from $S^3$ by 0-surgery on the positive trefoil is actually a $T^2$ bundle over $S^1$, because the trefoil is a fibered knot of genus 1. Its monodromy is well understood, and in fact this is one of the 3-manifolds considered by Ding and Geiges, which have contact structures $\xi_n$ similar to those on $T^3$ (the contact structures twist n times in the direction of the monodromy). These contact structures are all weakly symplectically fillable (the filling is a Lefschetz fibration over a disk with tori as regular fibers, and the argument that this is a weak filling is similar to the argument that $T^2\times D^2$ is a weak filling of $T^3$ above).

So we have a weak symplectic filling of 0-surgery on the positive trefoil with any of the contact structures $\xi_n$, but we would like to also do $-n-1$ surgery on a meridian of this trefoil to obtain the Brieskorn spheres of interest. Ghiggini shows that this meridian can be realized as a Legendrian knot whose its contact framing twists $-n$ times around it, therefore performing $-n-1$ surgery on this meridian corresponds to Legendrian surgery. Since the above result said that Legendrian surgery preserves weak fillability, the resulting Brieskorn sphere with its corresponding contact structure $\eta_0$ is weakly symplectically fillable.

Furthermore, because the result of the Legendrian surgery gives an integer homology sphere, a weak symplectic filling can be perturbed into a strong symplectic filling, as mentioned above (since the restriction of the symplectic structure to the boundary is an exact form). Therefore we have strongly symplectically fillable contact structures on these Brieskorn spheres.

On the other hand, these contact manifolds $(-\Sigma(2,3,6n+5), \eta_0)$ are not Stein fillable. Ghiggini proves this by showing that the contact invariant is in the fixed point set of an involution on Heegaard Floer homology, and then studying generators of this fixed point set. Properties of these generators imply that all of them are sent to zero by a map induced by a Stein cobordism between $(S^3,\xi_{std})$ and $(-\Sigma(2,3,6n+5),\eta_0)$. Therefore the contact invariant of $\eta_0$, which is a linear combination of these generators is sent to zero by the map induced by a Stein cobordism, which is a contradiction. (One of the first theorems proved about the Heegaard Floer contact invariant is that it is sent to the generator of $\widehat{HF}(S^3)$ by any map induced by a Stein cobordism from $(S^3,\xi_{std})$, and thus the contact invariant of a Stein fillable contact structure is non-zero.)

More specifically, the generators of the fixed point set can be written as $c(\eta_i)+c(\eta_{-i})$ where $\eta_i$ is the contact structure obtained by Legendrian surgery on $S^3$ as in the picture below, where the Legendrian unknot has $(n-i)/2$ cusps on one side and $(n+i)/2$ cusps on the other side.

Note that each of these contact structures is Stein fillable (since it is obtained from Legendrian surgery), so their contact invariants are non-vanishing. By showing that $\overline{\eta_i}=\eta_{-i}$, Ghiggini proved that when $n$ is even, the fixed point set of the involution on Heegaard Floer homology is generated by $c(\eta_i)+c(\eta_{-i})$. Furthermore, any map induced by a 4-manifold cobordism to $S^3$ sends $c(\eta_i)$ and $c(\eta_{-i})$ to the same element, so in $\mathbb{Z}/2$ coefficients, the image of each generator of the fixed point set is zero.

In conclusion, these examples distinguish these three notions of fillability, but I think there is still a lot left to understand about when such examples can arise. Given an arbitrary 3-manifold, there is not usually much one can say about whether it supports contact structures that are strongly symplectically fillable but not Stein fillable, and unless it is a homology sphere it is hard to tell whether there could be weakly but not strongly symplectically fillable contact strucutres. The Heegaard Floer contact invariant has been a useful probe to obstruct Stein (and with twisted coefficients, symplectic) fillability, so a related question is to understand geometric conditions under which the contact invariant vanishes for tight contact structures (see Honda, Kazez, and Matic’s work for some situations when this occurs). Because there was a significant period of time where it was unknown whether weak, strong, and Stein fillability were equivalent notions, it seems that examples where these notions diverge are rare, but I don’t think we really have any idea how prevalent these examples can be, or what geometric properties can allow this phenomenon to occur.

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## Morse Homotopy and A-infinity; Part 2

Last time, I describe the first step in Fukaya’s proof that Morse homology has an $A-\infty$ structure, defined in terms of gradient flow trees.  This time I’ll describe how the higher relations ($m_3, m_4, \dots$) arise from Morse theory.

To describe $m_3$, we need to look for all trees connecting $x,y,z$ to some $w$.  At this level there are some complications.  First, not all trees we need to consider will be isomorphic (at least if we fix a cyclic ordering of the exterior vertices).  For the $m_2$, every tree was a Y.  But at higher levels, we can have nonisomorphic trees, such as the following:

So we need to make sure we look for all possible trees.  As we get to the higher $A – \infty$ maps the space of all possible trees gets complicated.

Secondly, we need to start keeping track of the length of interior edges.  Each parametrizes a flow line, so we need to know exactly how long the partial flow line is that we want to parametrize.  This hasn’t been a problem because up to now, every flow line we considered had at least one noncompact end, because one end was asymptotic to a critical point, so its length was infinite.

We can solve both these problems by realizing that we can form a moduli space of italics(metric trees), an approach originally due to Stasheff.  Luckily, if we are careful, the moduli space is just some affine space $\mathbb{R}^k$ for some dimension $k$.

Embed the tree in the unit disk with the exterior vertices (the 1-valent vertices which map to critical points) cyclically ordered along the boundary.  Furthermore, assume that no vertex has valence 2, as this corresponds to a broken flow line, which we will consider separately later on.  The [exterior] edges will be those connected to the boundary and [interior] edges will be any other edge.  Assign a positive real number to each interior edge (the exterior edges are assumed to have length $\infty$).  Then the lengths of the interior edges, of which there can be at most $k-3$, where $k$ is the number of exterior vertices, identify parametrize the moduli space and identify it with $\mathbb{R}^{k-3}$.  Call this space $\mathcal{T}_k$

For example, suppose $k = 4$, which is the relevant space for the $m_3$.  The tree on the left corresponds to $-m \in \mathbb{R}$ and the tree on the right corresponds to $m \in \mathbb{R}$.

The space of metric trees is not compact but it can also be compactified, in the sense that as the length of the interior edge goes off to $\pm \infty$, the tree breaks into two metric trees.  For $k = 4$, this is just the union of a two 3-leaf Y trees.

For higher $k$, it will break into two trees with $j$ and $k-j+2$ exterior vertices, respectively.  Again, we see the principle that a moduli space can be compactified using the product of lower-dimensional moduli spaces of the same type of object.

Choose four Morse functions $f_1,f_2,f_3,f_4$ such that their differences $f_i-f_j$ are collectively generic.  Let $\widetilde{\mathcal{M}}(x,y,z;w)$ denote the moduli of metric trees with 4 exterior vertices parametrizing flow lines of the difference functions $f_i-f_j$ in the following way:  Each tree can be thought of as embedding in the unit disk and thus separates the disk into 4 regions.  Cyclically label each region with a function $f_i$.  Then an edge of a tree parametrizes a flow line of $f_i-f_j$ if it separates the regions labeled by $f_i$ and $f_j$.  The trees can be oriented so  that every interior vertex has exactly one outgoing edge and exactly one exterior vertex has an incoming edge.  Assume that the oriented flows respect this orientation on edges.

The $m_3$ map is defined as follows.

$m_3: C(f_1,f_2) \otimes C(f_2,f_3) \otimes C(f_3,f_4) \rightarrow C(f_1,f_4)$
$m_3(x,y,x) = \sum_{w|[index]} |\widetilde{\mathcal{M}}(x,y,z;w)| w$

In other words, count all rigid trees connecting $x,y,z$ to $w$.

We’d now like to establish the $A-\infty$ relation
$m_3(d(x),y,z) + m_3(x, d(y),z) + m_3(x,y,d(z)) + m_2(m_2(x,y),z) + m_2(x,m_2(y,z)) + d(m_3(x,y,z))= 0$

As with the $m_2$, each term here describes one way a 1-dimensional tree could degenerate.  The first three correspond to an incoming flow line breaking:

The last term corresponds to the outgoing flow line breaking:

And the terms involving $m_2$ correspond to an interior flow line breaking:

Again, the relation follows because each possible combination of broken trees can be glued to form the boundary of a 1-dimensional moduli space.  Moreover, each 1-dimensional tree must break/degenerate in one of the above ways.  Either an exterior edge breaks, which corresponds to the familiar compactification of Morse flow lines, or an interior breaks, which corresponds to the compactification of the moduli of metric trees.

Fukaya notes that this $m_3 relation$ descends to Massey products on cohomology but I won’t go into that here.

The higher $A-\infty$ maps arise in the same way.  The map $m_k$ is defined by counting rigid trees with $k$ incoming and 1 outgoing edge.  The $A-\infty$ relation follows from the fact that each 1-dimensional tree breaks into two rigid trees.

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## Morse Homotopy and A-infinity; Part 1

I found it useful to thoroughly go through and understand why the differential in Legendrian Contact Homology squares to 0 and so did some others, so I’m going to continue and discuss where  higher A-infinity relations come from.  A-infinity things seem intimidating because of all the little details necessary to define them.  I hope it helps to have a geometric interpretation.  It also helps to open your mind to universal algebra and operads, but I won’t go into that here.

Recall the definition of an A-infinity algebra.  Let $A$ be a graded $k$-vector space.  Then there exists an infinite family of maps $\{m_k \}$

$m_k: A^{\otimes k} \rightarrow A$

that satisfy the A-infinity relations, which are a nightmare to state.  I’ll describe them pictorially as follows.  Each $m_k$ can be represented by a box with k strands entering on the top and 1 strand leaving on the bottom, i.e. there are k inputs and 1 output.  (Some people use trees to visualize this as well).

Then, we some over all possible ways to combine two of these maps into a map that has $l$ inputs and 1 output:

and require that this sum is 0, (if for simplicity we ignore signs and assume $k = \mathbb{F}_2$).  So, we sum over all $i,j$ such that $i + j -1 = l$ and we can move the $m_i$ map left and right so that it takes as inputs any adjacent $i$-tuple.

The simplest condition is that $m_1 \circ m_1 = 0$, since there is only one way to combine two maps in a manner that takes 1 input and yields 1 output.

As a consequence, this means that the $m_1$ map is a differential.

The new two conditions are $m_2 ( d(x), y) + m_2 ( x,d(y)) + d( m_2 (x,y)) = 0$.

and $m_2(m_2(x,y),z) + m_2(x,m_2(y,z)) + d(m_3(x,y,x)) + m_3(d(x),y,z) + m_3(x,d(y),z) + m_3(x,y,d(z)) = 0$

Using Morse theory, Fukaya proved that the cohomology ring of a real analytic manifold is actually an A-infinity algebra.  In the case of proving $d^2=0$, we used the fact that each term in $d^2$ corresponds to a union of two flowlines, called a broken flow.

Fukaya studied gradient flow trees.  Let T be the tree in figure 1, with 4 vertices and 3 edges in a Y pattern.  Now, pick 3 Morse functions $f_1,f_2,f_3$ such that their difference functions $f_i - f_j$ are generic.  This means that the (un)stable manifolds of all difference functions intersect transversely.  To define the $m_2$ map, we are going to look at the moduli space of flow trees corresponding to Y.  That is, each edge will parametrize a flow line of some $f_i-f_j$.  Let $\widetilde{\mathcal{M}}(x,y;z)$ denote the moduli space of gradient flow trees from $x,y$ to $z$ for $x$ a critical point of $f_1 - f_2$, $y$ a critical point of $f_2 - f_3$ and $z$ a critical point of $f_1 - f_3$,:

One way to get our hands on this space of trees is to take $W_u(x) \cap W_u(y) \cap W_s(z)$.  For each point in this space, there is a unique triple of oriented flow lines connecting it to $x,y$ and $z$.  Together, these form a tree of the form we are looking for.  The dimension of the moduli space is $I(x) + I(y) - I(z) - n$, which can easily be checked because this is assumed to be a transverse intersection.

Then the $m_2$ map can be defined as

$m_2:C(f_1 - f_2) \otimes C(f_2 - f_3) \rightarrow C(f_1 - f_3)$
$m_2(x,y) = \sum_{z: I(z) = I(x) + I(y) - n} |\widetilde{\mathcal{M}}(x,y;z)| z$

We need to show that this satisfies the A-$\infty$ relation
$d(m_2(x,y)) + m_2(d(x),y) + m_2(x,d(y)) = 0$

In Morse theory, all moduli spaces, of flow lines and trees and of all dimensions, can be compactified.  We just need to know how trees/flows degenerate as they head off to the open end.  But as always, the principle here is that it can only degenerate into a union of trees you already know about.

For instance, take the Y.  Since everything is finite dimensional, any open end of the moduli space must come from an open end of the moduli of the individual flow lines.  So degeneration for trees looks exactly like degeneration for flow lines.  Any of the three edges could break into pieces.

So, now we have a union of two trees, a segment and another Y.  But the segment is just a flow line from $x$ to $w$, and so algebraically shows up in the differential.  And the Y is another tree corresponding to an $m_2$ map.

Let’s work the other way.  The each of term of $m_2(d(x),y)$ corresponds to a pair of a rigid flow line from $x$ to some $w$ and a rigid tree connecting $w,y$ to $z$.  Similarly, each term of $m_2(y,d(y))$ corresponds to a pair of a rigid flow line from $y$ to some $w$ and a rigid tree connecting $x,w$ to $z$.  Finally, each term of $d(m_2(x,y))$ corresponds to a pair of a rigid tree connecting $x,y$ to some $w$ and a rigid flow line from $w$ to $z$.

As with the differential, these pairs can be glued together into a 1-dimensional tree and each pair corresponds to the endpoint of some 1-dimensional component of the moduli space of trees from $x,y$ to $z$.  Since this 1-dimensional space can be compactified in such a way that if we look at the boundary of all 1-dimensional moduli of trees, we get pairs as in the figure above.

Thus, $d(m_2(x,y)) + m_2(d(x),y) + m_2(x,d(y)) = 0$ and we know that our chain complex is at least an $A_2$-algebra.

Now, we have been using 3 different Morse functions and then three other difference functions.  These critical points live in different chain complexes.  This is ok.  We already know that the chain homotopy type of the Morse complex is independent of the Morse function.  So the algebraic structure of the $d$ map is the same in all three chain complexes.  So it’s OK to think of this relation as living on a single chain complex

Fukaya also shows how this $m_2$ map descends to the familiar cup product on cohomology.  First, recall the chain homotopy equivalence between Morse homology and singular homology.  In one direction, the descending manifold of an index $i$ critical point, which is topologically a disk, is a singular $i$-chain, a continuous map of an $i$-dimensional simplex into the manifold $M$.  In the other direction, given a singular $i$-chain, its image will (generically) intersect the stable manifolds of an index $i$ critical points in a finite number of points.  Summing over all such intersections gives the algebraic image of the chain in the Morse complex.  In more suggestive terms, one can think of flowing the image of the singular chain down by the descending gradient flow.  The singular chain will “hang” on some of the index $i$ critical points and summing over these points gives the corresponding algebraic chain in the Morse complex.

In Morse cohomology (N.B the differential increases the grading, so we look at rigid, ascending flow lines), the cohomology complex is still generated by the critical points and their Poincare duals can be represented by the singular chains given by their ascending manifolds.  Thus, take two basis elements $x \in H^i(M,\partial_{\text{Morse}})$, $y \in H^j(M,\partial_{\text{Morse}})$.  These have Poincare duals $X,Y$ which are disks of dimension $n-i,n-j$.  The Poincare dual of $x \wedge y$ is $X \cap Y$, which has dimension $n-i-j$.  To determine which element this is in Morse cohomology, we flow this intersection upward by the gradient vector field and see which index $i+j$ critical points it gets caught by.  In terms of gradient trajectories, we look for all the gradient flow lines from $X \cap Y$ to some critical point $z$ of index $i + j$.  There are a finite number of such flow lines, each of which corresponds to a unique tree connecting $x,y$ to $z$.  In the chain complex, this is exactly the $m_2$ map and so passing to cohomology we recover the cup product from $m_2$.

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## Fillings of Contact Manifolds

There are various notions of fillability of a contact 3-manifold, $(Y,\xi)$: weak symplectic filliability, strong symplectic fillability, and Stein fillability. When these notions were initially studied, it was not yet known whether they all coincided. It is fairly straightforward to show that a Stein filling is a strong symplectic filling, and a strong symplectic filling is a weak symplectic filling so

$\{\text{Stein fillable}\} \subseteq \{\text{Strongly sympl. fill.}\} \subseteq \{\text{Weakly sympl. fill.}\}$
However, it took more time to find examples to show that each of these inclusions is strict. Now it seems useful to mention the examples used to prove this in one place. While looking up these examples I noticed there are a few equivalent definitions of each of these notions, which are interchangeably used throughout the literature, so I’ll start by discussing those equivalences. That much seems to fill up an entire post so I’ll keep this post about definitions and equivalences (which make the above inclusions easy to conclude), and next post I’ll summarize known examples that show the inclusions are strict. So this is just an intro post.

A weak symplectic filling of $(Y,\xi)$ is a symplectic manifold $(W,\omega)$ with boundary $\partial W=Y$ such that $\omega|_{\xi}>0$. This definition is only considered for fillings of contact 3-manifolds, since that is when the symplectic 2-form can be positive on the 2-dimensional contact planes. However, the other notions of fillability hold in higher dimensions.

A strong symplectic filling of $(Y,\xi)$ is a symplectic manifold $(W,\omega)$ satisfying certain conditions that can be defined in a few equivalent ways. The first way is to require that there exists a Liouville vector field V defined near the boundary of $W$ ($\mathcal{L}_V\omega = \omega$) which is transverse to $\partial W=Y$ and a contact form for $\xi$ is given by $\alpha = i^*(\iota_V\omega)$.

One can verify any such an $\alpha$ is a contact form (i.e. $\alpha\wedge d\alpha >0$) since

$\alpha \wedge d\alpha = i^*(\iota_V\omega)\wedge d(i^*(\iota_V\omega)) = i^*(\iota_V\omega) \wedge i^*(\mathcal{L}_V\omega) = i^*(\iota_V\omega \wedge \omega)$

($i$ denotes the inclusion map of $\partial W$ into $W$ and $\iota_V$ denotes contraction by the vector field $V$. We have used $\mathcal{L}_V = \iota_V\circ d +d\circ \iota_V$ and $d\omega=0$.) The right most expression is a volume form by nondegeneracy of $\omega$ and the fact that $V$ is everywhere transverse to $\partial W$.

Another way to define a strong filling independently of the Liouville vector field, is to ask that $\omega$ be exact in a neighborhood of $\partial W$ and it has a primitive $\eta$ defined on this neighborhood of the boundary such that $\xi = \ker(i^*\eta)$ and $d\eta|_{\xi}>0$ (i.e. $i^*\eta$ is a contact form for $\xi$).

The first definition of a strong filling implies the second since $\eta=\iota_V\omega$ is a primitive for $\omega$ near the boundary, and the other conditions are satsified since $i^*(\iota_V\omega)$ is a contact form for $\xi$. Conversely, since $\omega$ is nondegenerate, contraction into $\omega$ gives an isomorphism between 1-forms and vector fields so there is a unique vector field $V$ such that $\iota_V\omega = \eta$. Therefore $\mathcal{L}_V\omega = d(\iota_V\omega) = d\eta = \omega$ so $V$ is a Liouville vector field which induces the correct contact form on the boundary because of its relation to $\eta$. If it were not transverse to $\partial W$ at some point then $i^*(\eta\wedge d\eta) = i^*(\iota_V\omega \wedge \omega)$ could not be positive at that point, so $i^*\eta$ would not be a contact form.

Note there is a shorter way of stating the second definition which is again equivalent. We must only require that $\omega$ restricts to an exact form on $\partial W$ such that $i^*\omega = d\alpha$ where $\alpha$ is a contact form for $\xi$. This is because $\alpha$ can be extended to a primitive for $\omega$ in a small neighborhood of $\partial W$ (note such a primitive exists because if $\omega$ restricts to something trivial in cohomology in $\partial W$, then it will represent something trivial in cohomology when restricted to a tubular neighborhood diffeomorphic to $\partial W\times [0,\varepsilon)$).

A Stein filling of a contact manifold, is a complex manifold $(X,J)$ for which there is a strictly plurisubharmonic function $\phi: X\to [0,c]$, meaning the form $\omega_{\phi}:= -dd^{\mathbb{C}}\phi$ is a symplectic form compatible with $J$ (where $d^{\mathbb{C}}\phi = d\phi \circ J$) or equivalently $\omega_\phi(v,Jv)>0$ for all $v\neq 0$. Cieliebak and Eliashberg call these functions $J$-convex in their new book and give a thorough discussion of various aspects of Stein and Weinstein structures. The contact structure induced on the boundary of a Stein filling is the hyperplane field of complex tangencies to the boundary, namely $T(\partial X)\cap J(T(\partial X))$.

A Weinstein filling of a contact manifold is an exact symplectic manifold $(W,\omega)$ such that $\omega=d\eta$ and there is a Liouville vector field $V$ on all of $W$ for which $\iota_V\omega = \eta$ and $V$ is gradient-like for some Morse function on $W$ for which $\partial W$ is the maximal level set. It is clear from this definition that Weinstein fillable implies strongly symplectically fillable. It takes a lot more work to show that Weinstein fillability is equivalent to Stein fillability (see Cieliebak and Eliashberg’s book for a thorough explanation).

There are a few other characterizations of Stein fillings which are often more useful for constructions. Eliashberg proved that in dimensions greater than four, a $2n$-manifold can be given a Stein structure if and only if it has a Morse function with critical points of index at most $n$. Attaching the index $n$ handles is the trickiest part and in dimension four, additional hypotheses are needed. Specifically the 2-handles must be attached along Legendrian knots in the boundary of the 0- and 1-handles with framings on the attaching circles exactly one less than the contact framing. A standard way to keep track of the contact framing in $\partial(\natural_k S^1\times B^3) = \#_k S^1\times S^2$ in a Kirby diagram is given by Gompf here .

Finally if one thinks of contact structures in terms of open book decompositions, a contact manifold is Stein fillable if and only if it is supported by an open book decomposition whose monodromy can be factored into positive Dehn twists. A Stein filling is given by the corresponding Lefschetz fibration over a disk. This correspondence is in this paper by Loi and Piergallini and this paper by Akbulut and Ozbagci.

Discussion of how these notions differ coming soon.