2 More Simons Center Workshops

In addition to the workshop on Symplectic Homology in October, there are two more workshops this academic year at the Simons Center.

Symplectic and Low-Dimensional Topologies in Interaction

• December 3-7, 2012
• Organizers: Peter Ozsvath, Yasha Eliashberg and Robert Lipshitz
• Application deadline: November 15, 2012

Low-Dimensional Topology

• May 20-24, 2013
• Organizers: Peter Ozsvath and Dylan Thurston
• Application deadline: April 1, 2013.

You can apply to either/both here.

Simons Center workshops are open to the public and you only need to apply if you want funding support.

TGTC at Rice in November

The Texas Geometry and Topology Conference will be held at Rice University in Houston on November 9-11. Conference info is here:

TGTC at Rice

There’s a list of speakers up, and they intend to provide some lodging and travel support for grad students and postdocs. It’s easy travel if you live in Texas or Louisiana.

Lattice Homology for knots

Here is the picture relevant to Allison’s post on Lattice Homology (see post and comment below). Moving from left to right is just an isotopy of the diagram, and moving down a line corresponds to blowing down a -1 framed unknot (the effect of this on the diagram is to add a +1 twist in the strands going through the -1 unknot, and increase the framings by the square of their linking number with the -1 unknot).

Lattice Homology and Knots

I’m excited to be joining the blog. I’ve been consumed mainly by watching talks and meeting people at the Stanford conference, so this first post will consist only of my notes from Andras Stipsicz‘s talk on lattice homology. The content of the talk represents joint work of Stipsicz with Ozsváth and Szabó. I don’t have a lot of experience with lattice homology, so if I’ve missed or misunderstood something, please contribute your corrections in the comments section.

The talk began with a review of the standard Heegaard Floer package. Knot Floer homology was recently mentioned on the blog when Laura discussed Zoltan Szabó’s research talk at Budapest. So, if you’d like a refresher on notation, please check out that post (but feel free to skip the bordered part if you’re in a rush).

With an powerful invariant like Heegaard Floer (or knot Heggaard Floer), a lot of people are interested in how to compute it. Stipsicz gave an overview of some of the pertinent questions about computation and existing technology.

Computational aspects:

• How to compute the invariant in a simple way?
• How to prove invariance in a simple way? (Simple here means without infinite dimensional analysis or having to count pseduoholomorphic curves.)
• Can you find an effective way of computing invariants? (Effective depends on your personal opinion of an efficient algorithm.)

Existing technology:

• Sarkar-Wang (2006) – “Nice” diagrams allow you to compute HF-hat.
• Manolescu-Ozsváth-Sarkar and Manolescu-Ozsváth-Szabó -Thurston – Grid diagrams allow for a combinatorial descriptions of the knot and link invariants.
• Ozsváth-Sarkar-Szabó (2007) – Verification of independence with special nice diagrams for HF.
• New algorithms coming from Lipshitz-Ozsváth-Thurston’s bordered Floer homology.

The first two references mentioned are examples of simple ways to compute the invariants. The cost of theoretical simplicity, though, is a miserably inefficient algorithm. For example, the algorithm that computes HFK from a grid diagram has order of complexity somewhere in the neighborhood of . Yikes! The bordered techniques are computationally better, but require some rather difficult algebra.

So keeping these motivating issues in mind, let’s see what Stipsicz said about lattice homology and how it relates to Heegaard Floer.

Lattice homology
Suppose that is a tree with integer weights associated to its vertices . We consider the plumbing construction . This means that is the compact oriented 4-manifold obtained by plumbing bundles over with Euler numbers given by the weights and plumbing instructions indicated by the graph. The three dimensional boundary of the resulting manifold is our object of interest.

The construction that follows is due to Nèmethi in 2008:

Define to be mod 2 for all , i.e. the set of characteristic elements in the second integral cohomology group of .

Let denote the module freely generated over by pairs , where is a characteristic covector and is a subset of vertices in .
The differential of this complex is given by:

Those exponents and are integers which are defined with an auxiliary function . If , then . That second term represents a self-intersection in . Let:

Also note that since specifies a spin-c structure on , we associate to the spin-c structure .

OK. That sets up the lattice homology, more or less. I don’t really understand what the differential is actually doing, but its apparent benefit is that it is defined in a combinatorial manner (i.e. no counts of holomorphic curves). If anyone would like to really understand this, it would probably be best to check out the papers by Nèmethi and Ozsváth, Stipsicz, and Szabó.

Rational graphs
My notes became somewhat illegible at this point in the lecture, so I’ve attempted to vet this section with some information from this paper. Let’s assume that is a negative definite plumbing graph. We’d like to get a handle on when is rational, and as it turns out, there is a combinatorial algorithm that does this.

Algorithm. Let , and compute all of the products . If:

• any of the products is greater than or equal to 2, stop. is not rational.
• all of the products are less than or equal to 0, stop. is rational.
• any product , then set and repeat.

(I’m being a bit sloppy here with the notation; stands for both a vertex and the second homology class in the plumbed manifold corresponding to that vertex.) This algorithm produces a series of vectors , and terminates in finite time. The product, by the way, is a dot product computed in the intersection matrix of the four-manifold. I believe was a suggested example of a rational graph on which to perform the algorithm.

Additionally, we say that is of type-k if there exists vertices such that sufficiently decreasing the weights makes rational.

So why do we care about types of rational graphs? The following theorem of Nèmethi in 2008:
Theorem. Suppose is negative definite plumbing tree. Then,

• is a three-manifold invariant.
• is rational if and only if for all spin-c structures over .
• If is of type-1 this implies .

Moreover, there exists a surgery exact sequence for lattice homology, a result also obtained independently by Josh Greene. The theorem leads to the conjecture that for any plumbing tree . To this end, there is a theorem of Ozsváth, Stipsicz, and Szabó that tells us there is a spectral sequence on converging to . If is of type-2, the spectral sequence collapses and the homology groups are isomorphic.

Refinement to knots

Are you still hanging in there? Great! Onward to knots.

Now, we let denote a plumbing graph with a distinguished, unweighted vertex . Define the lattice chain complex on . The vertex induces a filtration on the lattice homology complex. For a generator of the lattice complex, the filtration level is given by a formula that is essentially the same as the auxiliary function above, except it use a specific extension of and to . The filtered complex is the subject of the main theorem:

Theorem. Suppose the tree with distinguished vertex is given, that negative definite, and that is negative definite* for . Then determines the lattice chain complex

(*I took to mean that we’ve put a framing of on the vertex , large enough to ensure that is an L-space.)

You might be asking yourself where does the knot fit into this picture? Well, when you do the plumbing, the unknot living at the distinguished vertex becomes knotted. That’s the knot. It’s also important to note that while knot Floer homology is defined for all knots, the construction described above is not. It works for connected sums of iterated torus knots, but certainly not all knots.

I do not have very detailed notes about the proof of the theorem, but it seemed to me to be a consequence of a complicated and formal similarity of homological algebra. In Heegaard Floer, is a doubly filtered complex. The filtrations can be used to chop up the complex into various subcomplexes, and then these these subcomplexes, the maps relating them, and their mapping cones are used to describe the Heegard Floer of surgeries along . Here is a paper that details this construction.

Apparently, lattice homology also enjoys such a structure. The ‘infinity’ lattice complex is doubly filtered by the action of and an extension of . It, too, can be chopped up into subcomplexes, the mapping cones of which are quasi-isomorphic to lattice complexes of .

Relating the Heegaard Floer and lattice complexes takes more work. Unfortunately, there isn’t much I am able to say about this. So, let me just say that lattice homology is really interesting and I’m looking forward to learning more about it.

I’ll sign-off by reporting the big result in Ozsváth, Stipsicz, and Szabó‘s paper.
Theorem. If is a plumbing graph with a distinguished vertex such that the components of are all rational, then the filtered chain complex of in lattice homology is filtered chain homotopy equivalent to the filtered complex for in Heegaard Floer.

…and it’s wonderful corollary:
Corollary. The lattice homology of a knot in is equal to its knot Floer homology.

Workshop at Simons Center in October

There is a graduate workshop at the Simons Center this coming October on symplectic and contact topology (focusing on symplectic cohomology). I’m not sure if I’ll be able to attend but I thought I’d share the information. Here is the program description and here is the registration/funding application.  The application deadline for funding is September 1.

Concordance

There are a lot of research talks this week at Stanford, so I’ll write about a few of them in the next few days.

Matt Hedden and Jen Hom both gave talks about knot concordance, so I’ll talk about what both of them said here. One goal in studying knot concordance is to try to understand something about 4-manifolds. Finding knots with certain concordance properties can be used to show the existence of exotic ‘s. While knot concordance may not be able to tell us everything about smooth 4-manifolds, it can lead to interesting results. The goal then is to learn as much as possible about the size and structure of the concordance group.

Concordance definitions:

Two knots are smoothly concordant if they are the boundary of a proper embedding . This is an equivalence relation, and the space of knots modulo concordance equivalence forms an abelian group, (addition is connected sum, the unknot is the identity, and negation is the mirror image with reversed orientation). An equivalent definition for and to be concordant is that is smoothly slice (bounds a disk in the 4-ball). We can also consider the topological concordance equivalence, which instead of requiring the above mapping of the cylinder to be smooth, we only require that it is continuous and extends to a continuous map on a tubular neighborhood. Topological concordance is a weaker equivalence than smooth concordance. Let denote the group of knots up to smooth concordance, and denote the group of knots up to topological concordance.

There is good motivation to focus on topologically slice knots (topologically concordant to the unknot), up to the equivalence relation given by smooth concordance. Denote this space by . Showing that is non-trivial implies the existence of exotic ‘s.

History of results regarding $latex\mathcal{C}_{TS}$:

Casson showed in 82 using Donaldson’s diagonalization theorem that there are knots with trivial Alexander polynomial, which are not smoothly slice. Freedman showed around the same time that knots with trivial Alexander polynomial are necessarily topologically slice. Combining these results shows that is non-trivial.

In 95 Endo showed that this group is big, namely there is a copy of . Livingston, and Manolescu-Owens more recently showed that contains a direct summand of distinguished using the and invariants from Heegaard Floer and Khovanov homologies, which are both concordance invariants.

Satellite operations and concordance:

One operation on knots that works compatibly with concordance is forming satellites. Given a pattern knot P embedded in a solid torus, one obtains a map from knots to knots sending a knot K, to its satellite with that pattern P(K) (embed the pattern solid torus in a neighborhood of the knot K). A particularly useful example is the Whitehead double whose pattern is:

It is a consequence of the Skein relation that the Whitehead double of a knot is trivial (resolve the crossing at the clasp), so by Freedman’s result above, all Whitehead doubles are topologically slice, and thus represent elements of . One may want to understand how many elements in can be represented by Whitehead doubles. Hedden and Kirk prove that there there is a . The knots are Whitehead doubles of torus knots, and the proof uses SO(3) gauge theory to show the knots are not smoothly concordant.

Another infinite family of independent topologically slice knots formed via satellite operations, which is independent of both Endo’s examples and the above examples, are (p,1) cables of the Whitehead double of the right-handed trefoil. Jen Hom distinguishes these in the concordance group using her concordance invariants from Heegaard Floer homology. She defines an invariant which can be computed from the chain complex through an algebraic process involving the invariant (which is also a concordance invariant), or more geometrically by looking at the triviality or nontriviality of cobordism maps on induced by large integer surgeries on the knot in . This is a concordance invariant of the knot, and it can be used to create a new equivalence relation on knots through their Heegaard Floer chain complexes. The idea is as follows. We can associate to a knot K, the complex and to its inverse in the concordance class, we get . The analog of addition in the knot concordance group is the tensor product of chain complexes by the following Kunneth formula: . If a knot K is smoothly slice, then . With the concordance equivalence relation, we started with a monoid of knots under connected sum, and mod out by the concordance equivalence relation to get a group. Similarly, the complexes form a monoid under tensor product and we obtain a group if we mod out by the equivalence relation . The resulting group has additional useful structure: a total ordering, a notion of much greater than, and a filtration. These structures can be used to show linear independence of knots in the concordance group.

Based on how is defined, there is definitely a relation to the invariant, but is a more powerful invariant. It turns out that the exact relation is related to the satellite operation. Jen proved that if and only if for every pattern P. Furthermore the satellite map descends to a well defined map on the group of knots up to equivalence.

More pieces of the topologically slice concordance group:

Since we have a lot of examples of independent concordance classes with trivial Alexander polynomial obtained by Whitehead doubles, one may ask whether the smallest subgroup generated by knots with trivial Alexander polynomial gives all topologically slice concordance classes, i.e. does ? The answer to this question is strongly no. Hedden and Livingston prove that there is an infinitely generated free abelian subgroup in the quotient: .

So now we know that there are lots of knot concordance classes with trivial Alexander polynomial, lots with nontrivial Alexander polynomial, but each of these constructions produce concordance classes of infinite order. We can also ask about torsion in the knot concordance group. The easiest kind of torsion to understand is 2-torsion. In this case so , i.e. the knot is isotopic to its reverse mirror image. Such knots have been studied for awhile, and are called amphichiral. There are lots of such knots, so it is reasonable to expect some of the topologically slice knots to have this property.

Indeed there are lots of amphichiral knots which are smoothly concordance independent, but also topologically slice. The theorem is due to Hedden, S.G. Kim, and Livingston:
.

The knots is this family are constructed by starting with an amphichiral knot that is not topologically slice, and then performing satellite operations with different knots, and taking connected sums to obtain topologically slice knots that are amphichiral. Next one needs to show that these knots are not smoothly slice, and that they represent independent concordance classes. Here you need Heegaard Floer homology. The obstruction to K being slice comes from the d-invariant. The d-invariant, is keeping track of the highest grading of the generator of the nontorsion elements in the (minus) Heegaard Floer homology of a homology 3-sphere. To use this to obstruct sliceness, first one notices that if K were smoothly slice, then its branched double cover would be a homology 3-sphere and it would bound a homology 4-ball (this comes from looking at the double branched cover of the 4-ball branched over the slice disk). This implies that the d-invariant for all spin-c structures on Y which are a restriction of a spin-c structure on Q. Since we are trying to obtain a contradiction, and show that such a Q does not exist, we don’t know exactly which spin-c structures will show up on the boundary. However such spin-c structures will satisfy certain properties (e.g. they form a subgroup of a certain size in . One can explicitly compute the d-invariants for the candidate 2-torsion knots, and look for possible spin-c subgroups satisfying the necessary conditions, and rule out the possibility that vanishes for all required .

Concordance genus:

The Seifert genus and 4-ball genus of a knot by definition satisfy the inequality . We can define an intermediate genus, called the concordance genus . One may ask what the possible size of the gaps can be in the inequality . The gap between concordance genus and slice genus can be made arbitrarily large by connect summing nontrivial slice knots, but it is more difficult to get gaps between and . The first result regarding this problem is due to Nakanishi who found knots with concordance genus arbitrarily larger than 4-ball genus. Livingston improved this result and found algebraically slice (though not topologically slice knots) with but arbitrarily large. Jen improved this result even further with her equivalence, finding examples of topologically slice knots that all have , but for each .

The moral seems to be, invariants defined through Heegaard Floer homology have been very useful in mapping out more of the concordance group, and providing lots of example of topologically slice, concordance-independent knots.

Stanford Holomorphic Curves: Pseudoholomorphic Quilts

Katrin Wehrheim gave a minicourse on pseudoholomorphic quilts. She explained the motivation behind these objects, described some of the analytic aspects involved, and discussed how to construct invariants of 3 and 4-manifolds. The notes she was using may show up on her website soon, but I couldn’t find them there yet. I hadn’t seen much of this before so I was convinced that these things are interesting, but don’t understand the details yet (and there are many analytic details). The point of this post is to pass on why this theory seems interesting.

The goal is to get invariants of 4-manifolds by associating symplectic constructions to data describing a generic function from the 4-manifold to a surface.

Quilts:

A generic function has a 1-dimensional submanifold of critical points, which map onto a 1-dimensional (almost) submanifold of Q, with finitely many cusps and crossings. This divides the base surface Q into “patches” (connected components of the complement of the critical values), divided by “seams” (the critical values except the cusps and crossings), plus some ends (discrete points at the crossings and cusps).

Over the patches the function is a fibration by a surface. As one passes over a seam from one patch to another the surface may change as a vanishing cycle on the surface collapses at the seam. At the ends, two different vanishing cycle singularities come together.

Pseudoholomorphic quilts take these marked base surfaces and associate a symplectic manifold to each patch , and a Lagrangian correspondence to each seam between patches . On the ends where multiple seams come together, one associates Floer homology classes. If you choose all of these things correctly, you can extract an invariant out of the 4-manifold. While it seems natural to me to build an invariant for a 4-manifold by gluing together simple pieces, it is not obvious where these symplectic manifolds and Lagrangian correspondences come from. It turns out the motivation is by looking at Donaldson theory in limiting cases.

Motivation from Donaldson theory:

Donaldson invariants are constructed by counting (modulo gauge) anti-self dual connections on a 4-manifold X. If you look at what this means locally over square patches, you can write out the connection as a Lie algebra valued 1-form in terms of the coordinates on the patch and some coordinates on the surface fibers, and then see what constraints you get from the anti-self dual equation. The motivation for quilts comes from looking at the “large structure limit” of these constraints. Vary the metric on the product by a parameter that shrinks down the fiber surface: and look at the new anti-self dual equations with this metric as . It turns out that the solution space of connections in this limit is a symplectic space (there may be some singularities in general, but I think there are analytic assumptions one can make to avoid this). This is the motivation to associate a symplectic manifold to each patch.

As you go towards the edge of a patch, the effect in the 4-manifold is to attach a handle, so a stitch transverse to the seam has preimage which is a cobordism Y from the surface on one side to the surface on the other. Above a neighborhood of the seam is YxI. Analyzing the solutions to the anti-self dual equations near a seam put the additional condition that these connections must extend over the new handle. Looking at the connections on both sides of the seam that extend correctly, cuts out a Lagrangian in the product of the symplectic solution spaces associated to each patch.

Once you know what these symplectic manifolds and Lagrangian correspondences associated to patches and seams are you can extract Donaldson type invariants by purely working in the symplectic world. The idea with quilted invariants is just to forget the differential equations that gave you the symplectic manifolds, and generalize to any quilted surface marked by symplectic manifolds and Lagrangian correspondences that satisfy similar axioms to the limiting Donaldson solutions spaces.

Invariants of 4-manifolds:

Start with a quilted surface, namely a surface (which can have boundary components and also infinite ends), with a symplectic manifold label for each patch and Lagrangian correspondences associated to each seam. If we consider some infinite ends as incoming and some as outgoing, the quilted invariant defines a relative invariant mapping the quilted Floer homology associated to the Lagrangians going towards the incoming ends to the quilted Floer homology (a quilted version similar to Lagrangian intersection Floer homology) associated to the Lagrangians going towards the outgoing ends. This is an invariant of the quilted surface up to isotopy and the Lagrangians up to Hamiltonian isotopy. It is unchanged under homotopy through quilted surfaces, and satisfies a composition gluing theorem (if you glue together the outgoing ends of one to the incoming ends of the next the relative invariant of the glued up surface is the composition of the relative invariants of the unglued surfaces). It is also unchanged under adding a trivial seam through a patch labeled by the diagonal Lagrangian correspondence. There is another move you can do called strip shrinking where a strip bounded by two seams can be collapsed to a single seam labeled by the composition of the Lagrangians from the two seams. This operation commutes with the relative invariant provided you make some additional assumptions on your symplectic manifolds. To get a 4-manifold invariant, choose Floer classes to associate to the ends, so that the quilt invariant is invariant under Cerf moves. Checking invariance involves strip shrinking and other allowable moves.

I think that point is that this provides a pretty general framework to construct invariants that are modeled off of Donaldson invariants, so they have the potential to be good at detecting exotics, but we don’t know yet what kinds of applications these invariants will have to 4-manifold topology.

Stanford Holomorphic Curves continued

This is my summary of Yasha Eliashberg’s minicourse. The topic is contact and symplectic homology invariants of Weinstein manifolds, and the effects of handle attachments. I think the theories are too complicated to explain all the details in 3 talks, so if you really want to compute these things, look at the paper by Bourgeois, Ekholm, and Eliashberg here: http://arxiv.org/pdf/0911.0026v4.pdf

Part I: Weinstein manifolds

The goal is to compute invariants of Weinstein manifolds. A Weinstein manifold consists of the following data: where
– is an exact symplectic manifold with primitive ()
– Z is a complete vector field such that (note Z and determine the primitive )
– is an exhausting function (proper and bounded below)
– Z is gradient-like for

Note that and is equivalent to . We say in this case that Z is a Liouville vector field. Also note, X is necessarily noncompact since it is an exact symplectic manifold (Stokes theorem).

We will restrict to Weinstein manifolds of finite type, meaning has finitely many critical points. This means that there is some regular value c, such that all critical values are less than c. In this case, has a conical end, . Therefore it is frequently interesting to just look at , such things are called Weinstein domains. One can also consider Weinstein cobordisms: for regular values .

For any regular level set of the Weinstein structure function , the restriction of is a contact form: must be positive on a basis for since the span of Z is the kernel of , and Z is transverse to the regular level sets.

There is some question of whether you can generally perturb the Weinstein structure to ensure is a Morse function, but in the case where it is Morse, you can consider the stable manifold of a critical point. Such stable manifolds are isotropic for both and (i.e. the restriction of the primitive to the stable manifold is zero). This implies that the indices of all critical points of are at most half the dimension of X. The intersection of the level sets with the stable manifolds are isotropic in the contact sense.

A question we would like to understand is how does the Weinstein manifold change as we pass through critical points of . Topologically we are doing a handle attachment, and the geometric information we get is that the attaching sphere will be isotropic in the contact sense, and the core of the handle will be isotropic in the symplectic sense. The framing on a Legendrian attaching sphere (for middle dimensional handles) is completely determined when the contact structure is co-oriented.

We can understand Weinstein manifolds and their invariants piece by piece for each handle addition. To use this method it is useful to establish a lemma that allows one to reorder the critical points, without meaningfully changing the Weinstein structure. One main reason for this is that all of the interesting symplectic geometry happens when attaching the n handles in a 2n dimensional manifold, so we want to attach all the k<n handles first and then focus on what happens when attaching the n-handles. Here, noninteresting means that h-cobordism principles hold, so information about the tangent bundle determines the symplectic manifold. There are also certain Legendrian attaching spheres (called loose) for n-handles for which the Weinstein manifold is determined homotopically (information about the tangent bundle determines the symplectic manifold).
Part II: Invariants of Weinstein manifolds: Symplectic and Contact homology

Set up: Let be a Weinstein manifold of finite type, be a Weinstein domain such that X is union a conical end, and let . We will build up some chain complexes that are invariants of the Weinstein manifold X.

Reeb orbits: the first basic generators of the chain complex will be Reeb chords and periodic orbits of the Reeb vector field associated to on Y. It is convenient for the following to assume . For each periodic Reeb orbit there is a Conley-Zehnder index, there is also a well-defined index associated to Reeb chords. There is some distinction between “good” Reeb orbits and “bad” ones. A bad Reeb orbit is an even multiple of another Reeb orbit, such that the parity of the Conley-Zehnder index can change as one takes different multiples of the simple Reeb orbit. Orbits that are not bad are good. These bad orbits introduce some additional problems one needs to compensate for to get a well defined theory.

Linearized contact homology : While this invariant is meant to give information about the contact manifold Y, it depends on the choice of a Weinstein filling X. The chain complex for linearized contact homology is generated (over some field of characteristic 0) by the set of all good Reeb orbits.

The differential for contact homology: The coefficient is a count of holomorphic “cylinders” in the symplectization of Y connecting to . The quotations are because we must count curves that have additional ends on the negative end besides just , when these additional ends can be capped of by holomorphic disks in the filling , of Y. These curves are called cylinders anchored in the symplectic filling.

Note that this is where the dependence on X for contact homology comes from. Counting these anchored cylinders is necessary to get the differential to square to zero since these appear as ends of the moduli space.

Reduced Symplectic Homology, : While contact homology ignores the bad vertices, we can obtain additional information by including the bad vertices and compensating for their bad behavior in the differential. To do this we take two direct sum copies of the module generated by all Reeb orbits. We indicate orbits are in the first copy by putting a check over them, and in the second copy by putting a hat over them. Additionally, the elements in the second copy have grading shifted by 1. , where and .

The differential for plus symplectic homology: Since there are two direct summands in the chain complex , we write the differential in block matrix form:
.
Here are the differentials defined previously for CH by counting anchored cylinders. sends good hat orbits to zero, and for a bad orbit (this has something to do with the parity issues that bad orbits have). For each Reeb orbit, we will keep track of a marked point (so we have more data about the parametrization). The map counts holomorphic cylinders where the marks on the two Reeb orbits are aligned in the product, and also broken cylinders where the marked points show up in a chosen cyclic order as in the below picture.

Note that both of these are counts of a moduli space one codimension higher than the moduli space of unrestricted cylinders. I’m assuming all of these maps defined via counting curves are just summing over the 0-dimensional moduli spaces (or 1-dimensional moduli spaces mod an R-action).

Full Symplectic Homology, : For this theory, we combine the symplectic homology chain complex which keeps track of Reeb orbits and holomorphic cylinders connecting them, with the Morse chain complex for , which keeps track of the handle attachments. The chain complex is given by , where is the Morse chain complex generated by critical points of , whose differential counts gradient flowlines between critical points.

The differential for symplectic homology can again be defined by a block matrix in terms of the two summands:

where here is defined by and where counts the number of holomorphic disks bounding whose center marked point maps into the stable manifold of p.

Note that since the differential for SH is lower-triangular, there is a short exact sequence of the complexes: which induces an exact triangle on the level of homology: .

For , there is not quite an exact triangle between since the differential is not triangular, so neither inclusions of or into are chain maps. However since the map $d_M$ is zero on good orbits, we get an exact triangle when all Reeb orbits are good.

Part III: Computing Contact and Symplectic Homology

We want to compute these invariants via handlebody decompositions. Therefore we would like to know what the effect of attaching handle has on the Contact/Symplectic homology. The effects of adding subcritical handles (index <n) are basically trivial (the symplectic homology vanishes for Weinstein manifolds built from handles of index <n). Therefore we look at what happens to these homology invariants when attaching n-handles.

In the Weinstein manifold, the attaching spheres of the n-handles will be Legendrian submanifolds of the level set Y. There is an algebraic construction that captures the contribution of the n-handle to contact/symplectic homology in terms of the Legendrian attaching sphere. The construction is essentially a differential polynomial algebra generated by the Reeb chords, with some extra formal variables added in. The differential is computed by correctly counting holomorphic disks with boundary along the Legendrian except at finitely many punctures which map to the Reeb chords. The details are probably more accurately described in the paper by Bourgeois, Ekholm, and Eliashberg, but the point is that there is a clear prescription on how to compute this once you understand the Reeb chords and some holomorphic disk counts of the Legendrian attaching sphere.

Now suppose where h is an n-handle attached along a Legendrian . The theorem is that we get the complex for X, from the complex for and , and the differential is given in block matrix form by
.
Here the map is zero on the hat part of SH. On the check part it counts holomorphic punctured disks, which send the puncture in the center asymptotically to the Reeb orbit , and maps the boundary of the disk to the Legendrian, except along finitely many points which map to Reeb chords showing up as a term in . (There is also a particular ray from to the boundary, which is kept track of through the formal structure of .) Finally on the Morse summand of the SH complex, counts some holomorphic disks that intersect the stable manifold of a critical point.

The payoff here is that there is an exact triangle relating . In the case that has only handles of index <n, its symplectic homology vanishes, so we get an isomorphism . There are some additional techniques to obtain the computations of that involve looking at the corresponding algebra for the boundary of the cocore instead of the attaching sphere, and looking at some linearizations using augmentations handed to you by the Lagrangian filling of the Legendrian. Eliashberg thinks that such invariants are reasonable to compute, though that probably assumes one knows a lot about counting various kinds of holomorphic curves.

Stanford Holomorphic Curves and Low-dimensional topology

Over the last week I’ve been at Stanford for a workshop. The speakers and abstracts are here http://www.math.umn.edu/~akhmedov/Stanford2012.html. There were 6 minicourses, so I’ll post on some of them over the next few days.

One of the minicourses that finished up yesterday was given by Akbulut on studying 4-manifolds via handlebody decompositions. There is a long set of notes covering a lot more than was discussed in the 3 lectures, that may eventually turn into a book: http://www.math.msu.edu/~akbulut/papers/akbulut.lec.pdf. I’ll just summarize some of the things that were discussed in the lectures that I found interesting.

1. Suppose you have 2 handlebody decompositions for 4-manifolds X, X’ with diffeomorphic boundaries, and you think the 4-manifolds may be diffeomorphic. Looking for the right sequence of Kirby moves from one diagram to the other can be an intractable problem, so here is another way to try to extend the diffeomorphism from the boundary to the interior. Consider the cocores of each of the 2-handles in X, and look at their circle boundaries in . See where those circles are sent under the diffeomorphism of . Check if the image of those circles bound disks in X’. If they do we can extend the diffeomorphism on the boundary over a neighborhood of these cocores. Then we are left with just extending a diffeomorphisms of over . In the case that n=0, we are trying to extend a diffeomorphism of to itself over . The diffeomorphism probably extends, since if it doesn’t you get a counterexample to the smooth Poincare conjecture in dimension 4, so either way you should be happy.

2. The guiding principle during these talks was essentially that you can answer most questions about a 4-manifold by drawing the correct handlebody diagram. In Akbulut’s notes he describes a bunch of ways to get a handlebody diagram from various descriptions of 4-manifolds that I’ve seen before in the literature (e.g. Gompf-Stipsicz 4-manifolds and Kirby Calculus), but he also discussed some techniques for producing diagrams he has been using more recently that I hadn’t seen before.

One such technique is the “cylinder method” which allows one to take two 4-manifolds with diffeomorphic boundary, and glue them together without having to turn one of the manifolds upside-down. Turning a 4-manifold upside-down can get pretty complicated and it is hard to recognize the two pieces you started with when everything is glued up. This cylinder method allows you to keep the original handlebody diagrams for the two 4-manifolds with diffeomorphic boundary, and just adds on more handles to build a mapping cylinder (diffeomorphic to one boundary cross an interval) connecting the two diffeomorphic boundary components.

Conceptually it is a very natural idea, but there are some things to keep track of to draw the new handles correctly. Suppose X and Y are 4-manifolds with boundaries M and N respectively, and is a diffeomorphism. We glue the boundary of X to the boundary of Y handle by handle. First to glue the 0-handle of X to the 0-handle of Y requires an additional 1-handle connecting the two 0-handles. Since a standard diagram assumes a unique 0-handle in the background, we can cancel the new 1-handle with one of the 0-handles. To glue a 1-handle of X to a 1-handle of Y, we need to add a 2-handle that passes through each 1-handle. This would be straightforward if the diffeomorphism identifying the boundaries were simply the identity map, but in general the diffeomorphism may move the 1-handles around. To keep track of this Akbulut drops down “ropes” from some point up above the boundaries where the ropes are looped around the cores of the 1-handles that show up on the boundary. Then apply the diffeomorphism f and track how the ropes get tangled up. Now draw the attaching circles for the new 2-handles so they hook through the core of the 1-handle of X, both strands go straight up and then together follow the path through the tangle described by the ropes after applying f, and finally hook around the core of the 1-handle of Y to close up the circle. Akbulut has a good picture describing this in the notes on page 29. To connect 2- and 3-handles requires adding 3- and 4-handles but the attaching maps of 3- and 4-handles are canonically determined for a closed 4-manifold.

Akbulut said he used this cylinder method as part of producing useful pictures of the Akhmedov-Park manifolds, which are exotic copies of and . Such diagrams apparently made it easy to read off the fundamental group of the manifolds, to confirm that they were simply connected and the constructions were actually the exotic manifolds they were suspected of being.

3. Here is another interesting application of drawing pictures: the construction of infinitely many homeomorphic but non-diffeomorphic Stein fillings of a contact 3-manifold. Take the elliptic surface E(1) which is a torus fibration over with 12 singular fibers, and do a p-log transform on one regular fiber and a q-log transform on another parallel regular fiber. This gives a 4-manifold which is an exotic copy of E(1). Akbulut constructs diagrams for E(1) where you can see the regular torus fibers explicitly and performs the log transforms then notices that much of the diagram for is unaffected by changing the values of p and q. Find a domain which contains all the handles that change with p and q. Then the boundary of the domain is constant since the complement of this domain you chose is independent of choice of p and q. If the domain you chose was Stein, you get infinitely many Stein fillings of a 3-manifold. There are only finitely many contact structures on a 3-manifold which are Stein fillable, so there is some contact structure with infinitely many fillings. With some extra work Akbulut and Yasui can show that these fillings obtained by varying p and q are all homeomorphic but non-diffeomorphic.