Monthly Archives: August 2012

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.

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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.

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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).

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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:

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 O(n^2!). 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 G is a tree with integer weights m_i associated to its vertices v\in V(G). We consider the plumbing construction G \rightsquigarrow X_G. This means that X_G is the compact oriented 4-manifold obtained by plumbing D^2 bundles over S^2 with Euler numbers given by the weights m_i and plumbing instructions indicated by the graph. The three dimensional boundary Y_G = \partial X_G of the resulting manifold is our object of interest.

The construction that follows is due to Nèmethi in 2008:

Define Char(G) to be \{K:V\rightarrow \mathbb{Z} | K(v)=m_v mod 2 for all v\} , i.e. the set of characteristic elements in the second integral cohomology group of X_G.

Let \mathbb{CF}^-(G) denote the \mathbb{Z}_2[U] module freely generated over \mathbb{Z}_2[U] by pairs [K, E], where K is a characteristic covector and E is a subset of vertices in V.
The differential of this complex is given by:

\partial^-[K,E] = \Sigma_{v\in E} U^{a_v[K,E]} [K,E-v] + U^{b_v[K,E]} [K+2v^*,E-v]

Those exponents a_v and b_v are integers which are defined with an auxiliary function \mathcal{I}. If I\subset E, then \mathcal{I}[K,I] = \frac{1}{2}(\sum_{u\in I}K(u) + (\sum_{u\in I}u)^2 ). That second term represents a self-intersection in X_G. Let:

A_v[K,E] = \min\{ \mathcal{I} | I\subset E-v\}
B_v[K,E] = \min\{ \mathcal{I} | v\in I\subset E\}
a_v = A_v- \min \{A_v, B_v\}
b_v = B_v - \min \{ A_v, B_v \}

Also note that since K specifies a spin-c structure on X_G, we associate to K the spin-c structure \mathfrak{s}_K \in Spin^c(Y_G).

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 G is a negative definite plumbing graph. We’d like to get a handle on when G is rational, and as it turns out, there is a combinatorial algorithm that does this.

Algorithm. Let K_1 = \Sigma_{v\in V} v, and compute all of the products K_1\cdot v. If:

  • any of the products is greater than or equal to 2, stop. G is not rational.
  • all of the products are less than or equal to 0, stop. G is rational.
  • any product K_1\cdot v = 1, then set K_2 = K_1+ v and repeat.

(I’m being a bit sloppy here with the notation; v 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 K_1, K_2, \cdots, 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 E_8 was a suggested example of a rational graph on which to perform the algorithm.

Additionally, we say that G is of type-k if there exists vertices v_1, \cdots, v_k such that sufficiently decreasing the weights m_{v_1}, \cdots m{v_k} makes G' rational.

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

  • H_*( \mathbb{CF}^-(G, \mathfrak{s}), \partial^-) =\mathbb{HF}^-(G, \mathfrak{s}) is a three-manifold invariant.
  • G is rational if and only if \mathbb{HF}^-(G, \mathfrak{s}) \cong \mathbb{Z}_2[U] for all spin-c structures over Y_G.
  • If G is of type-1 this implies \mathbb{HF}^-(G, \mathfrak{s}) \cong HF^-(Y_G, \mathfrak{s}).

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 \mathbb{HF}^-(G, \mathfrak{s}) \cong HF^-(Y_G, \mathfrak{s}) for any plumbing tree G. To this end, there is a theorem of Ozsváth, Stipsicz, and Szabó that tells us there is a spectral sequence on \mathbb{HF}^-(G, \mathfrak{s}) converging to HF^-(Y_G, \mathfrak{s}). If G 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 \Gamma_{v_0} denote a plumbing graph with a distinguished, unweighted vertex v_0. Define the lattice chain complex on G=\Gamma_{v_0}-v_0. The vertexv_0 induces a filtration \mathcal{A} on the lattice homology complex. For a generator [K, E] of the lattice complex, the filtration level \mathcal{A}[K,E] is given by a formula that is essentially the same as the auxiliary function \mathcal{I} above, except it use a specific extension of K and E to \Gamma_{v_0}. The filtered complex (\mathbb{CF}^-(G, \mathfrak{s}), \partial^-, \mathcal{A}_{v_0}) is the subject of the main theorem:

Theorem. Suppose the tree \Gamma_{v_0} with distinguished vertex v_0 is given, that G=\Gamma_{v_0}-v_0 negative definite, and that G_{v_0}(k) is negative definite* for k\in \mathbb{Z}. Then \mathbb{CF}^-(G, \partial^-, \mathcal{A}_{v_0}) determines the lattice chain complex \mathbb{CF}^-(G_{v_0}(k), \partial^-).

(*I took G_{v_0}(k) to mean that we’ve put a framing of k on the vertex v_0, large enough to ensure that Y_{G_{v_0}} 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, CF^\infty 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 K. Here is a paper that details this construction.

Apparently, lattice homology also enjoys such a structure. The ‘infinity’ lattice complex \mathbb{CF}^\infty(G) = \mathbb{CF}^-(G)\otimes_{\mathbb{Z}_2} \mathbb{Z}_2[U^{-1}, U] is doubly filtered by the action of U and an extension of \mathcal{A}. It, too, can be chopped up into subcomplexes, the mapping cones of which are quasi-isomorphic to lattice complexes of G_{v_0}(k).

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 G is a plumbing graph with a distinguished vertex v such that the components of G-v are all rational, then the filtered chain complex of \Gamma_{v_0} in lattice homology is filtered chain homotopy equivalent to the filtered complex for Y_G in Heegaard Floer.

…and it’s wonderful corollary:
Corollary. The lattice homology of a knot in S^3 is equal to its knot Floer homology.

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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.

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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 \mathbb{R}^4‘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 (S^1\times[0,1], S^1\times\{0,1\})\to (S^3\times[0,1], S^3\times\{0,1\}). 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 K_1 and K_2 to be concordant is that K_1\#\overline{K_2}^r 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 \mathcal{C} denote the group of knots up to smooth concordance, and \mathcal{C}^{top} 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 \mathcal{C}_{TS}. Showing that \mathcal{C}_{TS} is non-trivial implies the existence of exotic \mathbb{R}^4‘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 \mathcal{C}_{TS} is non-trivial.

In 95 Endo showed that this group is big, namely there is a copy of \mathbb{Z}^{\infty}\subset \mathcal{C}_{TS}. Livingston, and Manolescu-Owens more recently showed that \mathcal{C}_{TS} contains a direct summand of \mathbb{Z}^3 distinguished using the \tau and s 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 \mathcal{C}_{TS}. One may want to understand how many elements in \mathcal{C}_{TS} can be represented by Whitehead doubles. Hedden and Kirk prove that there there is a \mathbb{Z}^{\infty}\subset Image(D) \subset \mathcal{C}_{TS}. 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 \varepsilon\in \{-1,0,1\} which can be computed from the chain complex CFK^{\infty}(K) through an algebraic process involving the \tau invariant (which is also a concordance invariant), or more geometrically by looking at the triviality or nontriviality of cobordism maps on \widehat{HF} induced by large integer surgeries on the knot in S^3. 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 CFK^{\infty}(K) and to its inverse in the concordance class, we get (CFK^{\infty}(K))^*. The analog of addition in the knot concordance group is the tensor product of chain complexes by the following Kunneth formula: CFK^{\infty}(K_1\#K_2) = CFK^{\infty}(K_1)\otimes CFK^{\infty}(K_2). If a knot K is smoothly slice, then \varepsilon(CFK^{\infty}(K))=0. 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 CFK^{\infty} complexes form a monoid under tensor product and we obtain a group if we mod out by the equivalence relation CFK^{\infty}(K_1)\sim_{\varepsilon} CFK^{\infty}(K_2) \iff \varepsilon(CFK^{\infty}(K_1)\otimes CFK^{\infty}(K_2)^* = 0. The resulting group \mathcal{F} = \{CFK^{\infty}(K): K\subset S^3\}/\sim_{\varepsilon} 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 \varepsilon is defined, there is definitely a relation to the \tau invariant, but \varepsilon is a more powerful invariant. It turns out that the exact relation is related to the satellite operation. Jen proved that CFK^{\infty}(K_1)\sim_{\varepsilon} CFK^{\infty}(K_2) if and only if \tau(P(K_1)) = \tau(P(K_2)) for every pattern P. Furthermore the satellite map descends to a well defined map on the group of knots up to \varepsilon 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 \mathcal{C}_{\Delta} := \langle \{[K]: \Delta_K=1\}\rangle = \mathcal{C}_{TS}? The answer to this question is strongly no. Hedden and Livingston prove that there is an infinitely generated free abelian subgroup in the quotient: \mathbb{Z}^{\infty}\subset \mathcal{C}_{TS}/\mathcal{C}_{\Delta}.

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 0=2[K] so [K]=-[K]=[\overline{K}^r], 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:
(\mathbb{Z}/2)^{\infty} \subset \mathcal{C}_{TS}.

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, d(Y,\mathfrak{s}) is keeping track of the highest grading of the generator of the nontorsion elements in the (minus) Heegaard Floer homology of a \mathbb{Z}/2 homology 3-sphere. To use this to obstruct sliceness, first one notices that if K were smoothly slice, then its branched double cover \Sigma(K) would be a \mathbb{Z}/2 homology 3-sphere and it would bound a \mathbb{Z}/2 homology 4-ball Q^4 (this comes from looking at the double branched cover of the 4-ball branched over the slice disk). This implies that the d-invariant d(Y,\mathfrak{s})=0 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 H_1(Y). 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 d(Y,\mathfrak{s}) vanishes for all required \mathfrak{s}.

Concordance genus:

The Seifert genus and 4-ball genus of a knot by definition satisfy the inequality g_4(K)\leq g(K). We can define an intermediate genus, called the concordance genus g_c(K) :=\min\{g(J): J\sim K\}. One may ask what the possible size of the gaps can be in the inequality g_4(K)\leq g_c(K) \leq g(K). 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 g_4(K) and g_c(K). 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 g_4(K)=1 but g_c(K) arbitrarily large. Jen improved this result even further with her \varepsilon equivalence, finding examples of topologically slice knots that all have g_4(K)=1, but g_c(K)=p for each p\geq 1.

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.

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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 f: X^4\to Q^2 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 M_i to each patch P_i, and a Lagrangian correspondence L_{ij} to each seam between patches P_i,P_j. 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: ds^2+dt^2+\varepsilon^2g_{\Sigma} and look at the new anti-self dual equations with this metric as \varepsilon \to 0. 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.

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