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.
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.