Tobias Ekholm gave a talk today entitled Exact Lagrangian immersions with a single double point. The relevant paper on the arXiv has the same name and is joint between Ekholm and Ivan Smith. The following is based on my notes from the talk and discussions that followed. The motivating question here is how much information we can obtain about the smooth structure of a manifold by looking at the symplectic topology of its cotangent space.
First, the definition of an exact Lagrangian immersion… A Lagrangian immersion is an immersion such that . An exact symplectic manifold has . Note that the standard symplectic structures on and are exact. Finally, an exact Lagrangian immersion is a Lagrangian immersion into an exact symplectic manifold such that .
Because the zero section (with the standard symplectic structure) is an exact embedded Lagrangian, studying the smooth topology of M via the symplectic topology of exact Lagrangian embeddings into standard symplectic manifolds. One simple property of a manifold that can be detected in this way is whether the manifold is closed (in the topologist sense i.e. compact without boundary). Gromov proved that there are no exact Lagrangian embeddings of a closed manifold into . You can see this easily in the n=1 case: if you had an embedded exact closed Lagrangian curve in the plane: , it encloses some region D. By Stokes’ theorem
therefore any closed exact Lagrangian in (or more generally ) must have self-intersection points. Ekholm and Smith try to push this further, to eliminate the possibility of exact Lagrangian immersions into with a single double point. There is a necessary restriction on Euler characteristic, but otherwise they show that the sphere with its standard smooth structure is the only manifold with such an immersion. Formally, the theorem is:
Theorem: If K is a closed oriented 2k-manifold for with , which admits an exact Lagrangian immersion then K is diffeomorphic to the standard 2k-sphere.
This means that we get information about exotic even dimensional spheres by studying the symplectic topology on .
Note, the Euler characteristic restriction is necessary (there are counterexamples).
Here are some of the main ideas that come up in the proof of the theorem.
Step 1: Use some older results coming from symplectic field theory, to say that K is necessarily a homotopy sphere. Somehow the exact Lagrangian immersion f gives rise to a dga generated by the unique double point. The Euler characteristic restriction is necessary at this point to switch to linearized contact homology. This is used to show K is a homology sphere, and doing a harder version of all this shows that K is actually a homotopy sphere.
Step 2: Resolve the double point by Lagrangian surgery to a new exact Lagrangian embedding of a manifold L. The resolution is locally modeled on the following setup. The double point is the origin at the intersection of with in . Remove a disk around the origin and smooth in one of two possible ways, so that the resulting manifold is K with a 1-handle attached. Of course you have to show that you can do this in a Lagrangian way (construction due to Polterovich).
Step 3: Form a moduli space of curves such that and u satisfies a perturbed version of the Cauchy-Riemann equations. These equations include a parameter j. When j=0, the solutions to the perturbed equations are just the constant solutions of maps into L. As j is sent to infinity, eventually the moduli space of solutions becomes empty. The idea is that the set of all solutions forms a symplectic filling of the exact Lagrangian L. Unfortunately, there are some singularities to deal with. Resolving these singularities is pretty subtle, but Ekholm and Smith manage to get some restrictions on how bad things can get, in particular they can only have bubbling phenomena with 2 bubble components. Eventually they are able to change this moduli space into a smooth stably parallelizable filling of L (stably parallelizable means that if you direct sum the tangent bundle with a trivial bundle, you get a trivial bundle).
Step 4: After adding a cancelling 2-handle to get back to a manifold with boundary our unsurgered manifold K, we obtain a stably parallelizable filling of K. By results that date back to 1967 by Kervaire and Milnor, this stably parallelizable filling suffices to show that K has standard diffeomorphism type.
This theorem is stated only for even spheres of dimension strictly greater than 4. It seems likely that the proof fails on many levels if you try to apply it to 4-manifolds, but it seems like an interesting question to ask whether any of these ideas can be applied to studying the smooth Poincare conjecture in dimension 4, or understanding other exotic 4-manifolds based on their embeddings/immersions into symplectic manifolds.