Proposal for a smooth 4-manifold invariant

Warning: the following is fantasy.

Given an oriented closed $ 4$ -manifold $ X$ and an indefinite Morse $ 2$ -function $ G \colon\thinspace X \to S^2$ , choose a (noncompact) oriented $ 3$ -manifold $ M$ with a smooth map $ \mu \colon\thinspace M \to X$ satisfying the following properties:

  1. For any compact codimension-0 submanifold $ M_0 \subset M$ , $ \mu\vert _{M_0}$ is an immersion.
  2. For every simple closed curve $ C \subset S^2$ that is transverse to the fold locus and avoids the cusps, consider $ X_C = G^{-1}(C) \subset X$ , $ M_C = (G \circ \mu)^{-1}(C) \subset M$ , the map $ \mu_C = \mu\vert _{M_C} \colon\thinspace M_C \to X_C$ , and the $ S^1$ -valued Morse function $ g_C = G\vert _{X_C} \colon\thinspace X_C \to C$ . We require that, for some gradient-like vector field $ V$ for $ g_C$ :
    1. $ \mu_C(M_C)$ contains descending disks for each of the index $ 2$ critical points of $ g_C$ and ascending disks for each of the index $ 1$ critical points of $ g_C$ ,
    2. $ V$ is tangent to $ \mu_C(M_C)$ (i.e. the vector field $ \mu_C^*(V)$ in the vector bundle $ \mu_C^*(TX)$ actually lies in $ TM_C$ ), and
    3. $ \mu_C^*(V)$ is a complete vector field on $ M_C$ , i.e. you can flow forever along it and never run off the ``end'' of $ M_C$ .
  3. For every simple closed curve $ C$ as above, there is an annulus neighborhood $ [-\epsilon,\epsilon] \times C$ embedded in $ S^2$ over which $ G$ looks like $ (t,p) \mapsto (t,g_C(p))$ (i.e. a constant homotopy of Morse functions), and such that over each $ \{t\} \times C$ , $ M$ is as described above for a smoothly varying gradient-like vector field $ V_t$ which has the genericity properties we expect from Cerf theory: transversely intersecting ascending and descending manifolds except at isolated handle slides.
  4. We don't care what $ M$ does over the cusps - in fact the above conditions probably guarantee that $ M$ can't hit the cusps at all, but that's fine. (We might need to add the condition that there aren't infinitely many handle slides as you move in towards a cusp?)

Now given this data, a $ \mu$ -multisection is a smoothly embedded surface $ \Sigma \subset X$ with $ \partial \Sigma = Z$ (where $ Z$ is the fold set in $ X$ ), such that in fact $ \Sigma$ lies entirely in the double point set for $ \mu$ , and such that $ G\vert _{\Sigma} \colon\thinspace \Sigma \to S^2$ is a ``boundary-branched cover''; i.e. branching happens on the boundary.

There is something to be figured out here about orientations on $ M$ and $ \Sigma$ that should assign a sign to each such $ \Sigma$

Then here is the invariant: For each spin $ ^\mathbb{C}$ structure $ s$ on $ X$ , consider the associated homology class $ A_s \in H_2(X,Z)$ , where $ Z$ is the fold set, as in Perutz. Count up the $ \mu$ -multisections representing $ A_s$ , with signs, and the total number $ N(s)$ is an invariant.

What needs to be done?

  1. Prove such an $ M$ always exists.
  2. Figure out the sign issue with $ \mu$ -multisections.
  3. Figure out what a cobordism between such $ M$ 's would be, given a generic homotopy $ G_s$ between two indefinite Morse $ 2$ -functions.
  4. Show that such a cobordism would give a cobordism between the sets of $ \mu$ -multisections representing a given $ A_s$ - and somehow use this to conclude that the count is an invariant.


David Gay 2010-05-19


Symplectic polydisk and ellipsoid embeddings

Some explicit questions about polydisk and ellipsoid embeddings that Olga Buse and I hope to tackle:

  • Consider the map $ f \colon\thinspace B^4(1) \to B^2(1)$ defined in complex coordinates by $ f(w,z) = wz$ . Pick a regular value, say $ z_0 = 1/2$ . Let $ C = f^{-1}(z_0)$ . First of all, what is the area $ A=\int_C \omega_0$ ? Secondly, given any area $ a < A$ , let $ w(a)$ be the supremum over all $ w$ such that there is a polydisk $ B^2(a) \times B^2(w(a))$ embedded in $ B^4(1)$ with $ B^2(a) \times \{0\} \subset C$ . Calculate $ w(a)$ . It should obviously be a decreasing function of $ a$ . Then, for which $ a$ is $ a w(a)$ largest?
  • Same thing for any other Lefschetz fibration, or any complex polynomial for that matter.
  • In principle you might be able to use these same ideas to understand ellipsoid embeddings. Maybe the best way to say it is this: Given the symplectic surface $ C = f^{-1}(z_0)$ , find a smooth positive function $ r \colon\thinspace C \to (0,\infty)$ with a symplectic embedding of $ P(C,r)=\{(p,x,y) \in C \times \mathbb{R}^2 \mid \sqrt{x^2+y^2} < r(p)\}$ into $ B^4(1)$ mapping $ P \times \{(0,0)\}$ to $ P$ by the identity. Hopefully you can find such a function and such an embedding so that it ``fully fills'' $ B^4(1)$ . Then by trimming back $ C$ and trimming back $ r$ just right, we can get embeddings of ellipsoids, and by trimming back $ C$ and trimming $ r$ back to be constant, we can get embeddings of polydisks.
  • My original idea about using Lefschetz fibrations was a little different, but should probably amount to the same thing. Take your favorite topological Lefschetz fibration of $ B^4$ over $ B^2$ . Build it symplectically, with convex boundary, in some standard way, starting with $ F \times B^2$ (where $ F$ is the fiber) and then adding very thin Weinstein handles. You end up with a symplectic structure $ \omega$ on $ B^4$ with convex boundary, with induced contact form $ \alpha$ on $ S^3$ (contact form because you have kept careful track of the form, not just the underlying contact structure, as you attached the handles). And this nonstandard symplectic ball is almost fully filled with a polydisk. Now we know that there exists some function $ f \colon\thinspace S^3 \to [1,\infty)$ so that $ f \alpha = k \alpha_0$ , where $ \alpha_0$ is the standard contact form on $ S^3 = \partial B^4(1)$ , and $ k$ is some constant. I think that then you want to measure $ \int_{S^3} f$ , or something like that - this should be a measure of how much volume you have to add to change $ (B^4,\omega)$ into the standard round ball. Somehow this function is an interesting object attached to a word in Dehn twists.
  • The other idea for polydisks, which ultimately is still really the same basic idea, namely extending a symplectic surface out to a polydisk as much as possible, is to consider ribbon disks for interesting transverse ribbon knots.


P.S. I just gave a talk at MSRI on indefinite Morse 2-functions, my main project with Rob Kirby, and the pdf slides can be viewed here.

David Gay 2010-04-28