Tag: spinC structure

2010-04-11




 

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






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