Proposal for a smooth 4-manifold invariant
Warning: the following is fantasy.
Given an oriented closed -manifold and an indefinite Morse -function , choose a (noncompact) oriented -manifold with a smooth map satisfying the following properties:
- For any compact codimension-0 submanifold , is an immersion.
- For every simple closed curve that is transverse to the fold locus and avoids the cusps, consider , , the map , and the -valued Morse function . We require that, for some gradient-like vector field for :
- contains descending disks for each of the index critical points of and ascending disks for each of the index critical points of ,
- is tangent to (i.e. the vector field in the vector bundle actually lies in ), and
- is a complete vector field on , i.e. you can flow forever along it and never run off the ``end'' of .
- For every simple closed curve as above, there is an annulus neighborhood embedded in over which looks like (i.e. a constant homotopy of Morse functions), and such that over each , is as described above for a smoothly varying gradient-like vector field which has the genericity properties we expect from Cerf theory: transversely intersecting ascending and descending manifolds except at isolated handle slides.
- We don't care what does over the cusps - in fact the above conditions probably guarantee that 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 -multisection is a smoothly embedded surface with (where is the fold set in ), such that in fact lies entirely in the double point set for , and such that is a ``boundary-branched cover''; i.e. branching happens on the boundary.
There is something to be figured out here about orientations on and that should assign a sign to each such
Then here is the invariant: For each spin structure on , consider the associated homology class , where is the fold set, as in Perutz. Count up the -multisections representing , with signs, and the total number is an invariant.
What needs to be done?
- Prove such an always exists.
- Figure out the sign issue with -multisections.
- Figure out what a cobordism between such 's would be, given a generic homotopy between two indefinite Morse -functions.
- Show that such a cobordism would give a cobordism between the sets of -multisections representing a given - and somehow use this to conclude that the count is an invariant.
David Gay 2010-05-19