arXiv subjects
Publications and source records attributed to David T Gay.
This is an expository account of the author's collaboration with Rob Kirby leading up to the theory of trisections of smooth 4-manifolds. This article was written for inclusion in an upcoming issue of Celebratio Mathematica dedicated to Rob Kirby's mathematical work.
We offer a new proof that two closed oriented 4-manifolds are cobordant if their signatures agree, in the spirit of Lickorish's proof that all closed oriented 3-manifolds bound 4-manifolds. Where Lickorish uses Heegaard splittings we use trisections. In fact we begin with a subtle recasting of Lickorish's argument: Instead of factoring the gluing map for a Heegaard splitting as a product of Dehn twists, we encode each handlebody in a Heegaard splitting in terms of a Morse function on the surface and build the 4-manifold from a generic homotopy between the two functions. This extends up a dimension by encoding a trisection of a closed 4-manifold as a triangle (circle) of functions and constructing an associated 5-manifold from an extension to a 2-simplex (disk) of functions. This borrows ideas from Hatcher and Thurston's proof that the mapping class group of a surface is finitely presented.
We construct four-dimensional symplectic cobordisms between contact three-manifolds generalizing an example of Eliashberg. One key feature is that any handlebody decomposition of one of these cobordisms must involve three-handles. The other key feature is that these cobordisms contain chains of symplectically embedded two-spheres of square zero. This, together with standard gauge theory, is used to show that any contact three-manifold of non-zero torsion (in the sense of Giroux) cannot be strongly symplectically fillable. John Etnyre pointed out to the author that the same argument together with compactness results for pseudo-holomorphic curves implies that any contact three-manifold of non-zero torsion satisfies the Weinstein conjecture. We also get examples of weakly symplectically fillable contact three-manifolds which are (strongly) symplectically cobordant to overtwisted contact three-manifolds, shedding new light on the structure of the set of contact three-manifolds equipped with the strong symplectic cobordism partial order.