Searcharxiv⌕ Search

arXiv subjects

David T. Gay

Publications and source records attributed to David T. Gay.

26 records · Page 2Linked to original sources
arXiv2008

Symplectic surgeries and normal surface singularities

We show that every negative definite configuration of symplectic surfaces in a symplectic 4--manifold has a strongly symplectically convex neighborhood. We use this to show that, if a negative definite configuration satisfies an additional negativity condition at each surface in the configuration, and if the complex singularity with resolution diffeomorphic to a neighborhood of the configuration has a smoothing, then the configuration can be symplectically replaced by the smoothing of the singularity. This generalizes the symplectic rational blowdown procedure used in recent constructions of small exotic 4--manifolds.

math.SG↗
arXiv2007

Constructing Lefschetz-type fibrations on four-manifolds

We show how to construct broken, achiral Lefschetz fibrations on arbitrary smooth, closed, oriented 4-manifolds. These are generalizations of Lefschetz fibrations over the 2-sphere, where we allow Lefschetz singularities with the non-standard orientation as well as circles of singularities corresponding to round 1-handles. We can also arrange that a given surface of square 0 is a fiber. The construction is easier and more explicit in the case of doubles of 4-manifolds without 3- and 4-handles, such as the homotopy 4-spheres arising from nontrivial balanced presentations of the trivial group.

math.GT↗
arXiv2006

Toric structures on near-symplectic 4-manifolds

A near-symplectic structure on a 4-manifold is a closed 2-form that is symplectic away from the 1-dimensional submanifold along which it vanishes and that satisfies a certain transversality condition along this vanishing locus. We investigate near-symplectic 4-manifolds equipped with singular Lagrangian torus fibrations which are locally induced by effective Hamiltonian torus actions. We show how such a structure is completely characterized by a singular integral affine structure on the base of the fibration whenever the vanishing locus is nonempty. The base equipped with this geometric structure generalizes the moment map image of a toric 4-manifold in the spirit of earlier work by the second author on almost toric symplectic 4-manifolds. We use the geometric structure on the base to investigate the problem of making given smooth torus actions on 4-manifolds symplectic or Hamiltonian with respect to near-symplectic structures and to give interesting constructions of structures which are locally given by torus actions but have nontrivial global monodromy.

math.SG↗
arXiv2004

Constructing symplectic forms on 4-manifolds which vanish on circles

Given a smooth, closed, oriented 4-manifold X and alpha in H_2(X,Z) such that alpha.alpha > 0, a closed 2-form w is constructed, Poincare dual to alpha, which is symplectic on the complement of a finite set of unknotted circles. The number of circles, counted with sign, is given by d = (c_1(s)^2 -3sigma(X) -2chi(X))/4, where s is a certain spin^C structure naturally associated to w.

math.GT↗
arXiv2004

Open books and configurations of symplectic surfaces and erratum

We study neighborhoods of configurations of symplectic surfaces in symplectic 4-manifolds. We show that suitably `positive' configurations have neighborhoods with concave boundaries and we explicitly describe open book decompositions of the boundaries supporting the associated negative contact structures. This is used to prove symplectic nonfillability for certain contact 3-manifolds and thus nonpositivity for certain mapping classes on surfaces with boundary. Similarly, we show that certain pairs of contact 3-manifolds cannot appear as the disconnected convex boundary of any connected symplectic 4-manifold. Our result also has the potential to produce obstructions to embedding specific symplectic configurations in closed symplectic 4-manifolds and to generate new symplectic surgeries. From a purely topological perspective, the techniques in this paper show how to construct a natural open book decomposition on the boundary of any plumbed 4-manifold. Erratum (added December 2003): We correct the main theorem and its proof. As originally stated, the theorem gave conditions on a configuration of symplectic surfaces in a symplectic 4-manifold under which we could construct a model neighborhood with concave boundary and describe explicitly the open book supporting the contact structure on the boundary. The statement should have included constraints on the areas of the surfaces.

math.GT↗
arXiv2001

Explicit concave fillings of contact three-manifolds

In this paper we give explicit, handle-by-handle constructions of concave symplectic fillings of all closed, oriented contact 3-manifolds. These constructions combine recent results of Giroux relating contact structures and open book decompositions of 3-manifolds, earlier results of the author on attaching 4-dimensional symplectic 2-handles along transverse links, and some tricks with mapping class groups of compact surfaces with non-empty boundary.

math.GT↗
arXiv2001

Symplectic 2-handles and transverse links

A standard convexity condition on the boundary of a symplectic manifold involves an induced positive contact form (and contact structure) on the boundary; the corresponding concavity condition involves an induced negative contact form. We present two methods of symplectically attaching 2-handles to convex boundaries of symplectic 4-manifolds along links transverse to the induced contact structures. One method results in concave boundaries and depends on a fibration of the link complement over S^1; in this case the handles can be attached with any framing larger than a lower bound determined by the fibration. The other method results in a weaker convexity condition on the new boundary (sufficient to imply tightness of the new contact structure), and in this case the handles can be attached with any framing less than a certain upper bound. These methods supplement methods developed by Weinstein and Eliashberg for attaching symplectic 2-handles along Legendrian knots.

math.GT↗