Final Report Summary - LTDBUD (Low Dimensional Topology in Budapest)
In the project we have delt with various aspects of low dimensional topology; below we list selected results from the main directions of research.
Contact Topology.
An important and hot research direction in manifold topology is to find further geometric strucutures on manifolds. In contact topology we consider specific hyperplane distributions on odd dimensional manifolds, which naturally serve as boundary values of symplectic structures on even dimensional manifolds. In a series of papers we have applied surgery theoretic methods to develope existence results for contact structures, and we also found an obstruction class for Stein fillable contact structures. In a slightly different direction, flexibility properties of Legendrian submanifolds have been examined in the complement of special contact submanifolds (so-called Plastikstufen). A number of the resulting publications appeared in top geometry journals (for example, in Geometry and Topology).
Heegaard Floer theory.
Heegaard Floer homology (and further derived invariants along similar lines) form the core of present day low dimensional topological research. The invariants are, however, rather hard to compute in general. We have found ways to interpret the original definitions (resting on highly complicated complex analytic methods and ideas) in terms of combinatorics, hence providing theoretically simple computational schemes for these invariants. Parallel to these efforts, applications of the invariants in knot theory and in the theory is rational, singular planar curves have been found. Indeed, the theory (and one of its variant, lattice homology) provides a nice and somewhat unexpected bridge between smooth topological properties of 3-dimensional manifolds and isolated surface singularities. Besides providing convenient computational tools, this approach also showed connection between rationality of a singularity and left-orderability of the fundamental group of its link. Indeed, lattice homology has been fruitfully extended from a certain class of 3-manifolds to pairs of (3-manifold, a knot in it). This extension can be used to show the conjectured isomorphism of lattice and Heegaard Floer homologies in many important instances.
Smooth 4-manifold topology.
The driving question of smooth 4-dimensional topology (the potential existence of an exotic 4-dimensional sphere, that is, a counterexampleto the smooth 4-dimensional Poincare conjecture) is still open. For some constructions (such as the Gluck construction along specific 2-spheres)we managed to show that the result is not exotic. Parallel to these findings, we also examined Lefschetz fibrations on 4-manifolds over the 2-dimensional torus, and found a lower bound on the number of singular fibers. Surprisingly, the manifold showing that the bound is sharp in certain cases is a complex surface which might play a prominent role in constructing exotic smooth structures on closely related 4-manifolds.
Summary.
The project had a significant impact on both the development of specific aspects of low dimensional topology, and also on the development of of topology in Budapest. A book on combinatorial aspects of Heegaard Floer homology has been published, and a number of research articles (about similar or related issues) were written, published and disseminated by members of the research group. The successful post-doctoral program (financially resting on the grant) provided an opportunity to build a strong group at the Renyi Institute working on low dimensional problems, and this had a direct effect on the position of topology within Hungarian mathematics.
Contact Topology.
An important and hot research direction in manifold topology is to find further geometric strucutures on manifolds. In contact topology we consider specific hyperplane distributions on odd dimensional manifolds, which naturally serve as boundary values of symplectic structures on even dimensional manifolds. In a series of papers we have applied surgery theoretic methods to develope existence results for contact structures, and we also found an obstruction class for Stein fillable contact structures. In a slightly different direction, flexibility properties of Legendrian submanifolds have been examined in the complement of special contact submanifolds (so-called Plastikstufen). A number of the resulting publications appeared in top geometry journals (for example, in Geometry and Topology).
Heegaard Floer theory.
Heegaard Floer homology (and further derived invariants along similar lines) form the core of present day low dimensional topological research. The invariants are, however, rather hard to compute in general. We have found ways to interpret the original definitions (resting on highly complicated complex analytic methods and ideas) in terms of combinatorics, hence providing theoretically simple computational schemes for these invariants. Parallel to these efforts, applications of the invariants in knot theory and in the theory is rational, singular planar curves have been found. Indeed, the theory (and one of its variant, lattice homology) provides a nice and somewhat unexpected bridge between smooth topological properties of 3-dimensional manifolds and isolated surface singularities. Besides providing convenient computational tools, this approach also showed connection between rationality of a singularity and left-orderability of the fundamental group of its link. Indeed, lattice homology has been fruitfully extended from a certain class of 3-manifolds to pairs of (3-manifold, a knot in it). This extension can be used to show the conjectured isomorphism of lattice and Heegaard Floer homologies in many important instances.
Smooth 4-manifold topology.
The driving question of smooth 4-dimensional topology (the potential existence of an exotic 4-dimensional sphere, that is, a counterexampleto the smooth 4-dimensional Poincare conjecture) is still open. For some constructions (such as the Gluck construction along specific 2-spheres)we managed to show that the result is not exotic. Parallel to these findings, we also examined Lefschetz fibrations on 4-manifolds over the 2-dimensional torus, and found a lower bound on the number of singular fibers. Surprisingly, the manifold showing that the bound is sharp in certain cases is a complex surface which might play a prominent role in constructing exotic smooth structures on closely related 4-manifolds.
Summary.
The project had a significant impact on both the development of specific aspects of low dimensional topology, and also on the development of of topology in Budapest. A book on combinatorial aspects of Heegaard Floer homology has been published, and a number of research articles (about similar or related issues) were written, published and disseminated by members of the research group. The successful post-doctoral program (financially resting on the grant) provided an opportunity to build a strong group at the Renyi Institute working on low dimensional problems, and this had a direct effect on the position of topology within Hungarian mathematics.