Topology and Contact and Symplectic Manifolds

University Of Arkansas, Fayetteville AR

Investigators

Abstract

Symplectic manifolds are spaces equipped with an additional structure coming from classical mechanics. Contact manifolds are in some sense a dimensional simplification, and have historically had connections to the differential equations of optics and dynamics. In recent years, mathematicians have found strong applications of the study of contact and symplectic manifolds to our understanding of three- and four-dimensional spaces. One foundational goal in topology is to understand the extent certain algebraic simplifications of a manifold determine the manifold itself. For example, the famous Poincare Conjecture asks whether the structure of a sphere is determined by a related algebraic entity known as its fundamental group; a four dimensional version of this question is still unanswered. Somewhat surprisingly, symplectic manifolds have played a strong role in answering such questions. In turn, as the field has progressed, mathematicians have used tools from topology, differential geometry and physics with the goal of better understanding contact and symplectic manifolds. This project aims to both use the tools from the study of contact and symplectic manifolds to further our understanding of three- and four-dimensional spaces, as well as to develop new tools to increase our understanding of contact and symplectic manifolds themselves. In one sense, contact and symplectic topology bridges the rigidity of Riemannian geometry and the flexibility of topology, showing traits of both: local flexibility and global rigidity. Modern contact topology began in the 1980s with Bennequin's work and was connected to symplectic topology by Gromov and Eliashberg. Giroux brought topology and contact geometry in 3-dimensions closely together by associating a topological object, a singular fiber bundle called an open book decomposition, to a contact structure, as well as a method for describing all open books compatible with that contact structure. This tool has been extremely effective at forming connections with low-dimensional topology, allowing for the construction of new contact invariants, surgery characterization of certain knots, and the classification of symplectic fillings, among much else. Open books additionally provide two new intrinsic invariants of the contact structure: the page, a fiber in the open book, and the monodromy, the gluing map of the bundle. We call the minimal genus of a compatible open book the page genus of the contact structure, and it is an extraordinarily interesting invariant. If the page genus is zero, then we can say a tremendous amount about the contact structure. If the page genus is not zero, then there are infinitely many compatible open books and the existing methods for describing them all are far from effective, which makes determining the page genus impossible. Indeed, it is unknown whether there are contact structures with minimal page genus greater than one. This project aims to simplify this picture, first by producing new invariants of contact manifolds that can be effectively calculated using a given open book; and second by producing effective mechanisms for listing all open books as well as determining whether two open books yield the same contact structure.