This project focuses on contact and symplectic topology in dimensions 3 and 4. Due to a result of Giroux, a topological insight into contact structures is offered by open book decompositions of 3-manifolds. Open books are related to Lefschetz fibrations on 4-manifolds; the latter provide a topological approach to Stein and symplectic structures. The PI plans to study these structures using low-dimensional techniques together with invariants from Heegaard Floer theory. Specific goals include, for example, the study (joint with M. Hedden) of Heegaard Floer contact invariants for rational open books. These will have important applications to surgeries on contact manifolds, and will allow to prove tightness for new classes of contact structures. In another project (joint with T. Mark and based on the previous work of the PI and J. Baldwin) Plamenevskaya will try to obtain invariants of Lefschetz fibrations on 4-manifolds from a certain spectral sequence in Heegaard Floer homology. She also hopes to develop a better understanding of the 4-dimensional Heegaard Floer invariants, and to use these (perhaps along with Lefschetz fibrations) to study certain exotic phenomena in 4 dimensions.
Understanding the shape of the space-time world and of other related objects is a fundamental problem. The space we live in is an example of a 3-dimensional manifold (in general, manifolds can be curved or twisted, or have holes in them). With the addition of time, we obtain a 4-dimensional manifold. Low-dimensional topology studies the shape of manifolds of dimension 3 and 4. Knots are another object of research in low-dimensional topology; they have practical importance for disciplines outside mathematics (for example, DNA and some polymers are knotted). The goal of the present project is to study 3- and 4-dimensional manifolds equipped with contact (for 3-manifolds) and respectively symplectic (for 4 manifolds) structures. These structures originate in mechanics, optics and hydrodynamics, and play a major role in mathematics. The study of sympectic and contact structures lies at the crossroads of several important subjects in topology and uses various tools, including gauge theory (originating from physics), knot theory, and various topological "building blocks" decompositions, such as those given by open books and Lefschetz fibrations. The present proposal touches upon many of these aspects; the PI's goal is both to construct new tools and to find new applications of the theory.