Open Books, Lefschetz Fibrations, and Related Questions in Low-Dimensional Topology

这个项目主要研究3维和4维的接触和辛拓扑。由于Giroux的结果，通过对3-流形的开卷分解提供了对接触结构的拓扑洞察。公开读物与4-流形上的Lefschetz纤颤有关；后者提供了一种关于Stein和辛结构的拓扑方法。PI计划使用低维技术以及Heegaard Floer理论中的不变量来研究这些结构。具体的目标包括，例如，理性开卷的Heegaard Floer接触不变量的研究(与M.Hedden联合)。这将在接触流形上的外科手术中有重要的应用，并将允许证明新类型的接触结构的紧密性。在另一个项目中，Plamenevskaya将尝试从Heegaard Floer同调的某个谱序列中获得4-流形上的Lefschetz纤颤的不变量。她还希望对四维Heegaard Floer不变量有更好的理解，并利用这些不变量(也许还有Lefschetz纤维)来研究四维空间中的某些奇异现象。理解时空世界和其他相关物体的形状是一个基本问题。我们生活的空间是一个三维流形的例子(一般来说，流形可以是弯曲的或扭曲的，或者在它们上面有洞)。随着时间的增加，我们得到了一个4维流形。低维拓扑学研究3维和4维流形的形状。纽结是低维拓扑学的另一个研究对象；它们对于数学以外的学科具有实际重要性(例如，DNA和一些聚合物是纽结的)。本项目的目标是研究具有接触(对于3-流形)和分别具有辛结构(对于4个流形)的3维和4维流形。这些结构起源于力学、光学和流体力学，并在数学中发挥着重要作用。渐近结构和接触结构的研究处于拓扑学中几个重要学科的十字路口，并使用了各种工具，包括规范理论(起源于物理学)、纽结理论和各种拓扑“积木”分解，如开放书籍和Lefschetz纤维给出的那些。目前的提议涉及到其中的许多方面；国际和平研究所的目标既是构建新的工具，也是寻找理论的新应用。

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.