FRG: Collaborative Research : Floer homotopy theory

FRG：合作研究：弗洛尔同伦理论

• 批准号: 1564289
• 负责人: Andrew Blumberg
• 金额: $ 19.93万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-07-01 至 2019-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1564289&HistoricalAwards=false
• 关键词: FRG; Collaborative; Research; Floer; homotopy

Topology is the study of those properties of shapes that are unchanged by stretching and bending. Over the years, mathematicians have developed various topological invariants, or in other words, quantities that are associated to shapes and can distinguish between those that have different properties. Homology is a well-known such invariant, which can be associated to any multi-dimensional shape, and which is a quantitative measure of the number of holes in a space. A circle, for instance, has only a one-dimensional hole, whereas the surface of a doughnut has two one-dimensional holes, a meridian and a longitude, that are not filled in by the surface itself, and an additional two-dimensional hole. Floer homology is a more refined invariant that is responsible for some of the most important recent advances in the study of knotted closed loops in space, three-dimensional shapes, and shapes with a geometry known as a symplectic structure that is exhibited by phase spaces in classical mechanics. This project brings together several researchers working in different areas of topology and geometry to study Floer homology. The main goal of the project is the following: To every knot, three-dimensional shape, or symplectic shape, one should associate a different object, called a Floer space or a Floer homotopy type, whose (ordinary) homology is the Floer homology of the initial shape. This has been accomplished so far in a limited number of cases. A general theory of Floer spaces will lead to new advances in several areas. Furthermore, the study of Floer spaces will be based on techniques from a subfield of topology called homotopy theory. This project will create a community of scholars at the interface of these current and extremely research active areas of mathematics.Floer homology is a fundamental tool in geometry and topology, whose applications range from the Arnold conjecture to the surgery characterization of various knots. Floer homology has also laid the basis for completely unexpected interconnections between algebraic and symplectic geometry in the form of homological mirror symmetry. Floer homotopy theory, an extension to spaces rather than homology groups, has been implemented in a small number of cases, leading to significant applications, for example, the resolution of the triangulation conjecture in high dimensions and work on immersed Lagrangian spheres. Further, the ideas behind Floer homotopy inspired the construction of a Khovanov homotopy type associated to knots in the three-sphere. The main scientific goal of this project is to give a general construction of Floer homotopy. The necessary foundational work will build upon recent advances in multiple areas. These include the conceptual advances in equivariant stable homotopy theory stemming from the resolution of Kervaire invariant one problem, and the development of new approaches to define virtual fundamental classes in Floer theory. The project aims to put the homotopical and homological variants of Floer theory on equal footing. As a consequence, new applications in both symplectic and low-dimensional topology are anticipated, for example: (i) a spectral Fukaya category associated to a symplectic manifold will be constructed; (ii) the Heegaard Floer theory of Ozsvath and Szabo will be used to produce a computable invariant parallel to the celebrated Bauer-Furuta invariant for four-manifolds; (iii) Seiberg-Witten Floer homotopy types will be studied using the tools of equivariant stable homotopy theory; and (iv) the Khovanov homotopy type will be extended to give invariants of knot cobordisms and tangles.

拓扑学是对那些不因拉伸和弯曲而改变的形状特性的研究。多年来，数学家开发了各种拓扑不变量，或者换句话说，与形状相关的数量，可以区分具有不同属性的形状。同源性是一种众所周知的不变量，它可以与任何多维形状相关联，并且是空间中孔洞数量的定量度量。例如，一个圆只有一个一维孔，而甜甜圈的表面有两个一维孔，一个子午线和一个经度，它们没有被表面本身填充，还有一个额外的二维孔。弗洛尔同调是一个更精致的不变量，它促成了空间中的打结闭环、三维形状以及具有称为辛结构的几何形状的研究中的一些最重要的最新进展，辛结构由相空间展示经典力学。该项目汇集了拓扑学和几何学不同领域的几位研究人员来研究弗洛尔同调性。该项目的主要目标如下：对于每个结、三维形状或辛形状，应该关联一个不同的对象，称为 Floer 空间或 Floer 同伦类型，其（普通）同源性是 Floer 同伦性初始形状。到目前为止，这已经在有限的案例中实现了。弗洛尔空间的一般理论将在几个领域带来新的进展。此外，Floer 空间的研究将基于称为同伦理论的拓扑学子领域的技术。该项目将在这些当前和极其活跃的数学领域的接口上创建一个学者社区。弗洛尔同调是几何和拓扑中的基本工具，其应用范围从阿诺德猜想到各种结的手术表征。弗洛尔同调还为代数几何和辛几何之间以同调镜像对称的形式实现完全意想不到的互连奠定了基础。弗洛尔同伦理论是对空间而不是同调群的扩展，已在少数情况下得到应用，产生了重要的应用，例如，解决高维三角测量猜想以及对浸没拉格朗日球体的研究。此外，弗洛尔同伦背后的想法启发了与三球体中的结相关的霍瓦诺夫同伦类型的构造。该项目的主要科学目标是给出弗洛尔同伦的一般构造。必要的基础工作将建立在多个领域的最新进展的基础上。其中包括等变稳定同伦理论的概念进步，源于 Kervaire 不变一问题的解决，以及在 Floer 理论中定义虚拟基本类的新方法的发展。该项目旨在将弗洛尔理论的同伦和同调变体置于平等的基础上。因此，预计在辛和低维拓扑中都有新的应用，例如：（i）将构造与辛流形相关的谱 Fukaya 类别； (ii) Ozsvath 和 Szabo 的 Heegaard Floer 理论将用于产生与著名的四流形 Bauer-Furuta 不变量平行的可计算不变量； (iii) 将使用等变稳定同伦理论工具来研究 Seiberg-Witten Floer 同伦类型； (iv) Khovanov 同伦类型将被扩展以给出结配边和缠结的不变量。