FRG: Collaborative Research: Floer Homotopy Theory

拓扑学是研究通过拉伸和弯曲保持不变的形状的属性。多年来，数学家已经开发出各种拓扑不变量，换句话说，与形状相关的量，可以区分那些具有不同性质的量。同调是众所周知的这种不变量，它可以与任何多维形状相关联，并且是对空间中空穴数量的定量测量。例如，圆只有一个一维的洞，而甜甜圈的表面有两个一维的洞，一个子午线和一个经度，这两个洞不是由表面本身填充的，另外还有一个二维的洞。Floer同调是一种更精细的不变量，它负责研究空间中打结的闭环、三维形状和具有经典力学中相空间所展示的辛结构的几何形状的一些最重要的最近进展。这个项目汇集了几个在不同的拓扑学和几何学领域工作的研究人员来研究Floer同调。该项目的主要目标如下：对于每个纽结、三维形状或辛形状，人们应该关联一个不同的对象，称为Floer空间或Floer同伦型，其(普通)同调是初始形状的Floer同调。到目前为止，这一点在数量有限的案例中已经实现。Floer空间的一般理论将在几个领域带来新的进展。此外，对Floer空间的研究将基于拓扑子领域中的技术，称为同伦理论。这个项目将在这些当前和非常活跃的数学研究领域之间建立一个学者社区。Floer同调是几何学和拓扑学中的一个基本工具，其应用范围从Arnold猜想到各种纽结的外科表征。Floer同调也为代数几何和辛几何之间以同调镜像对称的形式完全意想不到的相互联系奠定了基础。Floer同伦理论是对空间而不是同调群的推广，已经在少数情况下实现，导致了重要的应用，例如在高维三角剖分猜想的解决和在浸没的拉格朗日球面上的工作。此外，Floer同伦背后的思想启发了与三个球面中的纽结相关的Khovanov同伦型的构造。这个项目的主要科学目标是给出Floer同伦的一般构造。必要的基础工作将以最近在多个领域取得的进展为基础。其中包括由Kervaire不变问题的解决而产生的等变稳定同伦理论的概念进展，以及定义Floer理论中虚拟基本类的新方法的发展。该项目旨在使弗洛尔理论的同伦变体和同调变体处于平等的地位。因此，在辛拓扑和低维拓扑学中都有望得到新的应用，例如：(I)将构造与辛流形相关的谱Fukaya范畴；(Ii)Ozsvath和Szabo的Heegaard Floer理论将被用来产生与著名的Bauer-Furuta不变量平行的可计算不变量；(Iii)Seiberg-Witten Floer同伦类型将使用等变稳定同伦理论的工具来研究；以及(Iv)Khovanov同伦类型将被扩展以给出纽余切和缠结的不变量。

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.