FRG: Collaborative Research : Floer homotopy theory

拓扑学研究的是那些不受拉伸和弯曲影响的形状的性质。多年来，数学家们开发了各种拓扑不变量，或者换句话说，与形状相关的量，可以区分具有不同性质的量。同调性是一个众所周知的不变量，它可以与任何多维形状相关联，并且是空间中孔数的定量度量。例如，一个圆只有一个一维的洞，而一个甜甜圈的表面有两个一维的洞，一个子午线和一个经线，它们不是由表面本身填充的，还有一个额外的二维洞。花同调是一种更精细的不变量，它是研究空间中打结闭合回路、三维形状和几何形状的一些最重要的最新进展的原因，这些几何形状被称为辛结构，在经典力学中由相空间表现出来。该项目汇集了几位研究人员在不同领域的拓扑和几何研究花同源性。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）同调是初始形状的Floer同调。到目前为止，在数量有限的情况下已经做到了这一点。花空间的一般理论将导致几个领域的新进展。此外，花空间的研究将基于拓扑学的一个子领域称为同伦理论的技术。该项目将在这些当前和极其活跃的数学研究领域的界面上创建一个学者社区。花同源性是几何和拓扑学中的一个基本工具，其应用范围从阿诺德猜想到各种结的手术表征。花同调也以同调镜像对称的形式为代数几何和辛几何之间完全意想不到的相互联系奠定了基础。花同伦理论是对空间而非同伦群的一种扩展，在少数情况下得到了实现，并带来了重要的应用，例如高维三角剖分猜想的解决和浸入式拉格朗日球的研究。进一步，Floer同伦的思想启发了与三球结相关的Khovanov同伦型的构造。本课题的主要科学目标是给出花同伦的一般构造。必要的基础工作将建立在多个领域的最新进展之上。这包括由Kervaire不变量1问题的解决引起的等变稳定同伦理论的概念进展，以及花理论中定义虚基类的新方法的发展。该项目旨在将弗洛尔理论的同调变体和同调变体置于同等地位。因此，预计在辛维和低维拓扑中的新应用，例如：(i)将构建与辛流形相关的谱深谷范畴；（ii）将利用Ozsvath和Szabo的Heegaard Floer理论产生与著名的Bauer-Furuta不变量平行的可计算不变量；（iii）利用等变稳定同伦理论的工具研究Seiberg-Witten Floer同伦类型；（iv）将Khovanov同伦型推广到给出结协点和缠结的不变量。

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.