FRG: Collaborative Research : Floer homotopy theory

拓扑学研究的是形状的那些性质，这些性质不受拉伸和弯曲的影响。多年来，数学家已经开发出各种拓扑不变量，或者换句话说，与形状相关的量，可以区分具有不同属性的量。同调是一个众所周知的这样的不变量，它可以与任何多维形状相关联，并且是空间中孔的数量的定量度量。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度，它们不是由表面本身填充的，还有一个额外的二维洞。弗洛尔同调是一个更精细的不变量，它是最近在空间中的打结闭环、三维形状和具有被称为辛结构的几何形状的研究中取得的一些最重要的进展的原因，辛结构在经典力学中的相空间中表现出来。这个项目汇集了几个研究人员在不同领域的拓扑和几何研究弗洛尔同源。该项目的主要目标如下：对于每一个纽结、三维形状或辛形状，应该关联一个不同的对象，称为弗洛尔空间或弗洛尔同伦类型，其（普通）同调是初始形状的弗洛尔同调。到目前为止，这只在有限的几个案例中实现。弗洛尔空间的一般理论将导致在几个领域的新进展。此外，弗洛尔空间的研究将基于拓扑学中称为同伦理论的一个子领域的技术。这个项目将创建一个学者社区的接口，这些当前和非常活跃的研究领域的mathematics.Floer同调是一个基本的工具，在几何和拓扑，其应用范围从阿诺德猜想的外科手术表征的各种节点。弗洛尔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出乎意料的相互联系奠定了基础。弗洛尔同伦理论，一个扩展到空间，而不是同调群，已实施在少数情况下，导致重大的应用，例如，决议的三角猜想在高维和工作沉浸拉格朗日球。此外，弗洛尔同伦背后的思想启发了与三球中的节点相关的霍瓦诺夫同伦类型的构建。该项目的主要科学目标是给出Floer同伦的一般构造。必要的基础工作将建立在多个领域的最新进展基础上。这些包括概念上的进步等变稳定同伦理论源于解决Kervaire不变的一个问题，并制定新的方法来定义虚拟的基本类在Floer理论。该项目旨在将弗洛尔理论的同伦和同调变体置于同等地位。因此，在辛拓扑和低维拓扑中的新的应用是值得期待的，例如：（i）将构造一个与辛流形相关联的谱福谷范畴;（ii）Ozsvath和Szabo的Heegaard Floer理论将用于产生一个与著名的四维流形的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进行研究;（iv）Khovanov同伦类型将被扩展以给出结配边和缠结的不变量。

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.