Invariants of bordered 3-manifolds and contact structures in Floer homology, connections with Khovanov homology, and applications

Floer 同调中的有界 3 流形和接触结构的不变量、与 Khovanov 同调的联系以及应用

• 批准号: 1406383
• 负责人: John Baldwin
• 金额: $ 15.98万
• 依托单位: Boston College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-08-01 至 2017-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1406383&HistoricalAwards=false
• 关键词: Invariants; bordered; manifolds; contact; structures

The proposed project involves studying geometric structures on 3- and 4-dimensional manifolds. Understanding these spaces and structures is central to understanding the shape of our macroscopic universe, and many of the theories the PI intends to study have applications and origins in physics. A major goal of the PI's proposal is to develop methods which might be used to unify several existing and important tools for studying 3- and 4-dimensional manifolds. Such a unification would likely reveal interesting connections between quite disparate fields of mathematics. The PI also plans to use some of these tools to study contact structures and knots in new ways. Both serve as probes by which to study 3- and 4-dimensional manifolds. Contact structures are also important in classical mechanics, thermodynamics, dynamical systems, and in the study of liquid crystals, while knot theory has found many recent applications in the study of DNA knotting. This proposal also contains several explicit directions for student research, at both undergraduate and graduate levels, and will support the PI's teaching and mentoring activities.Over the last couple decades, Floer theory has revolutionized the study of the topology and geometry of smooth manifolds in dimensions 3 and 4. There are multiple distinct Floer theories for 3-manifolds defined using techniques from quite different branches of mathematics. Amazingly several of these theories appear to give the same information, indicating deep connections between fields like symplectic geometry and gauge theory. These connections have lead to beautiful results like Taubes' proof of the Weinstein conjecture. Despite remarkable progress in understanding these connections, one wonders whether there is a simpler, unifying explanation. Indeed, a fundamental open problem and a distant lodestar for the PI's proposal is to axiomatize Floer theory. One approach to this and to the related problem of computing Floer theories is to develop invariants of bordered 3-manifolds and a means of computing the invariant of a closed manifold from those of its pieces. The first such invariant was defined by Lipshitz, Ozsvath, and Szabo in Heegaard Floer homology. The PI's proposed bordered monopole Floer theory is novel in that it may provide methods for computing invariants of 4-manifolds as well (it is defined in the HM-from, "master" version, of monopole Floer homology), and can likely be ported to other Floer theories (an important ingredient for an approach to axiomatization). This proposal will support the further development of this theory. A closely related aspect of the proposal involves fleshing out connections between Floer theory and Khovanov homology, an invariant of links motivated by representation theory. Such connections were instrumental, for example, in Kronheimer and Mrowka's celebrated proof that Khovanov homology detects the unknot. Often these connections arise in the form of a spectral sequence from Khovanov homology and Floer thoery. A goal of this project is to better understand these spectral sequences. The PI proposes a means for combinatorially computing their terms and for proving that the sequences are functorial with respect to link cobordisms. The hope is to use these combinatorial methods and functoriality to better understand Floer invariants in dimensions 3 and 4. A final, complementary aspect of this proposal involves defining new invariants of contact structures. The PI has done so in recent work, using instanton and monopole Floer homology. An exciting future project is to use the instanton Floer invariant to uncover hitherto unexplored links between the Stein and symplectic fillings of a contact 3-manifold and its fundamental group.

拟议的项目涉及研究 3 维和 4 维流形上的几何结构。了解这些空间和结构对于理解宏观宇宙的形状至关重要，PI 打算研究的许多理论都在物理学中得到应用和起源。 PI 提案的一个主要目标是开发可用于统一研究 3 维和 4 维流形的几个现有重要工具的方法。这样的统一可能会揭示完全不同的数学领域之间有趣的联系。 PI 还计划使用其中一些工具以新的方式研究接触结构和结。两者都用作研究 3 维和 4 维流形的探针。接触结构在经典力学、热力学、动力系统和液晶研究中也很重要，而结理论最近在 DNA 打结研究中得到了许多应用。该提案还包含本科生和研究生水平的学生研究的几个明确方向，并将支持 PI 的教学和指导活动。在过去的几十年中，Floer 理论彻底改变了 3 维和 4 维光滑流形的拓扑和几何研究。对于 3 流形，有多种不同的 Floer 理论使用来自不同数学分支的技术定义。令人惊讶的是，其中一些理论似乎给出了相同的信息，表明辛几何和规范理论等领域之间存在深刻的联系。这些联系带来了美好的结果，比如陶布斯对韦恩斯坦猜想的证明。尽管在理解这些联系方面取得了显着进展，但人们想知道是否存在一种更简单、统一的解释。事实上，PI 提议的一个基本的开放问题和遥远的指南针是使弗洛尔理论公理化。解决这个问题以及计算 Floer 理论的相关问题的一种方法是开发有边界 3 流形的不变量以及从闭流形的各个部分计算闭流形的不变量的方法。第一个这样的不变量是由 Lipshitz、Ozsvath 和 Szabo 在 Heegaard Floer 同调中定义的。 PI 提出的有界单极 Floer 理论很新颖，因为它也可以提供计算 4-流形不变量的方法（它是在单极 Floer 同调的 HM-from“主”版本中定义的），并且很可能移植到其他 Floer 理论（公理化方法的重要组成部分）。该提议将支持该理论的进一步发展。该提案的一个密切相关的方面涉及充实弗洛尔理论和霍瓦诺夫同源性之间的联系，霍瓦诺夫同调性是由表示理论驱动的联系不变量。例如，克朗海默和姆罗卡证明了霍瓦诺夫同源性可以检测到未知结，这种联系发挥了重要作用。这些联系通常以霍瓦诺夫同调和弗洛尔理论的光谱序列的形式出现。该项目的目标是更好地理解这些光谱序列。 PI 提出了一种方法来组合计算它们的项并证明序列对于链接共边而言是函数的。希望使用这些组合方法和函数性来更好地理解维度 3 和维度 4 中的 Floer 不变量。该提案的最后一个补充方面涉及定义接触结构的新不变量。 PI 在最近的工作中使用瞬子和单极弗洛尔同源性来做到这一点。一个令人兴奋的未来项目是使用瞬子 Floer 不变量来揭示接触 3 流形及其基本群的 Stein 和辛填充之间迄今为止尚未探索的联系。