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.

该计划涉及研究三维和四维流形的几何结构。理解这些空间和结构对于理解我们宏观宇宙的形状至关重要，而且PI打算研究的许多理论在物理学中都有应用和起源。PI提议的一个主要目标是开发一种方法，可以用来统一研究三维和四维流形的几种现有的重要工具。这样的统一可能会揭示出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。两者都是用来研究三维和四维流形的探针。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结的研究中得到了许多应用。该提案还包含了本科生和研究生阶段学生研究的几个明确方向，并将支持PI的教学和指导活动。在过去的几十年里，花理论已经彻底改变了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，这些理论使用了来自不同数学分支的技术。令人惊讶的是，这些理论中有几个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着深刻的联系。这些联系导致了美妙的结果，比如陶布斯对温斯坦猜想的证明。尽管在理解这些联系方面取得了显著进展，但人们怀疑是否存在更简单、统一的解释。事实上，对于PI的提议来说，一个基本的开放问题和一个遥远的目标是使flower理论公理化。解决这一问题以及计算Floer理论的相关问题的一种方法是发展有边界3流形的不变量，以及从其块的不变量计算封闭流形的不变量的方法。第一个不变量是由Lipshitz， Ozsvath和Szabo在Heegaard flower同调中定义的。PI提出的有边单极花理论是新颖的，因为它可以提供计算4流形不变量的方法（它在单极花同调的“主”版本中定义），并且可能被移植到其他花理论（一种公化方法的重要组成部分）。这一建议将支持这一理论的进一步发展。该提议的一个密切相关的方面包括充实弗洛尔理论和霍瓦诺夫同调之间的联系，霍瓦诺夫同调是由表征理论激发的一种不变量。这种联系是有用的，例如，在Kronheimer和Mrowka著名的证明中，Khovanov同源性检测到解结。这些联系通常以Khovanov同调和Floer理论的谱序列的形式出现。这个项目的目标是更好地理解这些光谱序列。PI提出了一种组合计算它们的项的方法，并证明了序列对于连杆协是泛函的。希望使用这些组合方法和功能来更好地理解维3和维4中的花不变量。最后，本建议的补充方面涉及定义新的接触结构不变量。PI在最近的工作中使用了瞬子和单极子弗洛尔同源性。一个令人兴奋的未来项目是使用瞬时花不变量来揭示接触3流形及其基本群的斯坦和辛填充之间迄今未被探索的联系。