Dehn Surgery, Four-Manifolds, and Symplectic Topology

Dehn 手术、四流形和辛拓扑

• 批准号: 1709702
• 负责人: Tye Lidman
• 金额: $ 18万
• 依托单位: North Carolina State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-07-01 至 2021-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1709702&HistoricalAwards=false
• 关键词: Dehn; Surgery; Four; Manifolds; Symplectic

Topology is an area of mathematics that studies the intrinsic shapes of objects, such as a hula hoop, a massive data set, our universe, or a strand of DNA. For example, topology can be used to measure how biological processes change the shape of our DNA. Surprisingly, these shape changes can be analyzed through three- and four-dimensional manifolds, fundamental shapes in topology that appear throughout mathematics and physics. This leads to the fundamental problem of trying to completely understand and classify these three- and four-dimensional manifolds. While we understand many aspects of three-dimensional objects, in dimension four we are still mostly in the dark. One effective tool for studying these objects is called Floer homology, which comes from solving powerful equations from physics. In this project, the PI will use Floer homology to study the complexity of these shapes by analyzing the configurations of knotted loops and surfaces inside of them. The potential outcome of this project will be to strengthen connections between knots and three-/four-dimensional manifolds while discovering new structure in the topology of four-dimensional manifolds. This project will then be disseminated to the public through a collaboration with Black Box Dance Theater, a North Carolina non-profit dance company. By way of dance performances and public workshops, this will present to the public the fundamental notions of topology used in the proposed project. This joint project will also work to improve public perception of mathematics and enhance connections between STEM and the arts.Knots can be used to produce new three- and four-manifolds by an operation called Dehn surgery, where one removes a tubular neighborhood and reglues via a homeomorphism. Many invariants and tools in low-dimensional topology, especially Floer homology, are particularly well-behaved under Dehn surgery, and the PI will use these tools to improve our understanding of three- and four-manifolds and further their connections with knot theory. Three major goals of this project are to: 1) construct homology three-spheres which cannot be obtained by Dehn surgery on a two-component link, 2) find new constraints on the algebraic topology of four-manifolds admitting symplectic structures, and 3) further explore the algebraic structure of the concordance group of knots in homology spheres modulo concordance in homology cobordisms. Two potential outcomes would be a better understanding of the complexity of four-manifolds with boundary measured in terms of their handlebody structures and a stronger connection between properties of a knot and the topology of its Dehn surgeries.

拓扑学是数学的一个领域，它研究物体的内在形状，例如呼拉圈，大量数据集，我们的宇宙或DNA链。 例如，拓扑学可以用来测量生物过程如何改变我们DNA的形状。 令人惊讶的是，这些形状的变化可以通过三维和四维流形来分析，这些流形是拓扑学中的基本形状，出现在数学和物理学中。 这导致了试图完全理解和分类这些三维和四维流形的基本问题。 虽然我们了解三维物体的许多方面，但在四维中，我们仍然处于黑暗之中。 研究这些物体的一个有效工具是Floer同源性，它来自于求解物理学中的强大方程。 在这个项目中，PI将使用Floer同源性来研究这些形状的复杂性，通过分析打结环和它们内部的表面的配置。 该项目的潜在成果将是加强结和三维/四维流形之间的联系，同时发现四维流形拓扑中的新结构。 该项目随后将通过与北卡罗来纳州一家非营利舞蹈公司黑盒舞蹈剧院的合作向公众传播。 通过舞蹈表演和公共研讨会，这将向公众展示拟议项目中使用的拓扑学的基本概念。 这个联合项目还将致力于提高公众对数学的认知，加强STEM和艺术之间的联系。通过一种称为Dehn手术的操作，可以使用结来产生新的三维和四维流形，其中一个通过同胚去除管状邻域并重新调整。 低维拓扑中的许多不变量和工具，特别是Floer同调，在Dehn手术下表现得特别好，PI将使用这些工具来提高我们对三维和四维流形的理解，并进一步将它们与纽结理论联系起来。 本项目的主要目标是：1）构造不能通过Dehn外科手术在二分量连杆上获得的同调三球面; 2）发现允许辛结构的四流形的代数拓扑的新约束; 3）进一步探索同调球面中纽结协调群的代数结构。 两个潜在的结果将是更好地理解四流形的复杂性，其边界是根据其体结构来测量的，以及结的属性与其Dehn手术的拓扑结构之间的更强联系。

项目成果

Distance one lens space fillings and band surgery on the trefoil knot

• DOI: 10.2140/agt.2019.19.2439
• 发表时间: 2017-10
• 期刊: Algebraic & Geometric Topology
• 影响因子: 0.7
• 作者: Tye Lidman;Allison H. Moore;M. Vázquez

SIMPLY CONNECTED, SPINELESS 4-MANIFOLDS

简单连接、无骨架 4 歧管

• DOI: 10.1017/fms.2019.11
• 发表时间: 2019
• 期刊: Sigma
• 作者: LEVINE, ADAM SIMON;LIDMAN, TYE
• 通讯作者: LIDMAN, TYE

Lagrangian Cobordisms and Legendrian Invariants in Knot Floer Homology

结花同调中的拉格朗日配边和勒让德不变量

• DOI: 10.1307/mmj/20195786
• 发表时间: 2021
• 期刊: Michigan Mathematical Journal
• 影响因子: 0.9
• 作者: Baldwin, John A.;Lidman, Tye;Wong, C.-M. Michael
• 通讯作者: Wong, C.-M. Michael

APPLICATIONS OF INVOLUTIVE HEEGAARD FLOER HOMOLOGY

内卷Heegarard FLOER同源性的应用

• DOI: 10.1017/s147474801900015x
• 发表时间: 2019
• 期刊: Journal of the Institute of Mathematics of Jussieu
• 影响因子: 0.9
• 作者: Hendricks, Kristen;Hom, Jennifer;Lidman, Tye
• 通讯作者: Lidman, Tye

Khovanov homology detects the figure‐eight knot

霍瓦诺夫同源性检测数字——八结

• DOI: 10.1112/blms.12467
• 发表时间: 2021
• 期刊: Bulletin of the London Mathematical Society
• 影响因子: 0.9
• 作者: Baldwin, John A.;Dowlin, Nathan;Levine, Adam Simon;Lidman, Tye;Sazdanovic, Radmila
• 通讯作者: Sazdanovic, Radmila