RUI: New Approaches to Understanding the Four-Sphere

RUI：理解四个领域的新方法

• 批准号: 2006029
• 负责人: Jeffrey Meier
• 金额: $ 14.87万
• 依托单位: Western Washington University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-09-01 至 2024-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2006029&HistoricalAwards=false
• 关键词: RUI; New; Approaches; Understanding; Four

We inhabit a three-dimensional space. This means that, locally, one can move in three independent directions: left/right, forward/backward, and up/down. However, the global nature of our three-dimensional space is not known. For example, it is not known whether our universe extends infinitely far in every direction or if it curls back on itself. To complicate things, it becomes more natural to view the space we inhabit as a four-dimensional space by including time. The goal of low-dimensional topology is to study and classify three-dimensional and four-dimensional spaces. One important aspect of this program is to understand the topology (or "shape") of the four-dimensional sphere, which is a four-dimensional analog the spherical surface of a solid ball. Amazingly, despite over one hundred years of study, very little is known about the topology of the four-dimensional sphere. In contrast, the analogous objects in dimensions greater than or less than four are much better understood. The goal of this research project is to apply novel techniques and approaches better understand four-dimensional sphere. This work will help to inform and advance the general study of four-dimensional spaces, such as the spacetime expanse that we inhabit. The award provides funds to support research training for undergraduate students.Perhaps the most important open problem in low-dimensional topology, the Smooth Poincare Conjecture, ask whether any smooth four-manifold with the homotopy-type of the four-sphere is equivalent to the four-sphere. This problem has been open for more than 100 years, yet has seen almost no progress. Rather than approach this problem directly, this research program aims to study the smooth topology from four distinct, but related, angles. One angle is to study four-manifolds with the homotopy-type of the four-sphere that admit simple handle-decompositions. Recently, the PI and Zupan gave new results in this setting, offering the first general progress towards the Poincare Conjecture in 30 years. The first aim of this research is to build on this work to obtain results that apply in even broader settings. In another direction, the introduction of theory of trisections by Gay and Kirby in 2016, and the subsequent development and expansion, especially by the PI and Zupan, has ushered in a new era in four-manifold topology, with applications emerging to many interesting aspects of low-dimensional topology. This research project aims to advance and refine these theories, broaden their applicability, and bring to fruition many applications and connections. In particular, the PI will investigate what sort of trisections the four-sphere can admit. In the process, connections will be established with important open problems in low-dimensional topology, such as the Andrews-Curtis, Generalized Property R, and Slice-Ribbon Conjectures. The project will include an investigation of trisections of genus three, where the additivity of trisection genus will be explored in search for exotic copies of familiar four-manifolds. Finally, the PI will employ the theory of bridge trisections to study knotted surfaces in the four-sphere. This theory was introduced by the PI and Zupan and offers a powerful new way to study knot theory in dimension four. In particular, the PI will work to give connections between the braid group and potentially exotic knotted spheres.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

我们居住在一个三维空间。 这意味着，在本地，一个人可以在三个独立的方向上移动：左/右，前/后，和上/下。 然而，我们的三维空间的全球性质是未知的。 例如，我们不知道我们的宇宙是否在每个方向上无限延伸，或者它是否卷曲在自己身上。 更复杂的是，通过包括时间，将我们居住的空间视为四维空间变得更加自然。 低维拓扑学的目标是研究和分类三维和四维空间。 这个程序的一个重要方面是理解四维球体的拓扑（或“形状”），这是一个四维模拟的球形表面的固体球。 令人惊讶的是，尽管研究了一百多年，对四维球体的拓扑结构知之甚少。 相比之下，在大于或小于4维的维度上的类似物体更容易理解。 本研究项目的目标是应用新的技术和方法更好地理解四维球体。 这项工作将有助于通知和推进四维空间的一般研究，例如我们居住的时空广阔。该奖项提供资金支持本科生的研究培训。也许是低维拓扑学中最重要的开放问题，光滑庞加莱猜想，问是否任何光滑四流形与同伦型的四球是等价的四球。这个问题已经存在了100多年，但几乎没有取得任何进展。 而不是直接接近这个问题，这个研究计划的目的是从四个不同的，但相关的角度来研究光滑拓扑。一个角度是研究具有四球面同伦型的四流形，它允许简单的π-分解。最近，PI和Zupan在这个背景下给出了新的结果，提供了30年来对庞加莱猜想的第一个普遍进展。这项研究的第一个目的是在这项工作的基础上，获得适用于更广泛环境的结果。 在另一个方向上，Gay和Kirby在2016年引入了三分理论，以及随后的发展和扩展，特别是PI和Zupan，开创了四流形拓扑的新时代，应用出现在低维拓扑的许多有趣方面。本研究项目旨在推进和完善这些理论，扩大其适用性，并实现许多应用和联系。特别是，PI将研究什么样的三分四球可以承认。在这个过程中，将建立与低维拓扑学中重要的开放问题的联系，例如Andrews-Curtis，广义性质R和切片带猜想。该项目将包括研究三个亏格的三分性，其中三分亏格的可加性将被探索，以寻找熟悉的四维流形的奇异副本。 最后，PI将采用桥三分性理论来研究四球面中的纽结曲面。这个理论是由PI和Zupan提出的，它为四维纽结理论的研究提供了一种强有力的新方法。特别是，PI将致力于提供编织群和潜在的外来打结球体之间的连接。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

Branched covers bounding rational homology balls

限制有理同源球的分支覆盖

• DOI: 10.2140/agt.2021.21.3569
• 发表时间: 2021
• 期刊: Algebraic & Geometric Topology
• 影响因子: 0.7
• 作者: Aceto, Paolo;Meier, Jeffrey;Miller, Allison N;Miller, Maggie;Park, JungHwan;Stipsicz, András I
• 通讯作者: Stipsicz, András I

Bridge trisections and classical knotted surface theory

桥三等分和经典结面理论

• DOI: 10.2140/pjm.2022.319.343
• 发表时间: 2022
• 期刊: Pacific Journal of Mathematics
• 影响因子: 0.6
• 作者: Joseph, Jason;Meier, Jeffrey;Miller, Maggie;Zupan, Alexander
• 通讯作者: Zupan, Alexander