RUI: New Approaches to Understanding the Four-Sphere

我们居住在一个三维空间中。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、向前/向后和向上/向下。然而，我们的三维空间的全球性质尚不清楚。例如，我们不知道我们的宇宙是向各个方向无限延伸，还是卷曲回来。让事情变得复杂的是，通过将时间包括在内，将我们居住的空间视为四维空间变得更加自然。低维拓扑的目标是对三维和四维空间进行研究和分类。本程序的一个重要方面是了解四维球体的拓扑(或“形状”)，这是一个四维类似于实心球的球面。令人惊讶的是，尽管研究了一百多年，人们对四维球的拓扑结构知之甚少。相比之下，维度大于或小于四的相似物体更容易理解。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。这项工作将有助于提供信息并推动对四维空间的一般研究，例如我们居住的时空范围。该奖项提供资金支持本科生的研究培训。也许低维拓扑学中最重要的公开问题-光滑Poincare猜想-询问任何具有四球同伦型的光滑四流形是否等价于四球面。这个问题已经悬而未决100多年了，但几乎没有任何进展。这个研究项目不是直接探讨这个问题，而是从四个不同但又相互关联的角度来研究光滑拓扑。一个角度是研究具有同伦型的允许简单句柄分解的四球面的四流形。最近，Pi和Zupan在这一背景下给出了新的结果，这是30年来向Poincare猜想提供的第一个普遍进展。这项研究的第一个目的是在这项工作的基础上取得适用于更广泛环境的结果。另一方面，Gay和Kirby在2016年提出的三分理论，以及随后的发展和扩展，特别是Pi和Zupan的发展和扩展，开启了四流形拓扑学的新纪元，并出现了对低维拓扑的许多有趣方面的应用。这一研究项目旨在推进和完善这些理论，扩大它们的适用性，并实现许多应用和联系。特别是，PI将调查四个球体可以允许什么样的三角剖分。在这个过程中，将建立与低维拓扑中重要的公开问题的联系，如Andrews-Curtis猜想、广义性质R和切片-带状猜想。该项目将包括对三分属的三分集的调查，其中将探索三分分属的可加性，以寻找熟悉的四流形的奇异副本。最后，PI将使用桥三分理论来研究四球体中的结点曲面。这一理论是由Pi和Zupan提出的，为研究四维空间的纽结理论提供了一条强有力的新途径。特别是，PI将致力于在辫子群和潜在的异国情调的结球之间建立联系。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.