Interactions of 3- and 4-Dimensional Topology

拓扑学是研究抽象空间的性质，这些空间的性质不会因弯曲、拉伸和扭曲等变形而改变，但不会断裂，低维拓扑学涉及局部看起来像2维、3维或4维欧几里得空间的空间（称为流形）。 低维拓扑在现实世界中有着广泛的应用，包括与理论物理和量子计算的深刻联系。 此外，拓扑学在数据科学中也有应用;研究人员了解如何将大型数据集解释为拓扑空间，以便通过研究该数据的拓扑形状来获得见解。 虽然我们的世界本质上是三维的，但三维物体在一段时间内的演变产生了四维空间，其中时间构成了第四个自由度。 在过去的几十年里，三维拓扑学的研究活动激增，在2003年格里戈里·佩雷尔曼证明庞加莱猜想时达到顶峰，庞加莱猜想是克莱数学研究所提出的七个千年难题中唯一解决的问题。相比之下，四维流形的拓扑已经成为一个越来越活跃的研究领域，许多基本问题仍然悬而未决。 这个研究项目涉及从三维的想法，以发现这些四维问题的新方法。该奖项提供资金支持研究生的研究。该奖项还将支持大平原联盟（Great Plains Alliance），该计划由PI发起，旨在为研究生提供在附近机构演讲的机会。四维拓扑学中的两个基本开放问题是光滑四维庞加莱猜想，该猜想断言每个等价于标准四维球面的闭四维同伦都是标准四维球面的同胚，以及切片带猜想，其假定在4-球中界定嵌入盘的任何结也在3-球中界定浸没带状盘。 这个项目描述了一组不同的问题交织的想法，从纽结理论，3流形，光滑4流形拓扑。 PI之前的工作包括在这一领域的两个主要的合作贡献，证明了一个大的同伦4-球面族是标准光滑4-球面的同胚，并在3维结构和4-流形三分性之间建立了深远的联系。 当前的工作主要涉及四个方面的问题。 首先，PI将表征满足切片和带状之间的中间条件的结族。 第二，PI将寻求一种新的方法来解决鲍威尔猜想，关于3-球面的群同态的生成，留下一个Heegaard曲面不变。 第三，PI将使用桥三等分来解决与表示具有光滑嵌入的2-球面的4-流形的第二同调类相关的Kirby问题，最后，PI将从3-流形的Heegaard分裂到4-流形的三等分的想法，以便更好地理解奇异4-流形的处理分解。 这项工作的基本主题是同时整合三维和四维技术的方法，以获得对长期存在的开放问题的新见解。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Topology is the study of abstract spaces with properties that are unchanged by deformations such as bending, stretching, and twisting, but not breaking, and low-dimensional topology involves spaces (called manifolds) that locally look like 2-, 3-, or 4-dimensional Euclidean space. Low-dimensional topology has a wide range of real-world applications, including deep connections to theoretical physics and quantum computing. In addition, topology has applications to data science; researchers understand how to interpret a large data set as a topological space in order to gain insights by studying the topological shape of that data. While our world is 3-dimensional in nature, the evolution of a 3-dimensional object over a period of time gives rise to a 4-dimensional space, in which time constitutes a fourth degree of freedom. Topology in dimension three has seen an explosion of activity over the last several decades, reaching a peak with Grigory Perelman's proof of the Poincaré conjecture in 2003, the only resolved problem of the seven Millennium Problems posed by the Clay Mathematics Institute. In contrast, the topology of 4-dimensional manifolds has become an increasingly active area of research, with many fundamental problems still open. This research project involves adapting ideas from dimension three to discover novel approaches to these 4-dimensional problems. The award provides funds to support research by graduate students. The award will also support the Great Plains Alliance, a program initiated by the PI to connect graduate students with speaking opportunities at nearby institutions.Two foundational open problems in 4-manifold topology are the smooth 4-dimensional Poincaré Conjecture, which asserts that every closed 4-manifold homotopy equivalent to the standard 4-sphere is diffeomorphic to the standard 4-sphere, and the slice-ribbon conjecture, which posits that any knot bounding an embedded disk in the 4-ball also bounds an immersed ribbon disk in the 3-sphere. This project describes a varied set of problems interweaving ideas from knot theory, 3-manifolds, and smooth 4-manifold topology. The PI’s prior work includes two major collaborative contributions in this area, demonstrating that a large family of homotopy 4-spheres is diffeomorphic to the standard smooth 4-sphere, and developing far-reaching connections between 3-dimensional structures and 4-manifold trisections. The current work involves four main problems. First, the PI will characterize a family of knots satisfying an intermediate condition between being slice and being ribbon. Second, the PI will pursue a new approach to the Powell conjecture, concerning the generation of the group of diffeomorphisms of the 3-sphere leaving a Heegaard surface invariant. Third, the PI will use bridge trisections to address a Kirby problem related to representing second homology classes of 4-manifolds with smoothly embedded 2-spheres, and finally, the PI will adapt ideas from Heegaard splittings of 3-manifolds to trisections of 4-manifolds in order to better understand handle decompositions of exotic 4-manifolds. The underlying theme of this work is an approach that simultaneously integrates 3- and 4-dimensional techniques in order to obtain new insights into long-standing open problems.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.