FRG: Collaborative Research: Trisections -- New Directions in Low-Dimensional Topology

FRG：协作研究：三等分——低维拓扑的新方向

• 批准号: 1664587
• 负责人: Abigail Thompson
• 金额: $ 25.93万
• 依托单位: University of California-Davis
• 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=1664587&HistoricalAwards=false
• 关键词: FRG; Collaborative; Research; Trisections; New

Topology is the study of spaces in a broad sense, from the three-dimensional space and four-dimensional space-time in which we live, to very high dimensional spaces such as the space of all possible configurations of a robot with numerous complicated joints. Smooth topology uses the tools of calculus to understand and classify these spaces; intriguingly, different dimensions behave very differently when looked at through the lens of calculus. Most surprisingly, foundational problems have been solved in dimensions less than and greater than four, but stubbornly resist attack in the space-time in which we actually live. This project brings together a group of researchers, with a diverse set of skills and experience, to help tackle these fundamental problems in smooth four-dimensional topology, by utilizing a key new idea about how to decompose (trisect) four-dimensional spaces into elementary building blocks. In particular, the study of trisections allows exporting many successful ideas from three-dimensional topology to four-dimensional topology. Along with the study of four-dimensional spaces in their own right, the investigators will also study the ways in which lower-dimensional spaces can be embedded in dimension four, in analogy with the study of knots as embeddings of circles in three-dimensional space. Using these tools and analogies, this focused research group aims to develop new ways to distinguish four-dimensional objects, new four-dimensional constructions, and new applications of four-dimensional results to topology and geometry in other settings and dimensions.The smooth topology of four-dimensional manifolds remains one of the greatest mysteries in topology, as evidenced by open questions such as the Poincare and Schoenflies conjectures, which have been solved in all dimensions other than four. This focused research group aims to breathe new life into this important field of study by exploiting a striking new perspective on four-manifolds: Every four-manifold decomposes into three simple pieces, and this trisection is unique up to a natural stabilization. The setup exactly parallels the three-dimensional theory of Heegaard splittings, setting the table for an interesting and valuable exchange of ideas between dimensions three and four. Many extremely rich theories have been developed over the last few decades in low-dimensional topology, such as contact topology, Heegaard Floer homology, Heegaard splittings and bridge splittings, Khovanov homology, Dehn surgery, curve complexes, and thin position. These ideas now have the potential to interact with the theory of trisections. The focus of this project is the development of these connections into a comprehensive theory that solves important problems in four-dimensional topology.

拓扑学是广义上对空间的研究，从我们生活的三维空间和四维时空，到非常高维的空间，如具有许多复杂关节的机器人的所有可能构型的空间。光滑拓扑使用微积分工具来理解和分类这些空间；有趣的是，从微积分的角度来看，不同的维度表现得非常不同。最令人惊讶的是，基本问题已经在小于或大于4维的空间中得到了解决，但在我们实际生活的时空中却顽固地抵制攻击。该项目汇集了一组具有不同技能和经验的研究人员，通过利用关于如何将四维空间分解（三分）为基本构建块的关键新思想，帮助解决光滑四维拓扑中的这些基本问题。特别是，对三分线的研究可以将许多成功的思想从三维拓扑导出到四维拓扑。在研究四维空间本身的同时，研究人员还将研究低维空间嵌入四维空间的方法，类似于研究结在三维空间中作为圆的嵌入。利用这些工具和类比，这个专注的研究小组旨在开发新的方法来区分四维物体，新的四维结构，以及四维结果在其他设置和维度中的拓扑和几何的新应用。四维流形的光滑拓扑仍然是拓扑学中最大的谜团之一，正如庞加莱和舍恩菲斯猜想等开放问题所证明的那样，这些问题已经在除四维以外的所有维度上得到了解决。这个专注的研究小组旨在通过开发四流形的一个引人注目的新视角，为这一重要的研究领域注入新的活力：每个四流形分解成三个简单的部分，这种三分割是独特的，直到自然稳定。这种设置与heeggaard分裂的三维理论完全相似，为三维和四维之间有趣而有价值的思想交流奠定了基础。在过去的几十年里，在低维拓扑中发展了许多非常丰富的理论，如接触拓扑、Heegaard flower同调、Heegaard分裂和桥分裂、Khovanov同调、Dehn手术、曲线复形和薄位置。这些观点现在有可能与三分线理论相互作用。这个项目的重点是将这些连接发展成一个全面的理论，解决四维拓扑中的重要问题。

项目成果

Trisections and link surgeries

三等分和连接手术

• DOI: 10.53733/94
• 发表时间: 2021
• 期刊: New Zealand Journal of Mathematics
• 作者: Kirby, Robion;Thompson, Abigail
• 通讯作者: Thompson, Abigail

A new invariant of 4-manifolds

一个新的4流形不变量

• DOI: 10.1073/pnas.1718953115
• 发表时间: 2018
• 期刊: Proceedings of the National Academy of Sciences
• 作者: Kirby, Robion;Thompson, Abigail
• 通讯作者: Thompson, Abigail