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

基本信息

批准号：
1664578
负责人：
Alexander Zupan
金额：
$ 18.71万
依托单位：
University of Nebraska-Lincoln
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=1664578&HistoricalAwards=false
关键词：
FRG Collaborative Research Trisections New

项目摘要

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.

拓扑是从广义上讲空间的研究，从我们居住的三维空间和四维时空到非常高维空间，例如具有许多复杂关节的机器人的所有可能配置的空间。光滑的拓扑使用微积分的工具来理解和分类这些空间。有趣的是，当通过微积分镜头看时，不同的维度的行为会大不相同。最令人惊讶的是，基础问题在少于和大于四个的维度上解决了，但在我们实际生活的时空中顽固地抵抗攻击。该项目通过利用有关如何分解（Trisect）四维空间进入基本构件的关键新想法，将一群具有多种技能和经验的研究人员汇集在一起，以帮助解决这些基本问题。特别是，对三触角的研究允许将许多成功的想法从三维拓扑导出到四维拓扑。除了对四维空间本身的研究外，研究人员还将研究较低维空间可以嵌入尺寸四的方式，类似于将结作为圆圈嵌入三维空间的嵌入方式。 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在四个以外的所有维度上解决。这个重点的研究小组旨在通过利用对四个manifolds的惊人新观点来将新的生命注入这一重要的研究领域：每四个manifold分解为三个简单的部分，而这种三角则是独特的，可以自然稳定。该设置与Heegaard分裂的三维理论完全平行，设置了桌子，以在三个和四个方面之间进行有趣而宝贵的思想交换。在过去的几十年中，许多极为丰富的理论在低维拓扑中发展了，例如接触拓扑，Heegaard Floer同源性，Heegaard分裂和桥梁分裂，Khovanov同源性，Dehn手术，曲线复合物和较薄的位置。这些想法现在有可能与三分裂理论互动。该项目的重点是将这些联系的发展发展为一个综合理论，该理论解决了四维拓扑中的重要问题。