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: 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

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.

拓扑学是对广义空间的研究，从我们生活的三维空间和四维时空到非常高维的空间，例如具有许多复杂关节的机器人的所有可能配置的空间。光滑拓扑学使用微积分的工具来理解和分类这些空间;有趣的是，当通过微积分的透镜观察时，不同的维度表现得非常不同。最令人惊讶的是，在小于和大于四维的维度上，基础问题已经得到解决，但在我们实际生活的时空中，这些问题却顽强地抵抗着攻击。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（三等分）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，三分性的研究可以将许多成功的想法从三维拓扑导出到四维拓扑。沿着四维空间的研究，研究者们也将研究低维空间嵌入四维空间的方式，类似于研究三维空间中的圆嵌入的结。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的拓扑和几何中的新应用。四维流形的光滑拓扑仍然是拓扑学中最大的谜团之一，正如庞加莱和舍恩夫利的开放问题所证明的那样，除了四维以外的所有维度都已经解决了。 这个专注的研究小组旨在通过利用四维流形的惊人新视角为这一重要的研究领域注入新的活力：每个四维流形都可以分解为三个简单的部分，并且这种三分法在自然稳定化之前是独一无二的。该设置与Heegaard分裂的三维理论完全相似，为三维和四维之间有趣且有价值的思想交流奠定了基础。 在过去的几十年里，在低维拓扑中已经发展了许多非常丰富的理论，如接触拓扑、Heegaard Floer同调、Heegaard分裂和桥分裂、Khovanov同调、Dehn手术、曲线复形和薄位置。这些想法现在有可能与三分理论相互作用。这个项目的重点是发展这些连接成一个全面的理论，解决四维拓扑中的重要问题。

项目成果

Characterizing Dehn surgeries on links via trisections

通过三等分在链接上表征 Dehn 手术

• DOI: 10.1073/pnas.1717187115
• 发表时间: 2018
• 期刊: Proceedings of the National Academy of Sciences
• 作者: Meier, Jeffrey;Zupan, Alexander
• 通讯作者: Zupan, Alexander

Bridge trisections of knotted surfaces in 4-manifolds

• DOI: 10.1073/pnas.1717171115
• 发表时间: 2017-10
• 期刊: Proceedings of the National Academy of Sciences
• 作者: J. Meier;Alexander Zupan

The Powell conjecture and reducing sphere complexes

鲍威尔猜想和约化球复形

• DOI: 10.1112/jlms.12272
• 发表时间: 2019
• 期刊: Journal of the London Mathematical Society
• 作者: Zupan, Alexander