FRG: Collaborative Research: Trisections -- New Directions in Low-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，如具有无数复杂关节的机器人的所有可能构型的空间。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，当通过微积分的镜头观察不同维度时，不同维度的表现非常不同。最令人惊讶的是，基本问题已经在小于或大于四个维度的维度上得到了解决，但却顽固地抵抗着我们实际生活的时空中的攻击。这个项目汇集了一群拥有不同技能和经验的研究人员，通过利用如何将(Trisect)四维空间分解为基本构建块的关键新想法，帮助解决平滑四维拓扑中的这些基本问题。特别是，对三分块的研究可以将许多成功的想法从三维拓扑输出到四维拓扑。除了研究四维空间本身，研究人员还将研究低维空间可以嵌入四维空间的方法，类似于研究节点作为三维空间中圆的嵌入。利用这些工具和类比，这个重点研究小组旨在开发新的方法来区分四维对象，新的四维结构，以及四维结果在其他环境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神秘的问题之一，正如Poincare和Schoenfys猜想等公开问题所证明的那样，这些问题已经在除四维之外的所有维度都得到了解决。这个专注的研究小组旨在通过开发一个关于四个流形的惊人的新视角来为这个重要的研究领域注入新的活力：每个四个流形分解成三个简单的部分，并且这个三分式是独特的，直到自然稳定。这一设置与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极其丰富的理论，如接触拓扑、Heegaard Floer同调、Heegaard分裂和桥分裂、Khovanov同调、Dehn手术、曲线复形和稀疏位置。这些想法现在有可能与三分法理论相互作用。这个项目的重点是将这些联系发展成解决四维拓扑中重要问题的综合理论。

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.