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

University Of Iowa, Iowa City IA

Investigators

Abstract

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.