Surfaces and 4-Manifolds

本研究项目的一个主要部分是关于“大”基本群的四维拓扑外科猜想的有效性。PI计划使用他最近开发的4-流形的链接群理论，在A-B切片问题的背景下建立障碍，这是外科猜想的重新表述。除了开发新的4-流形不变量外，本程序还将阐明拓扑4-流形的低维性质。该项目的另一部分涉及映射类群的量子表示的渐近行为。特别是，目标是使用(2+1)维TQFT所提供的表示来研究映射类群的刚性性质。该项目适合于对局部看起来像通常的欧几里德空间的对象的可能的大规模形状进行分类的一般框架。三维和四维形状的分类是一个特别重要和具有挑战性的问题。到目前为止，在四个维度上的拓扑分类的结果与在更高维度上的发展是平行的。这项研究项目将有助于澄清更高维度的类比是否在总体上适用于具有“大”基本基团的形状。

A major part of this research project concerns the validity of the 4-dimensional topological surgery conjecture for "large" fundamental groups. The PI plans to use his recently developed theory of link groups of 4-manifolds to formulate an obstruction in the context of the A-B slice problem, a reformulation of the surgery conjecture. Besides developing new invariants of 4-manifolds, this program will clarify the low-dimensional nature of topological 4-manifolds. Another part of the project concerns the asymptotic behavior of quantum representations of mapping class groups. The goal, in particular, is to use the representations that are provided by (2+1) dimensional TQFTs to investigate the rigidity properties of mapping class groups.This project fits in the general framework of classifying the possible large scale shapes of objects that locally look like the usual Euclidean space. The classification of three- and four-dimensional shapes is a particularly important and challenging problem. The results to date on topological classification in dimension four have paralleled the developments in higher dimension. This research project will help to clarify whether the analogy with higher dimensions works in general, for shapes with "large" fundamental groups.