Problems in Low-Dimensional Topology

Long计划在低维和高维双曲流形的几何、代数和结构方面开展各种新旧项目。这些问题来自不同的领域：一类问题来自于关于表面群的问题，这些问题不仅对低维拓扑感兴趣，而且在数论的某些部分也有兴趣。作为一个例子，我们将研究双曲3-orbilold是否必须包含一个表面群，或者是否可以增加双曲丛的同调。另一类问题来自对性格变化的研究。在更高的维度中，提出者将考虑结构以及来自物理的几何边界问题。最后，提出者将致力于与双曲流形算术产生的广泛问题相关的项目。如果一个空间由小块组成，而这些小块都与我们所居住的普通3维空间“相似”，那么它被称为3-流形。这个建议指向最重要的一类3-流形，即所谓的双曲3-流形的结构研究的各个方面。这一类在数学和物理中无处不在，例如，尽管宇宙的本质尚未得到解决，但至少有一个群体提出，宇宙是皮卡德奥比博尔德--一种双曲奥比博尔德。提出者的结果限制了在某些宇宙物理模型中可能出现的流形的种类。他还打算包含一个著名的古老猜想的工作，关于哪些类型的二维物体可以在这些三维物体内生存。

Long plans to work on various old and new projects in geometric,algebraic and structural aspects of low and high dimensional hyperbolicmanifolds. The problems are drawn from a variety of areas: One family ofproblems comes from questions about surface groups, which are of interest not only in low dimensional topology, but also in certain parts of number theory. As an example, we will investigatewhether hyperbolic 3-orbifolds must contain a surface group, orwhether one can increase the homology of a hyperbolic bundle. Anotherfamily of questions come from the study of character varieties. Inhigher dimensions the proposer will consider constructions as well asgeometrical bordism questions coming from physics. Finally, theproposer will work on projects connected to a broad range of questionsarising from the arithmetic of hyperbolic manifolds.A space is called a 3-manifold if it is made of small chunks all ofwhich are ``like'' the ordinary 3-dimensional space that we live in.This proposal directs itself towards aspects of the structural studyof the most important class of 3-manifolds, the so-called hyperbolic3-manifolds. This class is ubiquitous in mathematics and physics, for example although the nature of the universe has yet to be resolved, there is at least one group who have suggested that the universe is the Picard orbifold - a hyperbolic orbifold. The proposer has results which restrict the sorts of manifolds which can arise in certain physical models of the universe. He also intends to contain work on a famous old conjecture about which sorts of two-dimensional objects canlive inside these 3-dimensional objects.