Problems in Low-Dimensional Topology

Long计划在低维和高维双曲流形的几何、代数和结构方面从事各种新旧项目。这些问题来自不同的领域：一类问题来自面群的问题，它不仅在低维拓扑中很有趣，而且在数论的某些部分也很有趣。作为一个例子，我们将研究双曲3-轨道是否必须包含一个表面群，或者是否可以增加双曲束的同调性。另一类问题来自于对性格变异的研究。在更高的维度，提议者将考虑结构以及来自物理学的几何边界问题。最后，申请人将从事与从双曲流形的算术产生的广泛问题相关的项目。如果一个空间是由小块组成的，那么它就被称为3流形，这些小块都“像”我们生活的普通三维空间。这个建议指向了最重要的一类3-流形的结构研究，即所谓的双曲3-流形。这类课程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，但至少有一组人认为宇宙是皮卡德轨道——一个双曲轨道。提出者的结果限制了在宇宙的某些物理模型中可能出现的流形的种类。他还打算研究一个著名的古老猜想，即哪些二维物体可以存在于这些三维物体中。

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.