Mathematical Sciences: Essential Laminations and the Topology of 3-Manifolds

这个项目的主要目标是探索3流形的拓扑结构和它所能支持的层和叶的类型之间的相互关系。研究者将研究2球上群体行为的性质及其在3球上的延伸。他计划考虑的主要问题如下。(1)若M不可约，N是层积的，且M与N是同胚等价的，它们是否同胚？(2)如果M是层合的，那么同伦同胚是同位素吗？(3)什么流形有必要的层合？(4)如果G是一个2球收敛群，它扩展到3球上的一个适当的不连续作用，该扩展是否唯一到共轭？三维流形的研究得益于这样一个事实，即我们生活在这样一个空间中的经验赋予我们一种强烈的直觉，即什么能发生，什么不能发生。研究高维流形的拓扑结构必然依赖于可用的代数机制来辅助它，但在三维中，我们也有一个直接的感知途径。因此，令人惊讶的是，一些具有自然推广到更高维度的问题已经在这些更高维度的情况下得到了回答，而激发它们的三维情况仍然是不容置疑的。庞加莱关于球体的一个经典猜想就是这种现象最臭名昭著的例子。这违反了我们对三维几何直觉所赋予的优势的自然高估，似乎是什么在起作用？这至少部分是对直觉的天真信仰和对计算的不信任，但这也是一个事实，即已知的计算方法在拥有可操作的额外维度的奢侈条件下表现得最好。它们在只有三维空间的情况下显得有些局促。在这种情况下，我们可以看到研究者利用薄片作为一种特别适应三维环境的工具，以及我们对它的直觉，构成了一个受欢迎的、非常有前途的发展。

The main goal of this project is to explore the interrelationships between the topology of a 3-manifold and the types of laminations and foliations it can support. The investigator will look into the nature of group actions on the 2- sphere and their extensions into the 3-ball. The main questions he plans to consider are the following. (1) If M is irreducible, N is laminated, and M and N are homotopy equivalent, are they homeomorphic? (2) If M is laminated, then are homotopic homeomorphisms isotopic? (3) What manifolds have essential laminations? (4) If G is a 2-sphere convergence group which extends to a properly discontinuous action on the 3-ball, is that extension unique up to conjugation? The study of 3-dimensional manifolds is aided by the fact that our experience of living in such a space endows us with a strong intuition for what can and cannot take place. Studying the topology of higher dimensional manifolds is necessarily dependent upon the algebraic machinery available to assist in it, but in 3 dimensions we also have at our disposal a direct avenue of perception. It is therefore surprising to learn that some questions that have natural generalizations to higher dimensions have been answered already for these higher cases, while the 3- dimensional case that inspired them remains obdurately unassailable. There is a classical conjecture of Poincare about spheres that is the most notorious example of this phenomenon. What appears to be at work in thus violating our natural over- estimate of the advantage that geometric intuition should confer in 3 dimensions? It is at least partly a naive faith in intuition and mistrust of computation, but it is also the fact that the known methods of computation perform best with the luxury of excess dimensions in which to maneuver. They are somehow cramped in the presence of only three dimensions. In this setting one can see that the investigator's exploitation of laminations as a tool particularly adapted to the 3-dimensional environment and our intuition about it constitutes a welcome and enormously promising development.