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

该项目的主要目标是探索 3-流形的拓扑与 它可以支持的叠片和叶理的类型。 的 调查人员将调查2- 3号公路上集体行动的性质 球体及其延伸到3球。 的主要问题 他计划考虑以下几点。 (1)如果M不可约， N是层合的，M和N是同伦等价的，是吗？ 同胚的？ (2)如果M是层合的，则是同伦的 同胚同位素？ (3)什么样的流形 叠片？ (4)如果G是一个2-球面收敛群， 延伸到3球上的适当不连续动作， 扩展唯一直到共轭？ 三维流形的研究得益于以下事实： 我们在这样一个空间中生活的经历赋予了我们 强烈的直觉告诉我们什么可以发生，什么不可以发生。 研究 高维流形的拓扑必然是 依赖于可用的代数机器来帮助它， 但在三维空间里我们也有一个直接的途径 感知力 因此，令人惊讶的是， 这些问题可以自然地推广到更高的维度 已经回答了这些较高的情况下，而3- 激发他们灵感的三维案例仍然顽固地 无懈可击 庞加莱有一个经典的猜想， 这是这种现象最臭名昭著的例子。 似乎是什么在起作用，从而违反了我们的自然过度- 估计的优势，几何直观应赋予 在三维空间里？ 这至少在一定程度上是一种天真的信念， 直觉和对计算的不信任，但事实上， 已知的计算方法最适合于 在其中操纵的多余尺寸的奢侈。 他们是 在只有三维的空间里显得有些局促。 在 这个设置可以看出，调查员的剥削 叠片作为特别适用于三维的工具， 环境和我们对它的直觉构成了一个欢迎， 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.