Essential Laminations in 3-Manifolds

小行星9704811 本质叠层是用来探测三维流形结构的大量对象之一。 它们是由于试图同时概括两个这样的经典对象;不可压缩的表面和绷紧的叶理。 该项目涉及研究基本叠层的结构以及它们如何位于3-流形内部，以提取有关“层状”3-流形的信息。 主要目标，研究者将追求的是表明，每一个不可约的3-流形同伦等价于一个层流流形同胚，并表明，层流流形是覆盖的哈肯流形。 他还将从层流的角度研究绷紧的叶理，用这个新的观点来研究这些老天体。 我们的目标是找到流形的例子，承认基本的叠层，但不包含任何两个'父'对象。 一个相关的目标是找到一个双曲3流形，不承认一个必要的层。 一个三维流形是一个看起来像我们普通的三维空间的物体，如果你不看得离你站的地方太远。 然而，这些物体在长距离上的行为可能会非常不同;例如，直线行走可以将您带回到您开始的地方（就像在球形地球表面上行走一样）。 3-流形拓扑学试图描述这种全局结构。 在这样做的过程中，思想被从许多其他科学分支中提取出来并应用于其中;它们被应用于化学，在那里它们帮助解开了DNA重组的一些秘密;它们既被从物理学中提取出来又被应用于物理学，旨在理解宇宙的基本组成部分。 在研究三维流形时，一种被证明是富有成效的技术是将三维流形看作是堆叠在一起的曲面（称为叶状）的集合（就像一本书的页面）;然后可以使用已知的曲面来探索三维流形的结构。 调查员计划继续开展旨在制定这一方法的工作。 ***

9704811 Brittenham Essential laminations are one of a large array of objects that are used to probe the structure of 3-manifolds. They were developed as a result of an attempt to generalize simultaneously two such classical objects; the incompressible surface and the taut foliation. This project involves studying the structure of essential laminations and how they sit inside of a 3-manifold, to extract information about the `laminar' 3-manifold. The main goals that the investigator will pursue are to show that every irreducible 3-manifold homotopy-equivalent to a laminar manifold is homeomorphic to it, and to show that laminar manifolds are finitely covered by Haken manifolds. He will also study taut foliations from the laminar point of view, using this new point of view to study these older objects. The goal is to find examples of manifolds that admit essential laminations but which do not contain either of the two `parent' objects. An associated goal is to find a hyperbolic 3-manifold which does not admit an essential lamination. A 3-dimensional manifold is an object which looks like our ordinary 3-dimensional space, if you don't look too far away from where you are standing. Such objects can behave very differently over large distances, though; walking in a straight line, for example, can bring you right back to where you started from (much like walking on the surface of the spherical Earth). 3-manifold topology attempts to describe this global structure. In doing so, ideas are drawn from, and applied to, many other branches of science; they have been applied to chemistry, where they have helped to unlock some of the secrets of DNA recombination, and they have been both drawn from and applied to physics, in work aimed at understanding the basic building blocks of the universe. One technique that has proved fruitful in studying 3-manifolds is to think of the 3-manifold as a collection of surfaces (called a foliation) stacked together (like the pages of a book); then one can use what is known about surfaces to explore the structure of the 3-manifold. The investigator plans to continue work aimed at developing this approach. ***