Essential Laminations in 3-Manifolds

9704811个Brittenham Essential Layers是用于探索3-流形结构的大量对象之一。它们的发展是由于试图同时概括两个这样的经典物体：不可压缩的表面和紧绷的叶理。这个项目涉及研究基本层片的结构以及它们如何位于3-流形内，以提取关于“层流”3-流形的信息。研究人员的主要目标是证明与层流流形等价的每个不可约3-流形同伦，并证明层流流形被Haken流形有限覆盖。他还将从层状物的角度研究紧绷的叶状物，用这个新的观点来研究这些较老的物体。我们的目标是找到流形的例子，这些流形允许基本的分层，但不包含两个‘父’对象中的任何一个。一个相关的目标是找到一个不允许本质叠层的双曲3-流形。三维流形是一种看起来像我们普通的三维空间的物体，如果你看起来不离你所站的地方太远的话。然而，这样的物体在很长的距离上表现得非常不同；例如，沿着直线行走可以把你带回你开始的地方(很像在球形地球的表面上行走)。3-流形拓扑试图描述这种全局结构。在这样做的过程中，想法从许多其他科学分支中提取并应用于其中；它们被应用于化学，在化学中它们帮助解开了DNA重组的一些秘密，它们既来自物理学，又应用于物理学，旨在理解宇宙的基本组成部分。一种在研究三维流形中被证明是卓有成效的技术是将三维流形想象成堆叠在一起(就像一本书的页面)的表面(称为叶状)的集合；然后，人们可以使用关于表面的已知知识来探索三维流形的结构。调查员计划继续开展旨在发展这一方法的工作。***

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. ***