Essential Laminations in 3-Manifolds

9704811 Brittenham Essential lamations是用于探测3流形结构的大量物体之一。它们是由于试图同时概括两个这样的经典对象而发展起来的；不可压缩表面和紧绷面理。该项目涉及研究基本层压的结构以及它们如何位于3-歧管内部，以提取有关“层流”3-歧管的信息。本研究的主要目标是证明每一个与层流形等价的不可约3流形同胚都与层流形同胚，并证明层流形被Haken流形有限覆盖。他还将从层流的角度研究紧绷的叶面，用这种新的观点来研究这些古老的物体。我们的目标是找到一些流形的例子，这些流形承认基本的分层，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基本层压的双曲3流形。一个三维流形是一个看起来像我们普通三维空间的物体，如果你不从你站的地方看太远的话。然而，这些物体在远距离上的表现会非常不同；例如，走直线可以把你带回到你出发的地方（很像在球形地球表面上行走）。3流形拓扑试图描述这种全局结构。在这样做的过程中，思想是从许多其他科学分支中汲取并应用的；它们被应用于化学，帮助解开了DNA重组的一些秘密；它们也被应用于物理学，旨在理解宇宙的基本组成部分。在研究3-流形时，一种被证明卓有成效的技术是将3-流形想象成一个表面的集合（称为叶状）堆叠在一起（就像一本书的书页）；然后我们可以利用已知的曲面来探索3流形的结构。研究者计划继续致力于开发这种方法。＊＊＊

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. ***