Mathematical Sciences: Essential Laminations in 3-Manifolds

数学科学：3-流形中的基本叠片

• 批准号: 9400651
• 负责人: Mark Brittenham
• 金额: $ 5.38万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-08-01 至 1997-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9400651&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Essential; Laminations; Manifolds

9400651 Brittenham Essential laminations generalize two objects important to 3-manifold topology: the incompressible surface and the taut foliation. Both of these more `classical' objects have proved themselves to be very useful in the past (and continue to be so, today), and essential laminations have recently shown similar power in attacking many of the fundamental problems in the theory of 3-manifolds. The investigator intends to continue his work on essential laminations, with three main goals in mind: (1) to show that they behave in many situations much like their far more manageable cousin, the incompressible surface, (2) to show that homotopy equivalent 3-manifolds, which contain essential laminations, are homeomorphic, and (3) to construct essential laminations in a wide variety of 3-manifolds, using the notion of Haken normal form for laminations. In other words, the investigator intends to show that essential laminations (1) behave nicely, (2) tell us interesting things about 3-manifolds, and (3) can be found in `most' 3-manifolds. It is somehow surprising that, in spite of the fact that we live in a 3-dimensional world, and therefore have a great deal of natural intuition about how things `work' in 3 dimensions, our understanding of 3-dimensional manifolds, objects modelled on our 3-dimensional space, is far from complete. Many of its basic problems, such as the celebrated Poincare conjecture, remain unsolved, even though the analogous problems in higher dimensions have been settled. Many new techniques have been developed, and continue to be developed, to try to unravel this familiar, though mysterious, dimension. The investigator intends to continue to delevop a new technique for studying 3-dimensional manifolds which uses (well-understood) 2-dimensional objects to study these (not so well-understood) 3-dimensional ones. The idea is that by thinking of a 3-dimensional object as being made up of a collection of 2-dimensionsal `sheets' , stacked together, we can sometimes use our (even better!) intuition and understanding about dimension 2 to `stitch together' a solution to a 3-dimensional problem. ***

9400651 Brittenham 基本叠片概括了对 3 流形拓扑重要的两个对象：不可压缩表面和拉紧的叶状结构。 这两个更“经典”的对象在过去都被证明是非常有用的（并且在今天仍然如此），并且本质叠层最近在解决 3 流形理论中的许多基本问题方面表现出了类似的力量。 研究人员打算继续他对基本叠层的研究，并牢记三个主要目标：(1) 表明它们在许多情况下的行为与它们更易于管理的表亲（不可压缩表面）非常相似，(2) 表明同伦等价 3 - 包含必要叠片的流形是同胚的，并且 (3) 使用叠片的 Haken 范式概念在各种 3-流形中构造必要叠片。 换句话说，研究者打算证明基本叠片 (1) 表现良好，(2) 告诉我们有关 3 流形的有趣的事情，以及 (3) 可以在“大多数”3 流形中找到。 令人惊讶的是，尽管我们生活在 3 维世界中，因此对事物如何在 3 维中“工作”有很多自然的直觉，但我们对 3 维流形、建模对象的理解在我们的 3 维空间中，还远未完成。 尽管更高维度的类似问题已经得到解决，但它的许多基本问题，例如著名的庞加莱猜想，仍未得到解决。 许多新技术已经被开发出来，并且还在继续开发中，试图解开这个熟悉但神秘的维度。 研究人员打算继续开发一种研究 3 维流形的新技术，该技术使用（充分理解的）2 维对象来研究这些（不太充分理解的）3 维对象。 这个想法是，通过将 3 维物体视为由堆叠在一起的 2 维“片”的集合组成，我们有时可以利用我们（甚至更好！）对 2 维的直觉和理解来“缝合在一起” ' 3 维问题的解决方案。 ***