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. ***

小行星9400651 本质层积推广了三流形拓扑中两个重要的对象：不可压缩曲面和绷紧的叶理。 这两个更“经典”的对象已被证明是非常有用的，在过去（并继续如此，今天），和基本叠层最近已显示出类似的权力，在攻击许多基本问题的理论3流形。 研究人员打算继续他在基本层压方面的工作，考虑到三个主要目标：（1）证明它们在许多情况下的行为很像它们更容易处理的表亲，不可压缩曲面，（2）证明包含本质层的同伦等价3-流形是同胚的，（3）在各种各样的3-流形中构造本质层，使用叠层的哈肯标准形的概念。 换句话说，研究人员打算表明基本叠层（1）表现良好，（2）告诉我们有关3-流形的有趣事情，以及（3）可以在“大多数”3-流形中找到。 令人惊讶的是，尽管我们生活在一个三维世界中，因此对事物在三维中如何“工作”有很多自然的直觉，但我们对三维流形的理解，即对我们三维空间的物体建模，还远远没有完成。 它的许多基本问题，如著名的庞加莱猜想，仍然没有解决，即使在更高的维度类似的问题已经解决。 许多新的技术已经开发出来，并将继续开发，试图解开这个熟悉的，虽然神秘的，维度。 研究人员打算继续发展一种研究三维流形的新技术，这种技术使用（很好理解的）二维物体来研究这些（不太好理解的）三维物体。 这个想法是，通过把一个三维物体看作是由一组二维的“片”堆叠在一起组成的，我们有时可以使用我们的（甚至更好！）直觉和对2维的理解来“缝合”一个3维问题的解决方案。 ***