Mathematical Sciences: Some Problems in Geometric Topology

数学科学：几何拓扑中的一些问题

• 批准号: 9626101
• 负责人: Steven Ferry
• 金额: $ 7.41万
• 依托单位: SUNY at Binghamton
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-08-01 至 1999-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9626101&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Some; Problems; Geometric

9626101 Ferry The investigator plans to study a number of topics in controlled topology and differential geometry. The most pressing of these concern the nonresolvable homology manifolds discovered by the investigator, J. Bryant, W. Mio, and S. Weinberger. Simply put, the main project is to discover to what extent the classical geometric theory of topological manifolds carries over to these newly discovered spaces. This study is closely related to the well-known Borel and Bing-Borsuk Conjectures. Other problems on the investigator's list of projects include the construction of controlled Gamma-surgery theory and its application to the study of topological embeddings in codimension two, the construction of manifold structures on acyclic Poincare duality spaces, the study of ``short'' maps defined on ``large'' Riemannian manifolds, and applications of controlled topology to the study of spaces of Riemannian manifolds. Much of topology is concerned with the study of mathematical objects called spaces. The surface of a sphere is a space, as is the surface of a donut. In the study of mathematical systems with many variables, it is common for spaces of very high or even infinite dimension to play important roles. A classical problem in topology is to find small checkable sets of axioms which characterize particular spaces. One popular set of axioms involves the notion of connectivity. Roughly speaking, a space is connected if it is all in one piece. It is locally connected if it can be chopped up into arbitrarily small connected pieces. (Be warned -- these plain English definitions are at best rough approximations to the actual mathematical definitions.) In 1978, James Cannon made an amazingly perceptive conjecture. He conjectured that if a space satisfied certain generalized connectivity axioms and had enough room in it to push certain sets apart, then it was a topological manifold -- a space which is assembled by gluing together small pieces of ordi nary Euclidean space. (Both the surface of a sphere and the surface of a donut are topological manifolds of dimension 2. The ``curved spaces'' appearing in general relativity are topological manifolds of dimension 4.) Combining work of F. Quinn and R. D. Edwards shows that Cannon's conjecture is true whenever a connected space contains even the tiniest manifold piece. This manifold piece acts as a sort of seed which determines the entire local structure of the space. Working with J. Bryant, W. Mio, and S. Weinberger, the investigator has shown that Cannon's conjecture is not true in complete generality. The main question the investigator will be studying is whether the ``seeding'' phenomenon from the Edwards-Quinn case holds in general -- whether the local structure of connected counterexamples is the same at every point. If this turns out to be true, these new spaces could become objects of interest paralleling manifolds. Cannon's axioms guarantee that from a large-scale point of view, these spaces look exceedingly like manifolds. If the seeding phenomenon holds, one imagines that there could eventually be theories speculating that we live in one of these new mathematical objects rather than in ``ordinary'' Euclidean space. ***

9626101渡轮研究人员计划研究受控拓扑和差异几何学的许多主题。这些最大的紧迫性关注的是研究人员J. Bryant，W。Mio和S. Weinberger发现的不可辨别的同源歧管。 简而言之，主要项目是发现拓扑歧管的经典几何理论在多大程度上延续到这些新发现的空间。 这项研究与众所周知的Borel和Bing-Borsuk猜想密切相关。 研究人员的项目清单上的其他问题包括构建受控伽马外科理论及其在编成二次拓扑嵌入的研究中的应用，在二级联繁殖型二元空间上的歧管结构的构建，``short''地图的研究根据``大型''riemannian流形的定义，以及受控拓扑的应用在Riemannian歧管空间的研究中。 拓扑的大部分内容与称为空间的数学对象有关。 球的表面是一个空间，甜甜圈的表面也是如此。 在对具有许多变量的数学系统的研究中，对于非常高甚至无限维度的空间而言，扮演重要角色是很常见的。 拓扑结构的一个经典问题是找到特征特定空间的公理的小型公理集。 一组流行的公理涉及连通性的概念。 粗略地说，如果一个全部都可以连接一个空间。 如果可以将其切碎成任意的小连接部分，则可以将其连接起来。 （请注意 - 这些普通的英语定义充其量是对实际数学定义的粗略近似。）1978年，詹姆斯·坎农（James Cannon）做出了惊人的敏锐猜想。 他猜想，如果一个空间满足某些广义连通性公理，并且在其中有足够的空间以将某些套件隔开，那么它是一个拓扑歧管 - 通过将小片段的欧几里得空间粘合在一起而组装的空间。 （一个球的表面和甜甜圈的表面都是维度2的拓扑歧管。在一般相对性中出现的``弯曲空间''是维度4的拓扑歧管。每当连接的空间都包含最小的歧管时，Cannon的猜想是正确的。 该歧管作用是一种种子，它决定了空间的整个局部结构。 研究人员与J. Bryant，W。Mio和S. Weinberger合作，表明Cannon的猜想在完全的一般性中并非如此。 研究人员将研究的主要问题是，来自Edwards-Quinn案的“播种”现象一般是否存在 - 连接的反例的局部结构在每个点是否相同。 如果事实证明这是正确的，那么这些新空间可能会成为与歧管平行的感兴趣的对象。 Cannon的公理可以保证，从大规模的角度来看，这些空间看起来非常像歧管。 如果播种现象成立，人们会想象最终可能会有理论猜测我们生活在这些新的数学对象之一中，而不是``普通''欧几里得空间中。 ***