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 Ferry 研究者计划研究受控拓扑和微分几何中的一些主题。其中最紧迫的是研究者 J. Bryant、W. Mio 和 S. Weinberger 发现的不可解析的同源流形。 简而言之，主要项目是发现拓扑流形的经典几何理论在多大程度上适用于这些新发现的空间。 这项研究与著名的 Borel 和 Bing-Borsuk 猜想密切相关。 研究者项目清单上的其他问题包括受控伽玛手术理论的构建及其在余维二拓扑嵌入研究中的应用、非循环庞加莱对偶空间上流形结构的构建、“短”映射的研究关于“大”黎曼流形的定义，以及受控拓扑在黎曼流形空间研究中的应用。 拓扑学的大部分内容都与对称为空间的数学对象的研究有关。 球体的表面是一个空间，就像甜甜圈的表面一样。 在多变量数学系统的研究中，非常高甚至无限维的空间通常发挥重要作用。 拓扑中的一个经典问题是找到表征特定空间的可检查的小公理集。 一组流行的公理涉及连通性的概念。 粗略地说，如果一个空间是一个整体，那么它就是相连的。 如果它可以被切成任意小的连接块，那么它就是局部连接的。 （请注意——这些简单的英语定义充其量只是对实际数学定义的粗略近似。） 1978 年，詹姆斯·坎农 (James Cannon) 做出了一个令人惊讶的敏锐猜想。 他推测，如果一个空间满足某些广义连通性公理，并且其中有足够的空间来将某些集合分开，那么它就是一个拓扑流形——一个通过将普通欧几里得空间的小块粘合在一起而组装而成的空间。 （球体表面和甜甜圈表面都是 2 维拓扑流形。广义相对论中出现的“弯曲空间”是 4 维拓扑流形。）F. Quinn 和 R. D. Edwards 的结合工作表明：只要连通空间包含哪怕是最微小的流形部件，坎农的猜想都是正确的。 这个流形部件就像一种种子，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，证明坎农的猜想并不完全正确。 研究者要研究的主要问题是爱德华兹-奎因案例中的“播种”现象是否普遍存在——相连的反例的局部结构是否在每个点都相同。 如果这是真的，这些新空间可能会成为平行流形的有趣对象。 坎农公理保证，从大尺度的角度来看，这些空间看起来非常像流形。 如果播种现象成立，人们可以想象，最终可能会有理论推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。 ***