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 调查员计划研究控制拓扑学和微分几何的一些主题。其中最紧迫的问题是由研究者J.布莱恩特，W. Mio和S.温伯格 简单地说，主要项目是发现在何种程度上经典几何理论的拓扑流形结转到这些新发现的空间。 这一研究与著名的Borel猜想和Bing-Borsuk猜想密切相关。 其他问题的调查员的项目清单包括建设控制伽玛手术理论及其应用研究的拓扑嵌入在余维二，建设流形结构的非循环庞加莱对偶空间，研究“短”映射定义的“大”黎曼流形，和应用控制拓扑空间的研究黎曼流形。 拓扑学的大部分内容都与研究称为空间的数学对象有关。 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，很高甚至无限维的空间通常会发挥重要作用。 拓扑学中的一个经典问题是寻找刻画特定空间的小的可检验公理集。 一组流行的公理涉及连通性的概念。 粗略地说，如果一个空间都是一片的，那么它就是连通的。 它是局部连通的，如果它可以被分割成任意小的连通块。 (Be警告--这些简明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）提出了一个令人惊讶的猜想。 他指出，如果一个空间满足某些广义连通性公理，并且有足够的空间将某些集合分开，那么它就是一个拓扑流形--一个由普通欧几里得空间的小块粘合在一起而形成的空间。 (Both球面和圆环面是二维拓扑流形。 广义相对论中出现的“弯曲空间”是四维的拓扑流形。） 结合F. Quinn和R. D.爱德华兹表明，坎农的猜想是真实的，只要一个连接的空间包含即使是最小的流形。 这个多方面的作品作为一种种子，它决定了整个当地 空间的结构。 与J. Bryant，W. Mio和S.温伯格， 研究者已经表明，坎农的猜想在完全普遍的情况下是不正确的。 研究人员将研究的主要问题是， 爱德华-奎因案中的“播种”现象普遍存在， 连接反例的局部结构是否相同， 每一点。 如果这被证明是真的，这些新的空间可能成为平行于流形的感兴趣的对象。 坎农公理保证了从大尺度的观点来看，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，而不是“普通的”欧几里得空间。 ***