Mathematical Sciences: Some Problems in Geometric Topology

9626101费里研究人员计划研究受控拓扑学和微分几何学中的一些课题。其中最紧迫的是研究人员J.Bryant、W.Mio和S.Weinberger发现的不可分解的同调流形。简单地说，主要的项目是发现经典的拓扑流形几何理论在多大程度上继承了这些新发现的空间。这项研究与著名的Borel和Bing-Borsuk猜想密切相关。研究人员的项目清单上的其他问题包括受控Gamma-Surgery理论的构造及其在余维2中拓扑嵌入的研究中的应用，非循环Poincare对偶空间上流形结构的构造，定义在‘大’黎曼流形上的‘短’映射的研究，以及受控拓扑在黎曼流形空间的研究中的应用。拓扑学的很大一部分内容是研究被称为空间的数学对象。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扮演着重要的角色。拓扑学中的一个经典问题是寻找表征特定空间的小的可检查公理集。一套流行的公理涉及连通性的概念。粗略地说，如果一个空间都是完整的，那么它就是相连的。如果它能被切成任意连接的小块，它就是本地连接的。(请注意--这些简单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·坎农做出了一个惊人的洞察力猜想。他猜想，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--一个由小块有序欧几里得空间粘合而成的空间。(球面和环面都是2维的拓扑流形。广义相对论中出现的“曲面空间”是4维的拓扑流形。)结合F·奎因和R·D·爱德华兹的工作表明，只要连通空间包含哪怕是最微小的流形部分，坎农的猜想就是正确的。这种多种多样的部分就像一种种子，决定了整个空间的局部结构。与J.Bryant、W.Mio和S.Weinberger合作，这位研究人员已经证明，坎农的猜想并不完全正确。调查者将研究的主要问题是爱德华兹-奎恩案件中的“种子”现象是否普遍存在--相关反例的局部结构是否在每一点都是相同的。如果事实证明这是真的，这些新空间可能会成为与流形并行的感兴趣的对象。大炮的公理保证，从大范围的角度来看，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中，而不是“普通的”欧几里德空间。***

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. ***