Mathematical Sciences: Some Problems in Geometric Topology

研究者计划研究一些受控拓扑学和微分几何的课题。其中最紧迫的是由研究者J. Bryant， W. Mio和S. Weinberger发现的不可解同源流形。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。这一研究与著名的Borel猜想和Bing-Borsuk猜想密切相关。研究者的项目清单上的其他问题包括控制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，无环庞加莱对偶空间上流形结构的构建，在“大”黎曼流形上定义的“短”映射的研究，以及控制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，非常高维甚至无限维的空间通常起着重要的作用。拓扑学中的一个经典问题是寻找表征特定空间的公理的小校验集。一组流行的公理涉及到连通性的概念。粗略地说，如果一个空间是一体的，那么它就是连通的。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实际数学定义的粗略近似值。）1978年，詹姆斯·坎农（James Cannon）提出了一个非常敏锐的猜想。他推测，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而成的空间。球体表面和甜甜圈表面都是2维的拓扑流形。广义相对论中出现的“弯曲空间”是4维的拓扑流形。结合F. Quinn和R. D. Edwards的工作表明，只要连通空间包含最小的流形块，Cannon的猜想就成立。这个流形块作为一种种子，决定了整个空间的局部结构。与J. Bryant， W. Mio和S. Weinberger合作，研究者已经证明Cannon的猜想在一般情况下是不正确的。研究者将研究的主要问题是，爱德华兹-奎因案例中的“播种”现象是否普遍存在——连接反例的局部结构在每个点上是否相同。如果这被证明是正确的，这些新的空间可能成为有趣的对象并行流形。Cannon的公理保证了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“普通的”欧几里得空间中。＊＊＊

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. ***