Mathematical Sciences: Some Problems in Geometric Topology

研究人员打算研究一些主题， 控制拓扑学和微分几何 最紧迫的 其中涉及不可分解的同调流形发现， 由调查员J·布莱恩特，W. Mio和S.温伯格 简单地说，他希望研究这些空间，看看有什么 在某种程度上，经典的流形理论延续到了这个新的理论 类空间。 这项研究与 著名的Borel和Bing-Borsuk猜想 其他问题 包括应用受控的伽玛外科手术， 余维2中拓扑嵌入的研究，Borel和 对某些群类的积分诺维科夫猜想，以及 控制拓扑学在微分几何中的应用。 这项工作有许多潜在的应用。 正在研究的新空间具有以下全局性质： 流形上模仿欧几里德空间，但有非常不同的 小规模的财产。 这些空间可能会引起人们的兴趣， 试图调和大规模和小规模模型的工作者， 宇宙 Borel猜想和Novikov猜想是一种新的猜想 涉及到某些流形的刚性。 典型 结果表明，在某些情况下， 拓扑意义是相同的。 这似乎是一个自然的 在几何学和拓扑学中发生量子化的例子。

The investigator intends 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, he wishes to study these spaces and see to what extent the classical theory of manifolds carries over to this new category of spaces. This study is closely related to the well-known Borel and Bing-Borsuk Conjectures. Other problems involved include applications of controlled Gamma-surgery to the study of topological embeddings in codimension two, the Borel and integral Novikov Conjectures for certain classes of groups, and applications of controlled topology to differential geometry. There are a number of potential applications for this work. The new spaces being studied have the global properties of manifolds modelled on euclidean space, but have very different small-scale properties. Such spaces could be of interest to workers trying to reconcile large-scale and small-scale models of the universe. The Borel and Novikov Conjectures are conjectures involving the rigidity of certain classes of manifolds. Typical results say that manifolds which are "close together" in some topological sense are the same. This appears to be a naturally occurring instance of quantization in geometry and topology.