Mathematical Sciences: Problems in Topology of Manifolds

9401086达弗曼这个项目的目的是刻画(1)低维流形和(2)定义在流形上的一类结构良好的函数--近似纤颤。这两个部分都涉及对这些对象上的连续函数的密切审查，并对点前像施加了严格的限制。第一部分是对流形进行拓扑学刻画的一项深远努力的最后几个方面；它首先要求确定一个适当的一般位置的几何概念，以实现不相交圆盘性质在高维流形刻画中所起的作用。第二个问题的出现是因为分段线性范畴的最新结果使得人们有理由期待发现广泛的附加条件，在这些条件下人们可以快速而容易地检测到近似纤颤；具体地说，对于许多n-流形N，从(n+k)-流形到合理空间X的任何映射都将是近似纤颤，并且每个点的预像同伦等价于N。对这一类的兴趣源于领域、范围和典型光纤之间的可计算关系，并利用它们进行交易。为此，人们寻求了新的工具，在第一种情况下，几何工具用于区分真流形和伪流形，在第二种情况下，代数工具用于改进识别标准和扩大应用范围。计划的研究活动集中在流形上，这是一个基础项目，因为这些对象是如此多的数学的核心。一个目标是刻画低维流形，这将是刻画流形拓扑特征的深远努力的结果，最近在维度大于4的情况下取得了完全的成功，并希望在感兴趣的其余两个维度取得相关的成功。第二个目的是刻画定义在流形上的某些结构良好的函数，近似纤颤。这一努力现在看来很有希望，因为过去两年的几个结果表明，有理由期待发现更多有用的条件，在这些条件下，人们可以迅速检测到这类功能。由此产生的关于手头各种对象的计算的简单性和多样性将是一个显著的好处。对于这些努力中的每一个，都在寻找新的工具，第一种情况下的几何工具用于区分真正的流形和伪造者，而第二种情况下的代数流形用于提供所考虑的函数的定义域、值域和纤维之间的附加关系，从而扩大应用范围。***

9401086 Daverman The aim of this project is to characterize (1) low-dimensional manifolds and (2) a class of nicely structured functions -approximate fibrations - defined on them. Both components involve close scrutiny of continuous functions on these objects, with severe restrictions imposed on the point preimages. The first comprises the final aspects of a far-reaching effort to characterize manifolds topologically; primarily it calls for determination of an appropriate geometric notion of general position to fulfill the role played by the Disjoint Disks Property in the characterization of high dimensional manifolds. The second arises because recent results in the piecewise linear category make it reasonable to expect discovery of widespread additional conditions under which one quickly and easily detects an approximate fibration; specifically, it seems that for many n-manifolds N , any map from an (n+k)-manifold onto a reasonable space X with each point preimage homotopy equivalent to N will be an approximate fibration. Interest in this class stems from andtrades upon computable relationships among domain, range, and typical fiber. New tools are sought for this effort, geometric ones in the first case, to distinguish genuine manifolds from pretenders, and algebraic ones in the second case, both to improve the recognition criteria and to expand the applications. The projected research activity focuses on manifolds, a foundational project in that these objects are central to so much mathematics. One aim is to characterize low-dimensional manifolds, which would crown the far-reaching effort to characterize manifolds topologically, with complete success recently achieved in dimensions greater than 4, and hopes for related success in the two remaining dimensions of interest. A second aim is to characterize certain nicely structured functions, approximate fibrations, defined on manifolds. This effort appears promising now because several resu lts from the past two years make it reasonable to expect discovery of additional, useful conditions under which one could quickly detect this class of functions. The resultant ease and variety of computations about the various objects at hand would be a significant benefit. New tools are sought for each of these efforts, geometric ones in the first case, to distinguish genuine manifolds from pretenders, and algebraic ones in the second case, to provide additional relationships among domain, range, and fiber of the functions under consideration and thereby to expand the applications. ***