Mathematical Sciences: Problems in Topology of Manifolds

[401086] [Daverman]这个项目的目的是描述(1)低维流形和(2)一类结构良好的函数-近似纤维-定义在它们上面。这两个组件都涉及对这些对象上的连续函数的仔细检查，并对点预像施加了严格的限制。第一部分包括对流形进行拓扑表征的深远努力的最后几个方面；首先，它要求确定一个适当的一般位置的几何概念，以实现不相交盘性质在高维流形表征中所起的作用。第二个原因是，最近分段线性分类的结果使人们有理由期望发现广泛存在的附加条件，在这些条件下，人们可以快速而容易地检测到近似的颤动；具体来说，对于许多N -流形N，任何从(N +k)-流形到一个合理空间X的映射，每个点的原像同伦等于N，都是一个近似的纤缩。对这门课的兴趣源于域、范围和典型光纤之间的可计算关系。这一努力寻求新的工具，在第一种情况下是几何工具，以区分真正的流形和伪装者，在第二种情况下是代数工具，以改进识别标准并扩大应用范围。计划的研究活动集中在流形上，这是一个基础项目，因为这些对象是许多数学的中心。一个目标是表征低维流形，这将使从拓扑上表征流形的深远努力成为可能，最近在大于4的维度上取得了完全的成功，并希望在其余两个感兴趣的维度上取得相关的成功。第二个目标是描述在流形上定义的某些结构良好的函数，近似的纤维。这种努力现在看起来很有希望，因为过去两年的几个结果使人们有理由期待发现额外的、有用的条件，在这些条件下，人们可以快速检测到这类函数。由此产生的关于手边各种对象的计算的便利性和多样性将是一个显著的好处。为每一种努力寻找新的工具，在第一种情况下是几何的，以区分真正的流形和假冒的流形，在第二种情况下是代数的，在考虑的函数的域、范围和纤维之间提供额外的关系，从而扩大应用。＊＊＊

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. ***