Mathematical Sciences: Problems in Topology of Manifolds

小行星9401086 这个项目的目的是刻画（1）低维流形和（2）一类定义在其上的结构良好的函数-近似纤维化。 这两个组成部分都涉及对这些对象上的连续函数的仔细审查，并对点原像施加了严格的限制。 第一个包括最后方面的一个深远的努力，以表征流形拓扑;主要是它要求确定一个适当的几何概念的一般立场，以履行所发挥的作用不相交的性质在表征高维流形。 第二个问题的出现是因为最近在分段线性范畴中的结果使得人们有理由期望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维化;具体地说，似乎对于许多n-流形N，任何从（n+k）-流形到合理空间X上的映射，每个点的原像同伦都等价于N，都将是近似纤维化。 对这门课的兴趣源于域、值域和典型纤维之间的可计算关系. 寻求新的工具，这方面的努力，几何的第一种情况下，真正的流形区分伪装者，和代数的第二种情况下，既要提高识别标准，扩大应用。 预计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。 其中一个目标是刻画低维流形，这将使刻画流形拓扑的深远努力获得圆满成功，最近在大于4的维度上取得了完全的成功，并希望在剩下的两个感兴趣的维度上取得相关的成功。 第二个目标是刻画某些结构良好的函数，近似纤维化，定义在流形上。 这项工作现在看来很有希望，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条件，在这些条件下，人们可以快速检测这类函数。 由此产生的关于手头各种对象的计算的容易性和多样性将是一个显著的好处。 新的工具寻求这些努力中的每一个，几何的第一种情况下，区分真正的流形伪装者，和代数的第二种情况下，提供更多的域之间的关系，范围和纤维的功能正在考虑中，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. ***