Mathematical Sciences: Localization and periodicity in unstable homotopy theory

研究者打算：(1)利用Adams谱序列技术和Bousfield局域化技术，继续研究不稳定周期同伦群；(2)将与Mark Mahowald在二次悬映射光纤的v2周期同伦群上的联合工作推广到更高周期；(3)利用同调理论E(n)研究不稳定球的局域性；(4)研究基于E(n)的不稳定Adams-Novikov谱序列；(5)计算自由迭代循环空间的mod-p k理论，并将其应用于v1个周期同伦群的计算。拓扑学最初是作为研究微分方程的一种方法而出现的。在这中间的一个世纪里，它发生了惊人的进化，似乎经常有一个完全独立于它的来源的生命。然而，它已经开始越来越多地偿还债务。关键是，知道一个方程的解是闭环还是打结，通常比精确地、数值地知道它所有点的坐标更重要。此外，尽管前一种定性信息隐含在后一种完整的描述中，但如何提取它却并不明显。拓扑学家开发的复杂机制，比如本项目概述的同伦理论的研究，现在已经发展到一定程度，通常允许以自然的方式处理这些自然问题。

The investigator intends: (1) to continue his study of unstable periodic homotopy groups, using Adams spectral sequence techniques and Bousfield localization; (2) to generalize joint work with Mark Mahowald on v2 periodic homotopy groups of the fiber of the secondary suspension map to higher periodicity; (3) to study the localization of an unstable sphere with respect to the homology theory E(n); (4) to study the unstable Adams-Novikov spectral sequence based on E(n); (5) to compute the mod-p K-theory of free iterated loop spaces and apply this to the calculation of v1 perioidic homotopy groups. Topology arose originally as an approach to the study of differential equations. Over the intervening century it has evolved prodigiously, often seeming to have a life entirely its own and independent of its source. Increasingly, however, it has begun to repay its debt. The point is that it is often more important to know of the solution of an equation whether it is a closed loop or whether it is knotted, for example, than to know precisely and numerically what are the coordinates of all of its points. Furthermore, although the former qualitative information is implicitly contained in the latter complete description, it is not at all obvious how to extract it. The sophisticated machinery that has been developed by topologists, such things as the investigations in homotopy theory outlined for this project, has now been developed to a degree that often permits these natural problems to be handled in a natural way.