Mathematical Sciences: Localization and Periodicity in Homotopy Theory

数学科学：同伦理论中的定域性和周期性

• 批准号: 9204508
• 负责人: Aldridge Bousfield
• 金额: $ 8.04万
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1992
• 资助国家: 美国
• 起止时间: 1992-08-15 至 1997-01-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9204508&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Localization; Periodicity; Homotopy

Bousfield will continue his research in homotopy theory and will focus particularly on the newly emerging theory of vn-periodic localizations of spaces. He will explore unstable versions of periodicity results and will work to analyze homotopy types by resolving them into vn-periodic parts for positive integers n. Bousfield is establishing strong links between the v1-periodic localization and the K-theoretic localization of spaces, and this is leading to solutions of old problems concerning the K-theory of iterated loop spaces. He will continue his work, using homotopical and K-theoretic techniques. For instance, he will work on the development of a K-theoretic unstable Adams spectral sequence. Bousfield will also work to extend his algebraic classification of K-local spectra to cover mapping classes and will continue other related research. The details of these parts vary, but all are concerned with reducing geometric information to a subject for calculation or to perfecting the algebraic machinery used for the calculations. The nature of the geometric information involved is the crux of the difficulty. While questions about lengths, areas, angles, volumes, and so forth virtually cry out to be reduced to calculations, it is far different with what are known as topological properties of geometric objects. These are properties such as connectedness (being all in one piece), knottedness, having no holes, and so forth. All systematic study of such properties, for example, how to tell whether two geometric objects really differ in respect to one of these properties or are only superficially different, or how to classify the variety of differences that can occur, all these have only truly been comprehended and mastered when they have been reduced to matters of calculation, and two of the principal tools for this are homotopy theory and K-theory.

布斯菲尔德将继续他在同伦理论方面的研究，并将特别关注新出现的空间 vn 周期局域化理论。 他将探索周期性结果的不稳定版本，并通过将同伦类型解析为正整数 n 的 vn 周期部分来分析同伦类型。 Bousfield 正在 v1 周期局部化和空间的 K 理论局部化之间建立牢固的联系，这导致了有关迭代循环空间 K 理论的老问题的解决。 他将使用同伦和 K 理论技术继续他的工作。 例如，他将致力于开发 K 理论不稳定亚当斯谱序列。 Bousfield 还将致力于扩展他的 K 局部谱的代数分类以涵盖映射类别，并将继续其他相关研究。 这些部分的细节各不相同，但都涉及将几何信息减少到计算主题或完善用于计算的代数机制。 所涉及的几何信息的性质是困难的关键。 虽然关于长度、面积、角度、体积等的问题实际上迫切需要简化为计算，但它与所谓的几何对象的拓扑性质有很大不同。 这些属性包括连通性（全部为一体）、打结性、无孔等。 所有对这些性质的系统研究，例如，如何判断两个几何对象在这些性质之一上是否确实不同或只是表面上的不同，或者如何对可能出现的各种差异进行分类，所有这些只有在简化为计算问题时才能真正被理解和掌握，而其中的两个主要工具是同伦理论和K理论。