Mathematical Sciences: Topics in Homotopy Theory

数学科学：同伦论专题

• 批准号: 8900272
• 负责人: Donald Davis
• 金额: $ 7.08万
• 依托单位: Lehigh University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1989
• 资助国家: 美国
• 起止时间: 1989-06-01 至 1991-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8900272&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Topics; Homotopy; Theory

Davis will work on problems in three general areas of homotopy theory. i) A primary goal is to prove that, in the Adams spectral sequence converging to the 2-primary stable homotopy groups of spheres, the line above which only the image of J survives has slope 1/5, or perhaps even 1/6, rather than the 3/10 already established. The odd primary analogue of this result will also be considered. Projects in unstable homotopy theory arising from the same circle of ideas include determining the v-one-periodic homotopy groups of the special orthogonal groups SO(n), the unstable Adams spectral sequence for SO, and a better bound on v-zero-torsion in the unstable Adams spectral sequence for spheres. ii) He hopes to study the classification of the stable homotopy types of smash products of stunted real projective spaces. iii) He will study a conjecture of Landweber relating the complex bordism of a finite group to its rank, by first attempting the abelian case. These problems all call upon expertise in computing with homotopy groups, but the computing will be mainly human rather than electronic, with only modest and occasional machine assistance. Flexible branching guided by seasoned intuition is expected to surpass brute force. Geometric applications are envisioned for the results of these calculations, which will provide new tools for the use of topologists.

戴维斯将研究同伦理论三个一般领域的问题。 i) 主要目标是证明，在收敛到球体 2 基稳定同伦群的 Adams 谱序列中，只有 J 的图像保留在其上方的线的斜率是 1/5，甚至可能是 1/6，而不是已经确定的 3/10。 还将考虑该结果的奇数主要类似物。 源自同一思想圈的不稳定同伦理论项目包括确定特殊正交群 SO(n) 的 v-单周期同伦群、SO 的不稳定 Adams 谱序列以及 v-零挠率的更好界限在球体的不稳定亚当斯谱序列中。 ii）他希望研究发育不良的实射影空间的粉碎积的稳定同伦类型的分类。 iii) 他将通过首先尝试阿贝尔案例来研究兰德韦伯的猜想，该猜想将有限群的复杂边界主义与其等级联系起来。 这些问题都需要同伦群计算方面的专业知识，但计算主要是人类而不是电子，只有适度和偶尔的机器辅助。 由经验丰富的直觉引导的灵活分支预计将超越蛮力。 设想这些计算结果的几何应用，这将为拓扑学家的使用提供新的工具。