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Mathematical Sciences: Topics in Homotopy Theory

数学科学：同伦论专题

基本信息

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

项目摘要

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.

戴维斯将致力于同伦理论的三个一般领域的问题。I)一个主要目标是证明，在收敛到球面的2-初等稳定同伦群的Adams谱序列中，只有J的像在其上存活的直线具有1/5的斜率，甚至可能是1/6，而不是已经确定的3/10。对这一结果的奇怪的初步类比也将被考虑。在不稳定同伦理论中，由同一思想圈产生的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，以及球面不稳定Adams谱序列中v-零挠率的一个更好的界。Ii)他希望研究矮实射影空间的Smash积的稳定同伦型的分类。Iii)他将通过首先尝试阿贝尔情形来研究Landweber的一个猜想，该猜想将有限群的复边论与其秩相联系起来。这些问题都需要同伦小组的计算专业知识，但计算将主要是人类而不是电子的，只有适度的和偶尔的机器辅助。在经验丰富的直觉指导下，灵活的分枝有望超越蛮力。这些计算的结果将被设想为几何应用，这将为拓扑学家的使用提供新的工具。