Mathematical Sciences: Homotopy Fixed Point Problems in K-Theory

数学科学：K 理论中的同伦不动点问题

• 批准号: 8907771
• 负责人: Gunnar Carlsson
• 金额: $ 12.07万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1989
• 资助国家: 美国
• 起止时间: 1989-07-15 至 1992-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8907771&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Homotopy; Fixed; Point

The investigator will work on the algebraic K-theory of group rings of infinite discrete groups, using the homotopy fixed point set of the action of the group on a "bounded K-group." He hopes to extend the class of groups to which the methods apply to all fundamental groups of non-positively curved manifolds with boundary, to S-arithmetic groups, and to discrete subgroups of p- adic groups. In addition, he plans to study the Dennis trace map from the stable pseudoisotopy space of a manifold M to the S1- homotopy fixed point set of its free loop space, the singular loci in representations of the symmetric group, and orbit spectra in equivariant stable homotopy theory. K-theory is one of the standard techniques for finding topological invariants of spaces and for determining properties of these invariants. Sophisticated algebra is involved. Typically one relates geometric objects to algebraic ones, calculates with the algebraic objects, and then translates the results back into geometry.

研究人员将利用群在“有界K-群”上作用的同伦不动点集，研究无限离散群的群环的代数K-理论。他希望把这类群推广到有边界的非正曲线流形的所有基本群，S算术群，以及p-进制群的离散子群。此外，他还计划研究从流形M的稳定伪同余空间到其自由环空间的S1-同伦不动点集的Dennis迹映射、对称群表示中的奇异轨迹以及等变稳定同伦理论中的轨道谱。K-理论是寻找空间的拓扑不变量和确定这些不变量的性质的标准技术之一。这涉及到复杂的代数。通常，人们将几何对象与代数对象相关联，使用代数对象进行计算，然后将结果转换回几何。