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Computational Motivic and Equivariant Homotopy Theory

计算动机和等变同伦理论

基本信息

批准号：
1710379
负责人：
Bertrand Guillou
金额：
$ 13.98万
依托单位：
University of Kentucky Research Foundation
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-07-15 至 2020-12-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1710379&HistoricalAwards=false
关键词：
Computational Motivic Equivariant Homotopy Theory

项目摘要

Spheres are simple yet important objects of study in topology. One of the central questions of algebraic topology is to classify all of the possible mappings of a sphere of a given dimension onto a sphere of different dimension. More recently, this question has received attention in other contexts: when the spheres are considered in the realm of algebraic geometry, or when the spheres have specified symmetries which must be preserved by the mappings in question. Even more recently, greater understanding of how these various contexts impact each other has emerged. The PI will endeavor to expand the range in which these questions are understood, especially in the setting of spheres with a twofold symmetry. Another component of the project focuses on an invariant known as K-theory, which lies at the interface of topology and algebra. The PI will develop a version of K-theory which is simultaneously equipped with symmetries and multiplications.The PI will continue his joint work with Dan Isaksen on computations in the motivic and C2-equivariant stable homotopy groups of spheres. The computation of motivic stable homotopy groups over C is fairly similar to that of classical stable homotopy groups. From there, the Bockstein spectral sequence recovers the motivic stable homotopy groups over R. Finally, accounting for the "negative cone" in the C2-equivariant homology of a point leads to the equivariant stable homotopy groups. In a different direction the PI, together with May, Merling, and Osorno, will continue to investigate the presentation of G-spectra as spectral Mackey functors. A centerpiece of this theory is an equivariant infinite loop space machine with good multiplicative properties. A number of tools that figure centrally in applications, such as equivariantly commutative multiplications, norm maps, and geometric fixed points, are not well understood in this model of G-spectra.

球面是拓扑学中简单而又重要的研究对象。代数拓扑学的核心问题之一是将给定维度的球面到不同维度的球面上的所有可能映射分类。最近，这个问题在其他情况下受到了关注：当球体被考虑在代数几何的领域中时，或者当球体指定了必须由所讨论的映射保持的对称性时。甚至在最近，对这些不同背景如何相互影响的理解也得到了更多的理解。PI将努力扩大理解这些问题的范围，特别是在具有双重对称性的球体的设置中。该项目的另一个组成部分专注于一个被称为K-理论的不变量，它位于拓扑学和代数的交界处。PI将发展一种同时配备对称和乘法的K-理论。PI将继续与Dan Isaksen在球面的Motivic和C2-等变稳定同伦群中的计算方面的联合工作。C上的动机稳定同伦群的计算与经典稳定同伦群的计算非常相似。从这里，Bockstein谱序列恢复了R上的动机稳定同伦群。最后，考虑了点的C2-等变同调中的“负锥”，得到了等变稳定同伦群。在一个不同的方向上，PI将与May、Merling和Osorno一起，继续研究G谱作为光谱Mackey函子的呈现。该理论的核心是一种具有良好乘性的等变无限循环空间机。在这个G谱模型中，一些在应用中居于中心位置的工具，如等变交换乘法、范数映射和几何不动点，并没有得到很好的理解。

项目成果

期刊论文数量（9）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

Multiplicative equivariant K-theory and the Barratt-Priddy-Quillen theorem

乘法等变 K 理论和 Barratt-Priddy-Quillen 定理

DOI：
10.1016/j.aim.2023.108865
发表时间：
2023
期刊：
Advances in Mathematics
影响因子：
1.7
作者：
Guillou, Bertrand J.;May, J. Peter;Merling, Mona;Osorno, Angélica M.
通讯作者：
Osorno, Angélica M.

The Klein four slices of $$\Sigma ^n H\underline{{\mathbb {F}}}_2$$

克莱因四片 $$Sigma ^n Hunderline{{mathbb {F}}}_2$$

DOI：
10.1007/s00209-019-02433-3
发表时间：
2020
期刊：
Mathematische Zeitschrift
影响因子：
0.8
作者：
Guillou, B.;Yarnall, C.
通讯作者：
Yarnall, C.

The cohomology of C2-equivariant ?(1) and thehomotopy of koC2

C2-等变式 ?(1) 的上同调和 koC2 的同伦

DOI：
10.2140/tunis.2020.2.567
发表时间：
2020
期刊：
Tunisian Journal of Mathematics
影响因子：
0.9
作者：
Guillou, Bertrand J.;Hill, Michael A.;Isaksen, Daniel C.;Ravenel, Douglas Conner
通讯作者：
Ravenel, Douglas Conner

A symmetric monoidal and equivariant Segal infinite loop space machine

对称幺半等变Segal无限循环空间机