Research on a localized stable homotopy category

局部稳定同伦范畴的研究

基本信息

批准号：
14540083
负责人：
SHIMOMURA Katsumi
金额：
$ 2.18万
依托单位：
KOCHI UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2002
资助国家：
日本
起止时间：
2002 至 2003
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14540083/
关键词：
stable homotopy groups spheres Bousfield localization spectra Johnson-Wilson spectrum Picard group finite complexes Adams-Novikov spectral sequence 安定ホモトピー圏 Morava K理論球面のホモトピー群可逆スペクトラム

项目摘要

In this research, we aimed to understand the stable homotopy category L_n, localized with respect to the Johnson-Wilson spectrum E(n)(in a sense of Bousfield). As a way of understanding it, we study a geometric properties of spaces in the unstable homotopy category, which is a base of the stable homotopy category, and study properties of cohomologies of groups. We also have a little more direct way of understanding L_n, in other words, the determination of the homotopy groups π_*(L_nS^O) of the sphere L_nS^O of the category L_n4 and the determination of the Picard group Pic(L_n). For the viewpoint of unstable homotopy theory, we have Hemmi's results, which showed how much we can say about the associativity of a finite complex with H-structure, by observing the cohomology classes after embedding it to relevant loop spaces. Komatsu studied a condition on dimensions of immersions of a. real projective space by observing the bundle structure on the space. Ohkawa studied the stable homotopy category from the viewpoint of the Bousfield classes. For the viewpoint of cohomologies of groups, Yagita investigated the realization map from a motivic cohomology to an ordinary cohomology on a compact group by using the Milnor operations. On stable homotopy groups, Nakai determined the E_2-term, which will be a base of a future computation, of the Adams-Novikov spectral sequence converging to homotopy groups of a spectrum relating to the spheres. For the homotopy groups of spheres, they are known for n<2 and n=2 and p> 3.Shimomura determined the homotopy groups and the Adams-Novikov E_2-term in the cases where n= 2 and p=3 and where n= and p=2, respectively. On the Picard group, we showed a relation between the E_r-terms of the E(n)-based Adams spectral sequence converging to π_*(L_nS^O) and the Picard group Pic(L_n), and gave an example of an element of Pic(L_n), that is not the sphere, in the case where n=2 and p=3.

在这项研究中，我们的目的是了解稳定同伦范畴 L_n，它相对于约翰逊-威尔逊谱 E(n)（在布斯菲尔德意义上）局部化。作为理解它的一种方式，我们研究了空间的几何性质。不稳定同伦范畴，它是稳定同伦范畴的基础，并研究群的上同调性质。我们还有一个更直接的方式来理解L_n，换句话说，同伦群的确定。范畴 L_n4 的球面 L_nS^O 的 π_*(L_nS^O) 和 Picard 群 Pic(L_n) 的确定对于不稳定同伦理论的观点，我们有 Hemmi 的结果，这表明我们可以说多少。具有 H 结构的有限复形的结合性，通过将其嵌入到相关循环空间后观察上同调类，研究了的浸入维数的条件。 a. 通过观察空间上的丛结构，Ohkawa 从 Bousfield 类的角度研究了稳定同伦范畴。对于群的上同调，Yagita 研究了从动机上同调到普通上同调的实现图。在稳定同伦群上，Nakai 确定了 Adams-Novikov 谱序列收敛的 E_2 项，这将成为未来计算的基础。对于与球体相关的谱的同伦群，它们已知为 n<2 和 n=2 且 p>3。Shimomura 确定了这些情况下的同伦群和 Adams-Novikov E_2 项。其中 n= 2 和 p=3 以及 n= 和 p=2 分别在 Picard 群上，我们展示了基于 E(n) 的 Adams 谱序列的 E_r 项之间的关系。收敛于 π_*(L_nS^O) 和 Picard 群 Pic(L_n)，并给出了在 n=2 和 p=3 的情况下 Pic(L_n) 的一个元素的示例，该元素不是球体。