Homotopy Theoretic Study of Higher Dimensional Categories

高维范畴的同伦理论研究

• 批准号: 16540061
• 负责人: NISHIDA Goro
• 金额: $ 2.37万
• 依托单位: Kyoto University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2004
• 资助国家: 日本
• 起止时间: 2004 至 2006
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-16540061/
• 关键词: formal group; simplicial set; symmetric group; cohomology; linear group; K-theory; homotopy type; 高次元カテゴリー; 自由ループ空間; 亜群; 表現論; 楕円コホモロジー; モラバK理論; カテゴリー; 分類空間空間

I tried to make a new construction of algebraic K-theory based on formal group laws of general heights, e.g., Honda group law. Formal group laws other than additive or multiplicative group, do not give algebraic groups, but can be regarded as a gamma set defined by Segal. Since a gamma set is a simplicial set, we have a homotopy type by taking the geometric realization. We consider the full matrix algebra over a ring of integers of a local field. Then the resulting homotopy type is a candidate for a space of algebraic K-theory of higher height. First I studied the cohomology group of the space. In the case of multiplicative group, the space is the classifying space of the general linear group. The cohomology is symmetric polynomials of the cohomology of the maximal torus. If the height of the formal group law is h, then the dimension of the maximal torus is h-times of the ordinary case. Then we obtain a multi-symmetric polynomials as cohomology of the space for the candidate. What we have to do next is to define or construct the representation theory based on a given formal group law of higher height. In particular I hope to get a group theoretic interpretation of Hokins-Kuhn-Ravenel character not using the Morava K-theory.

我试图使一个新的建设代数K理论的基础上正式组法律的一般高度，例如，本田集团法。除了加法群或乘法群之外的形式群律，不给出代数群，但可以看作是Segal定义的伽玛集。由于伽玛集是一个单纯集，我们有一个同伦类型，通过采取几何实现。考虑局部域的整数环上的全矩阵代数。然后由此产生的同伦类型是一个更高高度的代数K-理论空间的候选者。首先研究了空间的上同调群。在乘法群的情况下，空间是一般线性群的分类空间。上同调是极大环面上同调的对称多项式。如果形式群律的高度为h，则极大环面的维数是通常情况的h倍。然后我们得到一个多重对称多项式作为候选空间的上同调。我们接下来要做的是在给定的更高高度的形式群律的基础上定义或构造表示论。特别是我希望得到一个不使用摩拉瓦K理论的Hokins-Kuhn-Ravenel字符的群论解释。

项目成果

An application of unstable K-theory.

不稳定 K 理论的应用。

• 发表时间: 2004
• 期刊: J.Math.Kyoto Univ. 44・2
• 作者: Hamanaka;Hiroaki;Kono;Akira
• 通讯作者: Akira

Metric Riemannian geometry

公制黎曼几何

• 发表时间: 2006
• 期刊: Handbook of differential geometry Vol. II
• 作者: Fukaya;Kenji
• 通讯作者: Kenji

Self homotopy groups with large nilpotency classes.

具有大幂零类的自同伦群。

• 发表时间: 2006
• 期刊: Topology Appl. 153
• 作者: Hamanaka;Hiroaki;Kishimoto;Daisuke;Kono;Akira
• 通讯作者: Akira

A note on the Samelson products in π* (SO(2n)) and the group [SO(2n), SO(2n)]

关于 π* (SO(2n)) 和群 [SO(2n), SO(2n)] 中的萨梅尔森积的注释

• 发表时间: 2007
• 期刊: Topology Appl. 154
• 作者: J.Itch;K.Kiyohara;H.Tamura;N.Tanaka;Y.Agaoka;Y.Agaoka;Y.Agaoka;Y.Agaoka;Y.Agaoka;Y.Agaoka;Y.Agaoka;Y.Agaoka;Y.Agaoka;Y.Agaoka;Y.Agaoka;Y.Agaoka;Y.Agaoka;阿賀岡芳夫;Y.Agaoka;Y.Agaoka;阿賀岡 芳夫;Y.Agaoka;Y.Agaoka;Y.Agaoka;Y.Agaoka;Y.Agaoka;Masaaki Ue;Masaaki Ue;Michihiko Fujii;Masaaki Ue;Akira Kono;Michihiko Fujii;Akira Kono
• 通讯作者: Akira Kono

Level 0 monomial crystals

0级单项式晶体

• 发表时间: 2006
• 期刊: Nagoya Math. J. 184
• 作者: David Hernandez;Hiraku Nakajima
• 通讯作者: Hiraku Nakajima