Topology of spaces of conjugation-equivariant holomorphic maps

共轭等变全纯映射空间的拓扑

• 批准号: 15540087
• 负责人: KAMIYAMA Yasuhiko
• 金额: $ 1.92万
• 依托单位: The University of the Ryukyus
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2003
• 资助国家: 日本
• 起止时间: 2003 至 2005
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15540087/
• 关键词: rational function; conjugation; polynomial; multiple root; configuration space; loop space; stable splitting; generating variety; 対合; 充満; スペクトル系列; シュティーフェル多様体; 一般の位置; 正則写像; 共通根; ホモトピー同値; ホモトピーファイバー; 偶数次元局面

Let Rat_k(CP^n) be the space of basepoint-preserving holomorphic maps fromS^2 to CP^n. This is a subspace of Ω^2_kCP^n. A theorem by me and Cohen-Cohen-Mann-Milgram tells that the stable homotopy type of Rat_k(CP^n) is described in terms of stable summands of Ω^2S^<2n+1>. The purpose of this research is to generalize the theorem in two directions.(i)Let RRat_k(CP^n) be the subspace of Rat_k(CP^n) of maps which commute with an involution by complex conjugation. Brockett and Segal determined the homotopy type of RRat_k(CP^1). But the case n【greater than or equal】2 was unknown. The first achievement of this research is to determine the stable homotopy type of RRat_k(CP^n) completely. In this case, the corresponding continuous mapping space is ΩS^n×Ω^2S^<2n+1>.(ii)Let P_<k,n> be the space of polynomials such that the number of n-fold roots is at most l. In 1970, Arnold tried to determine the homology group of P^l_<k,n>, but most part was left unknown. The second achievement of research is to determine the stable homotopy type of P^l_<k,n> completely, and to show that the homology groups of P^l_<k,n> are determined from this. As a result, I solved Arnold's problem completely. Roughly, the main result is to prove a relationship between a space of single polynomials and a space of n-tuples of polynomials.The achievement in (ii) was highly evaluated. For example, I gave a plenary talk at the COE International Conference held at the University of Tokyo in July 2005.

设Rat_k（CP^n）是从S^2到CP^n的保基点全纯映射空间。这是Ω ^2_kCP ^n的一个子空间。我和Cohen-Cohen-Mann-Milgram的一个定理告诉我们Rat_k（CP^n）的稳定同伦类型是用Ω^2S^<2n+1>的稳定和项来描述的。本研究的目的是在两个方向上推广定理。（i）设Rat_k（CP^n）是Rat_k（CP^n）的复共轭对合映射的子空间。Brockett和Segal确定了RRat_k（CP^1）的同伦类型。但n[大于或等于]2的情况是未知的。本研究的第一个成果是完全确定了RRat_k（CP^n）的稳定同伦类型。在这种情况下，相应的连续映射空间是ΩS^n×Ω^2S^<2n+1>。（ii）设P_<k，n>是多项式空间，使得n重根的个数至多为l。1970年，Arnold试图确定P^l_<k，n>的同调群，但大部分未知。研究的第二个成果是完全确定了P^l_<k，n>的稳定同伦型，并证明了P^l_<k，n>的同调群是由此确定的。结果，我完全解决了阿诺德的问题。粗略地说，主要结果是证明了单多项式空间和n元多项式空间之间的关系，对（ii）中的成果给予了高度评价。例如，2005年7月在东京大学举行的COE国际会议上，我做了全体会议的发言。

项目成果

Yasuhiko Kamiyama: "Homology of the completion of instanton moduli spaces"Bull.Belgian Math.Soc.. 10. 169-178 (2003)

Yasuhiko Kamiyama：“瞬子模空间补全的同调”Bull.Belgian Math.Soc.. 10. 169-178 (2003)

A polynomial model for the double-loop space of an even sphere

偶球双环空间的多项式模型

• 发表时间: 2004
• 期刊: Proceedings of the Edinburgh mathematical Society Vol.47
• 作者: Yasuhiko Kamiyama;Yasuhiko Kamiyama;Yasuhiko Kamiyama;Yasuhiko Kamiyama;Yasuhiko Kamiyama;Yasuhiko Kamiyama;Yasuhiko Kamiyama;Yasuhiko Kamiyama;Yasuhiko Kamiyama;Yasuhiko Kamiyama;Yasuhiko Kamiyama;Yasuhiko Kamiyama;Yasuhiko Kamiyama;Yasuhiko Kamiyama
• 通讯作者: Yasuhiko Kamiyama

Yasuhiko Kamiyama: "On tangent bundles of certain homogeneous spaces"International Journal of Pure and Applied Mathematics. (印刷中).

Yasuhiko Kamiyama：“论某些齐次空间的切丛”国际纯粹与应用数学杂志（正在出版）。

Geometric construction of a classifying space for the fibre of the double suspension

双悬浮纤维分级空间的几何结构

• 发表时间: 2004
• 期刊: JP Journal of Geometry and Topology Vol.4
• 作者: Yasuhiko Kamiyama;Yasuhiko Kamiyama;Yasuhiko Kamiyama;Yasuhiko Kamiyama;Yasuhiko Kamiyama;Yasuhiko Kamiyama;Yasuhiko Kamiyama;Yasuhiko Kamiyama;Yasuhiko Kamiyama;Yasuhiko Kamiyama;Yasuhiko Kamiyama;Yasuhiko Kamiyama;Yasuhiko Kamiyama
• 通讯作者: Yasuhiko Kamiyama

Generating varieties for the triple loop space of classical Lie groups

经典李群三环空间的生成簇

• 发表时间: 2003
• 期刊: Fundamenta Mathematicae Vol.177
• 作者: Yasuhiko Kamiyama;Yasuhiko Kamiyama;Yasuhiko Kamiyama;Yasuhiko Kamiyama;Yasuhiko Kamiyama;Yasuhiko Kamiyama;Yasuhiko Kamiyama;Yasuhiko Kamiyama;Yasuhiko Kamiyama;Yasuhiko Kamiyama;Yasuhiko Kamiyama;Yasuhiko Kamiyama;Yasuhiko Kamiyama;Yasuhiko Kamiyama;Yasuhiko Kamiyama;Yasuhiko Kamiyama;Yasuhiko Kamiyama;Yasuhiko Kamiyama;Yasuhiko Kamiyama;Yasuhiko Kamiyama;Yasuhiko Kamiyama
• 通讯作者: Yasuhiko Kamiyama