Homotopy Theory and its Aplications

同伦理论及其应用

• 批准号: 13440020
• 负责人: MINAMI Norihiko
• 金额: $ 7.36万
• 依托单位: Nagoya Institute of Technology
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2003
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-13440020/
• 关键词: Stable Homotopy Category; Chromatic Tower; Obstruction Theory; Characteristic Classes; Equivariant Homotopy Theory; 4 dimensional Manifolds; Seiberg-Witten Invariants; Fredholm Universe; adjunction inequality; charomotic tower; Seiber-Witten不変量

When G is a topological group and X, Y be G-spaces, I defined an Euler class e(X,Y)∈[X_+,S^0*Y]^G_* as the very universal obstraction class for [X,Y]^G, the set of G-maps from X to Y, to be non-empty. I obstained some sufficient results when e(X,Y) becomes a faithful obstruction class. Concerning this problem, I also developed a more computable, but generally non faithful, obstruction class with Yasuhiro Hara.This sort of idea has been applied to consider Farber's topological complexity which concerns motion plannings such as robot arms, and, through the model categorical point of view, A^1-homotopy theory of Voevodsky and Morel. Also, some advance has been obtained in generalizing the Thorn polynomials in the singularity theory through HELP (=Homotopy Extension Lifting Property).Mikio Furuta pointed out that the stable homotopy Seiverg-Witten invariants for 4-manifolds with boundary are described by pro-spectra defined by Fredholm Universe, which is not a traditional universe in algebraic topology invented by May and his collaborators. Actually, this Fredholm universe is sort of "twisted" and appears to be very interesting.Also, during this period, we organized many meetings and lots of research interactions with mathematicians in other areas have been achieved :(a) International Conference on Algebraic Topology (July 27 -August 1, 2003)(b) International Workshop on Algebraic Topology at Nagoya Institute of Technology (August 3 -August 8, 2003)(c) Homotopy Nagoya Institute of Technology Homotopy Theory Meeting 01, 02(4 times!), 03, 04.

当G是拓扑群，X，Y是G-空间时，我定义了一个欧拉类e(X，Y)∈[X_+，S^0*Y]^G_*作为从X到Y的G-映射的集合[X，Y]^G的非常泛封闭类。当e(X，Y)成为忠实的障碍类时，我得到了一些充分的结果。关于这个问题，我还和Yasuhiro Hariro一起开发了一个更易于计算但通常不忠实的障碍类。这种想法已经被应用于考虑Farber的拓扑复杂性，它涉及到机器人手臂等运动规划，并通过模型范畴的观点，讨论了Voevodsky和Morel的A^1-同伦理论。同时，通过HELP(=Homotopy Extension Living Property)将Thorn多项式推广到奇点理论中也取得了一些进展。Mikio Furuta指出，带边界的4-流形的稳定同伦Seiverg-Witten不变量是由Fredholm宇宙定义的PRO谱描述的，这不是May及其合作者在代数拓扑学中发明的传统宇宙。在此期间，我们组织了许多会议，并与其他领域的数学家进行了大量的研究互动：(A)国际代数拓扑学会议(2003年7月27日至8月1日)(B)名古屋工业大学代数拓扑学国际研讨会(2003年8月3日至8月8日)(C)名古屋工业大学同伦理论会议01，02(4次！

项目成果

Yasuhiro Hara, N.Minami: "Borsuk-Ulam type theorems for compact Lie group actions"Proc.Amer.Math.Soc.. 132. 903-909 (2004)

Yasuhiro Hara，N.Minami：“紧李群作用的 Borsuk-Ulam 型定理”Proc.Amer.Math.Soc.. 132. 903-909 (2004)

古田 幹雄: "指数定理2(岩波講座 現代数学の展開)"岩波書店. 327 (2002)

古田干雄：“索引定理 2（岩波讲座：现代数学的发展）”岩波书店 327（2002 年）。

N. Minami: "Some mathematical influences of Stewart Priddy"Contemp. Math.. 293(印刷中). (2002)

N. Minami：“斯图尔特·普里迪的一些数学影响”Contemp.. 293（印刷中）。

N. Minami: "Fron K(n+1)_*X to K(n)_*X"Proc. Amr. Math. Soc.. 130. 1557-1562 (2002)

N. Minami：“从 K(n 1)_*X 到 K(n)_*X”过程。

Norihiko Minami: "A possible hierarchy of Morava K-theories"Isaac Newton Institute Proceedings. (掲載決定).

Norihiko Minami：“摩拉瓦 K 理论的可能层次”艾萨克·牛顿研究所论文集（决定出版）。