Homotopy Theory and its Aplications

同伦理论及其应用

基本信息

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

项目摘要

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是拓扑组和x时，y是g空间时，我定义了euler e（x，y）∈[x _+，s^0* y]^g_*是[x，y]^g的非常普遍的反对类，是从x到y的g映射集的集合。当e（x，y）成为忠实的反对类时，我获得了一些足够的结果。关于这个问题，我还开发了一种更可计算的，但通常不忠实的反对类别的Yasuhiro Hara。这种想法已应用于考虑Farber的拓扑复杂性，涉及机器人武器等运动策略，以及通过模型的观点，即Voevodsky和Morel的模型观点。 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 introduced 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)同型纳戈亚技术研究所同义理论会议01，02（4次！），03，04。