Homotopy Theory and its Aplications

当G是拓扑群，X，Y是G空间时，我定义了欧拉类e(X，Y)∈[X_+，S^0*Y]^G_*作为[X，Y]^G的非常普遍的抽象类，即G从X映射到Y的集合是非空的。当e（X，Y）成为一个忠实的障碍类时，我得到了一些充分的结果。关于这个问题，我也开发了一个更可计算的，但通常不忠实，阻碍类与Yasuhiro Hara。这种思想已被应用于考虑法伯的拓扑复杂性，它涉及运动规划，如机械臂，并通过模型范畴的观点，Voevodsky和Morel的A^1-同伦理论。此外，利用HELP（=同伦扩展提升性质）推广奇异理论中的Thorn多项式也取得了一些进展。Mikio Furuta指出，具有边界的4流形的稳定同伦Seiverg-Witten不变量是由Fredholm宇宙定义的亲谱来描述的，Fredholm宇宙不是May及其合作者发明的代数拓扑中的传统宇宙。实际上，这个弗雷德霍姆的宇宙有点“扭曲”，看起来很有趣。此外，在此期间，我们还组织了许多会议，并与其他领域的数学家进行了许多研究互动：(a)代数拓扑国际会议（2003年7月27日至8月1日）(b)名古屋工业大学代数拓扑国际研讨会（2003年8月3日至8月8日）(c)同伦名古屋工业大学同伦理论会议01、02（4次！），03、04。

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.