Homotopy Theory and its Aplications

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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.