Homotopy theory in group theoretic aspects

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Supported by the aid we are sure that we have made interesting contribution to homotopy theory in algebric toplology. In the following we summarize our results.First we study the invariance of the group of self homotopy euqivalences of a space on the genus set. We have obtained the result that certain subgroups associated with homotopy groups actually satisfy the invariance property for Hopf spaces. On this result, we gave a talk at the work shop in Italy in September 1999.Secondary, we studied the stability property, so called the Mittag-Leffler property, of normal series of homotopy sets associated with homotopy groups. We were able to show that the Mittag-Leffler property holds for rational spaces by using the our previous result which generalizes the results by Sullivan and Wilkerson. Further we have showed that good spaces such as products of spheres or Hopf space have the Mittag-Leffler property. We are now writing a paper on this result and also we are expecting more reslts in this direction.Thirdly we have observed the above phenomena concretely on spaces whose topologicl properties are well known such as Lie groups. But even among Lie grops of low ranks, it was not an easy task. In many case, the primary obstruction is that we do not have sufficient information on their homotopy groups. Therefore we have carried out computation of unstable homotopy groups of spheres along the methods of Toda.Other co-workers also have made contribution to this project and obtained excellent results on their fields as well.