Homotopy theory in group theoretic aspects

在此支持下，我们确信我们对代数拓扑学中的同伦理论做出了有趣的贡献。在下面我们总结我们的结果。首先，我们研究亏格集上空间的自同伦等叶群的不变性。我们得到了与同伦群相关的某些子群实际上满足Hopf空间的不变性。关于这一结果，我们于1999年9月在意大利的工作车间作了一次演讲。其次，我们研究了与同伦群相关的正规级数同伦集的稳定性，即所谓的Mittag-Leffler性质。我们利用我们以前的结果证明了Mittag-Leffler性质对有理空间成立，推广了Sullivan和Wilkerson的结果。此外，我们还证明了好的空间，如球的乘积或Hopf空间，具有Mittag-Leffler性质。我们现在正在写一篇关于这一结果的论文，我们也期待着在这个方向上有更多的结果。第三，我们在拓扑性质已知的空间上具体观察到了上述现象，如李群。但即使是在级别较低的谎言摸索中，这也不是一件容易的事情。在许多情况下，主要障碍是我们没有关于它们同伦群的足够信息。因此，我们沿着TOD方法对球面的不稳定同伦群进行了计算。其他同事也为这个项目做出了贡献，并在他们的领域取得了很好的结果。

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.