Study on rank filetration of finite H-spaces

研究结果总结如下：1知道群[X，X]对Hopf空间X的幂零性是很重要的，然而另一个重要的事情是研究群[X，X]的组成结构。对于X=SU(3)和Sp(2)， Morisugi确定了[X，X]的组成结构与基团结构之间的关系。这种结构看起来像鲍斯在德国研究过的“方环”。设M^n为摩尔空间的模2。对于n【大于或等于】3，已知Hopf映射的悬架存在一个抬升<η_n>^^^^, η_n:S^<n+1>→S^n。我们调查的顺序怀特黑德产品[<η_n”> ^ ^ ^ ^,<η_n”> ^ ^ ^ ^)在π_ < 2 n + 1 > (M ^ n)。2设G为经典型的单李群。Oshima证明了群[G，G]对于上述G的几乎所有情况都是不可交换的。对于G的某些情况，他确定了[G，G]的幂零类。设X是一个Hopf复形。在这种情况下，[X，X]是一个所谓的代数循环，也就是说，它具有左逆和右逆的二元运算。大岛渚还调查了它们彼此之间的区别Hemmi证明了有限Hopf空间的模3上同环可能的偶维生成只出现在8维或20维。他几乎确定了这种模3上同环的结构。他还证明了在某些条件下，不存在具有H(X;Z/p)≡Λ（X,p^1x，…p^<p-2> X）的Hopf空间X。

The summary of research results is as follows.1 It is important to know what nilpotency the group [X,X] has for a Hopf space X. However another important thing is to study the composition structure of the group [X,X]. Morisugi determined the relationship between the composition structure and the group structure of [X,X] for X=SU(3) and Sp(2). This structure looks like "Square ring" which Baues in Germany studied.Let M^n be the mod 2 Moore space. For n【greater than or equal】3 it is known that there exists a lift <η_n>^^^^of the suspension of the Hopf map, η_n:S^<n+1>→S^n. We investigated the order of the Whitehead product [<η_n>^^^^,<η_n>^^^^] in π_<2n+1>(M^n).2 Let G be the simple Lie group of classical type. Oshima showed that the group [G,G] is non-commutative for almost all cases of G mentioned above. And for some cases of G, he determined the nilpotency class of [G,G].Let X be a Hopf complex. In this case [X,X] is, so called, an algebraic loop, that is, it has a binary operation with both left and right inverses. Oshima also investigated how they differs from each other.3 Hemmi showed that the possible even dimensional generators of mod 3 cohomology rings of finite Hopf spaces occurs only in dimension 8 or 20. And he almost determined the structure of such mod 3 cohomology rings. He also showed that under some conditions, there is no Hopf space X with H(X;Z/p)≡Λ(x,p^1x,…p^<p-2>x)