Study on rank filetration of finite H-spaces

1.知道群[X，X]对Hopf空间X有什么幂零是很重要的，但另一件重要的事情是研究群[X，X]的合成结构。Morisugi确定了X=SU(3)和Sp(2)时[X，X]的组成结构和基团结构之间的关系。这种结构看起来像德国Baues研究的“正方形环”，设M^n是mod-2-Moore空间。对于n[大于或等于]3，已知存在η映射的悬挂的升力&lt；η_n&gt；^^，→_n：S^&lt；n+1&gt；ηS^n，我们研究了η_n&gt；^，π_n&gt；^]中Whitehead乘积[&lt；η_n&gt；^]的阶。Oshima证明了群[G，G]对于上述几乎所有的G情形都是非对易的。对于G的某些情形，他确定了[G，G]的幂零类，设X是Hopf复形。在这种情况下，[X，X]是所谓的代数循环，也就是说，它具有左逆和右逆的二元运算。3hemmi证明了有限Hopf空间的mod3上同调环的可能偶维生成元只出现在8或20维上，并且他几乎确定了这种mod3上同调环的结构。他还证明了在一定条件下，不存在具有H(X；Z/p)≡Λ(x，p^1X，…P^&lt；p-2&gt；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)