Algebraic structure of homotopy sets of self maps of Hopf spaces

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”。M^n取摩尔平方的模。当n大于或等于3时，已知Hopf映射的悬架存在一个升力<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)。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。让我们成为霍普夫情结。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之间的不同之处。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)≡∧(X, p^1x，【O!R】【O!R】【O！p^<p-2>x)

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 square. 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).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 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 inverse. Oshima also investigated how they differed from each other.3. Hemmi showed that the possible even dimensional generators of mod 3 cohomology rings of finite Hopf spaces occurred 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 was no Hopf space X with H (X; Z/p) ≡∧(x, p^1x, 【O!R】【O!R】【O!R】p^<p-2>x)