Algebraic structure of homotopy sets of self maps of Hopf 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。这个结构看起来像德国Baues研究的“正方形环”，设M^n是mod 2的Moore正方形。对于n[大于或等于]3，已知存在η_n的悬挂&lt；η_n&gt；^^-，→S^n.我们研究了Whitehead乘积[&lt；η_n&gt；^^-，&lt；η_n&gt；^^-]在π_&lt；2n+1&gt；(M^n)中的阶.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。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。大岛还调查了它们之间的区别。Hemmi证明了有限Hopf空间的mod-3上同调环的可能的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H(X；Z/p)≡∧(x，p^1x，[O！r][O！r]p^&lt；p-2&gt；x)的Hopf空间。

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)