Study of Hopf spaces by using higher order cohomology operations

已知2n+1维球面同伦群的非平凡p分量出现在n<i<p的2n+2i(p-1)-1维。对于n=1的情况，生成函数是由Toda元素的组合给出的，但对于n>1，我们证明了它们是由Toda括号或更大的Toda括号给出的。我们研究了检测这些发生器的高上同操作。对于3球，我们证明了它是在研究Hopf空间的高同伦结合性方面有用的二次运算

It is known that the non trivial p-component of the homotopy group of 2n+1 dimensional sphere occurs in dimension 2n+2i(p-1)-1 for n<i<p. The generators for the case of n=1 is given by the composition of the Toda's alpha elements, but for n>1 we showed that they are given by the Toda bracket or higher Toda brackets. We studied higher cohomology operatons detecting these generators. For 3-sphere we showed that it is the secoandary operation which has been useful in studying higher homotopy associativity of Hopf spaces