Study of Hopf spaces by using higher order cohomology operations

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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