In [1] the author introduced a method to construct a map [φ_1,φ_2, φ_3]:S~(r1+r2+r3-1)→X, if maps fi: S~(ri)→X(i=1, 2, 3) are given such that [f_i,f_i]～0 in X. Here he has extended this operation to construct an element, [φ_1,…,φ_n], of π_(r1+…+rn-1)(X) if elements a_i∈π_(ri) (X), (i=1,…,n) are given with appropriate properties. As a consequence a special cell complex, K_(r,n), is constructed which consists of one zero-dimensional cell, e~o, one n-dimensional cell, e~n, such that e~o ∪ e~n is an n-sphere, S~n, and one (kn)-dimensional cell if 1≤k≤r. Moreover, the (kn)-cell is attached by the map [φ_1,…,φ_k] which is so introduced that each a_i (i= 1,…, k) used in defining [φ_1,…, φ_k] is the class determined by the identity map of the sphere S~n (in K_(r,n)). The main theorem of this paper is π_(q+1) (S~(n+1))≈π_q(K_(r,n)), if n>1,q≤(r+1) n-2; π_(q+1)(S~2)≈π_q(K_(r,2)), if 11, for all q and particularly π_(q+1)(S~2)≈π_q(K_(∞,2)), q>1.Let E: π_q (S~n)→π_(q+1)(S~(n+1)) denote the suspension homomorphism. The author has successfully exhibited its kernel and its image. This implies many important results, e.g. (i) Pontrjagin theorem that π_5(S~3)≠0, (ii) simple proofs of the Frendenthal theorem that E is onto if q≦2n-1 and isomorphism onto if q≦2n-2, and (iii) extended G. W. Whitehead theorem about exact sequence regarding suspension, Hopf invariant and Whitehead product.
