Table of the homotopy groups pi_n+k(Sⁿ)

Cohen-Moore-Neisendorfer Theorem Let p be an odd prime and let x be any element in the p-primary torsion component of pi_k(S²ⁿ⁺¹). Then pⁿ x=0.

Wu Theorem For any n>2, the homotopy group pi_n(S³) is isomorphic to the center of the group G(n) defined as follows:

Let x₁,...,x_n be letters and let w(x₁,...,x_n) denote a word in the free group F(x₁,...,x_n). Let 1 denote the identity of a group. The group G(n) is defined combinatorially by

(1) the product element x₁x₂...x_n;

(2) the words w(x₁,...,x_n) that satisfy: w(x₁,...,x_i-1,1,x_i+1,...,x_n)=1 for each i=1,2,...,n.

Note.

	The second relation above consists of all those words that will collapses to the identity if one of the generators is replaced by the identity.
	There is a braid group action on G(n) induced by the canonical braid group action on free groups. The center of G(n), that is the n-th homotopy group of S3, is the fixed set of the pure braid group action on G(n).