The combinatorial description of homotopy groups of sphere (Q2717463)

A simplicial group \(G=\{G_m\}\) is a Kan complex with each \(G_m\) a group. In this case the Moore theorem holds: \(\pi_*(G,1)\) is isomorphic to the homology of the complex \((NG_m,d_0)\), where \(NG_m= \bigcap^m_{j=1} \text{Ker} (d_j: G_m\to G_{m-1})\) for the face operator \(d_j\). J. Wu shows that for a simplicial group with an additional condition, \(\pi_m(G,1)\) is isomorphic to a center of the group \(G_m/d_0 (NG_{m+1})\). In this paper, the author studies on the simplicial group \(FS^n\) whose geometric realization is \(\Omega S^{n+1}\), and describes \(\pi_{n+k+1} S^{n+1}\) for \(k\geq 2\) as a center of a group defined by more definite elements of \((FS^n)_m\).