Combinatorial descriptions of homotopy groups of certain spaces (Q2725002)

Let \(X\) be a wedge of \(2\)-spheres, a \(3\)-sphere, or the suspension of an Eilenberg-MacLane space of type \(K(\pi,1)\). Using simplicial methods, the author shows that the homotopy groups of \(X\) are isomorphic to the centres of certain groups with explicit presentations, in which the relators are iterated commutators. The results are applied to a construction of \textit{F. R. Cohen} [Contemp. Math. 188, 49-55 (1995; Zbl 0849.55015)]. As a corollary, there is a proof of an unpublished result of Milnor saying that a certain three-stage Postnikov system does not have the homotopy type of a minimal simplicial group.