On the second stable homotopy group of the Eilenberg-Maclane space and the Schur Multiplier

Original · EN

We prove that for a finitely generated group G, the second stable homotopy group π₂ˢ(K(G,1)) of the Eilenberg-Maclane space K(G,1) is completely determined by the Schur multiplier H₂(G). We also prove that the second stable homotopy group π₂ˢ(K(G,1)) is equal to the Schur multiplier H₂(G) for a torsion group G with no elements of order 2 and show that for such groups, π₂ˢ(K(G,1)) is a direct factor of π₃(SK(G,1)), where S denotes suspension and π₂ˢ the second stable homotopy group. We compute π₃(SK(G,1)) and π₂ˢ(K(G,1)) for symmetric, alternating, general linear groups over finite fields and some infinite general linear groups G. We also obtain a bound for the Schur multiplier of all finite groups G analogous to Green's bound for p-groups.