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

Abstract

We prove that for a finitely generated group $G$, the second stable homotopy group $\pi_2^S(K(G,1))$ of the Eilenberg-Maclane space $K(G,1)$ is completely determined by the Schur multiplier $H_2(G)$. We also prove that the second stable homotopy group $\pi_2^S(K(G,1))$ is equal to the Schur multiplier $H_2(G)$ for a torsion group $G$ with no elements of order $2$ and show that for such groups, $\pi_2^S(K(G,1))$ is a direct factor of $\pi_{3}(SK(G,1))$, where $S$ denotes suspension and $\pi_2^S$ the second stable homotopy group. We compute $\pi_{3}(SK(G,1))$ and $\pi_2^S(K(G,1))$ 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.