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

TL;DR

This paper explores the relationship between the second stable homotopy group of Eilenberg-MacLane spaces and the Schur multiplier, providing explicit computations and bounds for various classes of groups.

Contribution

It establishes that the second stable homotopy group is determined by the Schur multiplier for finitely generated groups and computes these groups for several important classes.

Findings

For torsion groups without elements of order 2, the stable homotopy group equals the Schur multiplier.

Explicit calculations of a_3(SK(G,1)) and a_2^S(K(G,1)) for classical groups.

A new bound for the Schur multiplier of finite groups analogous to Green's bound.

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.