On homotopy groups of the suspended classifying spaces

Abstract

In this paper, we determine the homotopy groups ${\pi }_4\left(\Sigma K\left(A,1\right)\right)$ and ${\pi }_5\left(\Sigma K\left(A,1\right)\right)$ for abelian groups $A$ by using the following methods from group theory and homotopy theory: derived functors, the Carlsson simplicial construction, the Baues–Goerss spectral sequence, homotopy decompositions and the methods of algebraic $K$–theory. As the applications, we also determine ${\pi }_i\left(\Sigma K\left(G,1\right)\right)$ with $i=4,5$ for some nonabelian groups $G={\Sigma }_3$ and $SL\left(\mathbb{Z}\right)$, and ${\pi }_4\left(\Sigma K\left(A_4,1\right)\right)$ for the $4$–th alternating group $A_4$.