On homotopy groups of the suspended classifying spaces

Abstract

In this paper, we determine the homotopy groups π4(∑K(A, 1)) and π5(∑K(A, 1))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 πi(∑K(G, 1)) with i =4; 5 for some nonabelian groups G =∑3 and SL(Z), and π4(∑K(A4, 1)) for the 4-th alternating group A4.