On homotopy groups of the suspended classifying spaces

TL;DR

This paper computes specific homotopy groups of suspended classifying spaces for abelian and some non-abelian groups using advanced algebraic and homotopy theoretical methods.

Contribution

It provides explicit calculations of C6_4 and C6_5 for C0 C6 K(A,1) with A abelian and extends results to certain non-abelian groups G=A3_3, SL(Z), and A_4.

Findings

Determined C6_4 and C6_5 for C0 C6 K(A,1) with A abelian.

Calculated C6_4 and C6_5 for G=A3_3 and SL(Z).

Found C6_4 for A_4, the 4th alternating group.

Abstract

In this paper, we determine the homotopy groups \pi_4(\Sigma K(A,1)) and \pi_5(\Sigma K(A,1)) for abelian groups A by using different facts and 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(\Sigma K(G,1)) with i=4,5 for some non-abelian groups G=\Sigma_3 and SL(Z), and \pi_4(\Sigma K(A_4,1)) for the 4-th alternating group A_4.