Volume 10, issue 1 (2010)

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On homotopy groups of the suspended classifying spaces

Roman Mikhailov and Jie Wu

Algebraic & Geometric Topology 10 (2010) 565–625
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 $SL\left(ℤ\right)$, and ${\pi }_{4}\left(\Sigma K\left({A}_{4},1\right)\right)$ for the $4$–th alternating group ${A}_{4}$.

Keywords
homotopy group, Whitehead exact sequence, spectral sequence, Moore space, suspension of $K(G,1)$ space, simplicial group
Mathematical Subject Classification 2000
Primary: 55Q52
Secondary: 55P20, 55P40, 55P65, 55Q35