On homotopy groups of the suspended classifying spaces

Mikhailov, Roman; Wu, Jie

doi:10.2140/agt.2010.10.565

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On homotopy groups of the suspended classifying spaces

Roman Mikhailov and Jie Wu

Algebraic & Geometric Topology 10 (2010) 565–625

DOI: 10.2140/agt.2010.10.565

Abstract

In this paper, we determine the homotopy groups $π_{4} (Σ K (A, 1))$ and $π_{5} (Σ K (A, 1))$ 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 $π_{i} (Σ K (G, 1))$ with $i = 4, 5$ for some nonabelian groups $G = Σ_{3}$ and $SL (ℤ)$ , and $π_{4} (Σ K (A_{4}, 1))$ 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

References

Publication

Received: 26 October 2009

Accepted: 31 January 2010

Published: 7 March 2010

Authors

Roman Mikhailov
Steklov Mathematical Institute Gubkina 8 Moscow 119991 Russia

Jie Wu
Department of Mathematics National University of Singapore 2Block S17 (SOC1), 06-02 10 Lower Kent Ridge Road Singapore 119076 Singapore
http://www.math.nus.edu.sg/~matwujie

Volume 10, issue 1 (2010)