On homotopy groups of the suspended classifying spaces

Mikhailov, Roman¹ ; Wu, Jie²

¹Steklov Mathematical Institute, Gubkina 8, Moscow 119991, Russia
²Department of Mathematics, National University of Singapore, 2Block S17 (SOC1), 06-02, 10 Lower Kent Ridge Road, Singapore 119076, Singapore

Algebraic and Geometric Topology, Tome 10 (2010) no. 1, pp. 565-625

Résumé

In this paper, we determine the homotopy groups π4(ΣK(A,1)) and π5(ΣK(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(ΣK(G,1)) with i = 4,5 for some nonabelian groups G = Σ3 and SL(ℤ), and π4(ΣK(A4,1)) for the 4–th alternating group A4.

Détail

DOI : 10.2140/agt.2010.10.565

Keywords: homotopy group, Whitehead exact sequence, spectral sequence, Moore space, suspension of $K(G,1)$ space, simplicial group

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