Algebraic & Geometric Topology

On homotopy groups of the suspended classifying spaces

Roman Mikhailov and Jie Wu

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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(), and π4(ΣK(A4,1)) for the 4–th alternating group A4.

Article information

Algebr. Geom. Topol., Volume 10, Number 1 (2010), 565-625.

Received: 26 October 2009
Accepted: 31 January 2010
First available in Project Euclid: 21 December 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55Q52: Homotopy groups of special spaces
Secondary: 55P20: Eilenberg-Mac Lane spaces 55P40: Suspensions 55P65: Homotopy functors 55Q35: Operations in homotopy groups

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


Mikhailov, Roman; Wu, Jie. On homotopy groups of the suspended classifying spaces. Algebr. Geom. Topol. 10 (2010), no. 1, 565--625. doi:10.2140/agt.2010.10.565.

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