On homotopy groups of the suspended classifying spaces

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.

[1]  G. Ellis,et al.  A colimit of classifying spaces , 2008, 0804.3581.

[2]  D. Puppe,et al.  Homologie nicht-additiver Funktoren. Anwendungen , 1961 .

[3]  Jie Wu.  Combinatorial descriptions of homotopy groups of certain spaces , 2001, Mathematical Proceedings of the Cambridge Philosophical Society.

[4]  Wu Jie.  ON COMBINATORIAL DESCRIPTIONS OF HOMOTOPY GROUPS OF CERTAIN SPACES , 1998 .

[5]  John Henry Constantine Whitehead,et al.  The homotopy type of a special kind of polyhedron , 1948 .

[6]  L. Breen.  On the functorial homology of abelian groups , 1999 .

[7]  G. Whitehead.  On Spaces with Vanishing Low-Dimensional Homotopy Groups. , 1948, Proceedings of the National Academy of Sciences of the United States of America.

[8]  N. Kuhn,et al.  Stable decompositions of classifying spaces of finite abelian p-groups , 1988, Mathematical Proceedings of the Cambridge Philosophical Society.

[9]  H. Baues,et al.  On the tensor algebra of a non-abelian group and applications , 1991 .

[10]  P. Goerss,et al.  A homotopy operation spectral sequence for the computation of homotopy groups , 2000 .

[11]  D. Quillen.  Spectral sequences of a double semi-simplical group , 1966 .

[12]  Ronald Brown,et al.  Van Kampen theorems for diagrams of spaces , 1987 .

[13]  Hans-Joachim Baues,et al.  Homotopy Type and Homology , 1996 .

[14]  戸田 宏,et al.  Composition methods in homotopy groups of spheres , 1962 .

[15]  Jie Wu.  Homotopy theory of the suspensions of the projective plane , 2003 .

[16]  D. Arlettaz.  Algebraic K-theory of rings from a topological viewpoint , 2000 .

[17]  H. Toda.  Order of the Identity Class of a Suspension Space , 1963 .

[18]  J. Neisendorfer.  Primary Homotopy Theory , 1980 .

[19]  J. C. Moore,et al.  Torsion in Homotopy Groups , 1979 .

[20]  Saunders MacLane,et al.  On the Groups H(Π, n), II: Methods of Computation , 1954 .

[21]  G. Carlsson.  A simplicial group construction for balanced products , 1984 .

[22]  J. Whitehead.  A Certain Exact Sequence , 1950 .