We prove that for any group G, π2S(K(G,1)), the second stable homotopy group of the Eilenberg–Maclane space K(G, 1), is completely determined by the second homology group H2(G, Z). We also prove that the second stable homotopy group π2S(K(G,1)) is isomorphic to H2(G, Z) for a torsion group G with no elements of order 2 and show that for such groups, π2S(K(G,1)) is a direct factor of π3(SK(G, 1)) , where S denotes suspension and π2S the second stable homotopy group. For radicable (divisible if G is abelian) groups G, we prove that π2S(K(G,1)) is isomorphic to H2(G, Z). We compute π3(SK(G, 1)) and π2S(K(G,1)) for symmetric, alternating, dihedral, general linear groups over finite fields and some infinite general linear groups G. For all finite groups G, we obtain a sharp bound for the cardinality of π2S(K(G,1)).
|Number of pages||16|
|State||Published - 1 Dec 2017|
Bibliographical notePublisher Copyright:
© 2017, Springer-Verlag Berlin Heidelberg.
- Eilenberg–Maclane space
- Group actions
- Non-abelian tensor square
- Schur multiplier
- Second stable homotopy group
ASJC Scopus subject areas
- Mathematics (all)