On the size of the third homotopy group of the suspension of an Eilenberg--MacLane space

Authors: PEYMAN NIROOMAND, FRANCESCO RUSSO

Abstract: The nonabelian tensor square G \otimes G of a group G of |G| = p^n and |G'| = p^m (p prime and n,m \ge 1) satisfies a classic bound of the form |G \otimes G|\le p^{n(n-m)}. This allows us to give an upper bound for the order of the third homotopy group \pi_3(SK(G,1)) of the suspension of an Eilenberg--MacLane space K(G,1), because \pi_3(K(G,1)) is isomorphic to the kernel of \kappa : x \otimes y \in G \otimes G \mapsto [x,y] \in G'. We prove that |G \otimes G| \le p^{(n-1)(n-m)+2}, sharpening not only |G \otimes G|\le p^{n(n-m)} but also supporting a recent result of Jafari on the topic. Consequently, we discuss restrictions on the size of \pi_3(SK(G,1)) based on this new estimation.

Keywords: Schur multipliers, p--groups, nonabelian tensor square, homotopy

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