Dergiler / Turkish Journal of Mathematics / 2014 / Cilt: 38 - Sayı: 4
On the size of the third homotopy group of the suspension of an Eilenberg--MacLane space
- Sayfa
- 664–671
- DOI
- —
Özet
The nonabelian tensor square G \otimes G of a group G of |G| = pn and |G'| = pm (p prime and n,m \ge 1) satisfies a classic bound of the form |G \otimes G|\le pn(n-m). This allows us to give an upper bound for the order of the third homotopy group p3(SK(G,1)) of the suspension of an Eilenberg--MacLane space K(G,1), because p3(K(G,1)) is isomorphic to the kernel of k : 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 pn(n-m) but also supporting a recent result of Jafari on the topic. Consequently, we discuss restrictions on the size of p3(SK(G,1)) based on this new estimation.
Abstract
The nonabelian tensor square G \otimes G of a group G of |G| = pn and |G'| = pm (p prime and n,m \ge 1) satisfies a classic bound of the form |G \otimes G|\le pn(n-m). This allows us to give an upper bound for the order of the third homotopy group p3(SK(G,1)) of the suspension of an Eilenberg--MacLane space K(G,1), because p3(K(G,1)) is isomorphic to the kernel of k : 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 pn(n-m) but also supporting a recent result of Jafari on the topic. Consequently, we discuss restrictions on the size of p3(SK(G,1)) based on this new estimation.