The symmetric action on secondary homotopy groups (Q2519135)

The secondary homotopy groups \(\Pi_{n,*} X\) of a space \(X\) are quadratic pair modules which form a model for \((n-1)\)-connected \((n+1)\)-types. They are a sort of crossed modules and in particular there is a group homomorphism \(\Pi_{n,1} X \rightarrow \Pi_{n,0} X\) with kernel \(\pi_n X\) and cokernel \(\pi_{n+1} X\) when \(n \geq 3\). These objects have been studied by the authors in [Forum Math. 20, No. 4, 631--677 (2008; Zbl 1158.18004)]. The symmetric group \(Sym(n)\) acts on the sphere \(S^n = S^1 \wedge \dots \wedge S^1\) by permutation of coordinates and there is a central extension by a cyclic group of order \(2\) acting on \(\Pi_{n,*} X\), as crossed modules. This extension is classified in \(H^2(Sym(n); \mathbb F_2)\) by the second Stiefel-Whitney class of the fiber bundle defined by the obvious representation of \(Sym(n)\) in \(O_n(\mathbb R)\). It is called ``symmetric track group'' in this article since its elements can be interpreted as certain homotopy classes of homotopies between self-maps of \(S^n\).