Symmetricity of the Whitehead element (Q1370800)

For an odd prime \(p\), the Whitehead element \(\omega_n\in\pi_{2np-3}(S^{2n-1})\) is given by \(\omega_{n}=\epsilon_{*}(z)\), where \(\epsilon :C(n)\to S^{2n-1}\) is the homotopy fiber of the double suspension map \(\Sigma ^{2}:S^{2n-1}\to\Omega ^{2}S^{2n+1}\), and \(z\) is a generator of \(\pi_{2np-3}(C(n))\cong\mathbb{Z}_{p}\). For any space \(X\), an element \(\alpha\in\pi_{2n+1}(X)\) is said to be symmetric if it has a representative which factors through the \(p\)-fold covering map \(\sigma :S^{2n+1}\to L^{2n+1}\), where \(L^{2n+1}\) is the standard lens space. In this paper, the authors show that \(\omega_{n}\) is symmetric only when \(n\) is a power of \(p\), and that \(\omega_{p^{i}}\) is symmetric for \(0\leq i\leq 4\). This is some of an odd prime version of the result of \textit{D. Randall} [Proc. Am. Math. Soc. 40, 609-611 (1973; Zbl 0271.55012)] and \textit{W.-H. Lin} [Topology 34, No. 2, 411-422 (1995; Zbl 0846.55011)] for the projectivity of the Whitehead product \([\iota_{2n-1},\iota_{2n-1}]\in\pi_{4n-3}(S^{2n-1})\).