The transfer and James-Hopf invariants
If X is a connected space, there is an equivalence \(\Sigma ^{\infty}QX=\bigvee _{q\geq 1}\Sigma ^{\infty}D_ qX\) arising from Hopf invariants \(j_ q: QX\to QD_ qX\), where \(QX=\Omega ^{\infty}\Sigma ^{\infty}X\) and \(D_ qX=E \Sigma _{q+} \bigwedge _{\Sigma _ q} X^{[q]}\). Let \(f^ n_{r,q}\) be the composite \[ \Sigma ^{\infty}D_ nX\quad \to \quad \Sigma ^{\infty}QX\quad \to ^{\Sigma ^{\infty}j_ q}\quad \Sigma ^{\infty}QD_ qX\quad \to \quad \Sigma ^{\infty}D_ rD_ qX\quad. \] We describe \(f^ n_{r,q}\) in terms of more familiar maps. In particular, \(f^ n_{r,q}\simeq *\) if either \(n>rq\) or \(r>\left( \begin{matrix} n\\ q\end{matrix} \right)\), and \(f^{rq}_{r,q}\) is the transfer associated to the wreath product inclusion \(\Sigma _ r\int \Sigma _ q\subset \Sigma _{rq}\). Some related results are also given.
scientific article; zbMATH DE number 3986180