Two-primary analogues of Selick's theorem and the Kahn-Priddy theorem for the 3-sphere
Assume all spaces tacitly localized at 2. Let \(\Omega^ 2S^ 5\{2\}\) be the fibre of the H-space squaring map on \(\Omega^ 2S^ 5\) and \(\omega_ 2\) the fibre of the double suspension map \(S^ 3\to \Omega^ 2S^ 5\); finally, let \(S^ 3<3>\) be the three-connected cover of \(S^ 3\) and \(\Omega^ 3_ 0S^ 3=\Omega^ 3(S^ 3<3>)\). The author shows (Theorem 1.2) that \(\Omega^ 2S^ 5\{2\}\) is homotopy equivalent to \(\Omega^ 2S^ 3<3>\times w_ 2\). In particular (Theorem 1.1) \(\Omega^ 3_ 0S^ 3\) is a retract of \(\Omega^ 3S^ 5\{2\}\). The relation to the Kahn-Priddy theorem is given by a commutative diagram in which \(\theta\), \(\theta\) ' induce split epimorphisms on homotopy groups. The construction of the maps entering these theorems is quite involved. The proof of their properties relies on extensive calculations in \(H_*\Omega^ 3S^ 3\).
scientific article; zbMATH DE number 3905457