A reformulation of the Arf invariant one mod p problem and applications to atomic spaces
The author gives detailed proofs of some equivalent formulations of the strong form of the Arf invariant. Namely, he discusses equivalent conditions for when the Whitehead square \([\iota_ n,\iota_ n]\) in \(\pi_{2n-1}S^ n\) is divisible by 2. Odd primary analogues are also given. Let W(n) denote the homotopy theoretic fibre of the double suspension \(E^ 2: S^{2n-1}\to \Omega^{2n+1}\). The author proves that if p is an odd prime with all spaces X localized at p, then a self-map f of X with \(X=\Omega^ 3S^{2n+1}\), \(n>1\), or \(X=W(n)\) is a p-local equivalence if and only if f induces an isomorphism on the first non- vanishing homotopy group of X. Similar results hold when X is the fibre of the pth power map on \(\Omega S^{2n+1}\) if \(p\geq 5\) and \(n\neq 1\) or p. Analogous results in case \(p=2\) are given by the author, \textit{E. Campbell}, \textit{F. Peterson}, and the reviewer in ''The space of maps of Moore spaces to spheres'' (to appear).
scientific article; zbMATH DE number 3897829