On the 2-primary \(v_{1}\)-periodic homotopy groups of spaces
The \(p\)-primary, \(v_1\)-periodic homotopy groups \(v^{-1}_1\pi_*(X)\) are defined in terms of localization of a portion of homotopy groups, see \textit{D. M. Davis} and \textit{M. Mahowald} [Adams Memorial Symposium, London Math Society Lecture Notes, vol. 176, 55--72 (1992; Zbl 0753.55004)]. \textit{A. K. Bousfield} [Topology 38, 1239--1264 (1999; Zbl 0933.57034)] showed that these groups are naturally isomorphic to \(\pi_*\tau_p(\Phi_1(X))\), with \(\tau_p\) being torsion and \(\Phi_1\) a \(v_1\)-stabilization, when \(p\) is an odd prime. The most modest description of this paper might be to call it a extension of those earlier results to the case \(p=2\). But in fact, it is a very impressive tour de force involving Atiyah's real \(K\)-theory, Brown-Comenetz duality, Bott sequences, stable Adams operations, etc. Of the many applications, the author recovers earlier results of Mahowald and Davis. The results are also used by Davis in his important paper [\textit{D. M. Davis}, Homology Homotopy Appl. 5, 297--324 (2003; Zbl 1031.55008)].
scientific article; zbMATH DE number 2142203