On the 2-adic \(K\)-localizations of \(H\)-spaces
\textit{M. Mahowald} and \textit{R. D. Thompson} [Topology 31, 133--141 (1992; Zbl 0759.55010)] determined the \(p\)-adic \(K\)-localization of the odd spheres at an arbitrary prime \(p\), expressing these localizations as homotopy fibers of maps between specified infinite loop spaces. The author generalized this result in [Topology 38, 1239--1264 (1999; Zbl 0933.57034)] to give the \(p\)-adic \(K\)-localizations for a large class of \(H\)-spaces and related spaces for an odd prime \(p\). In the paper under review, the author shows similar results for \(2\)-adic \(K\)-localizations of such spaces. Then he applies the results to the computations of the \(v_1\)-periodic homotopy groups of those spaces from their united \(2\)-adic \(K\)-cohomologies, and thus to the computations of the \(v_1\)-periodic homotopy groups for a large class of simply-connected compact Lie groups from their complex, real, and quaternionic representation theories. A \(2\)-adic \(\theta \Delta\)-module is a triple \(M=\{M_C,M_R,M_H\}\) of \(2\)-profinite abelian groups with some continuous additive operations. For any space \(X\), the triple \(K_{\Delta}^{-1}(X;\widehat {\mathbb Z}_2) =\{K^{-1}(X;\widehat {\mathbb Z}_2),KO^{-1}(X;\widehat {\mathbb Z}_2), KO^{-5}(X;\widehat {\mathbb Z}_2)\}\) is an example. A special \(2\)-adic \(\phi CR\)-algebra is a pair \(A=\{A_C,A_R\}\) of \(2\)-profinite algebras with some continuous additive operations. For any connected space \(X\), \(K^*_{CR}(X;\widehat {\mathbb Z}_2)=\{K^*(X;\widehat {\mathbb Z}_2),KO^*(X;\widehat {\mathbb Z}_2)\}\) is an example. Then the author constructs a functor \(\widehat L\) from the category of \(2\)-adic \(\theta \Delta\)-modules to the category of special \(2\)-adic \(\phi CR\)-algebras, extending the \(2\)-adic exterior algebra functor on complex components, and shows that if \(X\) is a connected space with \(K^*_{CR}(X;\widehat{\mathbb Z}_2)\cong \widehat LM\) for a \(2\)-adic \(\theta\Delta\)-module \(M\) equipped with Adams operations, then its \(K/2_*\)-localization \(X_{K/2}\) is the homotopy fiber of a map between specified infinite loop spaces with low dimensional modifications. The author shows that the result is applicable to any simply-connected compact simple Lie group other than \(E_6\) and \(Spin(4k+2)\) with \(k\) not a \(2\)-power. He also shows that the result is applicable to many other simply-connected finite \(H\)-spaces and related spaces such as the spheres \(S^{4k-1}\) for \(k\geq 1\).
scientific article; zbMATH DE number 5132168