The \(K\)-theory localizations and \(v_1\)-periodic homotopy groups of H-spaces
This paper sheds new light on the effect of localization with respect to mod-\(p\) complex \(K\)-homology \((K\mathbb Z/p)_*\) or, equivalently, with respect to \(p\)-adic complex \(K\)-cohomology \((K\mathbb Z^{\wedge}_{p})^*\); \(p\) an odd prime. For a space \(X\) whose \(K^*(X;\mathbb Z^{\wedge}_{p})\) is isomorphic, as a \(\mathbb Z/2\)-graded \(\lambda\)-ring, to an exterior algebra generated by a certain Adams-submodule \(M\) of \(K^1(X;\mathbb Z^{\wedge}_{p})\), the \((K\mathbb Z/p)_*\)-localization is constructed explicitly in terms of \(M\)-information. Spaces to which this method applies include finite \(H\)-spaces or odd-dimensional spheres. \smallskip\noindent Aided by these results, the author computes the \(v_1\)-periodic homotopy groups of such a space \(X\). Further he achieves a near complete classification of odd \((K\mathbb Z/p)_*\)-homology spheres.
scientific article; zbMATH DE number 1317257