THE K-THEORY LOCALIZATIONS AND v1-PERIODIC HOMOTOPY GROUPS OF H-SPACES

We determine the mod p K-theory localizations and v1-periodic homotopy groups of finite H-spaces and of other spaces with torsion-free exterior p-adic K-cohomology algebras at an odd prime p. Our localization results generalize those of Mahowald and Thompson (Topology 1992, 31, 133–141) for odd-dimensional spheres. We construct our mod p K-theory localizations as homotopy fibers of unstable maps between infinite loop spaces, and similarly construct a wide array of new spaces having torsion-free exterior p-adic K-cohomology algebras with prescribed Adams operations. This leads, for example, to a classification of the odd mod p K-homology spheres.