Introduction

Many of the hidden features of stable homotopy theory can be conveniently exposed by localizing the stable homotopy category with respect to appropriate homology theories. In this note we discuss the algebraic structure of K*-local stable homotopy theory, where K* is the usual complex K-homology theory. We work integrally, not just at a prime. Our main goal is to give a homotopy classification of the K*-local spectra, using a united K-homology theory which combines the complex, real, and self-conjugate theories. As an interesting byproduct of this work, we also obtain a Kunneth theorem for the united K-homology of a smash product of spectra or a product of spaces. Full technical details will appear in [9] and elsewhere.

We work in the stable homotopy category S of CW-spectra and begin by recalling the general theory of homological localizations of spectra for a homology theory E* determined by a spectrum E (see [2], [7], [11]). To expose the part of stable homotopy theory seen by E*, we may use the category of fractions S [ε–l] obtained from S by adjoining formal inverses to the E*-equivalences. However, to be more concrete and to avoid set theoretic difficulties, we call a spectrum Y E*-local if each E*-equivalence V → W of spectra induces an isomorphism [W, Y]* ≅ [v, Y]*, and we call a map X → XE of spectra an E*-localization if it is an E *-equivalence with XE E*-local.