A classification of K-local spectra

Abstract

When the stable homotopy category is localized with respect to ordinary topological K-theory, it becomes highly algebraic. In this paper, an algebraic classification of K ∗-local spectra is obtained using a ‘united K-homology theory’ K CRT ∗ which combines the complex, real, and self- conjugate theories. It has much better homological algebraic properties than its constituent homology theories and leads to a K CRT ∗-Adams spectral sequence for K ∗-local mapping class groups which always vanishes above homological degree 2. The main classification results of this paper hold without arithmetic localization and generalize results previously obtained at an odd prime.