BUNDLES OF SPECTRA AND ALGEBRAIC K-THEORY

A parametrized spectrum E is a family of spectra E-x continuously parametrized by the points x is an element of X of a topological space. We take the point of view that a parametrized spectrum is a bundle-theoretic geometric object. When R is a ring spectrum, we consider parametrized R-module spectra and show that they give cocycles for the cohomology theory determined by the algebraic K-theory K(R) of R in a manner analogous to the description of topological K-theory K-0(X) as the Grothendieck group of vector bundles over X. We prove a classification theorem for parametrized spectra, showing that parametrized spectra over X whose fibers are equivalent to a fixed R-module M are classified by homotopy classes of maps from X to the classifying space BAut(R)M of the topological monoid of R-module equivalences from M to M.