Vol. 285, No. 2, 2016

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Bundles of spectra and algebraic K-theory

John A. Lind

Vol. 285 (2016), No. 2, 427–452

A parametrized spectrum E is a family of spectra Ex continuously parametrized by the points x 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 K0(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 BAutRM of the topological monoid of R-module equivalences from M to M.

algebraic K-theory, parametrized spectra, bundle theory, classifying spaces
Mathematical Subject Classification 2010
Primary: 19D99, 55P43, 55R15, 55R65, 55R70
Received: 25 November 2015
Revised: 14 July 2016
Accepted: 15 July 2016
Published: 21 November 2016
John A. Lind
Reed College
3203 SE Woodstock Blvd.
Portland, Oregon 97202