#### Vol. 285, No. 2, 2016

 Recent Issues Vol. 304: 1  2 Vol. 303: 1  2 Vol. 302: 1  2 Vol. 301: 1  2 Vol. 300: 1  2 Vol. 299: 1  2 Vol. 298: 1  2 Vol. 297: 1  2 Online Archive Volume: Issue:
 The Journal Editorial Board Subscriptions Officers Special Issues Submission Guidelines Submission Form Contacts ISSN: 1945-5844 (e-only) ISSN: 0030-8730 (print) Author Index To Appear Other MSP Journals
Bundles of spectra and algebraic K-theory

### John A. Lind

Vol. 285 (2016), No. 2, 427–452
##### Abstract

A parametrized spectrum $E$ is a family of spectra ${E}_{x}$ continuously parametrized by the points $x\in X$ 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 $K\left(R\right)$ of $R$ in a manner analogous to the description of topological $K$-theory ${K}^{0}\left(X\right)$ 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 $B{Aut}_{R}M$ of the topological monoid of $R$-module equivalences from $M$ to $M$.

##### Keywords
algebraic K-theory, parametrized spectra, bundle theory, classifying spaces
##### Mathematical Subject Classification 2010
Primary: 19D99, 55P43, 55R15, 55R65, 55R70