Vol. 285, No. 2, 2016

Download this article
Download this article For screen
For printing
Recent Issues
Vol. 305: 1
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
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 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.

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