2013/04/30 by John Lind · 1 citation
Mathematics · #Advanced Topology and Set Theory #Algebraic Geometry and Number Theory #Algebraic number #Classifying space #Cohomology #Fibration #Homotopy #Homotopy and Cohomology in Algebraic Topology #Ring (chemistry) #Space (punctuation) #Spectral line #Spectrum (functional analysis) #Vector bundle #math.AT #math.KT
paper · pdf · doi:10.2140/pjm.2016.285.427
published as Pacific J. Math. 285 (2016) 427-452 · v3: simplified and shortened. journal version
openalex created_date 2016/06/24 · openalex publication_date 2016/11/21 · arxiv created 2017/02/24 · arxiv updated 2017/02/28 · openalex updated_date 2026/08/05
A parametrized spectrum E is a family of spectra Ex continuously parametrized by the points x of a topological space X. 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 BAutR(M) of the A_∞ space of R-module equivalences from M to M. In proving the classification theorem for parametrized spectra, we define of the notion of a principal G fibration where G is an A_∞ space and prove a similar classification theorem for principal G fibrations.