Units of ring spectra, orientations, and Thom spectra via rigid infinite loop space theory

Matthew Ando, Andrew J. Blumberg, David Gepner, Michael J. Hopkins, Charles Rezk

Research output: Contribution to journalArticle

Abstract

We extend the theory of Thom spectra and the associated obstruction theory for orientations in order to support the construction of the E string orientation of tmf, the spectrum of topological modular forms. Specifically, we show that, for an E∞ ring spectrum A, the classical construction of gl1A, the spectrum of units, is the right adjoint of the functor Σ: ho(connective spectra) -→ ho(E ring spectra). To a map of spectra f : b -→ bgl1A, we associate an E A-algebra Thom spectrum Mf, which admits an E A-algebra map to R if and only if the composition b -→ bgl1A -→ bgl1R is null; the classical case developed by May, Quinn, Ray, and Tornehave arises when A is the sphere spectrum. We develop the analogous theory for A ring spectra: if A is an A ring spectrum, then to a map of spaces f : B -→ BGL1A, we associate an A-module Thom spectrum Mf, which admits an R-orientation if and only if B -→ BGL1A -→ BGL1R is null. Our work is based on a new model of the Thom spectrum as a derived smash product.

Original languageEnglish (US)
Article numberjtu009
Pages (from-to)1077-1117
Number of pages41
JournalJournal of Topology
Volume7
Issue number4
DOIs
StatePublished - Jan 21 2013

ASJC Scopus subject areas

  • Geometry and Topology

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