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- Title
Units of ring spectra, orientations, and Thom spectra via rigid infinite loop space theory.
- Authors
Ando, Matthew; Blumberg, Andrew J.; Gepner, David; Hopkins, Michael J.; Rezk, Charles
- Abstract
We extend the theory of Thom spectra and the associated obstruction theory for orientations in order to support the construction of the $E_{\infty }$ string orientation of $tmf$, the spectrum of topological modular forms. Specifically, we show that, for an $E_{\infty }$ ring spectrum $A$, the classical construction of $gl_{1} {A}$, the spectrum of units, is the right adjoint of the functor\[\Sigma ^{\infty }_{+ } \Omega ^{\infty } \colon \mathrm {ho} (\mbox {{connective spectra}}) \longrightarrow \mathrm {ho} ({\text {{$E_{\infty } $ ring spectra}}}).\]To a map of spectra\[f\colon b \longrightarrow bgl_{1} {A},\]we associate an $E_{\infty }$ $A$-algebra Thom spectrum $Mf$, which admits an $E_{\infty }$ $A$-algebra map to $R$ if and only if the composition\[b \longrightarrow bgl_{1} {A} \longrightarrow bgl_{1} {R}\]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 $A_{\infty }$ ring spectra: if $A$ is an $A_{\infty }$ ring spectrum, then to a map of spaces\[f\colon B \longrightarrow B{ GL}_{1} {A},\]we associate an $A$-module Thom spectrum $Mf,$ which admits an $R$-orientation if and only if\[B \longrightarrow B{ GL}_{1} {A} \longrightarrow B{ GL}_{1} {R}\]is null. Our work is based on a new model of the Thom spectrum as a derived smash product.
- Subjects
OBSTRUCTION theory; ALGEBRAIC topology; LOOP spaces; TOPOLOGICAL spaces; TOPOLOGY
- Publication
Journal of Topology, 2014, Vol 7, Issue 4, p1077
- ISSN
1753-8416
- Publication type
Article
- DOI
10.1112/jtopol/jtu009