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Units of ring spectra, orientations, and Thom spectra via rigid infinite loop space theory.
- Source :
-
Journal of Topology . Dec2014, Vol. 7 Issue 4, p1077-1117. 41p. - Publication Year :
- 2014
-
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. [ABSTRACT FROM AUTHOR]
- Subjects :
- *OBSTRUCTION theory
*ALGEBRAIC topology
*LOOP spaces
*TOPOLOGICAL spaces
*TOPOLOGY
Subjects
Details
- Language :
- English
- ISSN :
- 17538416
- Volume :
- 7
- Issue :
- 4
- Database :
- Academic Search Index
- Journal :
- Journal of Topology
- Publication Type :
- Academic Journal
- Accession number :
- 99750042
- Full Text :
- https://doi.org/10.1112/jtopol/jtu009