Units of ring spectra, orientations, and Thom spectra via rigid infinite loop space theory. (1st July 2014)

• Record Type: Journal Article
• Authors: Ando, Matthew; Blumberg, Andrew J.; Gepner, David; Hopkins, Michael J.; Rezk, Charles
• Is Part Of: Journal of topology. Volume 7:Part 4(2014)
• Page Start: 1077
• Page End: 1117
• Publication Date: 2014-07-01
• Subjects: Topology -- Periodicals; 514.05
• DOI: 10.1112/jtopol/jtu009
• Languages: English
• ISSNs: 1753-8416

Abstract

Abstract : We extend the theory of Thom spectra and the associated obstruction theory for orientations in order to support the construction of theE ∞ string orientation oft m f, the spectrum of topological modular forms. Specifically, we show that, for anE ∞ ring spectrumA, the classical construction ofg l 1 A, the spectrum of units, is the right adjoint of the functorΣ + ∞ Ω ∞ : ho ( connective spectra ) ⟶ ho ( E ∞ ring spectra ) . To a map of spectraf : b ⟶ b g l 1 A, we associate anE ∞ A ‐algebra Thom spectrumM f, which admits anE ∞ A ‐algebra map toR if and only if the compositionb ⟶ b g l 1 A ⟶ b g l 1 R is null; the classical case developed by May, Quinn, Ray, and Tornehave arises whenA is the sphere spectrum. We develop the analogous theory forA ∞ ring spectra: ifA is anA ∞ ring spectrum, then to a map of spacesf : B ⟶ B G L 1 A, we associate anA ‐module Thom spectrumM f, which admits anR ‐orientation if and only ifB ⟶ B G L 1 A ⟶ B G L 1 R is null. Our work is based on a new model of the Thom spectrum as a derived smash product.