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

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 t m f , the spectrum of topological modular forms. Specifically, we show that, for an E ∞ ring spectrum A, the classical construction of g l 1 A , the spectrum of units, is the right adjoint of the functor Σ + ∞ Ω ∞ : ho ( connective spectra ) ⟶ ho ( E ∞ ring spectra ) . To a map of spectra f : b ⟶ b g l 1 A , we associate an E ∞ A-algebra Thom spectrum M f , which admits an E ∞ A-algebra map to R if and only if the composition b ⟶ b g l 1 A ⟶ b g l 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 ∞ ring spectra: if A is an A ∞ ring spectrum, then to a map of spaces f : B ⟶ B G L 1 A , we associate an A-module Thom spectrum M f , which admits an R-orientation if and only if B ⟶ 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.