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

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^∞+Ω^∞: 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.