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.
ASJC Scopus subject areas
- Geometry and Topology