Units of ring spectra and Thom spectra

Understand

We review and extend the theory of Thom spectra and the associated obstruction theory for orientations.

• We recall (from May, Quinn, and Ray) that a commutative ring spectrum A has a spectrum of units gl(A).
• To a map of spectra f: b -> bgl(A), we associate a commutative A-algebra Thom spectrum Mf, which admits a commutative A-algebra map to R if and only if b -> bgl(A) -> bgl(R) is null.
• If A is an associative ring spectrum, then to a map of spaces f: B -> BGL(A) we associate an A-module Thom spectrum Mf, which admits an R-orientation if and only if B -> BGL(A) -> BGL(R) is null.