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 Units of ring spectra and Thom spectra

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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 glA. To a map of spectra f: b - bglA, we associate a commutative A-algebra Thom spectrum Mf, which admits a commutative A-algebra map to R if and only if b - bglA - bglR is null. If A is an associative ring spectrum, then to a map of spaces f: B - BGLA we associate an A-module Thom spectrum Mf, which admits an R-orientation if and only if B - BGLA - BGLR is null. We also note that BGLA classifies the twists of A-theory. We develop and compare two approaches to the theory of Thom spectra. The first involves a rigidified model of A-infinity and E-infinity spaces. Our second approach is via infinity categories. In order to compare these approaches to one another and to the classical theory, we characterize the Thom spectrum functor from the perspective of Morita theory.

Author: Matthew Ando; Andrew J. Blumberg; David J. Gepner; Michael J. Hopkins; Charles Rezk

Source: https://archive.org/

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