Units of ring spectra and Thom spectra

Abstract

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. We also note that BGL(A) 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.

Various corrections and minor emendations, notably to Section 5

References in corpus (9)

• Higher Topos Theory
• Algebraic topology and modular forms
• Derived Algebraic Geometry III: Commutative Algebra
• Derived Algebraic Geometry V: Structured Spaces
• Topological Hochschild homology of Thom spectra and the free loop space
• Morita theory in abelian, derived and stable model categories
• Units of equivariant ring spectra
• Parametrized spaces model locally constant homotopy sheaves
• Morita Theory For Derived Categories: A Bicategorical Perspective

Cited by in corpus (25)

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• All $(\infty ,1)$-toposes have strict univalent universes
• Units of equivariant ring spectra
• Twisted Spin cobordism and positive scalar curvature
• Quantization via Linear homotopy types
• Spectra of units for periodic ring spectra and group completion of graded E-infinity spaces
• p-adic J-homomorphisms and a product formula
• A universal coefficient theorem for twisted K-theory
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• Gauge theory and string topology
• The Becker-Gottlieb Transfer Is Functorial
• Adjoining roots in homotopy theory
• Residue fields for a class of rational ${\mathbf{E}}_{\infty }$-rings and applications
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• Homotopy automorphisms of R-module bundles, and the K-theory of string topology
• Higher symmetries in abstract stable homotopy theories
• Strict units of commutative ring spectra
• An exposition of the topological half of the Grothendieck-Hirzebruch-Riemann-Roch theorem in the fancy language of spectra
• Morita bundle gerbes