[Paper Review] Units of ring spectra and Thom spectra

Matthew Ando, Andrew J. Blumberg+3 more|ArXiv.org|Oct 24, 2008

Homotopy and Cohomology in Algebraic Topology35 references43 citations

TL;DR

This paper establishes a unified framework for Thom spectra and orientations in both $E_inity$ and $A_inity$ ring spectra using modern homotopy theory. It proves that the units spectrum $gl_1A$ is right adjoint to the functor $Σ^\infty_+\Omega^\infty$, and shows that an $A$-module Thom spectrum $Mf$ admits an $R$-orientation if and only if the composition $B \to BGL_1A \to BGL_1R$ is null, generalizing classical obstruction theory for orientations.

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.

Motivation & Objective

• To extend the classical obstruction theory for orientations of Thom spectra to $A_\infty$ ring spectra.
• To provide a modern, homotopically coherent formulation of the units of a ring spectrum as a right adjoint to $Σ^\infty_+\Omega^\infty$.
• To unify the theory of Thom spectra across $E_\infty$ and $A_\infty$ settings using $∞$-categories and parametrized spectra.
• To characterize the Thom spectrum functor via Morita theory, linking it to the geometry of line bundles and invertible modules.

Proposed method

• Use the $∞$-categorical framework from [HTT] to model parametrized spectra and define Thom spectra via left Kan extensions.
• Construct the spectrum of units $gl_1A$ as the right adjoint to $Σ^\infty_+\Omega^\infty$, establishing a derived adjunction in the homotopy category of connective spectra and $E_\infty$ ring spectra.
• For $A$ an $A_\infty$ ring spectrum, define an $A$-module Thom spectrum $Mf$ via the colimit of a diagram $B \to BGL_1A \to \mathrm{Mod}_A$, where $f: B \to BGL_1A$ classifies a line bundle.
• Characterize the space of $R$-orientations of $Mf$ as a pullback involving mapping spaces in the $∞$-category of $R$-modules, using adjunctions and homotopy pullbacks.
• Establish two equivalent approaches: one using rigidified models of $A_\infty$ spaces (from [Blum05, BCS08]), and another using $∞$-categories and symmetric monoidal structures.
• Apply Morita theory to show that the Thom spectrum functor is characterized by its universal property as a left adjoint to the forgetful functor from $R$-algebras to $R$-line bundles.

Experimental results

Research questions

• RQ1 How can the classical obstruction theory for orientations of Thom spectra be reformulated using modern $∞$-categorical and homotopical algebra?
• RQ2 What is the precise homotopical nature of the spectrum of units $gl_1A$ for an $E_\infty$ ring spectrum $A$, and how does it relate to the $Σ^\infty_+\Omega^\infty$ functor?
• RQ3 Can the theory of Thom spectra and orientations be extended from $E_\infty$ to $A_\infty$ ring spectra in a coherent and computable way?
• RQ4 How do the two different approaches—rigidified spaces and $∞$-categories—yield equivalent constructions of the Thom spectrum functor?
• RQ5 What is the role of Morita theory in characterizing the Thom spectrum functor as a universal construction?

Key findings

• The spectrum of units $gl_1A$ is the right adjoint to the functor $Σ^\infty_+\Omega^\infty$ from connective spectra to $E_\infty$ ring spectra, establishing a derived adjunction.
• An $E_\infty$ $A$-algebra Thom spectrum $Mf$ admits a map to an $E_\infty$ $R$-algebra if and only if the composition $b \to bgl_1A \to bgl_1R$ is null, generalizing the classical obstruction condition.
• For $A$ an $A_\infty$ ring spectrum, the $A$-module Thom spectrum $Mf$ associated to $f: B \to BGL_1A$ admits an $R$-orientation if and only if $B \to BGL_1A \to BGL_1R$ is null, extending the obstruction theory to $A_\infty$ settings.
• The space of $R$-orientations of $Mf$ is equivalent to the space of lifts of $f$ to the trivial line bundle, realized as a homotopy pullback in the $∞$-category of $R$-modules.
• The Thom spectrum $MR$ associated to the identity on $R$-lines is equivalent to $R^\circ / \mathrm{Aut}(R^\circ)$, the homotopy quotient of the unit sphere by its automorphisms.
• The theory is unified via Morita theory: the Thom spectrum functor is characterized as the left adjoint to the forgetful functor from $R$-algebras to $R$-line bundles, providing a universal property.