2014/03/18 by Matthew Ando, Andrew J. Blumberg, David Gepner +2 · 3 citations
Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #Black Holes and Theoretical Physics #Computer science #Homotopy and Cohomology in Algebraic Topology #Library science #Mathematics #math.AT #msc:55
paper · pdf · doi:10.1112/jtopol/jtu009
arXiv admin note: substantial text overlap with arXiv:0810.4535
arxiv created 2014/03/18 · openalex publication_date 2014/07/01 · arxiv updated 2017/05/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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 t m f , the spectrum of topological modular forms. Specifically, we show that, for an E ∞ ring spectrum A, the classical construction of g l 1 A , 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 ⟶ b g l 1 A , we associate an E ∞ A-algebra Thom spectrum M f , which admits an E ∞ A-algebra map to R if and only if the composition b ⟶ b g l 1 A ⟶ b g l 1 R 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 ⟶ B G L 1 A , we associate an A-module Thom spectrum M f , which admits an R-orientation if and only if B ⟶ B G L 1 A ⟶ B G L 1 R is null. Our work is based on a new model of the Thom spectrum as a derived smash product.