Units of ring spectra, orientations, and Thom spectra via rigid infinite loop space theory

By M. Ando, Andrew Justin Blumberg, D. Gepner, M. Hopkins, Charles Rezk. 2014

81 citations recorded by the source corpus.

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 tmf , the spectrum of topological modular forms. Specifically, we show that, for an E∞ ring spectrum A , the classical construction of gl1A , the spectrum of units, is the right adjoint of the functor Σ+∞Ω∞:ho(connectivespectra)⟶ho(E∞ringspectra). To a map of spectra f:b⟶bgl1A, we associate an E∞ A ‐algebra Thom spectrum Mf , which admits an E∞ A ‐algebra map to R if and only if the composition b⟶bgl1A⟶bgl1R 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⟶BGL1A, we associate an A ‐module Thom spectrum Mf, which admits an R ‐orientation if and only if B⟶BGL1A⟶BGL1R is null. Our work is based on a new model of the Thom spectrum as a derived smash product.

Subfields: Geometry and Topology, Mathematical Physics, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/59f33033553367fbb7ddbe2e3faa42c8ecc40aa5

Citations in the corpus

• A simple universal property of Thom ring spectra (2014, unknown): “This point of view admits a generalization to En-ring spectra, as we briefly summarize now; see [ABG14a] and [ABG14b] for a comparison of the various notions of orientations in the E1 and E∞ cases, as well as [ABG11].”