Units of ring spectra and Thom spectra

By M. Ando, Andrew Justin Blumberg, D. Gepner, M. Hopkins, Charles Rezk. 2008

67 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c89a97b7672d56774765c2eafcdd667442e4d69b

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Citations in the corpus

Twists of K-theory and TMF (2010, methodology): “ATTHEW ANDO, ANDREW J. BLUMBERG, AND DAVID GEPNER Abstract. We explore an approach to twisted generalized cohomology from the point of view of stable homotopy theory and ∞-category theory provided by [ABGHR]. We explain the relationship to the twisted K-theory provided by Fredholm bundles. We show h”
Parametrized spectra, multiplicative Thom spectra and the twisted Umkehr map (2011, methodology): “egories 23 9. Multiplicative structures on generalized Thom spectra 27 10. Twisted cohomology theories and the twisted Umkehr map 28 References 37 1. Introduction In recent work with Hopkins and Rezk [1], we introduced an ∞-categorical approach to parametrized spaces and spectra and showed that it p”
An ∞‐categorical approach to R‐line bundles, R‐module Thom spectra, and twisted R‐homology (2014, unknown): “…immediately from the construction of the Thom spectrum, since by deﬁnition the quotient in the statement is the colimit of the map of ∞ -categories Turning to the comparisons, the deﬁnitions of [ 2 ] and this paper give two constructions of an R -module Thom spectrum from a map f : X → BGL 1 R .”