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Units of ring spectra and Thom spectra | | Matthew Ando
; Andrew J. Blumberg
; David J. Gepner
; Michael J. Hopkins
; Charles Rezk
; | | Date: |
24 Oct 2008 | | 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. | | Source: | arXiv, 0810.4535 | | Services: | Forum | Review | PDF | Favorites | |
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