Rational algebraic<i>K</i>–theory of topological<i>K</i>–theory

Ausoni, Christian; Rognes, John

doi:10.2140/gt.2012.16.2037

Cited by 8 publications

(1 citation statement)

References 33 publications

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“…Much like complex topological K-theory (Example 2.14) is the K-theory of topological C-module bundles, K(R)-cohomology theory is the K-theory of suitable R-module ∞-bundles [Li13]. Specifically, for R = ku the connective spectrum for topological K-theory, its algebraic K-theory K(ku) [Au09][AR02] [AR07] has been argued to be the K-theory of certain categorified complex vector bundles [BDR03][BDRR09]. Moreover, K(R) is itself a suitable ring spectrum, so that the construction may be iterated to yield iterated algebraic K-theories…”

Section: Coefficients H(x; Bg) Examplesmentioning

confidence: 99%

The character map in (twisted differential) non-abelian cohomology

Fiorenza,

Sati,

Schreiber

2020

Preprint

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The Chern character on K-theory has a natural extension to arbitrary generalized cohomology theories known as the Chern-Dold character. Here we further extend this to (twisted, differential) non-abelian cohomology theories, where its target is a non-abelian de Rham cohomology of twisted L ∞ -algebra valued differential forms. The construction amounts to leveraging the fundamental theorem of dg-algebraic rational homotopy theory to a twisted non-abelian generalization of the de Rham theorem. We show that the non-abelian character reproduces, besides the Chern-Dold character, also the Chern-Weil homomorphism as well as its secondary Cheeger-Simons homomorphism on (differential) non-abelian cohomology in degree 1, represented by principal bundles (with connection); and thus generalizes all these to higher (twisted, differential) non-abelian cohomology, represented by higher bundles/higher gerbes (with higher connections). As a fundamental example, we discuss the twisted non-abelian character map on twistorial Cohomotopy theory over 8-manifolds, which can be viewed as a twisted non-abelian enhancement of topological modular forms (tmf) in degree 4. This turns out to exhibit a list of subtle topological relations that in high energy physics are thought to govern the charge quantization of fluxes in M-theory.

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