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comparison map between algebraic and topological K-theory

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

For suitable setups, there is a canonical homomorphism from algebraic K-theory to topological K-theory with given coefficients.

This usually goes just by “the comparison map”. (e.g. Rosenberg, theorem 2.1)

Statement

For complex K-theory

There is a canonical homomorphism of spectra

KKU K \mathbb{C} \longrightarrow KU

from the algebraic K-theory spectrum of the complex numbers to KU (e.g. Paluch 01, lemma 2.6).

In fact in terms of cohesion of smooth spectra, this is a component of a natural transformation, this we discuss below.

Formulation in cohesive homotopy-type theory

Given a cohesive (infinity,1)-topos and a symmetric monoidal (∞,1)-category VCMon (Cat (H))V\in CMon_\infty(Cat_\infty(\mathbf{H})), internal to H\mathbf{H} write

𝒦(V)Stab(H)TH \mathcal{K}(V)\in Stab(\mathbf{H})\hookrightarrow T \mathbf{H}

for its K-theory of a symmetric monoidal (∞,1)-category (formed locally and then ∞-stackified). This is a stable homotopy type in the tangent cohesive (∞,1)-topos.

The flat modality part 𝒦(V)\flat \mathcal{K}(V) is the algebraic K-theory of VCMon (Cat )\flat V \in CMon_\infty(Cat_\infty). The shape modality part on the other hand is a “topological” version of this

The points-to-pieces transform Π\flat \to \Pi provides a natural comparison map

(𝒦(V))Π(𝒦(V)). \flat (\mathcal{K}(V)) \longrightarrow \Pi (\mathcal{K}(V)) \,.

For instance for H=\mathbf{H}= Smooth∞Grpd and V=Vect V = \mathbf{Vect}^{\oplus} the stack of smooth vector bundles with direct sum monoidal structure, then 𝒦(V)K\flat \mathcal{K}(V)\simeq K \mathbb{C} is the algebraic K-theory of the complex numbers and Π𝒦(V)\Pi \mathcal{K}(V)\simeq KU is complex topological K-theory.

See at differential cohomology diagram for more on this.

References

  • Henri Gillet, Comparing algebraic and topological K-theory, in Higher Algebraic K-Theory: an overview Lecture Notes in Mathematics Volume 1491, 1992, pp 55-99

Discussion relating algebraic K-theory of varieties to complex topological K-theory is in

  • Michael Paluch, Algebraic KK-theory and topological spaces, K-theory 471 (2001) (web)

and for sheaves of spectra of twisted K-theory in

Discussion in terms of Banach algebras is in

  • Jonathan Rosenberg, Comparison between algebraic and topological K-theory for Banach algebras and C *C^\ast-algebras (pdf)

Last revised on May 1, 2014 at 00:55:44. See the history of this page for a list of all contributions to it.