nLab K-theory spectrum

Context

Stable Homotopy theory

stable homotopy theory

Contents

Higher algebra

higher algebra

universal algebra

Contents

Idea

The K-theory spectrum $KU$ (for complex K-theory) or $KO$ (for orthogonal K-theory) in the strict sense is the spectrum that represents the generalized (Eilenberg-Steenrod) cohomology theory topological K-theory. For complex topological K-theory this is periodic with period 2 (reflect Bott periodicity) of the form

$\mathbb{Z} \times B U ,\; U ,\; \cdots \,.$

More generally, to every stable (infinity,1)-category $C$ is associated a K-theory space which in good cases, such as when the category is presented by a Waldhausen category is the degree 0 piece of a corresponding algebraic K-theory spectrum. The detailed construction is known as the Waldhausen S-construction.

Properties

Periodicity

$KU$ is a 2-periodic ring spectrum. This is the original Bott periodicity.

As a localization of an $\infty$-group $\infty$-ring

Snaith's theorem asserts that the K-theory spectrum for periodic complex K-theory is the ∞-group ∞-ring of the circle 2-group localized away from the Bott element $\beta$:

$KU \simeq \mathbb{S}[B U(1)][\beta^{-1}] \,.$

Relation between $KU$, $KO$ and $KR$-

$KO$ as homotopy-fixed points of $KU$

Complex conjugation on complex vector bundles induces on the complex K-theory spectrum $KU$ an involutive automorphism. KR-theory is the corresponding $\mathbb{Z}_2$-equivariant cohomology theory.

In particular, the homotopy fixed point of KU under this automorphism is KO

$KO \simeq (KU)^{\mathbb{Z}/2}$

and this way where in complex K-theory one has KU-modules (∞-modules), so in KR-theory one has $KO$-modules.

Wood’s theorem

$KO \wedge \Sigma^{-2}\mathbb{CP}^2 \simeq KU$

a proof in terms of moduli stacks is given in Mathew 13, section 3

Relation to Clifford algebras

There are close relations between K-theory and Clifford algebras. One conceptual statement relating them is this:

KO-theory is the first Weiss-derivative (in orthogonal calculus) of the K-theory of Clifford algebras. (Charles Rezk, MO comment, Sept ‘13)

cohomology theories of string theory fields on orientifolds

string theoryB-field$B$-field moduliRR-field
bosonic stringline 2-bundleordinary cohomology $H\mathbb{Z}^3$
type II superstringsuper line 2-bundle$Pic(KU)//\mathbb{Z}_2$KR-theory $KR^\bullet$
type IIA superstringsuper line 2-bundle$B GL_1(KU)$KU-theory $KU^1$
type IIB superstringsuper line 2-bundle$B GL_1(KU)$KU-theory $KU^0$
type I superstringsuper line 2-bundle$Pic(KU)//\mathbb{Z}_2$KO-theory $KO$
type $\tilde I$ superstringsuper line 2-bundle$Pic(KU)//\mathbb{Z}_2$KSC-theory $KSC$

References

Discussion of $KO$ in analogy to the construction of tmf is in

with a summary in

• Akhil Mathew, The homotopy groups of $TMF$ (pdf)

Revised on December 16, 2016 13:44:26 by Urs Schreiber (46.183.103.8)