nLab
K-theory spectrum

Context

Stable Homotopy theory

Higher algebra

Contents

Idea

The K-theory spectrum KUKU (for complex K-theory) or KOKO (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

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

More generally, to every stable (infinity,1)-category CC 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

KUKU 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 β\beta:

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

Relation between KUKU, KOKO and KRKR-

KOKO as homotopy-fixed points of KUKU

Complex conjugation on complex vector bundles induces on the complex K-theory spectrum KUKU an involutive automorphism. KR-theory is the corresponding 2\mathbb{Z}_2-equivariant cohomology theory.

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

KO(KU) /2 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 KOKO-modules.

Wood’s theorem

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

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

cohomology theories of string theory fields on orientifolds

string theoryB-fieldBB-field moduliRR-field
bosonic stringline 2-bundleordinary cohomology H 3H\mathbb{Z}^3
type II superstringsuper line 2-bundlePic(KU)// 2Pic(KU)//\mathbb{Z}_2KR-theory KR KR^\bullet
type IIA superstringsuper line 2-bundleBGL 1(KU)B GL_1(KU)KU-theory KU 1KU^1
type IIB superstringsuper line 2-bundleBGL 1(KU)B GL_1(KU)KU-theory KU 0KU^0
type I superstringsuper line 2-bundlePic(KU)// 2Pic(KU)//\mathbb{Z}_2KO-theory KOKO
type I˜\tilde I superstringsuper line 2-bundlePic(KU)// 2Pic(KU)//\mathbb{Z}_2KSC-theory KSCKSC

References

Discussion of KOKO in analogy to the construction of tmf is in

with a summary in

Revised on November 1, 2014 22:45:43 by Urs Schreiber (141.0.9.61)