nLab K-theory spectrum

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

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

The connective cover is denoted in lower case: ku.

Definition

As a symmetric spectrum: Schwede 12, Example I.2.10

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)

Related concepts

• Pontrjagin character

• Eilenberg-MacLane spectrum

• tmf

• cohomology theory

• KO-theory

• Tate K-theory

• splitting principle

• comparison map between algebraic and topological K-theory

• KU-local stable homotopy theory

cohomology theories of string theory fields on orientifolds

string theory	B-field	$B$-field moduli	RR-field
bosonic string	line 2-bundle	ordinary cohomology $H\mathbb{Z}^3$
type II superstring	super line 2-bundle	$Pic(KU)//\mathbb{Z}_2$	KR-theory $KR^\bullet$
type IIA superstring	super line 2-bundle	$B GL_1(KU)$	KU-theory $KU^0$
type IIB superstring	super line 2-bundle	$B GL_1(KU)$	KU-theory $KU^1$
type I superstring	super line 2-bundle	$Pic(KU)//\mathbb{Z}_2$	KO-theory $KO$
type $\tilde I$ superstring	super line 2-bundle	$Pic(KU)//\mathbb{Z}_2$	KSC-theory $KSC$

References

The weak homotopy eqivalence $B U \to \Omega U$ is discussed in detail in

• Marcelo Aguilar, Samuel Gitler, Carlos Prieto, appendix B.2 of Algebraic topology from a homotopical viewpoint, Springer (2002) (toc pdf)

The incarnation of $KU$ and $KO$ as a symmetric spectrum is discussed in

• Stefan Schwede, Example I.2.10 in Symmetric spectra, 2012 (pdf)

and the structure of a symmetric ring spectrum on $KO$ is discussed in

• Michael Joachim, A symmetric ring spectrum representing $KO$-theory, Topology 40 (2001) 299-308

The structure of a homotopy ring spectrum on $KU$ is discussed in

• Robert Switzer, section 13.90 of Algebraic Topology - Homotopy and Homology, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Vol. 212, Springer-Verlag, New York, N. Y., 1975.

The E-infinity ring structure of $KU$ is discussed in

• Peter May, section VIII §2 of $E_\infty$-Ring spaces and $E_\infty$ ring spectra (pdf)

and the underlying H-infinity ring spectrum structure in

• James McClure, $H_\infty$-ring spectra via space-level homotopy theory (pdf), chapter VII in R. Bruner, Peter May, James McClure, M. Steinberger, $H_\infty$-Ring Spectra and their Applications, Lecture Notes in Mathematics 1176, Springer 1986 (pdf)

The uniqueness of the E-infinity ring structure on $KU$ is due to

• Andrew Baker, Birgit Richter, $\Gamma$-cohomology of rings of numerical polynomials and $E_\infty$ structures on K-theory (arXiv:math/0304473)

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

• Akhil Mathew, The homology of $tmf$ (arXiv:1305.6100)

with a summary in

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