nLab K-theory spectrum



Stable Homotopy theory

Higher algebra



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

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

The connective cover is denoted in lower case: ku.


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



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

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-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


The weak homotopy eqivalence BUΩUB 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 KUKU and KOKO as a symmetric spectrum is discussed in

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

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

The structure of a homotopy ring spectrum on KUKU 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 KUKU is discussed in

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

and the underlying H-infinity ring spectrum structure in

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

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

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

with a summary in

Last revised on March 11, 2024 at 03:36:36. See the history of this page for a list of all contributions to it.