KR cohomology theory




Special and general types

Special notions


Extra structure



Stable Homotopy theory

Representation theory



What is called KR-theory (Atiyah 66) is variant of topological K-theory on spaces equipped with a 2\mathbb{Z}_2-action (by homeomorphism, hence equipped with one involutive homeomorphism – a “real space”).

In terms of cocycle models, classes of KR-theory are represented by complex vector bundles over XX on which the involution on their base space lifts to an anti-linear involution of the total space. Over manifolds with trivial involution these are precisely the complexification of real vector bundles and hence over such spaces KRKR-theory reduces to KO-theory. Conversely, over two copies XXX \cup X lof XX equipped with the involution that interchanges the two, KRKR-theory reduces to KU-theory. Finally over X×S 1X \times S^1 with the involution the antipodal identification on the second (circle) factor , KRKR-theory reduces to the self-conjugate KSC-theory (Anderson 64). So in general KRKR-theory interpolates between all these cases. For instance on X×S 1X \times S^1 with the reflection-involution on the circle (the real space denoted S 1,1S^{1,1}, the non-trivial 2\mathbb{Z}_2-representation sphere) it behaves like KOKO-theory at the two involution fixed points (the two O-planes) and like KUKU in their complement (a model that makes this very explicit is given in DMR 13, section 4), schematically:

KR(S 1,1)=(KOKUKO) KR(S^{1,1}) = ( KO --- KU --- KO )

More abstractly, complex conjugation of complex vector bundles induces on the complex K-theory spectrum KU an involutive automorphism. KRKR is the corresponding 2\mathbb{Z}_2-equivariant spectrum, and KRKR-theory 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}

(e.g.Karoubi 01, Dugger 03, corollary 7.6, Hill-Hopkins-Ravenel, section 7.3) and this way where in complex K-theory one has KU-modules (∞-modules), so in KR-theory one has KOKO-modules.

KR is an example of a real-oriented cohomology theory, together with for instance MR-theory and BPR-theory.

Remark on terminology

An involution on a space by a homeomorphism (or diffeomorphism) as they appear in KR theory may be thought of as a “non-linear real structure”, and therefore spaces equipped with such involutions are called “real spaces”. Following this, KRKR-theory is usually pronounced “real K-theory”. But beware that this terminology easily conflicts with or is confused with KO-theory. For disambiguation the latter might better be called “orthogonal K-theory”. But on abstract grounds maybe KRKR-theory would best be just called 2\mathbb{Z}_2-equivariant complex K-theory.


As the Grothendieck group of complex vector bundles with real structure

…(Atiyah 66)…

As a genuine 2\mathbb{Z}_2-Spectrum

The following gives KRKR as a genuine G-spectrum for G= 2G = \mathbb{Z}_2.

Using that every orthogonal representation of 2\mathbb{Z}_2 is contained in an n\mathbb{C}^n with its complex conjugation action, one can restrict attention to these. Write

P 1=S 2,1=S . \mathbb{C}P^1 = S^{2,1} = S^{\mathbb{C}} \,.

The reduced canonical line bundle over this (the Hopf fibration) is classified by a map

S 2,1=P 1×BU S^{2,1}= \mathbb{C}P^1 \to \mathbb{Z}\times BU

to the classifying space for topological K-theory. The homotopy-associative multiplication on this space then yields the structure map of a 2\mathbb{Z}_2-spectrum

S 2,1(×BU)×BU. S^{2,1} \wedge (\mathbb{Z} \times BU)\to \mathbb{Z}\times BU \,.

This is in fact an Omega spectrum, by equivariant complex Bott periodicity (for instance in Dugger 03, p. 2-3).



As any genuine equivariant cohomology theory KRKR-theory is naturally graded over the representation ring RO( 2)RO(\mathbb{Z}_2). Write \mathbb{R} for the trivial 1-dimensional representation and \mathbb{R}_- for that given by the sign involution. Then the general orthogonalrepresentation decomposes as a direct sum

V= + q. V = \mathbb{R}^+\oplus \mathbb{R}_-^q \,.

The corresponding representation sphere is

S V=(someconvention). S^V = (some\; convention) \,.

As induced from the derived moduli stack of tori

The relation between KUKU, KOKO and KRKR naturally arises in chromatic homotopy theory as follows.

Inside the moduli stack of formal group laws sits the moduli stack of one dimensional tori 𝔾 m\mathcal{M}_{\mathbb{G}_m} (Lawson-Naumann 12, def. A.1, A.3). This is equivalent to the quotient stack of the point by the group of order 2

𝔾 mB 2 \mathcal{M}_{\mathbb{G}_m}\simeq \mathbf{B}\mathbb{Z}_2

(Lawson-Naumann 12, prop. A.4). Here the 2\mathbb{Z}_2-action is the inversion involution on abelian groups.

Using the Goerss-Hopkins-Miller theorem this stack carries an E-∞ ring-valued structure sheaf 𝒪 top\mathcal{O}^{top} (Lawson-Naumann 12, theorem A.5); and by the above equivalence this is a single E-∞ ring equipped with a 2\mathbb{Z}_2-∞-action. This is KU with its involution induced by complex conjugation, hence essentially is KRKR.

Accordingly, the global sections of 𝒪 top\mathcal{O}^{top} over 𝔾 m\mathcal{M}_{\mathbb{G}_m} are the 2\mathbb{Z}_2-homotopy fixed points of this action, hence is KOKO. This is further amplified in (Mathew 13, section 3) and (Mathew, section 2).

As suggested there and by the main (Lawson-Naumann 12, theorem 1.2) this realizes (at least localized at p=2p = 2) the inclusion KOKUKO \to KU as the restriction of an analogous inclusion of tmf built as the global sections of the similarly derived moduli stack of elliptic curves.

chromatic homotopy theory

chromatic levelcomplex oriented cohomology theoryE-∞ ring/A-∞ ringreal oriented cohomology theory
0ordinary cohomologyEilenberg-MacLane spectrum HH \mathbb{Z}HZR-theory
0th Morava K-theoryK(0)K(0)
1complex K-theorycomplex K-theory spectrum KUKUKR-theory
first Morava K-theoryK(1)K(1)
first Morava E-theoryE(1)E(1)
2elliptic cohomologyelliptic spectrum Ell EEll_E
second Morava K-theoryK(2)K(2)
second Morava E-theoryE(2)E(2)
algebraic K-theory of KUK(KU)K(KU)
3 …10K3 cohomologyK3 spectrum
nnnnth Morava K-theoryK(n)K(n)
nnth Morava E-theoryE(n)E(n)BPR-theory
n+1n+1algebraic K-theory applied to chrom. level nnK(E n)K(E_n) (red-shift conjecture)
\inftycomplex cobordism cohomologyMUMR-theory

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

Substructure of the moduli stack of curves and the (equivariant) cohomology theory associated with it via the Goerss-Hopkins-Miller-Lurie theorem:

covering moduli spaceof level-n structures (modular curve)
*=Spec()\ast = Spec(\mathbb{Z})\toSpec([[q]])Spec(\mathbb{Z}[ [q] ])\to ell¯[n]\mathcal{M}_{\overline{ell}}[n]
structure group /2\downarrow^{\mathbb{Z}/2\mathbb{Z}} /2\downarrow^{\mathbb{Z}/2\mathbb{Z}} SL 2(/n)\downarrow^{SL_2(\mathbb{Z}/n\mathbb{Z})} (modular group)
1dTori\mathcal{M}_{1dTori}\hookrightarrow Tate\mathcal{M}_{Tate}\hookrightarrow ell¯\mathcal{M}_{\overline{ell}}\hookrightarrow cub\mathcal{M}_{cub}\to FG\mathcal{M}_{FG}
moduli stackof 1d toriof Tate curvesof elliptic curvesof cubic curvesof formal groups
𝒪 Σ top\mathcal{O}^{top}_{\Sigma}KUKU[[q]]KU[ [q] ]elliptic spectrumcomplex oriented cohomology theory
Γ(,𝒪 top)=\Gamma(-, \mathcal{O}^{top}) = (KO \hookrightarrow KU) = KR-theoryTate K-theory (KO[[q]]KU[[q]]KO[ [q] ] \hookrightarrow KU[ [q] ])(Tmf \to Tmf(n)) (modular equivariant elliptic cohomology)tmf𝕊\mathbb{S}


KR theory was introduced in

  • Michael Atiyah, K-theory and reality, The Quarterly Journal of Mathematics. Oxford. Second Series 17 (1) (1966),: 367–386, doi:10.1093/qmath/17.1.367, ISSN 0033-5606, MR 0206940 (pdf)

The version of KSCKSC-theory was introduced in

  • D. W. Anderson, The real K-theory of classifying spaces Proc. Nat. Acad. Sci. U. S. A., 51(4):634–636, 1964.

The dual concept of KR-homology was defined in

  • Gennady Kasparov, The operator K-functor and extensions of C *C^\ast-algebras, Izv. Akad. Nauk. SSSR Ser. Mat. 44, 571-636 (1980).

Discussion in the general context of real oriented cohomology theory is in

  • Igor Kriz, Real-oriented homotopy theory and an analogue of the Adams}Novikov spectral sequence, Topology 40 (2001) 317-399 (pdf)

Further discussion includes

Reviews include

  • Wikipedia, KR-theory

  • Paolo Masulli, Equivariant homotopy: KRKR-theory, Master thesis (2011) (pdf)

Remarks on homotopy-theoretic KR in the context of algebraic K-theory are in

Explicit groupoid/stack models for equivariant and twisted KR-theory theory are discussed in

  • El-kaïoum M. Moutuou, Twistings of KR for Real groupoids (arXiv:1110.6836)

  • Daniel Freed, Lectures on twisted K-theory and orientifolds, lectures at ESI Vienna, 2012 (pdf)

This is with motivation from orientifolds, see the references given there for more. A long list of computations of twisted KRKR-classes on tori with applications to T-duality on orientifolds is in

Discussion of KOKO as the 2\mathbb{Z}_2-homotopy fixed points of KUKU (or KRKR) is in

Discussion of KUKU with its 2\mathbb{Z}_2-action as the E-∞ ring-valued structure sheaf of the moduli stack of tori is due to

which is reviewed and amplified further in
























Revised on May 21, 2014 02:09:02 by Urs Schreiber (