nLab real oriented cohomology theory





Special and general types

Special notions


Extra structure



Stable Homotopy theory

Representation theory



The concept of real oriented cohomology theory (Araki 78, Araki 79) is the analog of that of complex oriented cohomology theory in the context of 2\mathbb{Z}_2-equivariant cohomology theory.

Where complex oriented cohomology theory singles out the classifying space BS 1=B S^1 = B U ( 1 ) B U(1) of the circle group, in real-oriented cohomology theory its role is played by this same space but equipped with the 2\mathbb{Z}_2-∞-action induced by the 2\mathbb{Z}_2-action S 1S 1S^1 \to S^1 on the circle that identifies the two semi-circles.

If one thinks of the canonical embedding S 1S^1 \hookrightarrow \mathbb{C} of the circle as the unit circle in the complex plane, then this involution is induced by the complex conjugation involution on the complex plane. In this sense real oriented cohomology theory is in a sense “complex oriented cohomology theory with complex conjugation divided out”, and that motivates the term real cohomology theory.

The original and motivating example is the real version of complex K-theory called KR-theory. The real-oriented version of complex cobordism is MR-theory and that of Morava E-theory is BPR-theory.

A modern review is in (Kriz 01, section 2).

Beware the terminological issue expanded on also at KR-theory: real-oriented versions of “complex” cohomology theory, being genuinely 2\mathbb{Z}_2-equivariant, are richer than just the “orthogonal” homotopy fixed point theory. In particular on 2\mathbb{Z}_2-spaces of the form XXX\cup X on which 2\mathbb{Z}_2 acts by swapping the two copies, the real version reduces to the original version on XX. Hence real-oriented cohomology is a genuine generalization of complex-oriented cohomology. For instance KR-theory subsumes both complex K-theory as well as KO-theory as well as KSC-theory.


In physics (string theory) where (gauge) fields are cocycles in some (differential) cohomology theory, real-oriented cohomology theory is that of orientifold spacetimes.


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


The definition is originally due to:

Discussion of the Adams-Novikov spectral sequence in this context is in

  • Igor Kriz, Real-oriented homotopy theory and an analogue of the

    Adams-Novikov spectral sequence_, Topology 40 (2001) 317-399 (pdf)

Last revised on March 5, 2024 at 00:22:49. See the history of this page for a list of all contributions to it.