group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
geometric representation theory
representation, 2-representation, ∞-representation
Grothendieck group, lambda-ring, symmetric function, formal group
principal bundle, torsor, vector bundle, Atiyah Lie algebroid
Eilenberg-Moore category, algebra over an operad, actegory, crossed module
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 $\mathbb{Z}_2$-equivariant cohomology theory.
Where complex oriented cohomology theory singles out the classifying space $B S^1 = B U(1)$ of the circle group, in real-oriented cohomology theory its role is played by this same space but equipped with the $\mathbb{Z}_2$-∞-action induced by the $\mathbb{Z}_2$-action $S^1 \to S^1$ on the circle that identifies the two semi-circles.
If one thinks of the canonical embedding $S^1 \hookrightarrow \mathbb{C}$ 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 $\mathbb{Z}_2$-equivariant, are richer than just the “orthogonal” homotopy fixed point theory. In particular on $\mathbb{Z}_2$-spaces of the form $X\cup X$ on which $\mathbb{Z}_2$ acts by swapping the two copies, the real version reduces to the original version on $X$. 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 level | complex oriented cohomology theory | E-∞ ring/A-∞ ring | real oriented cohomology theory |
---|---|---|---|
0 | ordinary cohomology | Eilenberg-MacLane spectrum $H \mathbb{Z}$ | HZR-theory |
0th Morava K-theory | $K(0)$ | ||
1 | complex K-theory | complex K-theory spectrum $KU$ | KR-theory |
first Morava K-theory | $K(1)$ | ||
first Morava E-theory | $E(1)$ | ||
2 | elliptic cohomology | elliptic spectrum $Ell_E$ | |
second Morava K-theory | $K(2)$ | ||
second Morava E-theory | $E(2)$ | ||
algebraic K-theory of KU | $K(KU)$ | ||
3 …10 | K3 cohomology | K3 spectrum | |
$n$ | $n$th Morava K-theory | $K(n)$ | |
$n$th Morava E-theory | $E(n)$ | BPR-theory | |
$n+1$ | algebraic K-theory applied to chrom. level $n$ | $K(E_n)$ (red-shift conjecture) | |
$\infty$ | complex cobordism cohomology | MU | MR-theory |
The definition is originally due to
S. Araki, $\tau$-cohomology theories, Japan Journal of Mathematics (N.S.) 4 (2) (1978) 363-416.
S. Araki, Forgetful spectral sequences, Osaka Journal of Mathematics 16 (1) (1979) 173-199.
S. Araki, Orientations in $\tau$-cohomology theories, Japan Journal of Mathematics (N.S.) 5 (2) (1979) 403-430.
Discussion of the Adams-Novikov spectral sequence in this context is in