nLab real oriented cohomology theory

Idea

The concept of real oriented cohomology theory (Araki-Murayama 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 (Hu & 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.

Applications

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

Examples

• real space

• KR-theory

• MR-theory

• BPR-theory

• ER-theory

Related concepts

• orientifold

• complex oriented cohomology theory, quaternionic oriented cohomology theory

• equivariant complex oriented cohomology theory

chromatic homotopy theory

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

References

The definition is originally due to:

• Shôrô Araki, Mitutaka Murayama, $\tau$-Cohomology Theories, Japanese Journal of Mathematics 4 2 (1978) 363-416 [doi:10.4099/math1924.4.363]

• Shôrô Araki, Forgetful spectral sequences, Osaka Journal of Mathematics 16 1 (1979) 173-199 [Euclid]

  (on the relevant spectral sequences)

• Shôrô Araki, Orientations in $\tau$-cohomology theories, Japan Journal of Mathematics (N.S.) 5 2 (1979) 403-430 [doi:10.4099/math1924.5.403]

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

• Po Hu, Igor Kriz: Real-oriented homotopy theory and an analogue of the Adams-Novikov spectral sequence, Topology 40 (2001) 317-399 [doi:10.1016/S0040-9383(99)00065-8, pdf]