nLab
MR cohomology theory

Idea

What is called $MR$ cohomology theory or real cobordism (Landweber 68, Landweber 69) is the $\mathbb{Z}_2$-equivariant cohomology theory version of complex cobordism $MU$.

There is an evident action of $\mathbb{Z}_2$ on formal group laws given by negation in the formal group (the inversion involution), and this lifts to an involutive automorphism $MU \stackrel{\simeq}{\to} MU$ of the spectrum MU. This induces an $\mathbb{Z}_2$-equivariant spectrum $M\mathbb{R}$, and real cobordism is the cohomology theory that it represents. This is directly analogous to how complex K-theory KU gives $\mathbb{Z}_2$-equivariant KR-theory, both are examples of real-oriented cohomology theories.

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

Properties

Relation to real manifolds

While $\pi_\bullet(M \mathbb{R})$ is not the cobordism ring of real manifolds, still every real manifold does give a class in $M \mathbb{R}$ (Kriz 01, p. 13). For details see here: pdf.

$E_\infty$-structure

$M \mathbb{R}$ is naturally an E-∞ ring spectrum. (reviewed as Kriz 01, prop. 3.1)

Universal real orientation

In direct analogy with the situation for complex cobordism theory in complex oriented cohomology theory, $M \mathbb{R}$ is the univeral real oriented cohomology theory:

Equivalence classes of real orientations of a $\mathbb{Z}/2\mathbb{Z}$-equivariant E-∞ ring $E$ are in bijection to equivalence classes of E-∞ ring homomorphisms

$$M \mathbb{R}\longrightarrow E \,.$$

(Kriz 01, theorem 2.25)

Related concepts

• real-oriented cohomology theory

  • KR-theory

  • BPR-theory

References

The definition is originally due to

• Peter Landweber, Fixed point free conjugations on complex manifolds, Annals of Mathematics 86 (2) (1967) 491-502.

• Peter Landweber, Conjugations on complex manifolds and equivariant homotopy of $MU$;, Bulletin of the American Mathematical Society 74 (1968) 271-274.

The Adams spectral sequence for real cobordism is discussed in

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