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 $M \mathbb{R}$ (Hu 99, Kriz 01, p. 13).

$E_\infty$-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

• equivariant cobordism cohomology theory

• real-oriented cohomology theory

  • KR-theory

  • BPR-theory

flavors of bordism homology theories/cobordism cohomology theories, their representing Thom spectra and cobordism rings:

bordism theory$\;$M(B,f) (B-bordism):

• MFr

• MO, MSO, MSpin, MString, …

• MU, MSU, …

  MΩΩSU(n)

  MP, MR

• MSpin^c

• MSp

relative bordism theories:

• MOFr, MUFr, MSUFr

equivariant bordism theory:

• equivariant MFr

• equivariant MO

• equivariant MU

global equivariant bordism theory:

• global equivariant mO,

• global equivariant mU

algebraic:

• algebraic cobordism

References

The definition of Real cobordism cohomology goes back to:

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

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

and in the broader context of real-oriented cohomology theories:

• Shôrô Araki, $\tau$-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]

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

The Adams spectral sequence for Real cobordism:

• Po Hu, The cobordism of Real manifolds and calculations with the Real Adams-Novikov spectral sequence, (1998) [hdl:2027.42/130996, pdf]

• 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]

Further discussion:

• Po Hu, The cobordism of Real manifolds, Fundamenta Mathematicae (1999) Volume: 161, Issue: 1-2, page 119-136 [dml:212395, pdf]

• Po Hu, Igor Kriz, Some Remarks on Real and Algebraic Cobordism, K-Theory 22 (2001) 335–366 [pdf, doi:10.1023/A:1011196901303]

• Po Hu, Igor Kriz, Topological Hermitian Cobordism, Journal of Homotopy and Related Structures, 11 (2016) 173–197 [arXiv:1110.5608, doi:10.1007/s40062-014-0100-9]