nLab
MR cohomology theory

Contents

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Stable Homotopy theory

Representation theory

Contents

Idea

What is called MRMR cohomology theory or real cobordism (Landweber 68, Landweber 69) is the 2\mathbb{Z}_2-equivariant cohomology theory version of complex cobordism MUMU.

There is an evident action of 2\mathbb{Z}_2 on formal group laws given by negation in the formal group (the inversion involution), and this lifts to an involutive automorphism MUMUMU \stackrel{\simeq}{\to} MU of the spectrum MU. This induces an 2\mathbb{Z}_2-equivariant spectrum MM\mathbb{R}, and real cobordism is the cohomology theory that it represents. This is directly analogous to how complex K-theory KU gives 2\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 π (M)\pi_\bullet(M \mathbb{R}) is not the cobordism ring of real manifolds, still every real manifold does give a class in MM \mathbb{R} (Kriz 01, p. 13). For details see here: pdf.

E E_\infty-structure

MM \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, MM \mathbb{R} is the univeral real oriented cohomology theory:

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

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

(Kriz 01, theorem 2.25)

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 MUMU;, 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)

Last revised on October 9, 2018 at 04:02:44. See the history of this page for a list of all contributions to it.