nLab
MR cohomology theory

Contents

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} (Hu 99, Kriz 01, p. 13).

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)


flavors of cobordism homology/cohomology theories and representing Thom spectra

bordism theoryMB\,M B (B-bordism):

equivariant bordism theory:

global equivariant bordism 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 MUMU, Bulletin of the American Mathematical Society 74 (1968) 271-274

Computations in:

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

The Adams spectral sequence for real cobordism:

Last revised on November 21, 2020 at 13:37:17. See the history of this page for a list of all contributions to it.