nLab MR cohomology theory

Contents

Context

Cobordism theory

Concepts of cobordism theory

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

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

relative bordism theories:

algebraic:

Contents

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

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

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

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

relative bordism theories:

algebraic:

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:

The Adams spectral sequence for Real cobordism:

Further discussion:

Last revised on September 17, 2022 at 10:53:03. See the history of this page for a list of all contributions to it.