nLab MU

Contents

Context

Cobordism theory

Concepts of cobordism theory

Contents

Idea

$MU$ is the universal Thom spectrum for complex vector bundles. It is the spectrum representing complex cobordism cohomology theory. It is the complex analog of MO.

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

The $M U$ spectrum

The spectrum denoted $M U$ is, as a sequential spectrum, in degree $2 n$ given by the Thom space of the underlying real vector bundle of the complex universal vector bundle: the vector bundle that is associated by the defining representation of the unitary group $U(n)$ on $\mathbb{C}^n$ to the $U(n)$-universal principal bundle:

$M U(2n) = Thom \left( standard\;associated\;bundle\;to\;universal\;bundle \array{ E U(n) \\ \downarrow \\ B U(n) } \right)$

A priori this yields a sequential S2-spectrum, which is then turned into a sequential $S^1$-spectrum by taking the component spaces in odd degree to be the smash product of the circle $S^1$ with those in even degree.

This represents a complex oriented cohomology theory and indeed the universal one among these, see at universal complex orientation on MU.

The periodic complex cobordism theory is given by adding up all the even degree powers of this theory:

$M P = \vee_{n \in \mathbb{Z}} \Sigma^{2 n} M U$

The cohomology ring $M P({*})$ is the Lazard ring which is the universal coefficient ring for formal group laws, see at Milnor-Quillen theorem on MU .

The periodic version is sometimes written $PMU$.

Properties

Homotopy groups: Cobordism and Lazard ring

The graded ring given by evaluating complex cobordism theory on the point is both the complex cobordism ring as well as the Lazard ring classifying formal group laws.

Theorem

Evaluation of $MU$ on the point yields the complex cobordism ring, whose underlying group is

$\pi_\ast MU \simeq MU_\ast(pt) \simeq \mathbb{Z}[x_1, x_2, \cdots] \,,$

where the generator $x_i$ is in degree $2 i$.

This is due to (Milnor 60, Novikov 60, Novikov 62). A review is in (Ravenel theorem 1.2.18, Ravenel, ch. 3, theorem 3.1.5).

The formal group law associated with $MU$ as with any complex oriented cohomology theory is classified by a ring homomorphism $L \longrightarrow \pi_\bullet(MU)$ out of the Lazard ring.

Theorem

This canonical homomorphism is an isomorphism

$L \stackrel{\simeq}{\longrightarrow} \pi_\bullet(MU)$

between the Lazard ring and the $MU$-cohomology ring, hence by theorem with the complex cobordism ring.

This is Quillen's theorem on MU. (e.g Lurie 10, lect. 7, theorem 1)

Universal complex orientation on $M U$

There is a canonical complex orientation on $MU$ obtained from the map

$\omega : \mathbb{C}P^\infty \stackrel{\simeq}{\to} M U(1) \;\;\;\; M U(\mathbb{C}P^\infty)$

For $E$ an E-infinity ring there is a bijection between complex orientation of $E$ and E-infinity ring maps of the form

$MU \longrightarrow E \,.$

$MU$-homology of a manifold: Cobordisms in $X$

For $X$ a manifold, the $MU$-homology group $MU_\ast(X)$ of its underlying homotopy type is the group of equivalence classes of maps $\Sigma \to X$ from manifolds $\Sigma$ with complex structure on the stable normal bundle, modulo suitable complex cobordisms.

(…)

$MU$-homology of $MU$: Hopf algebroid structure on dual Steenrod algebra

Moreover, the dual $MU$-Steenrod algebra $MU_\bullet(MU)$ forms a commutative Hopf algebroid over the Lazard ring. This is the content of the Landweber-Novikov theorem.

Snaith’s theorem

Snaith's theorem asserts that the periodic complex cobordism spectrum is the ∞-group ∞-ring of the classifying space for stable complex vector bundles (the classifying space for topological K-theory) localized at the Bott element $\beta$:

$PMU \simeq \mathbb{S}[B U][\beta^{-1}] \,.$

$p$-Localization and Brown-Peterson spectrum

The p-localization of $MU$ decomposes into the Brown-Peterson spectra.

flavors of cobordism homology/cohomology theories and representing Thom spectra

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

References

General

For general discussion of equivariant complex oriented cohomology see at equivariant cohomology – References – Complex oriented cohomology

On the Chern-Dold character on complex cobordism:

Relation to CFT

A relation to 2d CFT over Spec(Z) was suggested in

• Toshiyuki Katsura, Yuji Shimizu, Kenji Ueno, Complex cobordism ring and conformal field theory over $\mathbb{Z}$, Mathematische Annalen March 1991, Volume 291, Issue 1, pp 551-571 (journal)

Relation to divisors

Relation of complex cobordism cohomology with divisors, algebraic cycles and Chow groups:

Last revised on November 20, 2020 at 10:47:09. See the history of this page for a list of all contributions to it.