# nLab cobordism cohomology theory

### Context

#### Manifolds and cobordisms

manifolds and cobordisms

cohomology

# Contents

## Definition

### Homology groups

For $X$ a manifold, the group $MU_\ast(X)$ 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.

(…)

### As represented by a spectrum

Cobordism cohomology theory, denoted $M O$ for oriented cobordism cohomology , $M U$ for complex cobordism cohomology etc, is the generalized (Eilenberg-Steenrod) cohomology theory represented by the universal Thom spectrum.

This spectrum, also denoted $M U$ is the spectrum is in degree $2 n$ given by the Thom space of the vector bundle that is associated by the defining representation of the unitary group $U(n)$ on $\mathbb{C}^n$ to the universtal $U(n)$-principal bundle:

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

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

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

There is a canonical orientation? on this obtained from the map

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

(???)

this is the universal even periodic cohomology theory with orientation

The cohomology ring $M P({*})$ is the Lazard ring which is the universal coefficient ring for formal group laws.

The periodic version is sometimes written $PMU$.

### Differential cohomology refinement

The refinement of cobordism cohomology theory to differential cohomology is differential cobordism cohomology.

## Properties

### On the point: 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 1 with the complex cobordism ring.

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

### On itself: 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.

### Universal complex orientation

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

$MU \longrightarrow E \,.$

### 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 $MU$ decomoses into the Brown-Peterson spectra.

chromatic homotopy theory

chromatic levelcomplex oriented cohomology theoryE-∞ ring/A-∞ ringreal oriented cohomology theory
0ordinary cohomologyEilenberg-MacLane spectrum $H \mathbb{Z}$HZR-theory
0th Morava K-theory$K(0)$
1complex K-theorycomplex K-theory spectrum $KU$KR-theory
first Morava K-theory$K(1)$
first Morava E-theory$E(1)$
2elliptic cohomologyelliptic spectrum $Ell_E$
second Morava K-theory$K(2)$
second Morava E-theory$E(2)$
algebraic K-theory of KU$K(KU)$
3 …10K3 cohomologyK3 spectrum
$n$$n$th Morava K-theory$K(n)$
$n$th Morava E-theory$E(n)$BPR-theory
$n+1$algebraic K-theory applied to chrom. level $n$$K(E_n)$ (red-shift conjecture)
$\infty$complex cobordism cohomologyMUMR-theory

## References

### General

Original articles include

• John Milnor, On the cobordism ring ­$\Omega^\bullet$ and a complex analogue, Amer. J. Math. 82 (1960), 505–521.
• Sergei Novikov, Some problems in the topology of manifolds connected with the theory of Thom spaces, Dokl. Akad. Nauk. SSSR. 132 (1960), 1031–1034 (Russian).
• Sergei Novikov, Homotopy properties of Thom complexes, Mat. Sb. (N.S.) 57 (99) (1962), 407–442.

Textbooks accounts include

with an emphasis on the use of $MU$ in the Adams-Novikov spectral sequence, and

• Yuli Rudyak, On Thom spectra, orientability and cobordism, Springer Monographs in Mathematics, 1998 (pdf)

Lecture notes include