# nLab cobordism cohomology theory

### Context

#### Manifolds and cobordisms

manifolds and cobordisms

cohomology

# Contents

## Definition

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

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

$MU\left(2n\right)=\mathrm{Thom}\left(\mathrm{standard}\mathrm{associated}\mathrm{bundle}\mathrm{to}\mathrm{universal}\mathrm{bundle}\begin{array}{c}EU\left(n\right)\\ ↓\\ BU\left(n\right)\end{array}\right)$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:

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

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

$\omega :ℂ{P}^{\infty }\stackrel{\simeq }{\to }MU\left(1\right)\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}MU\left(ℂ{P}^{\infty }\right)$\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 $MP\left(*\right)$ is the Lazard ring which is the universal coefficient ring for formal group laws.

## Differential cohomology refinement

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

## Properties

### 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 $\beta$:

$\mathrm{MU}\simeq 𝕊\left[BU\right]\left[{\beta }^{-1}\right]\phantom{\rule{thinmathspace}{0ex}}.$MU \simeq \mathbb{S}[B U][\beta^{-1}] \,.

## References

for further context see the discussion at

Lecture 5 in

Revised on June 17, 2013 11:47:52 by Urs Schreiber (89.204.155.252)