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 $U(n)$ on ${ℂ}^n$ to the universtal $U(n)$-principal bundle:

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

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

(???)

this is the universal even periodic cohomology theory with orientation

The cohomology ring $MP(*)$ 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.

References

for further context see the discussion at

• A Survey of Elliptic Cohomology - cohomology theories