Landweber exact formal group laws and smooth cohomology theories

Abstract

The main aim of this paper is the construction of a smooth (sometimes called differential) extension $\widehat{MU}$ of the cohomology theory complex cobordism $MU$, using cycles for $\widehat{MU}\left(M\right)$ which are essentially proper maps $W\to M$ with a fixed $U$–structure and $U$–connection on the (stable) normal bundle of $W\to M$.

Crucial is that this model allows the construction of a product structure and of pushdown maps for this smooth extension of $MU$, which have all the expected properties.

Moreover, we show that $\widehat{R}\left(M\right):=\widehat{MU}\left(M\right){\otimes }_{{MU}^{\ast }}R$ defines a multiplicative smooth extension of $R\left(M\right):=MU\left(M\right){\otimes }_{{MU}^{\ast }}R$ whenever $R$ is a Landweber exact ${MU}^{\ast }$–module, by using the Landweber exact functor principle. An example for this construction is a new way to define a multiplicative smooth $K$–theory.