Landweber exact formal group laws and smooth cohomology theories

The main aim of this paper is the construction of a smooth ( sometimes called differential) extension (MU) over cap of the cohomology theory complex cobordism MU, using cycles for (MU) over cap (M) which are essentially proper maps W -> M with a fixed U-structure and U-connection on the (stable) normal bundle of W -> 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 (R) over cap (M): = (MU) over cap (M)circle times(MU*) R defines a multiplicative smooth extension of R(M): = MU(M)circle times(MU*) R whenever R is a Landweber exact MU*-module, by using the Landweber exact functor principle. An example for this construction is a new way to define a multiplicative smooth K-theory.