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The main aim of this paper is the construction of a smooth (sometimes calleddifferential) extension \hat{MU} of the cohomology theory complex cobordism MU,using cycles for \hat{MU}(M) which are essentially proper maps W\to M with afixed U(n)-structure and U(n)-connection on the (stable) normal bundle of W\toM. Crucial is that this model allows the construction of a product structureand of pushdown maps for this smooth extension of MU, which have all theexpected properties. Moreover, we show, using the Landweber exact functorprinciple, that \hat{R}(M):=\hat{MU}(M)\otimes_{MU^*}R defines a multiplicativesmooth extension of R(M):=MU(M)\otimes_{MU^*}R whenever R is a Landweber exactMU*-module. An example for this construction is a new way to define amultiplicative smooth K-theory.

Landweber exact formal group laws and smooth cohomology theories