## Landweber exact formal group laws and smooth cohomology theories.(English)Zbl 1181.55006

A smooth extension of a generalized cohomology theory $$h^*(\;)$$ is a functor $$\widehat{h}^*(\;)$$ from smooth compact manifolds to graded groups that is related to $$h^*(\;)$$ in the way that deRham cohomology is related to ordinary integral cohomology. The paper under review gives a precise definition of this idea and shows how to construct such an extension in the case $$h^*(\;) = MU^*(\;)$$, complex cobordism, using cycles which are essentially proper maps $$W\rightarrow M$$ of smooth manifolds with a fixed $$U$$-structure and $$U$$-connection on the (stable) normal bundle. The authors establish an extension of the Landweber exact functor theorem showing that $$\widehat{MU}^* (\;)\otimes_{MU^*}{R}$$ will be a smooth cohomology theory if the $$MU_*$$ module $$R$$ satisfies the exactness conditions in Landweber’s theorem. They also give conditions under which the extensions obtained are multiplicative and develop a smooth version of $$MU$$-orientation.

### MSC:

 55N20 Generalized (extraordinary) homology and cohomology theories in algebraic topology 57R19 Algebraic topology on manifolds and differential topology
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### References:

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