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.


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