The main aim of this paper is the construction of a smooth (sometimes called differential) 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 a fixed U(n)-structure and U(n)-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, using the Landweber exact functor principle, that \hat{R}(M):=\hat{MU}(M)\otimes_{MU^*}R defines a multiplicative smooth extension of R(M):=MU(M)\otimes_{MU^*}R whenever R is a Landweber exact MU*-module. An example for this construction is a new way to define a multiplicative smooth K-theory.
Deep Dive into 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 \hat{MU} of the cohomology theory complex cobordism MU, using cycles for \hat{MU}(M) which are essentially proper maps W\to M with a fixed U(n)-structure and U(n)-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, using the Landweber exact functor principle, that \hat{R}(M):=\hat{MU}(M)\otimes_{MU^}R defines a multiplicative smooth extension of R(M):=MU(M)\otimes_{MU^}R whenever R is a Landweber exact MU*-module. An example for this construction is a new way to define a multiplicative smooth K-theory.
Smooth (also called differentiable) extensions of generalized cohomology theories recently became an intensively studied mathematical topic with many applications ranging from arithmetic geometry to string theory. Foundational contributions are [CS85], [Bry93] (in the case of ordinary cohomology) and [HS05]. The latter paper gives among many other results a general construction of smooth extensions in homotopy theoretic terms. For cohomology theories based on geometric or analytic cycles there are often alternative models. This applies in particular to ordinary cohomology whose smooth extension has various different realizations ( [CS85], [Gaj97], [Bry93], [DL05], [HS05], [BKS]). The papers [SS] or [BS09] show that all these realizations are isomorphic.
An example of a cycle model of a smooth extension of a generalized cohomology theory is the model of smooth K -theory introduced in [BS07], see also [Fre00], [FH00].
The present paper contributes geometric models of smooth extensions of cobordism theories, where the case of complex cobordism theory MU is of particular importance. In [BS09] we obtain general results about uniqueness of smooth extensions which in particular apply to smooth K -theory and smooth complex cobordism theory MU . In detail, any two smooth extensions of complex cobordism theory or complex Ktheory which admit an integration along : S 1 × M → M are isomorphic by a unique isomorphism compatible with . In case of multiplicative extensions the isomorphism is automatically multiplicative. Note that the extension MU constructed in the present paper has an integration and is multiplicative.
We expect that our model MU of the smooth extension of MU is uniquely isomorphic to the one given by [HS05]. So far this fact can not immediately be deduced from the above uniqueness result since for the latter model the functorial properties of the integration map have not been developed yet in sufficient detail. However, for the uniqueness of the even part we do not need the integration. Therefore in even degrees our extension MU is uniquely isomorphic to the model in [HS05].
An advantage of geometric or analytic models is that they allow the introduction of additional structures like products, smooth orientations and integration maps with good properties. These additional properties are fundamental for applications. In [HS05,4.10] methods for integrating smooth cohomology classes were discussed, but further work will be required in order to turn these ideas into constructions with good functorial properties.
In the case of smooth ordinary cohomology the product and the integration have been considered in various places (see e.g. [CS85], [DL05], [Bry93]) (here smooth orientations are ordinary orientations). The case of smooth K -theory, discussed in detail in [BS07], shows that in particular the theory of orientations and integration is considerably more complicated for generalized cohomology theories.
In the present paper we construct a multiplicative extension of the complex cobordism cohomology theory MU . Furthermore, we introduce the notion of a smooth MUorientation and develop the corresponding theory of integration. The same ideas could be applied with minor modifications to other cobordism theories.
For an MU * -module R one can try to define a new cohomology theory R * (X) := MU * (X) ⊗ MU * R for finite CW -complexes X . By Landweber’s famous result [Lan76] this construction works and gives a multiplicative complex oriented cohomology theory provided R is a ring over MU * which is in addition Landweber exact. In Theorem 2.5 we observe that by the same idea one can define a multiplicative smooth extension R(X) := MU(X) ⊗ MU * R of R. It immediately follows that this smooth extension admits an integration for smoothly MU -oriented proper submersions.
In this way we considerably enlarge the class of examples of generalized cohomology theories which admit multiplicative extensions and integration maps. The construction can e.g. be applied to Landweber exact elliptic cohomology theories [LRS95], [Fra92] and complex K -theory 1 .
In Section 2 we review the main result of Landweber [Lan76] and the definition of a smooth extension of a generalized cohomology theory. We state the main result asserting the existence of a multiplicative smooth extension of MU with orientations and integration. Then we realize the idea sketched above and construct a multiplicative smooth extension for every Landweber exact formal group law.
In Section 3 we review the standard constructions of cobordism theories using homotopy theory on the one hand, and cycles on the other. Furthermore, we review the notion of a genus.
In Section 4 we construct our model of the multiplicative smooth extension of complex cobordism. Furthermore, we introduce the notion of a smooth MU -orientation and construct the integration map.
Thomas Schick was partially funded by the Courant Research Center “Higher order structures in Mathem
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