Landweber exact formal group laws and smooth cohomology theories

TL;DR

This paper constructs a smooth extension of complex cobordism cohomology theory, enabling product and pushdown maps, and extends this to Landweber exact modules, including a novel approach to smooth K-theory.

Contribution

It introduces a new smooth (differential) extension of MU using geometric cycles and extends this to Landweber exact modules, including a new method for smooth K-theory.

Findings

Constructed a smooth MU extension with product and pushdown maps.

Extended the smooth extension to Landweber exact modules R.

Provided a new approach to defining multiplicative smooth K-theory.

Abstract

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.