#### Volume 9, issue 3 (2009)

 Recent Issues
Author Index
 The Journal About the Journal Editorial Board Subscriptions Editorial Interests Editorial Procedure Submission Guidelines Submission Page Ethics Statement ISSN (electronic): 1472-2739 ISSN (print): 1472-2747 To Appear Other MSP Journals
Landweber exact formal group laws and smooth cohomology theories

### Ulrich Bunke, Thomas Schick, Ingo Schröder and Moritz Wiethaup

Algebraic & Geometric Topology 9 (2009) 1751–1790
##### Abstract

The main aim of this paper is the construction of a smooth (sometimes called differential) extension $\stackrel{̂}{MU}$ of the cohomology theory complex cobordism $MU$, using cycles for $\stackrel{̂}{MU}\left(M\right)$ which are essentially proper maps $W\to M$ with a fixed $U$–structure and $U$–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 that $\stackrel{̂}{R}\left(M\right):=\stackrel{̂}{MU}\left(M\right){\otimes }_{{MU}^{\ast }}R$ defines a multiplicative smooth extension of $R\left(M\right):=MU\left(M\right){\otimes }_{{MU}^{\ast }}R$ whenever $R$ is a Landweber exact ${MU}^{\ast }$–module, by using the Landweber exact functor principle. An example for this construction is a new way to define a multiplicative smooth $K$–theory.

##### Keywords
differential cohomology, generalized cohomology theory, Landweber exact, formal group law, smooth cohomology, bordism, geometric construction of differential cohomology
##### Mathematical Subject Classification 2000
Primary: 55N20, 57R19