Volume 9, issue 3 (2009)

 Download this article For screen For printing
 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
Publication
Received: 24 September 2008
Revised: 15 July 2009
Accepted: 19 July 2009
Published: 26 September 2009
Authors
 Ulrich Bunke NWF I - Mathematik Universität Regensburg 93040 Regensburg Germany http://www.mathematik.uni-regensburg.de/Bunke/ Thomas Schick Mathematisches Institut Georg-August-Universität Göttingen Bunsenstr 3 37073 Göttingen Germany http://www.uni-math.gwdg.de/schick Ingo Schröder Mathematisches Institut Georg-August-Universität Göttingen Bunsenstr 3 37073 Göttingen Germany Moritz Wiethaup Mathematisches Institut Georg-August-Universität Göttingen Bunsenstr 3 37073 Göttingen Germany