Landweber exact formal group laws and smooth cohomology theories

Bunke, Ulrich; Schick, Thomas; Schröder, Ingo; Wiethaup, Moritz

doi:10.2140/agt.2009.9.1751

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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

DOI: 10.2140/agt.2009.9.1751

Abstract

The main aim of this paper is the construction of a smooth (sometimes called differential) extension $\hat{M U}$ of the cohomology theory complex cobordism $M U$ , using cycles for $\hat{M U} (M)$ 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 $M U$ , which have all the expected properties.

Moreover, we show that $\hat{R} (M) : = \hat{M U} (M) \otimes_{{M U}^{*}} R$ defines a multiplicative smooth extension of $R (M) : = M U (M) \otimes_{{M U}^{*}} R$ whenever $R$ is a Landweber exact ${M U}^{*}$ –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

References

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

Volume 9, issue 3 (2009)