Volume 9, issue 3 (2009)

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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
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 MÛ of the cohomology theory complex cobordism MU, using cycles for MÛ(M) which are essentially proper maps W M with a fixed U–structure and U–connection on the (stable) normal bundle of W 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 R̂(M) := MÛ(M) MUR defines a multiplicative smooth extension of R(M) := MU(M) MUR whenever R is a Landweber exact MU–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