Algebraic & Geometric Topology

Landweber exact formal group laws and smooth cohomology theories

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

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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 WM with a fixed U–structure and U–connection on the (stable) normal bundle of WM.

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.

Article information

Algebr. Geom. Topol., Volume 9, Number 3 (2009), 1751-1790.

Received: 24 September 2008
Revised: 15 July 2009
Accepted: 19 July 2009
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55N20: Generalized (extraordinary) homology and cohomology theories 57R19: Algebraic topology on manifolds

differential cohomology generalized cohomology theory Landweber exact formal group law smooth cohomology bordism geometric construction of differential cohomology


Bunke, Ulrich; Schick, Thomas; Schröder, Ingo; Wiethaup, Moritz. Landweber exact formal group laws and smooth cohomology theories. Algebr. Geom. Topol. 9 (2009), no. 3, 1751--1790. doi:10.2140/agt.2009.9.1751.

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