Uniqueness of smooth extensions of generalized cohomology theories

Details Uniqueness of smooth extensions of generalized cohomology theories Uniqueness of smooth extensions of generalized cohomology theories

2009-01-28

arxiv.org/abs/0901.4423v2

J. Topol. 3 (2010), no. 1, 110-156

Mathematics

Algebraic Topology

63 pages, revised version, to appear in Journal of Topology

Scientific paper

10.1112/jtopol/jtq002

We provide an axiomatic framework for the study of smooth extensions of generalized cohomology theories. Our main results are about the uniqeness of smooth extensions, and the identification of the flat theory with the R/Z-theory. In particular, we show that there is a unique smooth extension of K-theory and of MU-cobordism with a unique multiplication, and that the flat theory in these cases is naturally isomorphic to the homotopy theorist's version of the cohomology theory with R/Z-coefficients. For this we only require a small set of natural compatibility conditions.

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