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We provide an axiomatic framework for the study of smooth extensions of generalized cohomology theories. Our main results are about the uniqueness of smooth extensions, and the identification of the flat theory with the associated cohomology theory with ℝ/ℤ‐coefficients. 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.

Uniqueness of smooth extensions of generalized cohomology theories