Uniqueness of smooth extensions of generalized cohomology theories

Abstract

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 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.

Landweber exact formal group laws and smooth cohomology theories

“…If the coefficients groups R is degree-wise finitely generated (see the corresponding remark in Section 2.2.1), then we obtain the same notion as in [5].…”

“…Let .B; N / WD .B/˝RN , where .B/ denotes the smooth real differential forms on B . Note that this definition only coincides with the corresponding definition of N -valued forms in [5] if N is degree-wise finite-dimensional. By dRW dD0 .B; N / !…”

“…These are all proven in Section 4.4. Here, we use again that y K.B/ is uniquely determined as a multiplicative extension of K -theory [5].…”

“…This applies in particular to ordinary cohomology whose smooth extension has various different realizations (see Cheeger and Simons [7], Gajer [12], Brylinski [2], Dupont and Ljungmann [8], Hopkins and Singer [14] and Bunke, Kreck and Schick [3]). The papers by Simons and Sullivan [21] or Bunke and Schick [5] show that all these realizations are isomorphic.…”

“…The present paper contributes geometric models of smooth extensions of cobordism theories, where the case of complex cobordism theory MU is of particular importance. In [5] we obtain general results about uniqueness of smooth extensions which in particular apply to smooth K -theory and smooth complex cobordism theory b MU . In detail, any two smooth extensions of complex cobordism theory or complex K -theory which admit an integration along R W S 1 M !…”