The restriction of a generalized cohomology theory to the category of smooth manifolds admits a smooth refinement. The first example has been introduced by Cheeger-Simons in the case of ordinary cohomology with integral coefficients. One motivation of the definition of Cheeger-Simons was that in the presence of connections the integral Chern classes of a vector bundle can naturally be lifted to classes in the smooth extension of integral cohomology. It also has important applications in arithmetic geometry. Recently, smooth extensions of other generalized cohomology theories came into the focus of research in mathematics and mathematical physics. Smooth K-theory plays a role in the description of fields in string theory. Classes in smooth extensions of other cohomology theories appeared as Lagrangians of field theories, e.g. the WZW-model. The key paper paper [HS05] provides a general approach to smooth extensions of generalized cohomology theories.The current understanding of the theory of smooth extensions of generalized cohomology theories is a patchwork of examples and partial results. Missing elements of a theory are:1. an axiomatic set-up2. uniqueness of smooth extensions and their additional structures like products3. the theory of Umkehr maps4. a theory of natural transformations.
