Differential K-theory and its Characters

One begins with how complex bundles with connections define, via the Chern-Simons equivalence relation on connections, the extension (called differential K-theory) of usual K-theory by all total odd forms modulo pull backs from the unitary group of the canonical primitive closed forms...

One also defines the analog of differential characters for these objects. A K-character assigns to each odd dimensional closed manifold with additional geometrical structure mapping into the base a number in R/Z. These numbers satisfy a deformation property and a product property.

The first theorem gives a bijection between K-characters and the elements of differential K-theory. One corollary is a differential geometric construction of the known (but unpublished) set of complete numerical invariants for elements in usual complex K-theory.

The first theorem also yields a natural push forward in differential K-theory for families of almost complex manifolds geometrized as a riemannian submersion. A non trivial computation shows that the natural product connection on the total space used to define the push forward is chern-simons equivalent to the limit of the levi civita connections on the total space as the fibre are scaled down to points.

Finally the stage is set to apply the Atiyah-Patodi-Singer theorem to obtain an analytic computation of the differential K-theory push forward at each value of the K-character as a limit of eta-invariants.