The asymptotic Connes–Moscovici characteristic map and the index cocycles
We show that the (even and odd) index cocycles for theta-summable Fredholm modules are in the image of the Connes–Moscovici characteristic map. To show this, we first define a new range of asymptotic cohomologies, and then we extend the Connes–Moscovici characteristic map to our setting. The ordinary periodic cyclic cohomology and the entire cyclic cohomology appear as two instances of this setup. We then construct an asymptotic characteristic class, defined independently from the underlying Fredholm module. Paired with the $K$-theory, the image of this class under the characteristic map yields a non-zero scalar multiple of the index in the even case, and the spectral flow in the odd case.