# The asymptotic Connes-Moscovici characteristic map and the index cocycles

## TL;DR

This paper demonstrates that index cocycles for theta-summable Fredholm modules are captured by the Connes-Moscovici characteristic map through a new asymptotic cohomology framework, linking cyclic cohomology to index theory.

## Contribution

It introduces a novel asymptotic cohomology and extends the Connes-Moscovici map, unifying index cocycles with cyclic cohomology in a broader setting.

## Key findings

- Index cocycles lie in the image of the extended characteristic map.
- Constructs an asymptotic characteristic class independent of Fredholm modules.
- Paired with K-theory, yields the index and spectral flow as scalar multiples.

## Abstract

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.