On the Topological Interpretation of Gravitational Anomalies

We consider the mixed gravitational-Yang-Mills anomaly as the coupling between the $K$-theory and $K$-homology of a $C^{\ast }$-algebra crossed product. The index theorem of Connes-Moscovici allows to compute the Chern character of the $K$-cycle by local formulae involving connections and curvatures. It gives a topological interpretation to the anomaly, in the sense of noncommutative algebras.