Index Theorems in Non-commutative Geometry

Abstract

The index formula on topological manifolds may not be expressed by de Rham-type forms, because this leads to making products of distributions which in classical geometry may not be performed. The solution to this problem consists of replacing the de Rham forms by quasi-local objects, i.e. objects living on a neighbourhood of the diagonal in the different powers of the space. This may be done in non-commutative geometry; one obtains the local index theorem of Connes, Moscovici. In the specific case of topological manifolds, this leads to the Connes, Hilsum, Sullivan, Teleman index formulae.