A GEOMETRIC APPROACH TO K-HOMOLOGY FOR LIE MANIFOLDS

We show that the computation of the Fredholm index of a fully elliptic pseudodifferential operator on an integrated Lie manifold can be reduced to the computation of the index of a Dirac operator, perturbed by a smoothing operator, canonically associated, via the so-called clutching map. To this end we adapt to our framework ideas coming from Baum-Douglas geometric K-homology and in particular we introduce a notion of geometric cycles, that can be categorized as a variant of the famous geometric K-homology groups, for the specific situation here. We also define a comparison map between this geometric K-homology theory and a relative K-theory group, directly associated to a fully elliptic pseudodifferential operator.