Longitudinal b-operators, Blups and Index theorems

Important references added, an important imprecision was corrected, many thanks to the colleague that pointed out this mistakeUsing recently introduced Debord-Skandalis Blup's groupoids we study index theory for a compact foliated manifold with boundary inducing a foliation in its boundary. For this we consider first a blup groupoid whose Lie algebroid has sections consisting of vector fields tangent to the leaves in the interior and tangent to the leaves of the foliation in the boundary. In particular the holonomy $b$-groupoid allows us to consider the appropriate pseudodifferential calculus and the appropriate index problems. We further use the blup groupoids as the one above, and in particular its functoriality properties, to actually get index theorems. In this situtation there are two index morphisms, one related to ellipticity and a second one related to fully ellipticity. For the first one, we are able to extend to this setting the longitudinal Connes-Skandalis index theorem and to use it to get that a $b$-longitudinal elliptic operator can be perturbed (up stable homotopy within elliptic operators) with a regularizing operator in the calculus to get a fully elliptic operator if and only if a certain boundary topological index vanishes. For example in the case of a fibration (family case) this topological obstruction is always zero. For the second index morphism, the one related to fully elliptic operators (families of Fredholm operators), we restrict ourselves to the case of families of manifolds with boundary and we prove a new K-theoretical index theorem, i.e. construct a topological index and prove the equality with the analytic-Fredholm index, and use it to get a cohomological index formula for every fully elliptic operator. In particular, for a perturbed family of generalized Dirac operators we can compare our formula with the one by Melrose-Piazza to get a new geometric expression for the eta form of the family