Blowup constructions for Lie groupoids and a Boutet de Monvel type calculus

We present natural and general ways of building Lie groupoids, by using the classical procedures of blowups and of deformations to the normal cone. Our constructions are seen to recover many known ones involved in index theory. The deformation and blowup groupoids obtained give rise to several extensions of $C^*$-algebras and to full index problems. We compute the corresponding K-theory maps. Finally, the blowup of a manifold sitting in a transverse way in the space of objects of a Lie groupoid leads to a calculus, quite similar to the Boutet de Monvel calculus for manifolds with boundary.