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

## TL;DR

This paper introduces new methods for constructing Lie groupoids using blowups and deformations, leading to advanced index theory tools and a Boutet de Monvel type calculus for manifolds with boundary.

## Contribution

It develops general blowup and deformation procedures for Lie groupoids, unifying and extending known constructions in index theory and calculus on manifolds.

## Key findings

- Constructed Lie groupoids via blowups and deformations
- Derived extensions of $C^*$-algebras and computed K-theory maps
- Established a Boutet de Monvel type calculus for manifolds with boundary

## Abstract

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.