Desingularization of Lie groupoids and pseudodifferential operators on singular spaces

Abstract

We introduce and study a "desingularization" of a Lie groupoid G along an "A(G)-tame" submanifold L of the space of units M . An A(G)-tame submanifold L ⊂ M is one that has, by definition, a tubular neighborhood on which A(G) becomes a thick pull-back Lie algebroid. The construction of the desingularization [[G : L]] of G along L is based on a canonical fibered pull-back groupoid structure result for G in a neighborhood of the tame A(G)submanifold L ⊂ M . This local structure result is obtained by integrating a certain groupoid morphism, using results of Moerdijk and Mrcun (Amer. J. Math. 2002). Locally, the desingularization [[G : L]] is defined using a construction of Debord and Skandalis (Advances in Math., 2014). The space of units of the desingularization [[G : L]] is [M : L], the blow up of M along L. The space of units and the desingularization groupoid [[G : L]] are constructed using a gluing construction of Gualtieri and Li (IMRN 2014). We provide an explicit description of the structure of the desingularized groupoid and we identify its Lie algebroid, which is important in analysis applications. We also discuss a variant of our construction that is useful for analysis on asymptotically hyperbolic manifolds. We conclude with an example relating our constructions to the so called "edge pseudodifferential calculus." The paper is written such that it also provides an introduction to Lie groupoids designed for applications to analysis on singular spaces.

Fredholm Conditions on Non-compact Manifolds: Theory and Examples

“…A tame submersion h between two manifolds with corners M 1 and M is a smooth map h : M 1 → M such that its differential dh is surjective everywhere and Clearly, if h : M 1 → M is a tame submersion of manifolds with corners, then x and h(x) will have the same depth. We have the following well known lemma (see for instance [69]). We now define Lie groupoids roughly by replacing our spaces with manifolds with corners and the continuity conditions with smoothness conditions.…”

“…Step 1. We first consider the adiabatic groupoid H ad of H := B × B (this is the tangent groupoid of [16]; see [69] for more details and references). The groupoid structure of H ad is such that A(H) × {0} has the Lie groupoid structure of a bundle of Lie groups and H×(0, ∞) has the product Lie groupoid structure with (0, ∞) the groupoid associated to a space (that is (0, ∞) has only units, and all orbits are reduced to a single point).…”

“…We then glue the reduction H ′ := π ↓↓ (H ad ) ⋊ R * + U U with M 0 × M 0 by identifying the reduction of H ′ to U ∂M (which is the pair groupoid of ∂M ×(0, 1), as we have seen), with its image in M 0 ×M 0 . This glueing construction is, of course, nothing but a particular case of the glueing construction in [35] and [69].…”

“…Recall the fibered pull-back π ↓↓ (B) of Example 2.19. We proceed in a slightly greater generality than in [69], to which we refer for more details and for the unexplained arguments.…”

Resolutions of Proper Riemannian Lie Groupoids

“…The blow-up construction for Lie groupoids has been studied extensively in the literature, e.g. [12,18,6]. In this paper, we restrict our attention to full subgroupoids of a proper Lie groupoid.…”