K-theory and Index Theory for Some Boundary Groupoids

We consider Lie groupoids of the form G(M, M-1) := M-0 x M-0 (sic) H x M-1 x M-1. M-1 paired right arrows where M-0 = M \ M-1 and the isotropy subgroup H is an exponential Lie group of dimension equal to the codimension of the manifold M-1 in M. The existence of such Lie groupoids follows from integration of almost injective Lie algebroids by Claire Debord. They correspond to (the s-connected version of) the problem of the existence of a holonomy groupoid associated to the singular foliation whose leaves are the connnected components of M-1 and the connected components of M-0. We study the Lie groupoid structure of these groupoids, and verify that they are amenable and Fredholm in the sense recently introduced by Carvalho, Nistor and Qiao. We compute explicitly the K-groups of these groupoid's C*-algebras, we obtain K-0(C* (G(M, M-1))) congruent to Z, K-1(C* (G(M, M-1))) congruent to Z for M-1 of odd codimension, and K-0(C*(G(M, M-1))) congruent to Z circle plus Z, K-1(C* (G(M, M-1))) congruent to {0} for M-1 of even codimension. When M and M1 are compact we obtain, as an application of our previous K-theory computations, that in the odd codimensional case every elliptic operator (in the groupoid pseudodifferential calculus) can be perturbed (up to stabilization by an identity operator) with a regularizing operator, such that the perturbed operator is Fredholm; and in the even case, given an elliptic operator there is a topological obstruction to satisfy the previous Fredholm perturbation property given by the Atiyah-Singer topological index of the restriction operator to M-1.