K-theory and Index Theory for Some Boundary Groupoids

Abstract

We consider Lie groupoids of the form $${\mathcal {G}}(M,M_1) := M_0 \times M_0 \sqcup H \times M_1 \times M_1 \rightrightarrows M,$$ where $$M_0 = M \setminus M_1$$ 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 $$ K _0 (C^* ({\mathcal {G}}(M,M_1))) \cong {\mathbb {Z}}, K _1 (C^* ({\mathcal {G}}(M,M_1))) \cong {\mathbb {Z}}$$ for $$M_1$$ of odd codimension, and $$ K _0 (C^* ({\mathcal {G}}(M,M_1))) \cong {\mathbb {Z}}\oplus {\mathbb {Z}}, K _1 (C^* ({\mathcal {G}}(M,M_1))) \cong \{ 0 \}$$ for $$M_1$$ of even codimension. When M and $$M_1$$ 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 $$M_1$$ .