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 $$M_1$$ .

中文翻译：

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一些边界群体的K理论和指数理论

我们考虑形式为 $${\mathcal {G}}(M,M_1) := M_0 \times M_0 \sqcup H \times M_1 \times M_1 \rightrightarrows M,$$ 的李群群，其中 $$M_0 = M \setminus M_1$$ 和各向同性子群 H 是一个指数李群，其维数等于 M 中流形 $$M_1$$ 的余维。这种李群群的存在源于克莱尔·德波 (Claire Debord) 对几乎单射李代数体的积分。它们对应于（s 连接版本）存在与奇异叶理相关的完整类群的问题，其叶子是 $$M_1$$ 的连通分量和 $$M_0$$ 的连通分量。我们研究了这些 groupoids 的 Lie groupoid 结构，并验证了它们在最近由 Carvalho、Nistor 和 Qiao 引入的意义上的服从和 Fredholm。我们明确地计算这些 groupoid 的 $$C^*$$ -algebras 的 K 群，我们得到 $$ K _0 (C^* ({\mathcal {G}}(M,M_1))) \cong {\mathbb {Z}}, K _1 (C^* ({\mathcal {G}}(M,M_1))) \cong {\mathbb {Z}}$$ 用于 $$M_1$$ 的奇数余维，以及 $$ K _0 (C^* ({\mathcal {G}}(M,M_1))) \cong {\mathbb {Z}}\oplus {\mathbb {Z}}, K _1 (C^* ({\mathcal {G}}(M,M_1))) \cong \{ 0 \}$$ 为 $$M_1$$ 的偶数。当 M 和 $$M_1$$ 是紧致的时，作为我们之前的 K 理论计算的应用，我们获得，在奇数共维情况下，每个椭圆算子（在群状伪微分演算中）都可以被扰动（直到通过恒等式稳定运算符）与正则化运算符，使得扰动运算符是 Fredholm；在偶数情况下，