K-theory and Index Theory for Some Boundary Groupoids
- PDF / 458,361 Bytes
- 20 Pages / 439.37 x 666.142 pts Page_size
- 95 Downloads / 193 Views
Results in Mathematics
K-theory and Index Theory for Some Boundary Groupoids Paulo Carrillo Rouse and Bing Kwan So
Abstract. We consider Lie groupoids of the form G(M, M1 ) := M0 × M0 H × M1 × M1 ⇒ M, where M0 = M \ M1 and the isotropy subgroup H is an exponential Lie group of dimension equal to the codimension of the manifold M1 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 M1 and the connected components of M0 . 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 K0 (C ∗ (G(M, M1 ))) ∼ = Z, K1 (C ∗ (G(M, M1 ))) ∼ =Z for M1 of odd codimension, and K0 (C ∗ (G(M, M1 ))) ∼ = Z ⊕ Z, K1 (C ∗ (G(M, M1 ))) ∼ = {0} for M1 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 M1 . Mathematics Subject Classification. 46L80. Keywords. K-theory, groupoids, pseudodifferential operators.
0123456789().: V,-vol
172
Page 2 of 20
P. Carrillo Rouse amd B. K. So
Results Math
1. Introduction The classic Atiyah-Singer index theorems [5–7] have been extended and generalized to many different situations and directions. One particularly hard direction is the index theory on manifolds with singularities in a broad sense. Many authors have worked in several different cases and with very different methods. One of the tools that have produced interesting results in the last few years is the use of Lie groupoids (Lie manifolds [1,2], singular foliations [3,4,12], manifolds with corners [18,19], stratified pseudomanifolds [14] and implicitly in cases where there are nice integrable Lie algebroids to mention some cases). These examples shows that having a “good” groupoid for a particular geometric situation is a very good first step to start doing index theory, in the sense that it allows one to construct appropriate algebra of differential and pseudodifferential operators and consider pseudodifferential calculus. Then one can apply various tools such as (K)K-theory, cyclic cohomology and general Lie groupoid/algebroid theory. In fact, one of the main problems when dealing with index or analytic problems for some geometric situations is to settle on a nice and appropriate K-theoretical (coho
Data Loading...
