Fredholm Conditions on Non-compact Manifolds: Theory and Examples

Abstract

We give explicit Fredholm conditions for classes of pseudodifferential operators on suitable singular and non-compact spaces. In particular, we include a "user's guide" to Fredholm conditions on particular classes of manifolds including asymptotically hyperbolic manifolds, asymptotically Euclidean (or conic) manifolds, and manifolds with poly-cylindrical ends. The reader interested in applications should be able read right away the results related to those examples, beginning with Section 5. Our general, theoretical results are that an operator adapted to the geometry is Fredholm if, and only if, it is elliptic and all its limit operators, in a sense to be made precise, are invertible. Central to our theoretical results is the concept of a Fredholm groupoid, which is the class of groupoids for which this characterization of the Fredholm condition is valid. We use the notions of exhaustive and strictly spectral families of representations to obtain a general characterization of Fredholm groupoids. In particular, we introduce the class of the so-called groupoids with Exel's property as the groupoids for which the regular representations are exhaustive. We show that the class of "stratified submersion groupoids" has Exel's property, where stratified submersion groupoids are defined by glueing fibered pull-backs of bundles of Lie groups. We prove that a stratified submersion groupoid is Fredholm whenever its isotropy groups are amenable. Many groupoids, and hence many pseudodifferential operators appearing in practice, fit into this framework. This fact is explored to yield Fredholm conditions not only in the above mentioned classes, but also on manifolds that are obtained by desingularization or by blow-up of singular sets.

The Fredholm Property for Groupoids is a Local Property

“…The operators P α should be thought of as "limit operators" giving some control on the behaviour of P at infinity. Note that Theorem 1.1 remains true if we replace P by an operator in a suitable pseudodifferential calculus or if we consider operators acting between vector bundles [5,35]. Theorem 1.1 recovers in a unified setting many similar results that were previously known in particular cases [9,10,12,21,22,26,27,43].…”

“…Most results giving sufficient conditions for a groupoid G to be Fredholm assume that G is Hausdorff, which sometimes is not so easy to check [5,34]. This is not a requirement for Theorem 1.2, which states that it is enough to look at the local structure of G (i.e.…”

“…A natural and important question is to seek extensions of this Fredholm characterization for open manifolds. In [5], Carvalho, Nistor and Qiao introduced a very large class of manifolds, called manifolds with amenable ends. To any manifold M 0 belonging to this class, we can associate a family of manifolds M α , for some α ∈ A, which are acted upon by Lie groups G α , and such that the following theorem holds.…”

“…This was made possible by noting that, in many cases, the relevant differential operators are generated by the action of a Lie groupoid G on the manifold M 0 , in a sense made more precise below. Obtaining Fredholm conditions can then be reduced to studying the representations of the reduced C * -algebra of G [5,34].…”

“…This motivated the definition of Fredholm groupoids in [5]. Roughly speaking, a Fredholm groupoid is a Lie groupoid G whose unit space is a compact manifold with boundary M , and such that a ∈ 1 + C * r (G) is Fredholm ⇔ π x (a) is invertible for any x ∈ ∂M .…”