M ay 2 00 4 Index theory with bounded geometry , the uniformly finite  class , and infinite connected sums

We work primarily in the category of manifolds of bounded geometry. The objects are manifolds with bounds on the curvature tensor, its derivatives, and on the injectivity radius. The morphisms are diffeomorphisms of bounded distortion. We think of these manifolds as having a chosen bounded distortion class of metrics. Unless otherwise stated, all manifolds in the paper are assumed of this type. These definitions are designed to reflect the restrictions imposed on a non-compact manifold which is controlled in some way by a compact manifold. The most common example of this is a covering of a compact manifold. Any metric on the base gives a metric of bounded geometry on the cover, and any two such metrics are bounded distortion equivalent. Similarly, leaves of foliations of compact manifolds have a canonical bounded distortion class of metric of bounded geometry. We try to understand index theory for these manifolds, and in particular, questions of positive scalar curvature. Generally, the appropriate notion here is uniformly positive scalar curvature, meaning the scalar curvature is bounded away from zero from below. When we say that M admits a metric of positive scalar curvature, we mean that within the chosen bounded distortion class of metrics there is a metric of uniformly positive scalar curvature. To understand positive scalar curvature we need an appropriate generalization of the  class. One interesting feature which emerges is that this class lives in a non-Hausdorff homology group, and thus standard C algebra methods do not apply. It turns out that to understand this class requires rather delicate spectral estimates for Dirac operators on the boundaries of certain compact submanifolds. The resulting theorems have unexpected applications to compact manifolds. The motivation for this work comes from some interesting infinite connected sum examples studied in [BW1] and [Ro]. As these examples provide