Index theory, coarse geometry, and topology of manifolds

The Atiyah-Singer index theorem is one of the most powerful tools for relating geometry, analysis, and topology. In its original form, however, it applies only to compact manifolds. This book describes a version of index theory which works for noncompact spaces with appropriate control, such as complete Riemannian manifolds. The relevant ``control'' is provided by the large scale geometry of the space, and basic notions of large scale geometry are developed. Index theory for the signature operator is related to geometric topology via surgery theory. And, paralleling the analytic development, ``controlled'' surgery theories for noncompact spaces have been developed by topologists. This book explores the connections between these theories, producing a natural transformation from surgery to ``analytic surgery''. The analytic foundations of the work come from the theory of $C^*$-algebras, and the properties of the $C^*$-algebra of a coarse space are developed in detail. The book is based on lectures presented at a conference held in Boulder, Colorado, in August 1995 and includes the author's detailed notes and descriptions of some constructions that were finalized after the lectures were delivered. Also available from the AMS by John Roe is Lectures on Coarse Geometry.