CHAPTER 1 Introduction This paper is a contribution to the study of geometry and analysis for com- plete non-compact Riemannian manifolds. The central result is a version of the Atiyah-Singer Index Theorem for such manifolds. Such a theorem is of intrinsic interest, but also provides a powerful method for studying compact manifolds, since non-compact manifolds can be obtained from compact ones in a variety of ways: for instance, as covering spaces or leaves of foliations, by adding cones or cylinders to boundary components, or by blowing up the metric on a neighbour- hood of some interesting subset. Several applications to compact manifolds will be discussed below. The main technical tool is a new 'cohomology' theory, called coarse cohomo- logy, for complete metric spaces. The word 'cohomology' is in quotes because this theory certainly is not a generalized cohomology theory in the usual sense. For instance, the coarse cohomology of any compact space is the same as that of a point. The theory measures the behaviour at infinity of a space more specifi- cally, it measures the way in which uniformly large bounded sets fit together. The link to index theory comes about as follows. For each coarse cohomology class (p and each Dirac-type operator D on a complete Riemannian manifold M, one can define a 'higher index' of D twisted by (p, which one might think of as an analogue of a Novikov higher signature. These higher indices have many of the stability and vanishing properties of the usual Atiyah-Singer index, to which they reduce if the manifold M is compact. Our main theorem (4.42 and 4.47) provides a topological formula for the higher index in terms of three pieces of data, two of which are familiar: the fundamental cycle of M, the cohomology class QD that represents the index of D according to the Atiyah-Singer formula, and the 'topological character' c[p] of the coarse class £ , which is an element of the ordinary cohomology of M with compact supports. Up to a numerical constant, the higher index is simply equal to (Sp ^ c[y?],[M]). Here are some applications of this index theorem, of which further details are given in chapter 6. Many of the results apply to all manifolds admitting a 'coarse fundamental class' in a certain sense, the w-large manifolds in the language of chapter 6. For simplicity they are stated here for nonpositively curved manifolds 1
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