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;

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