COMPLETE RIEMANNIAN MANIFOLDS

3

bounded neighbourhood of the diagonal. We compute some simple examples of

coarse cohomology, in particular working out the relationship of

HXl

to the ends

of a Riemannian manifold. Finally, we show that the natural primary product

in coarse cohomology is trivial, but that it has a secondary product arising from

the triviality of the primary product, which maps HXP g ) HXq to HXp~*~q~l.

This is one of the first appearances of the dimension shift that will come up

when relating coarse cohomology to the cohomology of an appropriate 'ideal

boundary'.

Chapter 3 introduces more powerful methods for computing coarse cohomo-

logy. The key idea is to relate coarse cohomology to the inverse limit of the Cech

cohomologies of succesively coarser coverings of the space. Of course one must

rule out the maximally coarse covering consisting of just one open set, and this

is done by insisting that all the coverings be by sets of uniformly bounded diam-

eter. As the diameter bound tends to infinity, one obtains an inverse system of

Cech cohomology groups, which is related to coarse cohomology by a short exact

sequence. Using this sequence, one can compute the coarse cohomology of Eu-

clidean spaces, cones, universal covers of compact A'(7r, 1) manifolds, free groups

(considered as metric spaces), and similar examples. One can also prove the

invariance of coarse cohomology under 'bornotopy': this is the concept, in our

categories of metric spaces, that is analogous to homotopy in ordinary topology.

Chapter 4 contains the index theorem. We begin by reformulating the con-

struction of the operator algebra X(M) of [57], and of the index of a Dirac

operator in the /^-theory of this algebra, in the language of Hilbert modules.

Recall that X is the algebra of bounded operators on

L2(M)

that are represent-

ed by smoothing kernels supported within a finite distance of the diagonal. The

Dirac operator is invertible modulo X, and so it has an index in the K-iheoiy of

this algebra. We then construct a character map from the coarse cohomology of

M to the cyclic cohomology of X(M). Formally, the definition of this character

map is the same as that given by Connes and Moscovici [22, 23]; the point to

notice is that the formal expression does not converge a priori, as it is give by

an integral over the non-compact space

Mq+1.

The definition of coarse cohomo-

logy is arranged so that the integrand should be compactly supported; in fact,

this was the original motivation for the definition of coarse cohomology. The

index theorem, which is proved next, computes the expression (Ind(jD), X&]) m

topological terms; here, D is a Dirac operator, Ind(D) is its index in K*(X), p

is a coarse cohomology class, and \ ls the character map from coarse to cyclic

theory. Once the definitions have been set up, this result follows directly from

the 'Localized Index Theorem' of [22, 23]. The point is, though, that coarse co-

homology has provided a new global analytic context in which to set Connes and

Moscovici's local computations. Such a global context is necessary if the 'index'

is to enjoy the stability and vanishing properties of the ordinary Atiyah-Singer

index.

In chapter 5 we begin to exploit this global context to prove vanishing theo-