4

JOHN ROE

rems. The main construction of this chapter is of the Higson corona [39] of a

non-compact metric space M. This space, denoted t/M, is a sort of 'ideal bound-

ary' which reflects the topology of conical ends of M (of arbitrarily small positive

growth) but not of cylindrical ends. In [39], Higson constructed a map from the

if-theory of the algebra X to the K-homology of i/M, shifting dimensions by 1.

Dually, we construct in this chapter a transgression map T from the cohomology

of i/M (or more precisely of certain quotients of it) to the coarse cohomology

of M. These two maps are dual in a natural way, as one can prove using the

index theorem of chapter 4. Using C*-algebra A'-theory, one knows that as soon

as the operator D is invertible, the image of its index is zero in K*(yM). This

leads to vanishing theorems for the index paired with coarse classes coming by

trangression from vM.

Finally, chapter 6 gives some geometric and analytic applications of all this

machinery, including more general versions of the theorems cited above. The

index theorem for partitioned manifolds of [57] is obtained as a corollary of the

general index theorem of this paper; in fact partitions of a manifold correspond

exactly to 1-dimensional coarse cohomology classes. Chapter 6 also contains a

brief discussion of the Novikov Conjecture. There are natural 'coarse' formu-

lations of the Novikov Conjecture and even of the more general Baum-Connes

Conjecture, and these imply the usual versions for certain group cohomology

classes. (They are more closely related to the quasi-isometric versions of [70,

§18-20].) Moreover there is a remarkable parallel between coarse cohomology and

the 'bounded algebraic A-theory' which was developed by Pedersen, Weibel, and

others [52, 60, 16] with a view to doing controlled surgery [27], What little I

know about this relationship will mostly be found in the final section.

Several recent preprints discuss index theorems and applications that overlap

with the results of this paper. In this regard the reader's attention is particularly

drawn to the work of Block and Weinberger [14], Hurder [43], Lott [47], and Yu

[72].

Acknowledgements: The definition of coarse cohomology occurred to me

while I was thinking about Henri Moscovici's exposition, in a lecture at the

Bowdoin conference on elliptic operators in 1988, of his joint work with Alain

Connes. As stated above, one way to think about coarse cohomology is as a new

'homologous' setting for their local index theorem. The influence of their work

is pervasive.

The main ideas of chapters 2,3 and 4 of this paper were presented at a con-

ference held at Boulder in 1989. I am very grateful to all the participants in this

conference, and in particular I should like to thank Steve Hurder, who organized

the conference and spurred me on to compute a lot of examples, and Nigel Hig-

son, who shared with me his definition of the corona space which now forms the

central theme of chapter 5.

I have also had helpful conversations with many other mathematicians, in-