John Roe

Index theory, coarse geometry, and topology of manifolds

Index theory (Chapter 1) Coarse geometry (Chapter 2)

Novikov Conjectures, Index Theorems and Rigidity: On the coarse Baum-Connes conjecture

C*

A coarse Mayer-Vietoris principle

-algebras (Chapter 3) An example of a higher index theorem (Chapter 4) Assembly (Chapter 5) Surgery (Chapter 6) Mapping surgery to analysis (Chapter 7) The coarse Baum-Connes conjecture (Chapter 8) Methods of computation (Chapter 9) Coarse structures and boundaries (Chapter 10) References Index.

C*-Algebras and Controlled Topology

The Baum-Connes conjecture [2, 3] concerns the K-theory of the reduced group C-algebra C r (G) for a locally compact group G. One can define a map from the equivariant K-homology of the universal proper G-space EG to K∗(C ∗ r (G)): each K-homology class defines an index problem, and the map associates to each such problem its analytic index. The conjecture is that this map is an isomorphism. The injectivity of the map has geometric and topological consequences, implying the Novikov conjecture for example; the surjectivity has consequences for C-algebra theory and is related to problems in harmonic analysis. In geometric topology it has proved to be very useful to move from studying classical surgery problems on a compact manifold M to studying bounded surgery problems over its universal cover (see [8, 24] for example). In terms of L-theory, one replaces the classical L-theory of Zπ by the L-theory, bounded over |π|, of Z (here |π| denotes π considered as a metric space, with a word length metric). Now the authors, motivated by considerations of index theory on open manifolds, have studied a C-algebra C(X) associated to any proper metric space X , and it has recently become quite clear that the passage from C r (π) to C (|π|) is an analytic version of the same geometric idea. Moreover, various descent arguments have been given [4, 9, 5, 17, 27], both in the topological and analytic contexts, which show that a ‘sufficiently canonical’ proof of an analogue of the Baum-Connes conjecture in the bounded category will imply the classical version of the Novikov conjecture. The purpose of this paper is to give a precise formulation of the Baum-Connes conjecture for the C-algebras C(X) (filling in the details of the hints in the last section of [26]) and to prove the conjecture for spaces which are non-positively curved in some sense, including affine buildings and hyperbolic metric spaces in the sense of Gromov. Notice that while the classical Novikov conjecture has been established for the analogous class of groups, the Baum-Connes conjecture has not. Unfortunately there does not seem to be any descent principle for the surjectivity side of the Baum-Connes conjecture as there is for the injectivity side. The main tool that we will use in this paper is the invariance of K∗(C (X)) under coarse homotopy, established by the authors in [15]. Coarse homotopy is a rather weak equivalence relation on metric spaces, weak enough that (for example) many spaces are coarse homotopy equivalent to open cones OY on compact spaces Y . The idea of ‘reduction to a cone on an ideal boundary’ is also used in some topological approaches to the Novikov conjecture, but the notion of coarse homotopy

A characterization of the sine function

In [1], [4], and [6] the authors have studied index problems associated with the ‘coarse geometry’ of a metric space, which typically might be a complete noncompact Riemannian manifold or a group equipped with a word metric. The second author has introduced a cohomology theory, coarse cohomology, which is functorial on the category of metric spaces and coarse maps, and which can be computed in many examples. Associated to such a metric space there is also a C *-algebra generated by locally compact operators with finite propagation. In this note we will show that for suitable decompositions of a metric space there are Mayer–Vietoris sequences both in coarse cohomology and in the K -theory of the C *-algebra. As an application we shall calculate the K -theory of the C *-algebra associated to a metric cone. The result is consistent with the calculation of the coarse cohomology of the cone, and with a ‘coarse’ version of the Baum–Connes conjecture.

Hyperbolic groups have finite asymptotic dimension

This paper is an attempt to explain some aspects of the relationship between the K-theory of C-algebras, on the one hand, and the categories of modules that have been developed to systematize the algebraic aspects of controlled topology, on the other. It has recently become apparent that there is a substantial conceptual overlap between the two theories, and this allows both the recognition of common techniques, and the possibility of new methods in one theory suggested by those of the other. In this first part we will concentrate on defining the C-algebras associated to various kinds of controlled structure and giving methods whereby their K-theory groups may be calculated in a number of cases. From a ‘revisionist’ perspective, this study originates from an attempt to relate two approaches to the Novikov conjecture. The Novikov conjecture states that a certain assembly map is injective. The process of assembly can be thought of as the formation of a ‘generalized signature’ [38], and therefore to understand the connection between different approaches to the conjecture is to understand the connection between different definitions of the ‘generalized signature’. Now, broadly speaking, there are two approaches to the Novikov conjecture in the literature. One approach considers the original assembly map of Wall in L-theory, and attempts to prove it to be injective by investigating homological properties of the L-theory groups (which properties themselves may be derived algebraically, or geometrically, by relating them to surgery problems). The other approach proceeds via analysis, considering the assembly map to be the formation of a generalized index of the Atiyah-Singer signature operator [3]. This approach ultimately leads to the consideration of assembly on the K-theory of C-algebras. Nevertheless it can be shown that the injectivity of assembly on the C-algebra level implies (modulo 2-torsion) the injectivity of assembly on the L-theory level. All this is explained in the paper of Rosenberg [37], to which we also refer for an extensive bibliography on the Novikov conjecture. To proceed with the background to this paper. In the late seventies and eighties it occurred both to the topologists and independently to the analysts that a more flexible generalized signature theory might be developed if one neglected the group structure of the fundamental group π in question and considered only its “large scale” or “coarse” structure induced by some translation invariant metric; up to “large scale equivalence” the choice of such a metric is irrelevant. Some references are [26, 29, 11, 12, 14, 16, 9, 33, 34, 35, 36, 41]. Since the two theories were based on the same idea, it was inevitable that they would eventually come into interaction, but this did not happen for some while. It was the insight of Shmuel Weinberger, and especially his note [40] relating the index theory of [36] to Novikov’s theorem on the topological invariance of the rational Pontrjagin classes, which provoked the discussions among the present authors which eventually led to the writing of this paper. This paper is intended to be foundational, setting down some of the language in which one can talk about the relationship between analysis and controlled topology. Many of its ideas have already been worked out in the special case of bounded control in the work of the first and third authors and G. Yu [18, 20, 19]. But it seems that there is much to be gained by considering more general kinds of control, more general coefficients, “spacification” of the theory and so on, and it

COARSE TOPOLOGY, ENLARGEABILITY, AND ESSENTIALNESS

If a function and all its derivatives and integrals are absolutely uniformly bounded, then the function is a sine function with period 2π.

Spaces with Vanishing ℓ2-Homology and their Fundamental Groups (after Farber and Weinberger)

We detail a proof of a result of Gromov, that hyperbolic groups (and metric spaces) have finite asymptotic dimension. This fact has become important in recent work on the Novikov conjecture.

On the localization algebra of Guoliang Yu

Using methods from coarse topology we show that fundamental classes of closed en- largeable manifolds map non-trivially both to the rational homology of their fundamental groups and to the K-theory of the corresponding reduced C -algebras. Our proofs do not depend on the Baum- Connes conjecture and provide independent confirmation for specific predictions derived from this conjecture.

A homotopy invariance theorem in coarse cohomology and

We characterize those groups which can occur as the fundamental groups of finite CW-complexes with vanishing ℓ2-homology (the first examples of such groups were obtained by Farber and Weinberger).