C*-Algebras and Controlled Topology

Abstract

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