Abstract. The principal aim of this paper is to give a dynamical presentation of the Jiang-Su algebra. Originally constructed as an inductive limit of prime dimension drop algebras, the Jiang-Su algebra has gone from being a poorly understood oddity to having a prominent positive role in George Elliott's classification programme for separable, nuclear C * -algebras. Here, we exhibit anétale equivalence relation whose groupoid C * -algebra is isomorphic to the Jiang-Su algebra. The main ingredient is the construction of minimal homeomorphisms on infinite, compact metric spaces, each having the same cohomology as a point. This construction is also of interest in dynamical systems. Any self-map of an infinite, compact space with the same cohomology as a point has Lefschetz number one. Thus, if such a space were also to satisfy some regularity hypothesis (which our examples do not), then the LefschetzHopf Theorem would imply that it does not admit a minimal homeomorphism. IntroductionThe fields of operator algebras and dynamical systems have a long history of mutual influence. On the one hand, dynamical systems provide interesting examples of operator algebras and have often provided techniques which are successfully imported into the operator algebra framework. On the other hand, results in operator algebras are often of interest to those in dynamical systems. In ideal situations, significant information is retained when passing from dynamics to operator algebras, and vice versa. This relationship has been particularly interesting for the classification of C * -algebras. An extraordinary result in this setting is the classification, up to strong orbit equivalence, of the minimal dynamical systems on a Cantor set and the corresponding K-theoretical classification of the associated crossed product C * -algebras [9,24]. Classification for separable, simple, nuclear C * -algebras remains an interesting open problem. To every simple separable nuclear C * -algebra one assigns a computable set of invariants involving K-theory, tracial state spaces, and the pairing between these objects. George Elliott conjectured that for all such C * -algebras,

We show that the homoclinic C * -algebras of mixing Smale spaces are classifiable by the Elliott invariant. To obtain this result, we prove that the stable, unstable, and homoclinic C * -algebras associated to such Smale spaces have finite nuclear dimension. Our proof of finite nuclear dimension relies on Guentner, Willett, and Yu's notion of dynamic asymptotic dimension.2000 Mathematics Subject Classification. 46L35, 37D20.

We construct a geometric analog of the analytic surgery group of Higson and Roe for the assembly mapping for free actions of a group with values in a Banach algebra completion of the group algebra. We prove that the geometrically defined group, in analogy with the analytic surgery group, fits into a six term exact sequence with the assembly mapping and also discuss mappings with domain the geometric group. In particular, given two finite dimensional unitary representations of the same rank, we define a map in the spirit of η-type invariants from the geometric group (with respect to assembly for the full group C * -algebra) to the real numbers.

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We construct an isomorphism between the geometric model and Higson-Roe's analytic surgery group, reconciling the constructions in the previous papers in the series on "Realizing the analytic surgery group of Higson and Roe geometrically" with their analytic counterparts. Following work of Lott and Wahl, we construct a Chern character on the geometric model for the surgery group; it is a "delocalized Chern character", from which Lott's higher delocalized ρ-invariants can be retrieved. Following work of Piazza and Schick, we construct a geometric map from Stolz' positive scalar curvature sequence to the geometric model of Higson-Roe's analytic surgery exact sequence.