Guido Mislin

Proper group actions and the Baum-Connes conjecture

Equivariant K-Homology of the Classifying Space for Proper Actions.- On the Baum-Connes Assembly Map for Discrete Groups.

Equivariant K -Homology of the Classifying Space for Proper Actions

These notes are a compendium to a series of lectures concerning the topological aspects of the Baum-Connes Conjecture — the left hand side of the equation K * G (E G) ≅ K * top (C r * (G)) — the equivariant K-homology of E G. Besides of a presentation of the material needed to compute * G (E G,the reader will find an extensive discussion of many conjectures related to the Baum-Connes Conjecture.

On co-H-spaces.

This paper is concerned with two aspects of the theory of cot-H-spaces, which we interrelate in our final result. First, let W be a finite connected complex and let X be a nilpotent space of finite type. It was proved in [HMR] that if W is a suspension or a 1-connected co-H-space, then there exists a cofinite set of primes Q such that the rationalizing map Xs---~X0 induces an injection of homotopy sets [W, Xs] ~ [W, X~], provided that S ~_ Q. We show in Section 3 that the class ~f of finite connected complexes W for which this conclusion holds is much broader than the result above indicates, and indeed that it properly contains all finite connected co-H-spaces. A set M with a binary operation, written additively, is called a loop if it admits a two-sided zero, and if the equations

Topological localization and nilpotent groups

I. The theory of topological localization has been developed in various forms by several authors (cf. for example [1], [2], [8], [10], [12]). In this note, we place ourselves in the framework of [10] and announce some foundational results in the theory. The importance and usefulness of these results has already been well illustrated, for example, by the applications in [4], [5], [9]. Before stating our main result, we fix some notation. Let X be a connected, simple CW-complex of finite type. If P is a collection of primes, we denote by XP the localization of X at P, by eP : X -» XP the localization map, by X0 the rationalization of X, and by rP : XP -• X0 the rationalization map, characterized by the equation rP ° eP = e0 [10]. We then have

Algebraic Topology: New Trends in Localization and Periodicity

Central to this collection of papers are new developments in the general theory of localization of spaces. This field has undergone tremendous change of late and is yielding new insight into the mysteries of classical homotopy theory. The present volume comprises the refereed articles submitted at the Conference on Algebraic Topology held in Sant Feliu de Guixols, Spain, in June 1994. Several comprehensive articles on general localization clarify the basic tools and give a report on the state of the art in the subject matter. The text is therefore accessible not only to the professional mathematician but also to the advanced student.