This book contains a collection of articles summarizing together the state of knowl edge in a broad portion of modern homotopy theory. These articles were assembled during 1998 and 1999, on the occasion of an emphasis semester organized by the Centre de Recerca Matematica (CRM) and its highlight, the 1998 Barcelona Con ference on Algebraic Topology (BCAT). First of all, we are indebted to all the authors for submitting their work, and to the referees for their help in the selec tion and for their generous contribution to the content of the articles. Many talks given during the CRM semester or at the conference focused on aspects of the following topics: abstract stable homotopy, model categories, homotopical localizations and cellular approximations, p-compact groups, mod ules over the Steenrod algebra, classifying spaces for proper actions of discrete groups, K-theory and other generalized cohomology theories, cohomology of fi nite and profinite groups, Hochschild homology, configuration spaces, Lusternik Schnirelmann category, stable and unstable splittings. Other talks treated multi disciplinary subjects related to quantum field theory, differential geometry, homo topical dynamics, tilings, and various aspects of group theory. In addition, an advanced course on Classifying Spaces and Cohomology of Groups was organized by the CRM in the days preceding the conference. Lecture notes from this course will be published by Birkhauser Verlag as the first volume of a newly created CRM Advanced Course series.

Inhalt
Etale approximations and the mod l cohomology of GLn.- On the Hurewicz map and Postnikov invariants of K?.- Recognition principle for generalized EilenbergMac Lane spaces.- Groups with infinite homology.- Unstable splittings related to BrownPeterson cohomology.- Stripping and splitting decorated mapping class groups.- Loop spaces of configuration spaces, braid-like groups, and knots.- On the homotopy type of infinite stunted projective spaces.- Stable splittings of ?SU(n).- Structure of mod p H-spaces with finiteness conditions.- Composition methods in the homotopy groups of ring spectra.- Tate cohomology in axiomatic stable homotopy theory.- Serre's theorem and the Nill filtration of Lionel Schwartz.- New relationships among loopspaces, symmetric products, and Eilenberg MacLane spaces.- Chern characters for the equivariant K-theory of proper G-CW-complexes.- The Thomified EilenbergMoore spectral sequence.- On the classifying space for proper actions.- Toric morphisms between p-compact groups.- On the vanishing of certain K-theory Nil-groups.- Lusternik-Schnirelmann cocategory: A Whitehead dual approach.- Rational self-equivalences for H-spaces.- Cellular approximations using Moore spaces.- Configuration spaces with summable labels.- Kaleidoscoping LusternikSchnirelmann category type invariants.- Essential category weight and phantom maps.