Survey on Classifying Spaces for Families of Subgroups
Abstract
We define for a topological group G and a family of subgroups F two versions for the classifying space for the family F, the GCWversion E_F(G) and the numerable Gspace version J_F(G). They agree if G is discrete, or if G is a Lie group and each element in F compact, or if F is the family of compact subgroups. We discuss special geometric models for these spaces for the family of compact open groups in special cases such as almost connected groups G and word hyperbolic groups G. We deal with the question whether there are finite models, models of finite type, finite dimensional models. We also discuss the relevance of these spaces for the BaumConnes Conjecture about the topological Ktheory of the reduced group C^*algebra, for the FarrellJones Conjecture about the algebraic K and Ltheory of group rings, for Completion Theorems and for classifying spaces for equivariant vector bundles and for other situations.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 December 2003
 arXiv:
 arXiv:math/0312378
 Bibcode:
 2003math.....12378L
 Keywords:

 Mathematics  Geometric Topology;
 55R35;
 57S99;
 20F65;
 18G99
 EPrint:
 60 pages (including index) In the revised version we have added the proof that the GCWversion and the Gnumerable version of the classifying space for the family of compact subgroups are always Ghomotopy equivalent