This book consists essentially of notes which were written for an
Advanced Course on Classifying Spaces and Cohomology of Groups. The
course took place at the Centre de Recerca Mathematica (CRM) in
Bellaterra from May 27 to June 2, 1998 and was part of an emphasis
semester on Algebraic Topology. It consisted of two parallel series
of 6 lectures of 90 minutes each and was intended as an
introduction to new homotopy theoretic methods in group cohomology.
The first part of the book is concerned with methods of decomposing
the classifying space of a finite group into pieces made of
classifying spaces of appropriate subgroups. Such decompositions
have been used with great success in the last 10-15 years in the
homotopy theory of classifying spaces of compact Lie groups and
p-compact groups in the sense of Dwyer and Wilkerson. For
simplicity the emphasis here is on finite groups and on homological
properties of various decompositions known as centralizer resp.
normalizer resp. subgroup decomposition. A unified treatment of the
various decompositions is given and the relations between them are
explored. This is preceeded by a detailed discussion of basic
notions such as classifying spaces, simplicial complexes and
homotopy colimits.
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