Homotopy Theoretic Methods in Group Cohomology

by Dwyer, William G.; Henn, Hans-Werner

• ISBN13: 9783764366056
• ISBN10: 3764366052
• Format: Paperback
• Copyright: 2002-10-01
• Publisher: Birkhauser

Summary

This book looks at group cohomology with tools that come from homotopy theory. These tools give both decomposition theorems (which rely on homotopy colimits to obtain a description of the cohomology of a group in terms of the cohomology of suitable subgroups) and global structure theorems (which exploit the action of the ring of topological cohomology operations).

Table of Contents

Preface	ix
Classifying Spaces and Homology Decompositions
William G. Dwyer
Introduction	1	(1)
Classifying spaces	2	(4)
Simplicial complexes and simplicial sets	6	(10)
Simplicial spaces and homotopy colimits	16	(6)
Nerves of categories and the Grothendieck construction	22	(4)
Homotopy orbit spaces	26	(2)
Homology decompositions	28	(5)
Sharp homology decompositions. Examples	33	(3)
Reinterpreting the homotopy colimit spectral sequence	36	(4)
Bredon homology and the transfer	40	(3)
Acyclicity for G-spaces	43	(2)
Non-identity p-subgroups	45	(1)
Elementary abelian p-subgroups	46	(3)
Appendix	49	(6)
References	51	(4)
Cohomology of Groups and Unstable Modules over the Steenrod Algebra
Hans-Werner Henn
Introduction	55	(2)
Cohomology of groups	57	(6)
Some fundamental structural results for cohomology groups and cohomology rings of finite groups	57	(1)
Quillen's F-isomorphism theorem	58	(5)
Quillen's F-isomorphism theorem from the point of view of unstable modules	63	(5)
Unstable modules over the Steenrod algebra	63	(3)
Quillen's F-isomorphism theorem from the point of view of unstable modules	66	(2)
Lannes' T-functor and Lannes' generalization of Quillen's theorem	68	(7)
Review of Lannes' T-functor	68	(3)
Lannes' generalization of Quillen's theorem	71	(3)
An outline of a proof of Quillen's Theorem	74	(1)
Approximations up to higher nilpotency	75	(8)
Localizations in abelian categories	75	(2)
Localization away from n-nilpotent modules	77	(6)
Commutative algebra of unstable H* BG-modules; approximations of H*BG up to torsion modules	83	(7)
The centralizer spectral sequence	90	(6)
References	96