Cohomology Rings of Finite Groups: With an Appendix: Calculations of Cohomology Rings of Groups of Order Dividing 64 (Algebra and Applications, 3)

Group cohomology has a rich history that goes back a century or more. Its origins are rooted in investigations of group theory and num­ ber theory, and it grew into an integral component of algebraic topology. In the last thirty years, group cohomology has developed a powerful con­ nection with finite group representations. Unlike the early applications which were primarily concerned with cohomology in low degrees, the in­ teractions with representation theory involve cohomology rings and the geometry of spectra over these rings. It is this connection to represen­ tation theory that we take as our primary motivation for this book. The book consists of two separate pieces. Chronologically, the first part was the computer calculations of the mod-2 cohomology rings of the groups whose orders divide 64. The ideas and the programs for the calculations were developed over the last 10 years. Several new features were added over the course of that time. We had originally planned to include only a brief introduction to the calculations. However, we were persuaded to produce a more substantial text that would include in greater detail the concepts that are the subject of the calculations and are the source of some of the motivating conjectures for the com­ putations. We have gathered together many of the results and ideas that are the focus of the calculations from throughout the mathematical literature.

this Text Offers Comprehensive Coverage Of Group Cohomology, From Introductory Material Through The Most Recent Developments In The Field. The Primary Motivation For This Book Is The Interaction Of Group Cohomology With Representation Theory, Especially The Geometry Of Support Varieties Over Cohomology Rings. The Appendices, Comprising Computer Calculations Of The Mod-2 Cohomology Rings Of The Groups Whose Orders Divide 64, Provide Information Useful For Further Developments In The Field. A Unique Feature Of This Text Is That It Includes The Concepts That Are The Subject Of The Calculations And Are The Source Of Some Of The Motivating Conjectures For The Computations. The Programs For Computing The Cohomology Rings Were Executed In The Magma Computer Algebra Language. The Text Is A Valuable Resource For Researchers In Group Cohomology And Related Disciplines. In Addition, The Book Could Be Used As The Text For An Advanced Graduate Class Or A Graduate Seminar.

Front Matter....Pages i-xv Homological Algebra....Pages 1-22 Group Algebras....Pages 23-46 Projective Resolutions....Pages 47-59 Cohomology Products....Pages 61-85 Spectral Sequences....Pages 87-110 Norms and the Cohomology of Wreath Products....Pages 111-128 Steenrod Operations....Pages 129-157 Varieties and Elementary Abelian Subgroups....Pages 159-177 Cohomology Rings of Modules....Pages 179-208 Complexity and Multiple Complexes....Pages 209-230 Duality Complexes....Pages 231-253 Transfers, Depth and Detection....Pages 255-282 Subgroup Complexes....Pages 283-312 Computer Calculations and Completion Tests....Pages 313-335 Back Matter....Pages 337-777