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Cohomology Rings of Finite Groups : With an Appendix: Calculations of Cohomology Rings of Groups of Order Dividing 64 - Jon F. Carlson

Cohomology Rings of Finite Groups

With an Appendix: Calculations of Cohomology Rings of Groups of Order Dividing 64

Hardcover Published: 31st October 2003
ISBN: 9781402015250
Number Of Pages: 776

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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.

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"The cohomology of groups was developed to be ... a powerful tool in the study of group representations. ... Some other books concerning the cohomology of finite groups have appeared, but they are at least ten years old, and the computational tools that have been developed since then, make this book to be really valuable for both professionals and students. ... This book is almost self-contained and it is very useful ... for all of those that are using the cohomology instruments in their work." (Viorel Mihai Gontineac, Zentralblatt MATH, Vol. 1056 (7), 2005)

Homological Algebra
Complexes and Sequences
Projective and Injective Models
Tensor Products and Tor
Group Algebras
Duality and Tensor Products
Induction and Restriction
Radicals, Socles and Projective Modules
Degree Shifting
The Stable Category
Group Cohomology and Change of Coefficients
Projective Resolutions
Minimal Resolutions
The Bar Resolution
Applications to Low Dimensional Cohomology
Restrictions, Inflations and Transfers
Cohomology Products
Yoneda Splices and Compositions of Chain Maps
Products and Group Algebras
Restriction, Inflation and Transfer
Cohomology Ring Computations
Shifted Subgroups and Restrictions
Automorphisms and Cohomology
Spectral Sequences
The Spectral Sequence of a Biocomplex
The Lyndon-Hochschild-Serre Spectral Sequence
Extension Classes
Minimal Resolutions and Convergence
Exact Couples and the Bockstein Spectral Sequence
Norms and the Cohomology of Wreath Products
Wreath Products
The Norm Map
Examples and Applications
Finite Generation of Cohomology
Steenrod Operations
The Steenrod Algebra and Modules
The Steenrod Operations on Cohomology
Cohomology and Modules Over the Steenrod Algebra
The Cohomology of Extraspecial 2-Groups
The Cohomology of Extraspecial p-Groups
Serre's Theorem on the Vanishing of Bocksteins
Varieties and Elementary Abelian Subgroups
Filtrations on Modules
Vanishing Products of Cohomology Elements
Minimal Primes in Cohomology Rings
The Stratification Theorem
Cohomology Rings of Modules
Generalized Bocksteins Over Elementary Abelian Groups
Rank Varieties and Cohomology Rings Over Elementary Abelian Groups
The Cohomological Support Variety of a Module
Equating the Rank and Cohomological Support Varieties
The Tensor Product Theorem
Properties of the Cohomological Support Varieties
Complexity and Multiple Complexes
Notes on Dimension and Rates of Growth
Complexity of Modules
Varieties for Modules with Other Coefficient Rings
Projective Resolutions as Multiple Complexes
Duality Complexes
Gaps in Cohomology
PoincarF Duality Complexes
Differentials in the HSS
Cohen Macaulay Cohomology Rings
Further Considerations
Transfers, Depth and Detection
Notes on Depth and Associated Primes
Depth and the p-Rank of the Center
Varieties and Transfers
Detection and Depth-Essential Cohomology
Special Cases
Associated Primes in Cohomology
Unstable Modules
Subgroup Complexes
Posets of Subgroups and Cell Complexes
Homotopy Equivalences and Equivariance
Complexes of Posets of Finite Groups
The Bouc Complex
Applications to Cohomology
Decompositions of Molecules
Additional Remarks
Cohomology Compositions
Computer Calculations and Completion Tests
The Visual Cohomology Ring: Generators and Relations
Resolutions, Maps and Homogeneous Parameters
Tests for Completion
Two Special Cases
Appendices: Calculations of the Cohomology Rings of Groups of Order Dividing 64
Notation and References
Groups of Order 8
Groups of Order 16
Groups of Order 32
Groups of Order 64
Tables of Krull Dimension and Depth
Tables of Hilbert/PoincarF Series
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9781402015250
ISBN-10: 1402015259
Series: Algebra and Applications
Audience: General
Format: Hardcover
Language: English
Number Of Pages: 776
Published: 31st October 2003
Publisher: Springer-Verlag New York Inc.
Country of Publication: NL
Dimensions (cm): 24.38 x 16.31  x 5.08
Weight (kg): 1.25

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