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Connective Real $K$-Theory of Finite Groups

Robert R. Bruner Wayne State University, Detroit, MI
J. P. C. Greenlees University of Sheffield, Sheffield, UK
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Hardcover ISBN: 978-0-8218-5189-0
Product Code: SURV/169
List Price: $103.00 MAA Member Price:$92.70
AMS Member Price: $82.40 Electronic ISBN: 978-1-4704-1396-5 Product Code: SURV/169.E List Price:$97.00
MAA Member Price: $87.30 AMS Member Price:$77.60
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List Price: $154.50 MAA Member Price:$139.05
AMS Member Price: $123.60 Click above image for expanded view Connective Real$K$-Theory of Finite Groups Robert R. Bruner Wayne State University, Detroit, MI J. P. C. Greenlees University of Sheffield, Sheffield, UK Available Formats:  Hardcover ISBN: 978-0-8218-5189-0 Product Code: SURV/169  List Price:$103.00 MAA Member Price: $92.70 AMS Member Price:$82.40
 Electronic ISBN: 978-1-4704-1396-5 Product Code: SURV/169.E
 List Price: $97.00 MAA Member Price:$87.30 AMS Member Price: $77.60 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$154.50 MAA Member Price: $139.05 AMS Member Price:$123.60
• Book Details

Mathematical Surveys and Monographs
Volume: 1692010; 318 pp
MSC: Primary 19; 55; 13; Secondary 20; 53;

This book is about equivariant real and complex topological $K$-theory for finite groups. Its main focus is on the study of real connective $K$-theory including $ko^*(BG)$ as a ring and $ko_*(BG)$ as a module over it. In the course of their study the authors define equivariant versions of connective $KO$-theory and connective $K$-theory with reality, in the sense of Atiyah, which give well-behaved, Noetherian, uncompleted versions of the theory. They prove local cohomology and completion theorems for these theories, giving a means of calculation as well as establishing their formal credentials. In passing from the complex to the real theories in the connective case, the authors describe the known failure of descent and explain how the $\eta$-Bockstein spectral sequence provides an effective substitute.

This formal framework allows the authors to give a systematic calculation scheme to quantify the expectation that $ko^*(BG)$ should be a mixture of representation theory and group cohomology. It is characteristic that this starts with $ku^*(BG)$ and then uses the local cohomology theorem and the Bockstein spectral sequence to calculate $ku_*(BG)$, $ko^*(BG)$, and $ko_*(BG)$. To give the skeleton of the answer, the authors provide a theory of $ko$-characteristic classes for representations, with the Pontrjagin classes of quaternionic representations being the most important.

Building on the general results, and their previous calculations, the authors spend the bulk of the book giving a large number of detailed calculations for specific groups (cyclic, quaternion, dihedral, $A_4$, and elementary abelian 2-groups). The calculations illustrate the richness of the theory and suggest many further lines of investigation. They have been applied in the verification of the Gromov-Lawson-Rosenberg conjecture for several new classes of finite groups.

Graduate students and research mathematicians interested in connective $K$-theory.

• Chapters
• 1. Introduction
• 2. $K$-theory with reality
• 3. Descent, twisting and periodicity
• 4. The Bockstein spectral sequence
• 5. Characteristic classes
• 6. Examples for cohomology
• 7. Examples for homology
• 8. Dihedral groups
• 9. The $ko$-cohomology of elementary abelian 2-groups
• 10. The $ko$-homology of elementary abelian groups (BSS)
• 11. The structure of $TO$
• 12. The $ko$-homology of elementary abelian groups (LCSS)
• 13. Ext charts
• 14. Conventions
• 15. Indices

• Reviews

• The book is very carefully written, including many diagrams and tables, and also a thorough review of the authors' previous work on the complex case.

Donald M. Davis, Mathematical Reviews
• Requests

Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Volume: 1692010; 318 pp
MSC: Primary 19; 55; 13; Secondary 20; 53;

This book is about equivariant real and complex topological $K$-theory for finite groups. Its main focus is on the study of real connective $K$-theory including $ko^*(BG)$ as a ring and $ko_*(BG)$ as a module over it. In the course of their study the authors define equivariant versions of connective $KO$-theory and connective $K$-theory with reality, in the sense of Atiyah, which give well-behaved, Noetherian, uncompleted versions of the theory. They prove local cohomology and completion theorems for these theories, giving a means of calculation as well as establishing their formal credentials. In passing from the complex to the real theories in the connective case, the authors describe the known failure of descent and explain how the $\eta$-Bockstein spectral sequence provides an effective substitute.

This formal framework allows the authors to give a systematic calculation scheme to quantify the expectation that $ko^*(BG)$ should be a mixture of representation theory and group cohomology. It is characteristic that this starts with $ku^*(BG)$ and then uses the local cohomology theorem and the Bockstein spectral sequence to calculate $ku_*(BG)$, $ko^*(BG)$, and $ko_*(BG)$. To give the skeleton of the answer, the authors provide a theory of $ko$-characteristic classes for representations, with the Pontrjagin classes of quaternionic representations being the most important.

Building on the general results, and their previous calculations, the authors spend the bulk of the book giving a large number of detailed calculations for specific groups (cyclic, quaternion, dihedral, $A_4$, and elementary abelian 2-groups). The calculations illustrate the richness of the theory and suggest many further lines of investigation. They have been applied in the verification of the Gromov-Lawson-Rosenberg conjecture for several new classes of finite groups.

Graduate students and research mathematicians interested in connective $K$-theory.

• Chapters
• 1. Introduction
• 2. $K$-theory with reality
• 3. Descent, twisting and periodicity
• 4. The Bockstein spectral sequence
• 5. Characteristic classes
• 6. Examples for cohomology
• 7. Examples for homology
• 8. Dihedral groups
• 9. The $ko$-cohomology of elementary abelian 2-groups
• 10. The $ko$-homology of elementary abelian groups (BSS)
• 11. The structure of $TO$
• 12. The $ko$-homology of elementary abelian groups (LCSS)
• 13. Ext charts
• 14. Conventions
• 15. Indices
• The book is very carefully written, including many diagrams and tables, and also a thorough review of the authors' previous work on the complex case.

Donald M. Davis, Mathematical Reviews
Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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