Hardcover ISBN:  9780821851890 
Product Code:  SURV/169 
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Electronic ISBN:  9781470413965 
Product Code:  SURV/169.E 
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Book DetailsMathematical Surveys and MonographsVolume: 169; 2010; 318 ppMSC: 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 wellbehaved, 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 2groups). The calculations illustrate the richness of the theory and suggest many further lines of investigation. They have been applied in the verification of the GromovLawsonRosenberg conjecture for several new classes of finite groups.ReadershipGraduate students and research mathematicians interested in connective \(K\)theory.

Table of Contents

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 2groups

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


Additional Material

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


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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 wellbehaved, 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 2groups). The calculations illustrate the richness of the theory and suggest many further lines of investigation. They have been applied in the verification of the GromovLawsonRosenberg 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 2groups

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