**Mathematical Surveys and Monographs**

Volume: 169;
2010;
318 pp;
Hardcover

MSC: Primary 19; 55; 13;
Secondary 20; 53

Print ISBN: 978-0-8218-5189-0

Product Code: SURV/169

List Price: $97.00

Individual Member Price: $77.60

**Electronic ISBN: 978-1-4704-1396-5
Product Code: SURV/169.E**

List Price: $97.00

Individual Member Price: $77.60

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#### Supplemental Materials

# Connective Real $K$-Theory of Finite Groups

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*Robert R. Bruner; J. P. C. Greenlees*

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.

#### Readership

Graduate students and research mathematicians interested in connective \(K\)-theory.

#### Reviews & Endorsements

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

#### Table of Contents

# Table of Contents

## Connective Real $K$-Theory of Finite Groups

- Contents iii4 free
- Chapter 1. Introduction 18 free
- Chapter 2. $K$-Theory with Reality 1926 free
- Chapter 3. Descent, Twisting and Periodicity 4956
- Chapter 4. The Bockstein Spectral Sequence 7178
- Chapter 5. Characteristic Classes 8390
- Chapter 6. Examples for Cohomology 115122
- Chapter 7. Examples for Homology 131138
- Chapter 8. Dihedral Groups 147154
- Chapter 9. The $ko$-Cohomology of Elementary Abelian 2-Groups 177184
- Chapter 10. The $ko$-Homology of Elementary Abelian Groups (BSS) 201208
- Chapter 11. The Structure of $TO$ 213220
- Chapter 12. The $ko$-Homology of Elementary Abelian Groups (LCSS) 235242
- Appendix A. Ext Charts 285292
- Appendix B. Conventions 303310
- Appendix C. Indices 305312
- Bibliography 315322