This paper is devoted to the connective K homology and cohomology of finite groups G. We attempt to give a systematic account from several points of view.
In Chapter 1, following Quillen [50, 51], we use the methods of algebraic geometry to study the ring ku*(BG) where ku denotes connective complex K-theory. We describe the variety in terms of the category of abelian p-subgroups of G for primes p dividing the group order. As may be expected, the variety is obtained by splicing that of periodic complex K-theory and that of integral ordinary homology, however the way these parts fit together is of interest in itself. The main technical obstacle is that the Künneth spectral sequence does not collapse, so we have to show that it collapses up to isomorphism of varieties.
In Chapter 2 we give several families of new complete and explicit calculations of the ring ku*(BG). This illustrates the general results of Chapter 1 and their limitations.
In Chapter 3 we consider the associated homology ku*(BG). We identify this as a module over ku*(BG) by using the local cohomology spectral sequence. This gives new specific calculations, but also illuminating structural information, including remarkable duality properties.
Finally, in Chapter 4 we make a particular study of elementary abelian groups V. Despite the group-theoretic simplicity of V, the detailed calculation of ku*(BV) and ku*(BV) exposes a very intricate structure, and gives a striking illustration of our methods. Unlike earlier work, our description is natural for the action of GL(V).
Readership: Graduate students and research mathematicians interested in algebra, algebraic geometry, geometry, and topology.
Author(s): R. R. Bruner, J. P. C. Greenlees
Series: Memoirs of the American Mathematical Society, 165/785
Publisher: American Mathematical Society
Year: 2003
Language: English
Pages: C+VII+127+B
Cover
S Title
The Connective K-Theory of Finite Groups
Copyright
© 2003 by the American Mathematical Society
ISBN 0-8218-3366-9
QA3.A57 2003 510--sdc21
LCCN 2003051902
Contents
Chapter 0. Introduction
0.1. Motivation
0.2. Highlights of Chapter 1
0.3. Highlights of Chapter 2
0.4. Highlights of Chapter 3
0.5. Highlights of Chapter 4
0.6. Reading guide
0.7. Acknowledgements
Chapter 1. General properties of the ku-cohomology of finite groups
1.1. Varieties for connective K-theory
1.2. Implications for minimal primes
1.3. Euler classes and Chern classes
1.4. Bockstein spectral sequences
1.5. The Kiinneth theorem
Chapter 2. Examples of ku-cohomology of finite groups
2.1. The technique
2.2. Cyclic groups
2.3. Nonabelian groups of order pq
2.4. Quaternion groups
2.5. Dihedral groups
2.6. The alternating group of degree 4
Chapter 3. The ku-homology of finite groups
3.1. General behaviour of ku[sub(*)](BG)
3.2. The universal coefficient theorem
3.3. Local cohomology and duality
3.4. The ku-homology of cyclic and quaternion groups
3.5. The ku-homology of BD[sub(8)]
3.6. Tate cohomology
Chapter 4. The ku-homology and ku-cohomology of elementary abelian groups
4.1. Description of results
4.2. The ku-cohomology of elementary abelian groups
4.3. What local cohomology ought to look like
4.4. The local cohomology of Q
4.5. The 2-adic filtration of the local cohomology of Q
4.6. A free resolution of T
4.7. The local cohomology of T
4.8. Hilbert series
4.9. The quotient P/T[sub(2)]
4.10. The local cohomology of R
4.11. The ku-homology of BV
4.12. Duality for the cohomology of elementary abelian groups
4.13. Tate cohomology of elementary abelian groups
Appendix A. Conventions
A.1. General conventions
A.2. Adams spectral sequence conventions
Appendix B. Indices
B.1. Index of calculations
B.2. Index of symbols
B.3. Index of notation
B.4. Index of terminology
Bibliography
Back Cover
