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The Connective K-Theory of Finite Groups
 
R. R. Bruner Wayne State University, Detroit, MI
J. P. C. Greenlees University of Sheffield, Sheffield, UK
The Connective K-Theory of Finite Groups
eBook ISBN:  978-1-4704-0383-6
Product Code:  MEMO/165/785.E
List Price: $62.00
MAA Member Price: $55.80
AMS Member Price: $37.20
The Connective K-Theory of Finite Groups
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The Connective K-Theory of Finite Groups
R. R. Bruner Wayne State University, Detroit, MI
J. P. C. Greenlees University of Sheffield, Sheffield, UK
eBook ISBN:  978-1-4704-0383-6
Product Code:  MEMO/165/785.E
List Price: $62.00
MAA Member Price: $55.80
AMS Member Price: $37.20
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 1652003; 127 pp
    MSC: Primary 19; 55; Secondary 20;

    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.

  • Table of Contents
     
     
    • Chapters
    • 0. Introduction
    • 1. General properties of the $ku$-cohomology of finite groups
    • 2. Examples of $ku$-cohomology of finite groups
    • 3. The $ku$-homology of finite groups
    • 4. The $ku$-homology and $ku$-cohomology of elementary abelian groups
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 1652003; 127 pp
MSC: Primary 19; 55; Secondary 20;

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.

  • Chapters
  • 0. Introduction
  • 1. General properties of the $ku$-cohomology of finite groups
  • 2. Examples of $ku$-cohomology of finite groups
  • 3. The $ku$-homology of finite groups
  • 4. The $ku$-homology and $ku$-cohomology of elementary abelian groups
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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