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Algebraic K-theory of Crystallographic Groups

The Three-Dimensional Splitting Case

  • Book
  • © 2014

Overview

Authors:
  1. Daniel Scott Farley

    1. Mathematics, Miami University, Oxford, USA

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    You can also search for this author in PubMed   Google Scholar

  2. Ivonne Johanna Ortiz

    1. Mathematics, Miami University, Oxford, USA

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    You can also search for this author in PubMed   Google Scholar

  • Contains an elementary classification of the arithmetic classes of three-dimensional crystallographic groups
  • Gives a clear construction, for a geometrically important class of groups, of the classifying spaces that are used in applications of the Farrell-Jones isomorphism conjecture
  • Shows how the Farrell-Jones isomorphism theorem is used in computations, assembling all of the required methods

Part of the book series: Lecture Notes in Mathematics (LNM, volume 2113)

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Table of contents (10 chapters)

  1. Front Matter

    Pages i-x
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  2. Introduction

    • Daniel Scott Farley, Ivonne Johanna Ortiz
    Pages 1-8

  3. Three-Dimensional Point Groups

    • Daniel Scott Farley, Ivonne Johanna Ortiz
    Pages 9-21

  4. Arithmetic Classification of Pairs (L, H)

    • Daniel Scott Farley, Ivonne Johanna Ortiz
    Pages 23-39

  5. The Split Three-Dimensional Crystallographic Groups

    • Daniel Scott Farley, Ivonne Johanna Ortiz
    Pages 41-43

  6. A Splitting Formula for Lower Algebraic K-Theory

    • Daniel Scott Farley, Ivonne Johanna Ortiz
    Pages 45-57

  7. Fundamental Domains for the Maximal Groups

    • Daniel Scott Farley, Ivonne Johanna Ortiz
    Pages 59-79

  8. The Homology Groups \(H_{n}^{\varGamma }(E_{\mathcal{F}\mathcal{I}\mathcal{N}}(\varGamma ); \mathbb{K}\mathbb{Z}^{-\infty })\)

    • Daniel Scott Farley, Ivonne Johanna Ortiz
    Pages 81-98

  9. Fundamental Domains for Actions on Spaces of Planes

    • Daniel Scott Farley, Ivonne Johanna Ortiz
    Pages 99-117

  10. Cokernels of the Relative Assembly Maps for \(\mathcal{V}\mathcal{C}_{\infty }\)

    • Daniel Scott Farley, Ivonne Johanna Ortiz
    Pages 119-136

  11. Summary

    • Daniel Scott Farley, Ivonne Johanna Ortiz
    Pages 137-141

  12. Back Matter

    Pages 143-150
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Keywords

About this book

The Farrell-Jones isomorphism conjecture in algebraic K-theory offers a description of the algebraic K-theory of a group using a generalized homology theory. In cases where the conjecture is known to be a theorem, it gives a powerful method for computing the lower algebraic K-theory of a group. This book contains a computation of the lower algebraic K-theory of the split three-dimensional crystallographic groups, a geometrically important class of three-dimensional crystallographic group, representing a third of the total number. The book leads the reader through all aspects of the calculation. The first chapters describe the split crystallographic groups and their classifying spaces. Later chapters assemble the techniques that are needed to apply the isomorphism theorem. The result is a useful starting point for researchers who are interested in the computational side of the Farrell-Jones isomorphism conjecture, and a contribution to the growing literature in the field.

Authors and Affiliations

  • Mathematics, Miami University, Oxford, USA

    Daniel Scott Farley, Ivonne Johanna Ortiz

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