Algebraic K-theory of Crystallographic Groups

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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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.

• 20H15,19B28,19A31,19D35
• Algebraic K-theory
• Classifying spaces
• Crystallographic groups
• Farrell-Jones isomorphism conjecture

• Mathematics, Miami University, Oxford, USA

  Daniel Scott Farley, Ivonne Johanna Ortiz

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