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Journal of Homotopy and Related Structures

, Volume 14, Issue 1, pp 281–291 | Cite as

On the topological computation of \(K_4\) of the Gaussian and Eisenstein integers

  • Mathieu Dutour Sikirić
  • Herbert GanglEmail author
  • Paul E. Gunnells
  • Jonathan Hanke
  • Achill Schürmann
  • Dan Yasaki
Article
  • 77 Downloads

Abstract

In this paper we use topological tools to investigate the structure of the algebraic K-groups \(K_4(R)\) for \(R=Z[i]\) and \(R=Z[\rho ]\) where \(i := \sqrt{-1}\) and \(\rho := (1+\sqrt{-3})/2\). We exploit the close connection between homology groups of \(\mathrm {GL}_n(R)\) for \(n\le 5\) and those of related classifying spaces, then compute the former using Voronoi’s reduction theory of positive definite quadratic and Hermitian forms to produce a very large finite cell complex on which \(\mathrm {GL}_n(R)\) acts. Our main result is that \(K_{4} ({\mathbb {Z}}[i])\) and \(K_{4} ({\mathbb {Z}}[\rho ])\) have no p-torsion for \(p\ge 5\).

Keywords

Cohomology of arithmetic groups Voronoi reduction theory Linear groups over imaginary quadratic fields K-theory of number rings 

Mathematics Subject Classification

Primary 19D50 Secondary 11F75 

Notes

Acknowledgements

We thank Ph. Elbaz-Vincent for very helpful discussions. We also thank an anonymous referee for suggesting numerous improvements and corrections to our paper. This research was conducted as part of a “SQuaRE” (Structured Quartet Research Ensemble) at the American Institute of Mathematics in Palo Alto, California in September 2013. It is a pleasure to thank AIM and its staff for their support, without which our collaboration would not have been possible.

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Copyright information

© Tbilisi Centre for Mathematical Sciences 2018

Authors and Affiliations

  • Mathieu Dutour Sikirić
    • 1
  • Herbert Gangl
    • 2
    Email author
  • Paul E. Gunnells
    • 3
  • Jonathan Hanke
    • 4
  • Achill Schürmann
    • 5
  • Dan Yasaki
    • 6
  1. 1.Rudjer Bošković InstituteZagrebCroatia
  2. 2.Department of Mathematical SciencesDurham UniversityDurhamUK
  3. 3.Department of Mathematics and Statistics, LGRT 1115LUniversity of MassachusettsAmherstUSA
  4. 4.PrincetonUSA
  5. 5.Institute of MathematicsUniversität RostockRostockGermany
  6. 6.Department of Mathematics and StatisticsUniversity of North Carolina at GreensboroGreensboroUSA

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