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


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 \(p\ge 5\).


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 



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.


  1. 1.
    Arlettaz, D.: The Hurewicz homomorphism in algebraic \(K\)-theory. J. Pure Appl. Algebra 71(1), 1–12 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ash, A., Mumford, D., Rapoport, M., Tai, Y.-S.: Smooth Compactifications of Locally Symmetric Varieties, 2nd edn. Cambridge Mathematical Library. Cambridge University Press, Cambridge (2010) (with the collaboration of Peter Scholze) Google Scholar
  3. 3.
    Bass, H., Tate, J.: The Milnor ring of a global field. In: Algebraic \(K\)-Theory, II: “Classical” Algebraic \(K\)-Theory and Connections with Arithmetic (Proceedings of the Conference, Seattle, Wash., Battelle Memorial Institute, 1972). Lecture Notes in Mathematics, vol. 342, pp. 349–446. Springer, Berlin (1973)Google Scholar
  4. 4.
    Borel, A., Serre, J.-P.: Corners and arithmetic groups. Comment. Math. Helv. 48, 436–491 (1973) (Avec un appendice: Arrondissement des variétés à coins, par A. Douady et L. Hérault) Google Scholar
  5. 5.
    Brown, K.S.: Cohomology of Groups. Graduate Texts in Mathematics, vol. 87. Springer, New York (1994) (corrected reprint of the 1982 original) Google Scholar
  6. 6.
    Dutour Sikirić, M., Gangl, H., Gunnells, P.E., Hanke, J., Schürmann, A., Yasaki, D.: On the cohomology of linear groups over imaginary quadratic fields. J. Pure Appl. Algebra 220(7), 2564–2589 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Elbaz-Vincent, P., Gangl, H., Soulé, C.: Quelques calculs de la cohomologie de \({\rm GL}_N({\mathbb{Z}})\) et de la \(K\)-théorie de \({\mathbb{Z}}\). C. R. Math. Acad. Sci. Paris 335(4), 321–324 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Elbaz-Vincent, P., Gangl, H., Soulé, C.: Perfect forms, K-theory and the cohomology of modular groups. Adv. Math. 245, 587–624 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Haesemeyer, C., Weibel, C.A.: The Norm Residue Theorem in Motivic Cohomology. Annals of Mathematics Studies. Princeton University Press, Princeton (to appear) Google Scholar
  10. 10.
    Koecher, M.: Beiträge zu einer Reduktionstheorie in Positivitätsbereichen I. Math. Ann. 141, 384–432 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kolster, M.: Higher relative class number formulae. Math. Ann. 323(4), 667–692 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Lee, R., Szczarba, R.H.: The group \(K_{3}({\bf Z})\) is cyclic of order forty-eight. Ann. Math. (2) 104(1), 31–60 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Lee, R., Szczarba, R.H.: On the homology and cohomology of congruence subgroups. Invent. Math. 33(1), 15–53 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Lee, R., Szczarba, R.H.: On the torsion in \(K_{4}({\mathbb{Z}})\) and \(K_{5}({\mathbb{Z}})\). Duke Math. J. 45(1), 101–129 (1978)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Quillen, D.: Finite generation of the groups \(K_{i}\) of rings of algebraic integers. In: Algebraic \(K\)-Theory, I: Higher \(K\)-Theories (Proceedings of the Conference, Battelle Memorial Institute, Seattle, Wash., 1972). Lecture Notes in Mathematics, vol. 341, pp. 179–198. Springer, Berlin (1973)Google Scholar
  16. 16.
    Quillen, D.: Higher algebraic \(K\)-theory I. In: Algebraic \(K\)-Theory, I: Higher \(K\)-Theories (Proceedings of the Conference, Battelle Memorial Institute, Seattle, Wash., 1972). Lecture Notes in Mathematics, vol. 341, pp. 85–147. Springer, Berlin (1973)Google Scholar
  17. 17.
    Rognes, J.: \(K_4({\mathbf{Z}})\) is the trivial group. Topology 39(2), 267–281 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Soulé, C.: On the \(3\)-torsion in \(K_4({\mathbf{Z}})\). Topology 39(2), 259–265 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Soulé, C.: The cohomology of \({\rm SL}_{3}({\mathbf{Z}})\). Topology 17(1), 1–22 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Staffeldt, R.E.: Reduction theory and \(K_{3}\) of the Gaussian integers. Duke Math. J. 46(4), 773–798 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Voevodsky, V.: On motivic cohomology with \({\bf Z}/l\)-coefficients. Ann. Math. (2) 174(1), 401–438 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Voronoi, G.: Nouvelles applications des paramètres continues à la théorie des formes quadratiques 1: Sur quelques propriétés des formes quadratiques positives parfaites. J. Reine Angew. Math. 133(1), 97–178 (1908)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Weibel, C.: Algebraic \(K\)-Theory of Rings of Integers in Local and Global Fields. Handbook of \(K\)-Theory, vols. 1, 2, pp. 139–190. Springer, Berlin (2005)Google Scholar
  24. 24.
    Yasaki, D.: Voronoi tessellation data. Accessed 2018

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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