Lower Algebraic K-Theory Groups of the Group Ring \(\mathbb Z[B_4(\mathbb S^{2})]\)

  • John GuaschiEmail author
  • Daniel Juan-Pineda
  • Silvia Millán López
Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)


As we mentioned in Sect. 2.1, \(B_n(\mathbb S^{2})\) is finite for all \(n\le 3\). For these values of n, the corresponding K-groups were given in Table 2.1. This chapter is devoted to the computation of the lower K-groups of \(\mathbb Z[B_{4}(\mathbb S^{2})]\). The aim is to prove Theorem 1, whose statement we recall here.


Infinite Direct Sum Quaternary Roots Cartesian Square Mayer-Vietoris Sequence Twisted Versions 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    D. Juan-Pineda, J.-F. Lafont, S. Millán-Vossler, S. Pallekonda, Algebraic \(K\)-theory of virtually free groups. Proc. R. Soc. Edinb. 141A, 1–22 (2011)MathSciNetzbMATHGoogle Scholar
  2. 2.
    D. Juan-Pineda, I. Leary, On classifying spaces for the family of virtually cyclic subgroups, in Recent Developments in Algebraic Topology. Contemporary Mathematics, vol. 407, pp. 135–145 (2001)Google Scholar
  3. 3.
    H. Bass, Algebraic K-theory (W. A. Benjamin Inc., New York-Amsterdam, 1968)zbMATHGoogle Scholar
  4. 4.
    F.T. Farrell, W.C. Hsiang, The Whitehead group of poly-(finite or cyclic) groups. J. Lond. Math. Soc. 24, 308–324 (1981)MathSciNetCrossRefGoogle Scholar
  5. 5.
    F. Waldhausen, Algebraic \(K\)-theory of generalized free products, Part 1. Ann. Math. 108, 135–204 (1978)MathSciNetCrossRefGoogle Scholar
  6. 6.
    J. Davis, K. Khan, A. Ranicki, Algebraic \(K\)-theory over the infinite dihedral group: an algebraic approach. Algebr. Geom. Topol. 11, 2391–2436 (2011)MathSciNetCrossRefGoogle Scholar
  7. 7.
    J.-F. Lafont, I.J. Ortiz, Relating the Farrell Nil-groups to the Waldhausen Nil-groups. Forum Math. 20, 445–455 (2008)MathSciNetCrossRefGoogle Scholar
  8. 8.
    F.T. Farrell, L.E. Jones, The lower algebraic \(K\)-theory of virtually infinite cyclic groups. K-theory 9 (1995), 13–30MathSciNetCrossRefGoogle Scholar
  9. 9.
    R.G. Swan, Projective modules over binary polyhedral groups. J. Reine Angew. Math. 342, 66–172 (1983)MathSciNetzbMATHGoogle Scholar
  10. 10.
    A.W. Chatters, Rings which are nearly principal ideal domains. Glasgow Math. J. 40, 243–351 (1998)MathSciNetCrossRefGoogle Scholar
  11. 11.
    S.C. Geller, C.A. Weibel, \(K_{1}(A, B, I)\). J. Reine Angew. Math. 342, 12–34 (1983)MathSciNetzbMATHGoogle Scholar
  12. 12.
    C. Weibel, \(NK_0\) and \(NK_1\) of the groups \(C_4\) and \(D_4\). Comment. Math. Helv. 84, 339–349 (2009). (addendum to [63])Google Scholar
  13. 13.
    J.-F. Lafont, S. Prassidis, K. Wang, Revisiting Farrell’s nonfiniteness of Nil. Ann. K-theory 1, 209–225 (2016)MathSciNetCrossRefGoogle Scholar
  14. 14.
    C. W. Curtis and I. Reiner, Methods of representation theory, vol. II, with applications to finite groups and orders (John Wiley & Sons Inc., New York, 1987)Google Scholar
  15. 15.
    J.-F. Lafont, I.J. Ortiz, Relative hyperbolicity, classifying spaces, and lower algebraic \(K\)-theory. Topology 46, 527–553 (2007)MathSciNetCrossRefGoogle Scholar
  16. 16.
    C. Weibel, The \(K\)-Book, An Introduction to Algebraic \(K\)-theory, Graduate Studies in Mathematics, vol. 145 (American Mathematical Society, 2013)Google Scholar
  17. 17.
    R.D. Martin, Nilgroups of finite abelian groups. Ph.D. Thesis, Columbia University, 1976. 40 pp, ProQuest LLCGoogle Scholar
  18. 18.
    V. Angeltveit, T. Gerhardt, M. Hill, A. Lindenstrauss, On the algebraic \(K\)-theory of truncated polynomial algebras in several variables. J. K-theory 13, 57–81 (2014)MathSciNetCrossRefGoogle Scholar
  19. 19.
    F. T. Farrell, The obstruction to fibering a manifold over a circle. Indiana Univ. Math. J. 21, 315–346 (1971/1972)MathSciNetCrossRefGoogle Scholar
  20. 20.
    A. Bartels, W. Lück, H. Reich, On the Farrell-Jones conjecture and its applications. J. Topol. 1, 57–86 (2008)MathSciNetCrossRefGoogle Scholar

Copyright information

© The Author(s), under exclusive license to Springer Nature Switzerland AG, part of Springer Nature 2018

Authors and Affiliations

  • John Guaschi
    • 1
    Email author
  • Daniel Juan-Pineda
    • 2
  • Silvia Millán López
    • 3
  1. 1.Laboratoire de Mathématiques Nicolas Oresme UMR CNRS 6139Université de Caen Normandie, Normandie UniversitéCaenFrance
  2. 2.Centro de Ciencias MatemáticasUniversidad Nacional Autónoma de MéxicoMoreliaMexico
  3. 3.Colegio de Bachilleres del Estado de TlaxcalaTlaxcalaMexico

Personalised recommendations