Advertisement

Lower Algebraic K-Theory of the Finite Subgroups of \(B_{n}(\mathbb S^{2})\)

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

Abstract

If G is a group that satisfies the Farrell-Jones fibred isomorphism conjecture, the lower algebraic \(K\)-theory of the group ring \(\mathbb Z[G]\) may be calculated in principle if one knows the lower algebraic \(K\)-theory of the group rings of the virtually cyclic subgroups of G (see Appendix A).

References

  1. 1.
    D.B.A. Epstein, Ends, in Topology of \(3\)-manifolds and related topics, in Proceedings of University of Georgia Institute, 1961 (Prentice-Hall, Englewood Cliffs, N.J., 1962) pp. 110–117Google Scholar
  2. 2.
    C.T.C. Wall, Poincaré complexes I. Ann. Math. 86, 213–245 (1967)MathSciNetCrossRefGoogle Scholar
  3. 3.
    D.L. Gonçalves, J. Guaschi, The classification and the conjugacy classes of the finite subgroups of the sphere braid groups. Algebr. Geom. Topol. 8, 757–785 (2008)MathSciNetCrossRefGoogle Scholar
  4. 4.
    A. Adem, J. Milgram, Cohomology of finite groups, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 309 (Springer, Berlin, 2004)Google Scholar
  5. 5.
    H.S.M. Coxeter, Regular Complex Polytopes, 2nd edn. (Cambridge University Press, Cambridge, 1991)zbMATHGoogle Scholar
  6. 6.
    H.S.M. Coxeter, W.O.J. Moser, Generators and relations for discrete groups, in Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 14, 4th edn. (Springer, Berlin, 1980)Google Scholar
  7. 7.
    J.A. Wolf, Spaces of Constant Curvature, vol. 372, 6th edn. (AMS Chelsea Publishing, 2011)Google Scholar
  8. 8.
    D.L. Gonçalves, J. Guaschi, The Virtually Cyclic Subgroups of the Braid Groups of the Sphere, Springer Briefs in Mathematics (Springer, 2013)Google Scholar
  9. 9.
    C.B. Thomas, Elliptic Structures on 3-manifolds, London Mathematical Society. Lecture Note Series, vol. 104 (Cambridge University, 1986)Google Scholar
  10. 10.
    C.T.C. Wall, Norms of units in group rings. Proc. Lond. Math. Soc. 29, 593–632 (1974)MathSciNetCrossRefGoogle Scholar
  11. 11.
    F. Ushitaki, A generalization of a theorem of Milnor. Osaka J. Math. 31, 403–415 (1994)MathSciNetzbMATHGoogle Scholar
  12. 12.
    R. Oliver, Whitehead Groups of Finite Groups, London Mathematical Society Lecture Note Series, vol. 132 (Cambridge University Press, Cambridge, 1988)Google Scholar
  13. 13.
    M. Kreck, W. Lück, The Novikov conjecture, Geometry and algebra, Oberwolfach Seminars 33 (Birkhäuser Verlag, Basel, 2005)zbMATHGoogle Scholar
  14. 14.
    C.W. Curtis, I. Reiner, Methods of Representation Theory: With Applications to Finite Groups and Orders, vol. II (John Wiley & Sons Inc., New York, 1987)Google Scholar
  15. 15.
    S. Endô, Y. Hironaka, Finite groups with trivial class groups. J. Math. Soc. Jpn. 31, 161–174 (1979)MathSciNetCrossRefGoogle Scholar
  16. 16.
    R.G. Swan, Projective modules over binary polyhedral groups. J. Reine Angew. Math. 342, 66–172 (1983)MathSciNetzbMATHGoogle Scholar
  17. 17.
    H. Bass, The Dirichlet unit theorem, induced characters, and Whitehead groups of finite groups. Topology 4, 391–410 (1966)MathSciNetCrossRefGoogle Scholar
  18. 18.
    D.A. Cox, Galois Theory, Pure and Applied Mathematics (Wiley-Interscience, Hoboken, NJ, 2004)CrossRefGoogle Scholar
  19. 19.
    Mathematica, Version 11.2 (Wolfram Research, Inc., Champaign, IL, 2017)Google Scholar
  20. 20.
    B.A. Magurn, Negative \(K\)-theory of generalized quaternion groups and binary polyhedral groups. Commun. Algebra 41, 4146–4160 (2013)MathSciNetCrossRefGoogle Scholar
  21. 21.
    C.W. Curtis, I. Reiner, Methods of Representation Theory: With Applications to Finite Groups and Orders, vol. I (John Wiley & Sons Inc., New York, 1981)Google Scholar
  22. 22.
    D.W. Carter, Lower \(K\)-theory of finite groups. Commun. Algebra 8, 1927–1937 (1980)MathSciNetCrossRefGoogle Scholar
  23. 23.
    J.-F. Lafont, B.A. Magurn, I.J. Ortiz, Lower algebraic \(K\)-theory of certain reflection groups. Math. Proc. Camb. Philos. Soc. 148, 193–226 (2010)MathSciNetCrossRefGoogle Scholar
  24. 24.
    J.-F. Lafont, I.J. Ortiz, Lower algebraic \(K\)-theory of hyperbolic \(3\)-simplex reflection groups. Comment. Math. Helv. 84, 297–337 (2009)MathSciNetCrossRefGoogle Scholar
  25. 25.
    T. Yamada, On the group algebras of metabelian groups over algebraic number fields I. Osaka J. Math. 6, 211–228 (1969)MathSciNetzbMATHGoogle Scholar
  26. 26.
    T. Yamada, On the group algebras of metacyclic groups over algebraic number fields. J. Fac. Sci. Univ. Tokyo Sect. I(15), 179–199 (1968)MathSciNetzbMATHGoogle Scholar
  27. 27.
    O. Broche Cristo, A. Konovalov, A. Olivieri, G. Olteanu, Á. del Río, Wedderga–Wedderburn Decomposition of Group Algebras, GAP package, Version 4.3.3 (2009). http://www.um.es/adelrio/wedderga.htm
  28. 28.
    J.-P. Serre, Corps locaux, deuxième édition, Publications de l’Université de Nancago, No. VIII (Hermann, Paris, 1968)Google Scholar
  29. 29.
    D.W. Carter, Localization in lower algebraic \(K\)-theory. Commun. Algebra 8, 603–622 (1980)MathSciNetCrossRefGoogle Scholar
  30. 30.
    M.A. Kervaire, M. Pavaman, Murthy, On the projective class group of cyclic groups of prime power order. Comment. Math. Helv. 52, 415–452 (1977)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