Skip to main content

Introduction

  • 993 Accesses

Part of the Lecture Notes in Mathematics book series (LNM,volume 2113)

Abstract

Algebraic K-theory is a branch of algebra dealing with linear algebra over a general ring R instead of a field. It associates to any ring R a sequence of abelian groups K n (R). The first two of these groups, K 0 and K 1, are easy to describe in concrete terms. For instance, a finitely generated projective R-module defines an element of K 0(R), and an invertible matrix over R has a “determinant” in K 1(R). The entire sequence of groups K n (R) behaves something like a homology theory of rings.

Keywords

  • Point Group
  • Short Exact Sequence
  • Cyclic Subgroup
  • Finite Index
  • Mapping Class Group

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.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (Canada)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   49.99
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. A. Alves, P. Ontaneda, A formula for the Whitehead group of a three-dimensional crystallographic group. Topology 45(1), 1–25 (2006)

    CrossRef  MATH  MathSciNet  Google Scholar 

  2. H. Bass, K-theory and stable algebra. Inst. Hautes ’Etudes Sci. Publ. Math. 22, 5–60 (1964)

    CrossRef  MATH  MathSciNet  Google Scholar 

  3. H. Bass, M. Murthy, Grothendieck groups and Picard groups of abelian group rings. Ann. Math. 86(2), 16–73 (1967)

    CrossRef  MATH  MathSciNet  Google Scholar 

  4. E. Berkove, D.J. Pineda, K. Pearson, The lower algebraic K-theory of Fuchsian groups. Comment. Math. Helv. 76(2), 339–352 (2001)

    CrossRef  MATH  MathSciNet  Google Scholar 

  5. E. Berkove, F.T. Farrell, D.J. Pineda, K. Pearson, The Farrell-Jones isomorphism conjecture for finite covolume hyperbolic actions and the algebraic K-theory of Bianchi groups. Trans. Am. Math. Soc. 352(12), 5689–5702 (2000)

    CrossRef  MATH  Google Scholar 

  6. E. Berkove, D.J. Pineda, Q. Lu, Algebraic K-theory of mapping class groups. K Theory 32(1), 83–100 (2004)

    Google Scholar 

  7. B. Bürgisser, On the projective class group of arithmetic groups. Math. Z. 184(3), 339–357 (1983)

    CrossRef  MATH  MathSciNet  Google Scholar 

  8. F. Connolly, T. Koźniewski, Rigidity and crystallographic groups, I. Invent. Math. 99(1), 25–48 (1990)

    CrossRef  MATH  Google Scholar 

  9. D. Farley, Constructions of \(E_{\mathcal{V}\mathcal{C}}\) and \(E_{\mathcal{F}\mathcal{B}\mathcal{C}}\) for groups acting on CAT(0) spaces. Algebr. Geom. Topol. 10(4), 2229–2250 (2010)

    CrossRef  MATH  MathSciNet  Google Scholar 

  10. F.T. Farrell, L. Jones, Isomorphism conjectures in algebraic K-theory. J. Am. Math. Soc. 6(2), 249–297 (1993)

    MATH  MathSciNet  Google Scholar 

  11. F.T. Farrell, L. Jones, The lower algebraic K-theory of virtually infinite cyclic groups. K Theory 9, 13–30 (1995)

    Google Scholar 

  12. F.T. Farrell, L. Jones, Algebraic K-theory of discrete subgroups of Lie groups. Proc. Nat. Acad. Sci. USA 84(10), 3095–3096 (1987)

    CrossRef  MATH  MathSciNet  Google Scholar 

  13. F.T. Farrell, W.C. Hsiang, The whitehead group of poly-(finite or cyclic) groups. J. Lond. Math. Soc. 24(2), 308–324 (1961)

    MathSciNet  Google Scholar 

  14. F.T. Farrell, W.C. Hsiang, Manifolds with π 1 = G × α T. Am. J. Math. 95(2), 813–848 (1973)

    Google Scholar 

  15. F.T. Farrell, W.C. Hsiang, A formula for K 1(R α  [T]), in Applications of Categorical Algebra. Proceedings of Symposia in Pure Mathematics, vol. 17 (American Mathematical Society, Providence, 1970), pp. 192–218

    Google Scholar 

  16. J.-F. Lafont, B. Magurn, I.J. Ortiz, Lower algebraic K-theory of certain reflection groups. Math. Proc. Camb. Philos. Soc. 148(2), 193–226 (2010)

    CrossRef  MATH  MathSciNet  Google Scholar 

  17. J.-F. Lafont, I.J. Ortiz, Lower algebraic K-theory of hyperbolic 3-simplex reflection groups. Comment. Math. Helv. 84, 297–337 (2009)

    CrossRef  MATH  MathSciNet  Google Scholar 

  18. W. Lück, R. Stamm, Computations of K- and L-theory of cocompact planar groups. K Theory 21(3), 249–292 (2000)

    Google Scholar 

  19. W. Lück, K- and L-theory of the semi-direct product of the discrete 3-dimensional Heisenberg group by \(\mathbb{Z}/4\). Geom. Topol. 9, 1639–1676 (2005)

    CrossRef  MATH  MathSciNet  Google Scholar 

  20. K. Pearson, Algebraic K-theory of two-dimensional crystallographic groups. K Theory 14(3), 265–280 (1998)

    Google Scholar 

  21. S.P. Plotnick, Vanishing of Whitehead groups for Seirfert manifolds with infinite fundamental group. Comment. Math. Helv. 55, 654–667 (1980)

    CrossRef  MATH  MathSciNet  Google Scholar 

  22. F. Quinn, Ends of maps II. Invent. Math. 68, 353–424 (1982)

    CrossRef  MATH  MathSciNet  Google Scholar 

  23. J. Ratcliffe, Foundations of Hyperbolic Manifolds. Graduate Texts in Mathematics, vol. 149 (Springer, New York, 1994)

    Google Scholar 

  24. R.L.E. Schwarzenberger, N-Dimensional Crystallography. Research Notes in Mathematics, vol. 41 (Pittman Advanced Publishing Program, Boston, Mass.-London 1980)

    Google Scholar 

  25. G. Tsapogas, On the K-theory of crystallographic groups. Trans. Am. Math. Soc. 347(8), 2781–2794 (1995)

    MATH  MathSciNet  Google Scholar 

  26. F. Waldhausen, Algebraic K-theory of generalized free products I, II. Ann. Math. 108, 135–256 (1978)

    CrossRef  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and Permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Farley, D.S., Ortiz, I.J. (2014). Introduction. In: Algebraic K-theory of Crystallographic Groups. Lecture Notes in Mathematics, vol 2113. Springer, Cham. https://doi.org/10.1007/978-3-319-08153-3_1

Download citation