Journal of Homotopy and Related Structures

, Volume 11, Issue 4, pp 885–891 | Cite as

An algebraic study of the Klein Bottle

  • Larry A. Lambe


We use symbolic computation (SC) and homological perturbation (HPT) to compute a resolution of the integers \(\mathbb {Z}\) over the integer group ring of G, the fundamental group of the Klein bottle. From this it is easy to read off the homology of the Klein bottle as well as other information.


Klein bottle Homological algebra Topology 

Mathematics Subject Classification

18G10 68W30 97U70 


  1. 1.
    Lambe, Larry, Stasheff, J.: Applications of perturbation theory to iterated fibrations. Manuscripta Math. 58(3), 363–376 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Brown, R.: The twisted Eilenberg-Zilber theorem. In Simposio di Topologia (Messina, 1964), pp. 33–37. Edizioni Oderisi, Gubbio (1965)Google Scholar
  3. 3.
    Gugenheim, V.K.A.M.: On the chain-complex of a fibration. Ill. J. Math. 16, 398–414 (1972)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Barnes, Donald W., Lambe, Larry A.: A fixed point approach to homological perturbation theory. Proc. Am. Math. Soc. 112(3), 881–892 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Mac Lane, S.: Homology. Classics in Mathematics. Springer, Berlin. Reprint of the 1975 edition (1995)Google Scholar
  6. 6.
    Cartan, H., Eilenberg, S.: Homological Algebra. Princeton University Press, Princeton (1956)zbMATHGoogle Scholar
  7. 7.
    Hilton, P.J., Wylie, S.: Homology theory: an introduction to algebraic topology. Cambridge University Press, New York (1960)CrossRefzbMATHGoogle Scholar
  8. 8.
    Lambe, L.A.: Entry on Homological Perturbation Theory in the Encyclopaedia of Mathematics. Supplement II. Kluwer Academic Press, Dordrecht (2000)Google Scholar
  9. 9.
    Lambe, Larry A.: Resolutions via homological perturbation. J. Symb. Comput. 12, 71–87 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Lambe, L.A.: Homological perturbation theory, Hochschild homology, and formal groups. In: Deformation theory and quantum groups with applications to mathematical physics (Amherst, MA, 1990), vol. 134 of Contemp. Math., pp. 183–218. Am. Math. Soc., Providence, RI (1992)Google Scholar
  11. 11.
    Lambe, Larry A.: Resolutions which split off of the bar construction. J. Pure Appl. Algebra 84(3), 311–329 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Lambe, L: Next generation computer algebra systems AXIOM and the scratchpad concept: applications to research in algebra. In: Analysis, algebra, and computers in mathematical research (Luleå, 1992), Lecture Notes in Pure and Appl. Math., vol. 156, pp. 201–222. Dekker, New York (1994)Google Scholar
  13. 13.
    Lambe, L.A., Grabmeier, J: Computing resolutions over finite \(p\)-groups. Algebraic combinatorics and applications. In: Betten, A., Kohnert, A., Laue, R., Wassermann, A. (eds.)Proceedings of the Euroconference (ALCOMA) held in Gweinstein, September 12–19 (1999)Google Scholar
  14. 14.
    Lambe, Larry A.: Scratchpad II as a tool for mathematical research. Not. Am. Math. Soc. 36, 141–148 (1989)Google Scholar
  15. 15.
    Lambe, Larry A.: The AXIOM system. Not. Am. Math. Soc. 41, 14–18 (1994)Google Scholar
  16. 16.
    Jenks, R.D., Sutor, R.S.: AXIOM Numerical Algorithms Group Ltd., Oxford (1992)Google Scholar
  17. 17.
    FriCAS. Accessed 04 Nov 2016
  18. 18.
    Asymptote: The Vector Graphics Language. Accessed 04 Nov 2016
  19. 19.
    Eilenberg, S., MacLane, S.: Relations between homology and homotopy groups of spaces. Ann. Math. 46, 480–509 (1945)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Eilenberg, S., MacLane, S.: Relations between homology and homotopy groups of spaces II. Ann. Math. 51, 514–533 (1950)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Tate, J.: Homology of Noetherian rings and local rings. Ill. J. Math. 1, 14–27 (1957)Google Scholar
  22. 22.
    Ellis, G.: HAP–Homological Algebra Programming. Accessed 04 Nov 2016
  23. 23.
    Brady, T: Free resolutions for semi-direct products. Tohoku Math. J. (2) 45(4) 535–537 (1993)Google Scholar

Copyright information

© Tbilisi Centre for Mathematical Sciences 2016

Authors and Affiliations

  1. 1.Multidisciplinary Software Systems Research CorporationHanover ParkUSA
  2. 2.Bangor UniversityBangorWales

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