Skip to main content
Springer Nature Link
Log in

Computing the first stages of the Bousfield-Kan spectral sequence

  • Published:
Applicable Algebra in Engineering, Communication and Computing Aims and scope

Abstract

In this paper, an algorithm computing the terms E 1 and E 2 of the Bousfield-Kan spectral sequence of a 1-reduced simplicial set X is defined. In order to compute the ordinary description of the first level E 1, some elementary operations of Homological Algebra are sufficient. On the contrary, to compute the stage E 2 it is necessary to know more information about the previous groups, in particular with respect to the generators. This additional information can be reached by computing the effective homology of RX, RX being the free simplicial Abelian group generated by X. The algorithm to get the effective homology of RX from the effective homology of X can be considered the main result in our paper. Moreover, we include a combinatorial proof of the convergence of the Bousfield-Kan spectral sequence, based on the tapered nature of the stage E 1.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

References

  1. Adams J.F.: On the Cobar construction. Proc. Natl. Acad. Sci. USA 42, 409–412 (1956)

    Article  MATH  Google Scholar 

  2. Adams J.F.: On the non-existence of elements of Hopf invariant one. Ann. Math. 72(1), 20–104 (1960)

    Article  Google Scholar 

  3. Baues, H.J.: Geometry of loop spaces and the cobar construction. Mem. Am. Math. Soc. 230 (1980)

  4. Bousfield A.K., Kan D.M.: Homotopy Limits, Completions and Localizations, Lecture Notes in Mathematics. Springer, Berlin (1972)

    Google Scholar 

  5. Bousfield A.K., Kan D.M.: The homotopy spectral sequence of a space with coefficients in a ring. Topology 11, 79–106 (1972)

    Article  MathSciNet  Google Scholar 

  6. Bousfield A.K., Kan D.M.: A second quadrant homotopy spectral sequence. Trans. Am. Math. Soc. 177, 305–318 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  7. Cartan, H.: Algèbres d’Eilenberg-MacLane et homotopie. Séminaire H. Cartan, École Normal Supérieure, Paris (1954–55). Exposés 2–11

  8. Dold A.: Lectures on Algebraic Topology. Springer, Berlin (1972)

    MATH  Google Scholar 

  9. Dousson, X., Rubio, J., Sergeraert, F., Siret, Y.: The Kenzo program. Institut Fourier, Grenoble (1999). http://www-fourier.ujf-grenoble.fr/~sergerar/Kenzo/

  10. Eilenberg S., Zilber J.A.: On products of complexes. Am. J. Math. 75, 200–204 (1953)

    Article  MATH  MathSciNet  Google Scholar 

  11. Kaczynski T., Mischaikow K., Mrozek M.: Computational Homology, Applied Mathematical Sciences. Springer, New York (2004)

    Google Scholar 

  12. MacLane S.: Homology, Grundleren der Mathematischen Wissenschaften. Springer, Berlin (1963)

    Google Scholar 

  13. May J.P.: Simplicial Objects in Algebraic Topology, Van Nostrand Mathematical Studies. Van Nostrand, Princeton (1967)

    Google Scholar 

  14. McCleary J.: User’s Guide to Spectral Sequences. Publish or Perish, Wilmington (1985)

    MATH  Google Scholar 

  15. Romero, A.: Effective homology and spectral sequences. Ph.D. thesis, Universidad de La Rioja (2007)

  16. Romero A., Rubio J., Sergeraert F.: Computing spectral sequences. J. Symb. Comput. 41(10), 1059–1079 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  17. Rubio J., Sergeraert F.: Constructive algebraic topology. Bull. Sci. Math. 126(5), 389–412 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  18. Rubio, J., Sergeraert, F.: Constructive Homological Algebra and Applications, Lecture Notes Summer School on Mathematics, Algorithms, and Proofs. University of Genova (2006). http://www-fourier.ujf-grenoble.fr/~sergerar/Papers/Genova-Lecture-Notes.pdf

Download references

Author information

Authors and Affiliations

  1. Departamento de Matemáticas y Computación, Universidad de La Rioja, Edificio Vives, c/ Luis de Ulloa s/n, 26004, Logroño, Spain

    Ana Romero

Authors
  1. Ana Romero
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Ana Romero.

Additional information

Partially supported by Ministerio de Educación y Ciencia, project MTM2009-13842-C02-01.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Romero, A. Computing the first stages of the Bousfield-Kan spectral sequence. AAECC 21, 227–248 (2010). https://doi.org/10.1007/s00200-010-0123-3

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00200-010-0123-3

Keywords

Mathematics Subject Classification (2000)