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.
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References
Adams J.F.: On the Cobar construction. Proc. Natl. Acad. Sci. USA 42, 409–412 (1956)
Adams J.F.: On the non-existence of elements of Hopf invariant one. Ann. Math. 72(1), 20–104 (1960)
Baues, H.J.: Geometry of loop spaces and the cobar construction. Mem. Am. Math. Soc. 230 (1980)
Bousfield A.K., Kan D.M.: Homotopy Limits, Completions and Localizations, Lecture Notes in Mathematics. Springer, Berlin (1972)
Bousfield A.K., Kan D.M.: The homotopy spectral sequence of a space with coefficients in a ring. Topology 11, 79–106 (1972)
Bousfield A.K., Kan D.M.: A second quadrant homotopy spectral sequence. Trans. Am. Math. Soc. 177, 305–318 (1973)
Cartan, H.: Algèbres d’Eilenberg-MacLane et homotopie. Séminaire H. Cartan, École Normal Supérieure, Paris (1954–55). Exposés 2–11
Dold A.: Lectures on Algebraic Topology. Springer, Berlin (1972)
Dousson, X., Rubio, J., Sergeraert, F., Siret, Y.: The Kenzo program. Institut Fourier, Grenoble (1999). http://www-fourier.ujf-grenoble.fr/~sergerar/Kenzo/
Eilenberg S., Zilber J.A.: On products of complexes. Am. J. Math. 75, 200–204 (1953)
Kaczynski T., Mischaikow K., Mrozek M.: Computational Homology, Applied Mathematical Sciences. Springer, New York (2004)
MacLane S.: Homology, Grundleren der Mathematischen Wissenschaften. Springer, Berlin (1963)
May J.P.: Simplicial Objects in Algebraic Topology, Van Nostrand Mathematical Studies. Van Nostrand, Princeton (1967)
McCleary J.: User’s Guide to Spectral Sequences. Publish or Perish, Wilmington (1985)
Romero, A.: Effective homology and spectral sequences. Ph.D. thesis, Universidad de La Rioja (2007)
Romero A., Rubio J., Sergeraert F.: Computing spectral sequences. J. Symb. Comput. 41(10), 1059–1079 (2006)
Rubio J., Sergeraert F.: Constructive algebraic topology. Bull. Sci. Math. 126(5), 389–412 (2002)
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
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Partially supported by Ministerio de Educación y Ciencia, project MTM2009-13842-C02-01.
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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
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DOI: https://doi.org/10.1007/s00200-010-0123-3
Keywords
- Constructive algebraic topology
- Bousfield-Kan spectral sequence
- Computation of homotopy groups
- Symbolic computation