Abstract
Homological Perturbation Theory [11,13] is a well-known general method for computing homology, but its main algorithm, the Basic Perturbation Lemma, presents, in general, high computational costs. In this paper, we propose a general strategy in order to reduce the complexity in some important formulas (those following a specific pattern) obtained by this algorithm. Then, we show two examples of application of this methodology.
Partially supported by the PAICYT research project FQM-296 and the UPV–EHU project 00127.310-E-15916.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Álvarez, V., Armario, A., Real, P., Silva, B.: HPT and computability of Hochschild and cyclic homologies of commutative DGA–algebras. In: Conference on Secondary Calculus and Cohomological Physics, EMIS Electronic Proceedings, Moscow (1997), http://www.emis.de/proceedings
Brown, R.: The twisted Eilenberg-Zilber theorem, Celebrazioni Archimedae del Secolo XX, Simposio di Topologia, pp. 34–37 (1967)
Burghelea, D., Vigué Poirrier, M.: Cyclic homology of commutative algebras I, Algebraic Topology Rational Homotopy. Lecture Notes in Mathematics, vol. 1318, pp. 51–72 (1986)
C.H.A.T.A. group. Computing “small” 1–homological models for Commutative Differential Graded Algebras. In: Proc. Third Workshop on Computer Algebra in Scientific Computing, pp. 87–100. Springer, Heidelberg (2000)
Cartan, H., Eilenberg, S.: Homological Algebra. Princeton Univ. Press, Princeton (1956)
Eilenberg, S., MacLane, S.: On the groups H(π,n), I. Annals of Math. 58, 55–106 (1953)
Eilenberg, S., MacLane, S.: On the groups H(π,n), II. Annals of Math. 60, 49–139 (1954)
Gugenheim, V.K.A.M.: On the chain complex of a fibration. Illinois J. Math. 3, 398–414 (1972)
Gugenheim, V.K.A.M.: On Chen’s iterated integrals. Illinois J. Math., 703–715 (1977)
Gugenheim, V.K.A.M., Lambe, L.: Perturbation theory in Differential Homological Algebra, I. Illinois J. Math. 33, 56–582 (1989)
Gugenheim, V.K.A.M., Lambe, L., Stasheff, J.: Perturbation theory in differential homological algebra, II. Illinois J. Math. 35(3), 357–373 (1991)
Gugenheim, V.K.A.M., Stasheff, J.: On perturbations and A ∞ –structures. Bull. Soc. Math. Belg. 38, 237–246 (1986)
Huebschmann, J., Kadeishvili, T.: Small models for chain algebras. Math. Zeit. 207, 245–280 (1991)
Jiménez, M.J., Real, P.: “Coalgebra” structures on 1–homological models for commutative differential graded algebras. In: Proc. Fourth Workshop on Computer Algebra in Scientific Computing, pp. 347–361. Springer, Heidelberg (2001)
Lambe, L.A., Stasheff, J.: Applications of perturbation theory to iterated fibrations. Manuscripta Math. 58, 363–376 (1987)
MacLane, S.: Homology, Classics in Mathematics. Springer, Berlin (1995); Reprint of the 1975 edition
Real, P.: Homological perturbation theory and associativity. Homology, Homotopy and Applications 2(5), 51–88 (2000)
Shih, W.: Homologie des espaces fibrés. Inst. Hautes Etudes Sci. 13, 93–176 (1962)
Weibel, C.A.: An introduction to Homological Algebra. Cambridge studies in advanced mathematics, vol. 38. Cambridge University Press, Cambridge (1994)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Berciano, A., Jiménez, M.J., Real, P. (2006). Reducing Computational Costs in the Basic Perturbation Lemma. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2006. Lecture Notes in Computer Science, vol 4194. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11870814_3
Download citation
DOI: https://doi.org/10.1007/11870814_3
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-45182-2
Online ISBN: 978-3-540-45195-2
eBook Packages: Computer ScienceComputer Science (R0)