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Programming before theorizing, a case study

Authors:

Francis SergeraertAuthors Info & Claims

ISSAC '12: Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation

Pages 289 - 296

https://doi.org/10.1145/2442829.2442871

Published: 22 July 2012 Publication History

Abstract

This paper relates how a "simple" result in combinatorial homotopy eventually led to a totally new understanding of basic theorems in Algebraic Topology, namely the Eilenberg-Zilber theorem, the twisted Eilenberg-Zilber theorem, and finally the Eilenberg-MacLane correspondance between the Classifying Space and Bar constructions. In the last case, it was an amazing lucky consequence of computations based on conjectures not yet proved. The key new tool used in this context is Robin Forman's Discrete Vector Fields theory.

References

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M. G. Barratt, V. K. A. M. Gugenheim, and J. C. Moore. On Semi-Simplicial Fibre Bundles. American Journal of Mathematics, 81:639,657, 1959.

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H. J. Baues. Geometry of loop spaces and the cobar construction, volume 230 of Memoirs of the American Mathematical Society. AMS, 1980.

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A. Berciano, J. Rubio, and F. Sergeraert. A case study of A _∞-structure. Georgian Mathematical Journal, 17:57--77, 2010.

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R. Brown. The twisted Eilenberg-Zilber theorem. In Celebrazioni Arch. Secolo XX, Simp. Top., pages 34--37, 1967.

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E. Brown Jr. Finite computability of Postnikov complexes. Annals of Mathematics, 65:1--20, 1957.

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E. Brown Jr. Twisted tensor products, I. Annals of Mathematics, 69:223--246, 1959.

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X. Dousson, J. Rubio, F. Sergeraert, and Y. Siret. The Kenzo program. www-fourier.ujf-grenoble.fr/~sergerar/Kenzo/.

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S. Eilenberg and S. MacLane. On the groups H(π, n), I. Annals of Mathematics, 58:55--106, 1953.

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S. Eilenberg and J. A. Zilber. Semi-simplicial complexes and singular homology. Annals of Mathematics, 51:499--513, 1950.

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R. Forman. Morse theory for cell complexes. Advances in Mathematics, 134:90--145, 1998.

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J. Milnor. Morse Theory, volume 51 of Annals of Mathematical Studies. Princeton University Press, 1963.

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P. Real. Algoritmos de Cálculo de Homología Efectiva de los Espacios Clasificantes. Thesis Memoir, Sevilla, 1993. http://fondosdigitales.us.es/media/thesis/1426/C_043-139.pdf.

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A. Romero and F. Sergeraert. Discrete Vector Fields and Fundamental Algebraic Topology. arxiv.org/abs/1005.5685/.

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J. Rubio and F. Sergeraert. Constructive Algebraic Topology. Bulletin des Sciences Mathématiques, 126:389--412, 2002.

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S. Saneblidze and R. Umble. Diagonals on the Permutahedra, Multiplihedra and Associahedra. Homology, Homotopy and Applications, 6:363--411, 2004.

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R. Schön. Effective algebraic topology, volume 451 of Memoirs of the American Mathematical Society. AMS, 1991.

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F. Sergeraert. The computability problem in algebraic topology. Advances in Mathematics, 104:1--29, 1994.

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W. Shih. Homologie des espaces fibrés. Publications Mathématiques de l'IHES, 13, 1962.

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Cited By

Sergeraert F(2018)The homological hexagonal lemmaGeorgian Mathematical Journal10.1515/gmj-2018-005525:4(603-622)Online publication date: 20-Sep-2018
https://doi.org/10.1515/gmj-2018-0055
Romero ASergeraert F(2018)A Combinatorial Tool for Computing the Effective Homotopy of Iterated Loop SpacesDiscrete & Computational Geometry10.1007/s00454-014-9650-153:1(1-15)Online publication date: 31-Dec-2018
https://dl.acm.org/doi/10.1007/s00454-014-9650-1
Romero ASergeraert F(2017)A Bousfield---Kan Algorithm for Computing the Effective Homotopy of a SpaceFoundations of Computational Mathematics10.1007/s10208-016-9322-z17:5(1335-1366)Online publication date: 1-Oct-2017
https://dl.acm.org/doi/10.1007/s10208-016-9322-z

Index Terms

Programming before theorizing, a case study
1. Mathematics of computing
  1. Mathematical software

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Published In

cover image ACM Other conferences

ISSAC '12: Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation

July 2012

390 pages

ISBN:9781450312691

DOI:10.1145/2442829

General Chair:
Joris van der Hoeven
École Polytechnique, France
,
Program Chair:
Mark van Hoeij
Florida State U.

Copyright © 2012 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Grenoble University: Grenoble University
INRIA: Institut Natl de Recherche en Info et en Automatique

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SIGSAM: ACM Special Interest Group on Symbolic and Algebraic Manipulation

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 22 July 2012

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Ministerio de Ciencia e Innovación

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ISSAC'12

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Grenoble University
INRIA

ISSAC'12: International Symposium on Symbolic and Algebraic Computation

July 22 - 25, 2012

Grenoble, France

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ISSAC '12 Paper Acceptance Rate 46 of 86 submissions, 53%;

Overall Acceptance Rate 395 of 838 submissions, 47%

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Cited By

Sergeraert F(2018)The homological hexagonal lemmaGeorgian Mathematical Journal10.1515/gmj-2018-005525:4(603-622)Online publication date: 20-Sep-2018
https://doi.org/10.1515/gmj-2018-0055
Romero ASergeraert F(2018)A Combinatorial Tool for Computing the Effective Homotopy of Iterated Loop SpacesDiscrete & Computational Geometry10.1007/s00454-014-9650-153:1(1-15)Online publication date: 31-Dec-2018
https://dl.acm.org/doi/10.1007/s00454-014-9650-1
Romero ASergeraert F(2017)A Bousfield---Kan Algorithm for Computing the Effective Homotopy of a SpaceFoundations of Computational Mathematics10.1007/s10208-016-9322-z17:5(1335-1366)Online publication date: 1-Oct-2017
https://dl.acm.org/doi/10.1007/s10208-016-9322-z

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