Homotopy groups of suspended classifying spaces: An experimental approach

Romero, Ana; Rubio, Julio

doi:10.1090/S0025-5718-2013-02680-4

Homotopy groups of suspended classifying spaces: An experimental approach
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by Ana Romero and Julio Rubio PDF

Math. Comp. 82 (2013), 2237-2244 Request permission

Abstract:

When the results of a computer program are compared to some theorems proved on a theoretical basis three situations can occur: there can be an agreement between both approaches, the computer program can obtain calculations not covered by the theorems, or a discrepancy can be found between both methods. In this paper we report on a work where the three above mentioned situations happen. We have enhanced the Computer Algebra called Kenzo to deal with the computation of homotopy groups of suspended classifying spaces, a problem tackled by Mikhailov and Wu in a paper published in the journal Algebraic and Geometric Topology. Our experimental approach, based on completely different methods from those by Mikhailov and Wu, has allowed us in particular to detect an error in one of their published theorems.

References

Jesús Aransay, Clemens Ballarin, and Julio Rubio, A mechanized proof of the basic perturbation lemma, J. Automat. Reason. 40 (2008), no. 4, 271–292. MR 2418280, DOI 10.1007/s10817-007-9094-x
Kenneth S. Brown, Cohomology of groups, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York-Berlin, 1982. MR 672956
César Domínguez and Julio Rubio, Effective homology of bicomplexes, formalized in Coq, Theoret. Comput. Sci. 412 (2011), no. 11, 962–970. MR 2796695, DOI 10.1016/j.tcs.2010.11.016
X. Dousson, J. Rubio, F. Sergeraert, and Y. Siret, The Kenzo program, Institut Fourier, Grenoble, 1999, http://www-fourier.ujf-grenoble.fr/~sergerar/Kenzo/.
G. Ellis, HAP – Homological Algebra Programming, a package for the GAP system, 2006, http://www.gap-system.org/Packages/hap.html.
Graham Ellis, James Harris, and Emil Sköldberg, Polytopal resolutions for finite groups, J. Reine Angew. Math. 598 (2006), 131–137. MR 2270569, DOI 10.1515/CRELLE.2006.071
Robin Forman, Morse theory for cell complexes, Adv. Math. 134 (1998), no. 1, 90–145. MR 1612391, DOI 10.1006/aima.1997.1650
J. Heras, Mathematical knowledge management in algebraic topology, Ph.D. thesis, Universidad de La Rioja, 2011, http://www.unirioja.es/cu/joheras/Thesis/.
J. Heras, V. Pascual, J. Rubio, and F. Sergeraert, fKenzo: a user interface for computations in algebraic topology, J. Symbolic Comput. 46 (2011), no. 6, 685–698. MR 2781947, DOI 10.1016/j.jsc.2011.01.005
J. Peter May, Simplicial objects in algebraic topology, Van Nostrand Mathematical Studies, No. 11, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967. MR 0222892
Roman Mikhailov and Jie Wu, On homotopy groups of the suspended classifying spaces, Algebr. Geom. Topol. 10 (2010), no. 1, 565–625. MR 2602844, DOI 10.2140/agt.2010.10.565
Ana Romero, Graham Ellis, and Julio Rubio, Interoperating between computer algebra systems: computing homology of groups with Kenzo and GAP, ISSAC 2009—Proceedings of the 2009 International Symposium on Symbolic and Algebraic Computation, ACM, New York, 2009, pp. 303–310. MR 2742718, DOI 10.1145/1576702.1576744
A. Romero and J. Rubio, Computing the homology of groups: the geometric way, Preprint. http://arxiv.org/abs/1107.3396, 2011.
A. Romero and F. Sergeraert, Discrete Vector Fields and fundamental Algebraic Topology, Preprint. http://arxiv.org/abs/1005.5685v1, 2010.
J. Rubio and F. Sergeraert, 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.
Ruben Jose Sanchez-Garcia, Equivariant K-homology of the classifying space for proper actions, ProQuest LLC, Ann Arbor, MI, 2005. Thesis (Ph.D.)–University of Southampton (United Kingdom). MR 2715961
M. Schönert et al., GAP – Groups, Algorithms, and Programming, Version 4.4.12, 2008, http://www.gap-system.org.
Francis Sergeraert, The computability problem in algebraic topology, Adv. Math. 104 (1994), no. 1, 1–29. MR 1272067, DOI 10.1006/aima.1994.1018
Christophe Soulé, The cohomology of $\textrm {SL}_{3}(\textbf {Z})$, Topology 17 (1978), no. 1, 1–22. MR 470141, DOI 10.1016/0040-9383(78)90009-5