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Effective homological computations on finite topological spaces

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Abstract

The study of topological invariants of finite topological spaces is relevant because they can be used as models of a wide class of topological spaces, including regular CW-complexes. In this work, we present a new module for the Kenzo system that allows the computation of homology groups with generators of finite topological spaces in different situations. Our algorithms combine new constructive versions of well-known results about topological spaces with combinatorial methods used on finite spaces. In the particular case of h-regular spaces, effective and reasonably efficient methods are implemented and the technique of discrete vector fields is applied in order to improve the previous algorithms.

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References

  1. Alexandroff, P.: Diskrete Räume. Mat. Sb. (N.S.) 2, 501–518 (1937)

  2. Barmak, J.A.: Algebraic topology of finite topological spaces and applications. Lecture Notes in Mathematics 2032, (2011)

  3. Barmak, J.A., Minian, E.G.: Strong homotopy types, nerves and collapses. Discrete Comput. Geom. 47(2), 301–328 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cianci, N., Ottina, M.: A new spectral sequence for homology of posets. Topol. Appl. 217, 1–19 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cuevas Rozo J.L.: Funciones submodulares y matrices en el estudio de los espacios topológicos finitos, Maestria Thesis, Universidad Nacional de Colombia, Bogotá (2016)

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

  7. Fernández X.L.: https://github.com/ximenafernandez/Finite-Spaces (2017)

  8. Forman, R.: Morse theory for cell complexes. Adv. Math. 134, 90–145 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  9. Heras, J.: A Kenzo module computing simplicial complexes. http://www.unirioja.es/cu/joheras/Thesis/Chapter%205/Kenzo/Simplicial%20Complexes.rar (2011)

  10. Jahn M.W., Bradley P.E., Al-Doori M., Breunig M.: Topologically consistent models for efficient big geo-spatio-temporal data distribution, ISPRS Annals of the Photogrammetry, Remote Sensing and Spatial Information Sciences IV-4/W5 (2017)

  11. Kovalevsky, V.A.: Finite topology as applied to image analysis. Comput. Vis. Graphics Image Process. 46, 141–161 (1989)

    Article  Google Scholar 

  12. Liu G., Kitazawa M., Eguchi M., Fuwa Y., Nakamura, Y.: Dilation and reduction processing in finite topological spaces and its application to inspection of printed boards. Electron. Commun. Jpn. (Part III: Fundam. Electron. Sci.) 85(12), 89–100 (2002)

  13. McCord, M.C.: Singular homology groups and homotopy groups of finite topological spaces. Duke Math. J. 33(3), 465–474 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  14. Minian, E.G.: Some remarks on Morse theory for posets, homological Morse theory and finite manifolds. Topol. Appl. 159(12), 2860–2869 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  15. Mischaikow, K., Nanda, V.: Morse theory for filtrations and efficient computation of persistent homology. Discrete Comput. Geom. 50(2), 330–353 (2013)

  16. Mori, F., Salvetti, M.: (Discrete) Morse theory for Configuration spaces. Mathe. Res. Lett. 18(1), 39–57 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. Renders, J.: Finite topological spaces in algebraic topology, Project of Master of Science in Mathematics, Ghent University (2019)

  18. Romero, A., Sergeraert, F.: Discrete Vector Fields and Fundamental Algebraic Topology, EPrint: https://arxiv.org/pdf/1005.5685.pdf (2010)

  19. SageMath, the Sage Mathematics Software System (Version 8.9), The Sage Developers, https://www.sagemath.org (2019)

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

    Article  MathSciNet  MATH  Google Scholar 

  21. Shiraki, M.: On finite topological spaces. Rep. Fac. Sci. Kagoshima Univ. 1, 1–8 (1968)

    MathSciNet  MATH  Google Scholar 

  22. Stong, R.E.: Finite topological spaces. Trans. Am. Math. Soc. 123(2), 325–340 (1966)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work was partially supported by the Spanish Ministry of Science, Innovation and Universities, project MTM2017-88804-P, and the COLCIENCIAS Scholarship Program No. 757.

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Authors and Affiliations

  1. Departamento de Matemáticas, Universidad Nacional de Colombia, Bogotá, Colombia

    Julián Cuevas-Rozo & Humberto Sarria

  2. Departamento de Matemáticas y Computación, Universidad de La Rioja, Logroño, Spain

    Laureano Lambán & Ana Romero

Authors
  1. Julián Cuevas-Rozo
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  2. Laureano Lambán
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  3. Ana Romero
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  4. Humberto Sarria
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Correspondence to Julián Cuevas-Rozo.

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Cuevas-Rozo, J., Lambán, L., Romero, A. et al. Effective homological computations on finite topological spaces. AAECC 34, 33–56 (2023). https://doi.org/10.1007/s00200-020-00462-8

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  • DOI: https://doi.org/10.1007/s00200-020-00462-8

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