Effective homological computations on finite topological spaces
Tóm tắt
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.
Tài liệu tham khảo
Alexandroff, P.: Diskrete Räume. Mat. Sb. (N.S.) 2, 501–518 (1937)
Barmak, J.A.: Algebraic topology of finite topological spaces and applications. Lecture Notes in Mathematics 2032, (2011)
Barmak, J.A., Minian, E.G.: Strong homotopy types, nerves and collapses. Discrete Comput. Geom. 47(2), 301–328 (2012)
Cianci, N., Ottina, M.: A new spectral sequence for homology of posets. Topol. Appl. 217, 1–19 (2017)
Cuevas Rozo J.L.: Funciones submodulares y matrices en el estudio de los espacios topológicos finitos, Maestria Thesis, Universidad Nacional de Colombia, Bogotá (2016)
Dousson X., Rubio J., Sergeraert, F., Siret, Y.: The Kenzo program, Institut Fourier, https://www-fourier.ujf-grenoble.fr/~sergerar/Kenzo/ (1999)
Fernández X.L.: https://github.com/ximenafernandez/Finite-Spaces (2017)
Forman, R.: Morse theory for cell complexes. Adv. Math. 134, 90–145 (1998)
Heras, J.: A Kenzo module computing simplicial complexes. http://www.unirioja.es/cu/joheras/Thesis/Chapter%205/Kenzo/Simplicial%20Complexes.rar (2011)
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)
Kovalevsky, V.A.: Finite topology as applied to image analysis. Comput. Vis. Graphics Image Process. 46, 141–161 (1989)
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)
McCord, M.C.: Singular homology groups and homotopy groups of finite topological spaces. Duke Math. J. 33(3), 465–474 (1966)
Minian, E.G.: Some remarks on Morse theory for posets, homological Morse theory and finite manifolds. Topol. Appl. 159(12), 2860–2869 (2012)
Mischaikow, K., Nanda, V.: Morse theory for filtrations and efficient computation of persistent homology. Discrete Comput. Geom. 50(2), 330–353 (2013)
Mori, F., Salvetti, M.: (Discrete) Morse theory for Configuration spaces. Mathe. Res. Lett. 18(1), 39–57 (2011)
Renders, J.: Finite topological spaces in algebraic topology, Project of Master of Science in Mathematics, Ghent University (2019)
Romero, A., Sergeraert, F.: Discrete Vector Fields and Fundamental Algebraic Topology, EPrint: https://arxiv.org/pdf/1005.5685.pdf (2010)
SageMath, the Sage Mathematics Software System (Version 8.9), The Sage Developers, https://www.sagemath.org (2019)
Rubio, J., Sergeraert, F.: Constructive algebraic topology. Bull. Sci. Math. 126(5), 389–412 (2002)
Shiraki, M.: On finite topological spaces. Rep. Fac. Sci. Kagoshima Univ. 1, 1–8 (1968)
Stong, R.E.: Finite topological spaces. Trans. Am. Math. Soc. 123(2), 325–340 (1966)