Publication: A combinatorial description of shape theory
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Abstract
We give a combinatorial description of shape theory using finite topological T0-spaces (finite partially ordered sets). This description may lead to a sort of computational shape
theory. Then we introduce the notion of core for inverse sequences of finite spaces and prove some properties.
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[1] P.S. Alexandroff. Diskrete R¨aume. Mathematiceskii Sbornik (N.S.), 2(3):501–519, 1937.
[2] M. Alonso-Morón and A. González Gómez. Homotopical properties of upper semifinite
hyperspaces of compacta. Topol. App., 155(9):927–981, 2008.
[3] M.A. Alonso-Morón, E. Cuchillo-Ibánez, and A. Luzón. �-connectedness, finite approxi�mations, shape theory and coarse graining in hyperspaces. Phys. D, 237(23):3109–3122,
2008.
[4] J. A. Barmak. Algebraic topology of finite topological spaces and applications. LecturevNotes in Mathematics, volume 2032. Springer, 2011.
[5] J. A. Barmak, M. Mrozek, and T. Wanner. A Lefschetz fixed point theorem for multi�valued maps of finite spaces. Math. Z., 294:1477–1497, 2020.
[6] P. Bilski. On the inverse limits of T0-Alexandroff spaces. Glas. Mat., 52(2):207–219,
2017.
[7] K. Borsuk. Theory of Shape, volume Lectures Notes Series 28. Matematisk Inst. Aarhus
Univ., 1971.
[8] G. Carlsson. Topology and data. AMS Bulletin, 46(2):255–308, 2009.
[9] P. J. Chocano, M. A. Morón, and F. R. Ruiz del Portal. Coincidence theorems for finite
topological spaces. arXiv:2010.12804, 2020.
[10] P. J. Chocano, M. A. Morón, and F. R. Ruiz del Portal. Shape of compacta as extension
of weak homotopy of finite spaces. arXiv:2110.02574, 2021.
[11] P. J. Chocano, M. A. Morón, and F. R. Ruiz del Portal. Computational approximations
of compact metric spaces. Phys. D, 433(133168), 2022.
[12] E. Clader. Inverse limits of finite topological spaces. Homol. Homotop. App., 11(2):223–
227, 2009.
[13] C. Conley. Isolated Invariant Sets and the Morse Index. volume 38 of CBMS Regional
Conference Series in Mathematics. American Mathematical Society, Providence, R.I.,
1978.
[14] J.M. Cordier and T. Porter. Shape theory: Categorical mehtods of approximation. Hor�wood Series: Mathematics and Its Applications, 1989.
[15] J. Curry, R. Ghrist, and V. Nanda. Discrete Morse Theory for Computing Cellular Sheaf
Cohomology. Found. Comput. Math., 16:875–897, 2015.
[16] T. Dey, M. Juda, T. Kapela, J. Kubica, M. Lipi´nski, and M. Mrozek. Persistent Homol�ogy of Morse Decompositions in Combinatorial Dynamics. SIAM J. Appl. Dyn. Syst.,
18(1):510–530, 2019.
[17] T. Dey, M. Mrozek, and R. Slechta. Persistence of the Conley index in combinatorial
dynamical systems. Proceedings of the 36th International Symposium on Computa- tional
Geometry, pages 37:1–37:17, 2020.
[18] T. Dey, M. Mrozek, and R. Slechta. Persistence of Conley-Morse Graphs in Combinatorial
Dynamical Systems. Preprint. arXiv:2107.02115, 2021.
[19] H. Edelsbrunner and J.L. Harer. Computational Topology: An Introduction. American
Mathematical Society, 2008.
[20] D. Fern´andez-Ternero, E. Mac´ıas-Virg´os, D. Mosquera-Lois, and J.A. Vilches. Morse�Bott Theory on posets and a homological Lusternik-Schnirelmann Theorem. Journal of
Topology and Analysis, 0(0):1–24, 2021.
[21] R. Forman. Combinatorial vector fields and dynamical systems. Math. Z., 228:629–681,
1998.
[22] A. Giraldo, M. A. Mor´on, F. R. Ruiz del Portal, and J. M. R Sanjurjo. Finite approxi�mations to ˇcech homology. J. Pure Appl. Algebra, 163:81–92, 2001.
[23] A. Giraldo and J. M. R. Sanjurjo. Density and finiteness. A discrete approach to shape.
Topol. App., 76:61–77, 1997.
[24] S. Harker, K. Mischaikow, M. Mrozek, and V. Nanda. Discrete Morse Theoretic Al�gorithms for Computing Homology of Complexes and Maps. Found. Comput. Math.,
14:151–184, 2013.
[25] A. Katok and Hasselblatt B. Introduction to the Modern Theory of Dynamical Systems.
Cambridge University Press, 1995.
[26] S. Lefschetz. On the fixed point formula. Ann. of Math., 38(4):819–822, 1937.
[27] M. Lipi´nski, J. Kubica, M. Mrozek, and T. Wanner. Conley–Morse–Forman the�ory for generalized combinatorial multivector fields on finite topological spaces.
arXiv:1911.12698, 2019.
[28] S. Mardeˇsi´c and J. Segal. Shape theory: the inverse system approach. North-Holland
Mathematical Library, 1982.
[29] J. P. May. Finite spaces and larger contexts. Unpublished book, 2016.
[30] M. C. McCord. Singular homology groups and homotopy groups of finite topological
spaces. Duke Math. J., 33(3):465–474, 1966.
[31] J. Milnor. Morse theory. Based on lectures notes by M. Spivak and R. Wells. Annals
of Mathematics Studies, No. 51. Princeton University Press, Princeto, N.J., New York�Berlin, 1963.
[32] E.G. Minian. Some remarks on Morse theory for posets, homological Morse theory and finite manifolds. Topol. App., 159(12):2860–2869, 2012.
[33] D. Mond´ejar Ruiz. Hyperspaces, Shape Theory and Computational Topology. PhD thesis,
Universidad Complutense de Madrid, 2015.
[34] K. Morita. The Hurewicz isomorphism theorem on homotopy and homology pro-groups.
Proc. Japan Acad., 50(7):453–457, 1974.
[35] M. Mrozek. Leray functor and cohomological Conley index for discrete dynamical sys�tems. Trans. Amer. Math. Soc., 318(1):149–178, 1990.
[36] S.B. Nadler. Hyperspaces of Sets: A Text with Research Questions. M. Dekker, 1978.
[37] N. Otter, A.M. Porter, U. Tillmann, and H.A. Harrington. A roadmap for the computa�tion of persistent homology. EPJ Data Sci., 6(17):101–128, 1978.
[38] A. Radunskaya and T. (eds) Jackson. Applications of dynamial systems in biology and
medicine, volume 158. Springer, 2015.
[39] J.W. Robbin and D. Salamon. Dynamical systems, shape theory and the conley index.
Ergod. Th. and Dynam. Sys., 8:375–393, 1988.
[40] J.J. S´anchez-Gabites. Dynamical systems and shapes. Rev. R. Acad. Cien. Serie A.
Mat., 102:127–159, 2008.
[41] J. M. R. Sanjurjo. An Intrinsic Description of Shape. Trans. Amer. Math. Soc., 329(2):625–636, 1992.
[42] S.H. Strogatz. Nonlinear Dynamics and Chaos: With Applications to Physics, Biology,Chemistry, and Engineering. 2nd ed. Westview Press, 2015.
[43] A. Szymczak. The Conley index for discrete semidynamical systems. Topol. App., 66:215–245, 1995