Discrete topological complexities of simplical maps

Subject Areas : Algebraic topology

M. İs ¹ , I. Karaca ²

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Received: 2024-05-25 Accepted : 2024-09-23 Published : 2024-09-26

Keywords: Discrete topological complexity‎, ‎higher topological complexity‎, ‎simplicial LS-category‎, ‎contiguity distance‎, ‎simplicial fibration,

Abstract :

‎In this study‎, ‎we delve into the discrete TC of surjective simplicial fibrations‎, ‎aiming to unravel the interplay between topological complexity‎, ‎discrete geometric structures‎, ‎and computational efficiency‎. ‎Moreover‎, ‎we examine the properties of the discrete TC number in higher dimensions and its relationship with scat‎. ‎We also touch on the basic properties of the notion of higher contiguity distance and show that it is possible to consider discrete TC computations in a simpler sense‎.

References:

[1] S. Aaronson, N. A. Scoville, Lusternik-Schnirelmann category for simplicial complexes, Illinois J. Math. 57 (3) (2013), 743-753.

[2] S. A. Aghili, H. Mirebrahimi, A. Babaee, On the targeted complexity of a map, Hacet. J. Math. Stat. 52 (3) (2023), 572-584.

[3] H. Alabay, A. Borat, E. Cihangirli, E. Dirican Erdal, Higher analogues of discrete topological complexity, Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. (2024), 118:125.

[4] J. A. Barmak, E. G. Minian, Strong homotopy types, nerves and collapses, Discrete Comput. Geom. 47 (2) (2012), 301-328.

[5] A. Borat, M. Pamuk, T. Vergili, Contiguity distance between simplicial maps, Turk. J. Math. 47 (2023), 664-677.

[6] O. Cornea, G. Lupton, J. Oprea, D. Tanre, Lusternik-Schnirelmann Category, Mathematical Surveys and Monographs, Vol. 103, AMS, 2003.

[7] D. Fernandez-Ternero, J. M. Garcia-Calcines, E. Macias-Virgos, J. A. Vilches, Simplicial fibrations, Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 115 (2) (2021), 1-25.

[8] D. Fernandez-Ternero, E. Macias-Virgos, E. Minuz, J. A. Vilches, Discrete topological complexity, Proc. Amer. Math. Soc. 146 (10) (2018), 4535-4548.

[9] D. Fernandez-Ternero, E. Macias-Virgos, E. Minuz, J. A. Vilches, Simplicial Lusternik-Schnirelmann category, Publ. Mat. 63 (1) (2019),265-293.

[10] J. Gonzalez, Simplicial complexity: piecewise linear motion planning in robotics, New York J. Math. 24 (2018), 279-292.

[11] M. Is, I. Karaca, Digital Topological Complexity of Digital Maps, arXiv:2103.00585, 2021.

[12] M. Is, I. Karaca, Higher topological complexity for fibrations, Filomat. 36 (20) (2022), 6885-6896.

[13] D. N. Kozlov, Combinatorial Algebraic Topology, Algorithms and Computation in Mathematics 21, Springer, Berlin, 2008.

[14] E. Macias-Virgos, D. Mosquera-Lois, Homotopic distance between maps, Math. Proc. Camb. Philos. Soc. 172 (1) (2022), 73-93.

[15] J. J. Rotman, An Introduction to Algebraic Topology, Graduate Texts in Mathematics, Vol. 119, Springer, New York, 2013.

[16] N. A. Scoville, W. Swei, On the Lusternik-Schnirelmann category of a simplicial map, Topol. Appl. 216 (2017), 116-128.

[17] E. H. Spanier, Algebraic Topology, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Company, 1966.