Discrete topological complexities of simplical maps
محورهای موضوعی : Algebraic topology
1 - Department of Mathematics, Faculty of Sciences, Ege University, Izmir, Turkiye
2 - Department of Mathematics, Faculty of Sciences, Ege University, Izmir, Turkiye
کلید واژه: Discrete topological complexity, higher topological complexity, simplicial LS-category, contiguity distance, simplicial fibration,
چکیده مقاله :
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.
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.
[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.
