Subdivision-based homotopy equivalence of digital circles

• Ashraf, Samia ^[1] ; Akram, Madiha ^[1] ; Ali, Amna Amanat ^[1]
  1. [1] Government College Women University Sialkot
• Localización: Applied general topology, ISSN-e 1989-4147, ISSN 1576-9402, Vol. 26, Nº. 1, 2025, págs. 431-445
• Idioma: inglés
• DOI: 10.4995/agt.2025.22288
• Enlaces
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• Resumen

  • A direct translation of the notion of homotopy equivalence from algebraic topology to digital images leads to a much more rigid definition in the context of digital topology. This results in two digital circles of different radii being not homotopic. A more suitable idea of equivalences, called "subdivision-based homotopy equivalence" which is introduced by Lupton et al., involves the concept of subdivision of digital images. We prove that, in this sense, any digital circle is subdivision-based homotopic to the Diamond, the prototypical digital circle of radius 1, which implies, as a consequence, that the fundamental group of any digital circle is isomorphic to Z.

• Referencias bibliográficas
  • L. Boxer, Multivalued functions in digital topology, Note di Mathematica, 37, no. 2 (2017), 61-76.
  • L. Boxer, Properties of digital homotopy, J. Math Imaging Vis. 22, no. 1 (2005), 19-26.
  • https://doi.org/10.1007/s10851-005-4780-y
  • L. Boxer, A classical construction for the digital fundamental group, J. Math Imaging Vis. 10, no. 1 (1999), 51-62.
  • L. Boxer and Staecker, Remarks on pointed digital homotopy, Topology Proceedings 51 (2018), 19-37.
  • C. Escribano, A. Giraldo, and M. Sastre, Digitally continuous multivalued functions, Discrete Geometry for Computer Imagery, Lecture Notes...
  • https://doi.org/10.1007/978-3-540-79126-3_9
  • T. Y. Kong, A digital fundamental group, Computers and Graphics 13, no. 2 (1989), 159-166.
  • https://doi.org/10.1016/0097-8493(89)90058-7
  • G. Lupton, J. Oprea, and N. A. Scoville, Subdivision of maps of digital images, Discrete Comput. Geom. 67 (2022), 698-742.
  • https://doi.org/10.1007/s00454-021-00350-z
  • G. Lupton and N. A. Scoville, Digital fundamental groups and edge groups of clique complexes, J. Appl. and Comput. Topology 6 (2022), 529-558.
  • https://doi.org/10.1007/s41468-022-00095-5
  • G. Lupton, J. Oprea, and N. A. Scoville, Homotopy theory in digital topology, Discrete and Comput. Geom. 67, no. 1 (2022), 112-165.
  • https://doi.org/10.1007/s00454-021-00336-x
  • G. Lupton, J. Oprea, and N. A. Scoville, A fundamental group for digital images, J. Appl. and Comput. Topology 5, no. 2 (2021), 249-311.
  • https://doi.org/10.1007/s41468-021-00067-1
  • A. Rosenfeld, "Continuous" functions on digital pictures, Pattern Recognit. Lett. 4, no. 3 (1986), 177-184.
  • https://doi.org/10.1016/0167-8655(86)90017-6