Email this article Homologies of transitive digraphs and discrete spaces
- Authors: Muranov Y.V.1, Jimenez R.B.2
-
Affiliations:
- University of Warmia and Mazury in Olsztyn
- National Autonomous University of Mexico, Institute of Mathematics
- Issue: Vol 214, No 8 (2023)
- Pages: 74-93
- Section: Articles
- URL: https://ogarev-online.ru/0368-8666/article/view/133544
- DOI: https://doi.org/10.4213/sm9842
- ID: 133544
Cite item
Open Access
Access granted
Subscription Access
Abstract
We prove that for transitive digraphs path homology, and therefore also Alexandroff homology, coincides with singular cubic homology. Also, discrete topological spaces are defined that are natural analogues of standard topological cubes. Using them, the singular cubic homology of discrete topological spaces is defined, and it is proved that these homology groups coincide with the Alexandroff homology groups.
About the authors
Yury Vladimirovich Muranov
University of Warmia and Mazury in Olsztyn
Email: ymuranov@mail.ru
Doctor of physico-mathematical sciences, Professor
Rolando Benitez Jimenez
National Autonomous University of Mexico, Institute of Mathematics
Author for correspondence.
Email: ymuranov@mail.ru
Candidate of physico-mathematical sciences
References
- A. Connes, “Non-commutative differential geometry”, Inst. Hautes Etudes Sci. Publ. Math., 62 (1985), 41–144
- A. Dimakis, F. Müller-Hoissen, “Discrete differential calculus: graphs, topologies, and gauge theory”, J. Math. Phys., 35:12 (1994), 6703–6735
- A. Dimakis, F. Müller-Hoissen, F. Vanderseypen, “Discrete differential manifolds and dynamics on networks”, J. Math. Phys., 36:7 (1995), 3771–3791
- A. Grigor'yan, Yong Lin, Yu. Muranov, Shing-Tung Yau, “Cohomology of digraphs and (undirected) graphs”, Asian J. Math., 19:5 (2015), 887–932
- G. Hochschild, “On the cohomology groups of an associative algebra”, Ann. of Math. (2), 46:1 (1945), 58–67
- M. Gerstenhaber, S. D. Schack, “Simplicial cohomology is Hochschild cohomology”, J. Pure Appl. Algebra, 30:2 (1983), 143–156
- A. Grigor'yan, Yu. Muranov, Shing-Tung Yau, “On a cohomology of digraphs and Hochschild cohomology”, J. Homotopy Relat. Struct., 11:2 (2016), 209–230
- A. Grigor'yan, Y. V. Muranov, Shing-Tung Yau, “Graphs associated with simplicial complexes”, Homology Homotopy Appl., 16:1 (2014), 295–311
- А. А. Григорьян, Йонг Лин, Ю. В. Муранов, Шинтан Яу, “Комплексы путей и их гомологии”, Фундамент. и прикл. матем., 21:5 (2016), 79–128
- A. Grigor'yan, Yong Lin, Yu. Muranov, Shing-Tung Yau, “Homotopy theory for digraphs”, Pure Appl. Math. Q., 10:4 (2014), 619–674
- A.A. Grigor'yan, R. Jimenez, Y. Muranov, Shing-Tung Yau, “On the path homology theory of digraphs and Eilenberg–Steenrod axioms”, Homology Homotopy Appl., 20:2 (2018), 179–205
- А. А. Григорьян, Ю. В. Муранов, Р. Хименес, “Гомологии орграфов”, Матем. заметки, 109:5 (2021), 705–722
- Ю. В. Муранов, “Гомологии Александрова и гомологии путей”, Матем. заметки, 112:1 (2022), 148–152
- P. Alexandroff, “Discrete Räume”, Матем. сб., 2(44):3 (1937), 501–519
- П. С. Александров, “О понятии пространства в топологии”, УМН, 2:1(17) (1947), 5–57
- J. W. Evans, F. Harary, M. S. Lynn, “On the computer enumeration of finite topologies”, Comm. ACM, 10:5 (1967), 295–297
- C. Marijuan, “Finite topologies and digraphs”, Proyecciones, 29:3 (2010), 291–307
- M. C. McCord, “Singular homology groups and homotopy groups of finite topological spaces”, Duke Math. J., 33:3 (1966), 465–474
- R. E. Stong, “Finite topological spaces”, Trans. Amer. Math. Soc., 123 (1966), 325–340
- M. L. Wachs, “Poset topology: tools and applications”, Geometric combinatorics, IAS/Park City Math. Ser., 13, Amer. Math. Soc., Providence, RI, 2007, 497–615
- J. A. Barmak, Algebraic topology on finite topological spaces and applications, Lecture Notes in Math., 2032, Springer, Berlin, 2011, xviii+170 pp.
- P. J Hilton, S. Wylie, Homology theory. An introduction to algebraic topology, Cambridge Univ. Press, New York, 1960, xv+484 pp.
- А. Хатчер, Алгебраическая топология, МЦНМО, М., 2011, 688 с.
- В. В. Прасолов, Элементы теории гомологий, МЦНМО, М., 2006, 448 с.
Supplementary files
