Abstract complexes, their homologies and applications

Cataranciuc Sergiu ; Soltan Petru

Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 2 (2010), pp. 31-58.

Résumé

The complex of multi-ary relations ${\mathcal{K}}^n$ is defined in a more natural way than it was defined in [18, 58, 59]. The groups of homologies and co-homologies of this complex over the group of integer numbers are constructed. The methods used for these constructions are for the most part analogous with classical methods [2, 32, 52], but sometimes they are based on methods from [18, 44, 58]. The importance and originality consist in application of the multi-ary relations of a set of objects in construction of homologies. This allows to extend areas of theoretical researches and non-trivial practical applications in a lot of directions. Other abstract structures, which are developed in a natural way from generalized complex of multi-ary relations are also examined. New notions such as the notions of abstract quasi-simplex and its homologies, the complex of abstract simplexes and the complex of the $n$-dimensional abstract cubes are introduced.

Bibliographie

[1] Alexandroff P. (Aleksandrov P. S.), Hopf H., Topologie, v. I, Springer, Berlin, 1935 | Zbl

[2] Aleksandrov P. S., Introduction to Homological dimension theory and general combinatorial topology, Moscow, 1975 (in Russian) | MR

[3] Baranyai Z., “On the factorization of the complete uniform hypergraph”, Infinite and finite sets, Proceeding of a Colloquium Dedicated to Paul Erdos on his 60th Birthday (Keszthely, June 25 – July 1, 1973), v. 1, eds. A. Hajnal, R. Rado, V. T. Sos, North Holland, Amsterdam, 1975, 91–108 | MR

[4] Bauese H. J., Combinatorial foundation of homology and homotopy, Springer, Berlin, 1999 | MR

[5] Berge C., Graphs and hypergraphs, North-Holland, Amsterdam, 1973 | MR | Zbl

[6] Berge G., Hypergraphs: combinatorics of finite sets, North Holland, Amsterdam, 1989 | MR | Zbl

[7] Berge C., Theorie des graphes et ses applications, DUNOD, Paris, 1958 | MR

[8] Björner A., “Topological methods”, Handbook of Combinatorics, eds. R. Grahfm, M. Grotschel, L. Lovasz, North Holland, Amsterdam, 1995, 1819–1872 | MR | Zbl

[9] Björner A., Ziegler G. M., “Introduction to Greedoids”, Matroid applications, ed. N. White, Cambridge Univ. Press, London, 1992, 284–357 | DOI | MR

[10] Björner A., Las Vergnas M., Sturmfels B., White N., Ziegler G. M., Oriented Matroids, Encyclopedia of Mathematics and its Applications, 46, Cambridge Univ. Press, London, 1993 | MR | Zbl

[11] Bollobas B., “On generalized graphs”, Acta Math. Acad. Sci. Hung., 16 (1965), 447–452 | DOI | MR | Zbl

[12] Bollobas B., Modern Graph Theory, Graduate text in Mathematics, 184, Springer, New-York, USA, 1998 | DOI | MR | Zbl

[13] Boltyanski V., The homotopic theory of continuous images and vector fields, Trudy Matematicheskogo instituta “V. A. Steklov” AN SSSR, 47, Moscow, 1955, 199 pp. (in Russian) | MR | Zbl

[14] Boltyanski V., Martini H., Soltan P. S., Excursions into Combinatorial Geometry, Springer, Berlin, 1997 | MR | Zbl

[15] Bujac M., “Classification of abstract multidimensional orientable manifolds without borders”, Analele Stiintifice ale USM. Seria “Stiinte fizico-matematice”, Chişinău, 2003, 247–250 (in Romanian)

[16] Cartan H., Eilenberg S., Homological Algebra, Princeton University Press, 1956 | MR | Zbl

[17] Cataranciuc S., “$\mathcal G$-complex of multi-ary relations”, Analele ştiinţifice ale USM, Seria “Ştiinţe fizico-matematice”, Chişinău, 2006, 119–122 (in Romanian)

[18] Cataranciuc S., Soltan P., “Hypergraphs and their homology”, Trends in the Development of the Information and Communication Technology in Education and Management, International Conference (March 20–21, 2003, Chişinău), 294–300 (in Romanian)

[19] Cataranciuc S., Bujac M., Soltan P., “On the Division of Abstract Manifolds in Cubes”, Buletinul Academiei de Ştiinţe a Republicii Moldova, Matematica, 2006, no. 2(51), 29–34 | MR | Zbl

[20] Cataranciuc S., Scripnic M., Soltan P., “About the median does not depend on the space metric”, The 30-th Annual Congres of the American Romanian Academy of Arts and Sciences (ARA), Proceedings (Chişinău, July 5–10, 2005), 58–61

[21] Cataranciuc S., Scripnic M., Soltan P., “A metric space with independent median”, Annals of Tiberiu Popoviciu, Seminar of Functional Equations, Approximation and Convexity, v. 3, Cluj-Napoca, 2005, 3–12

[22] Cataranciuc S., “The complex of parallelepipeds and the 3-dimensional tree”, The 33-rd Annual Congres of the American Romanian Academy of Arts and Sciences (ARA), Proceedings (Sibiu, Romania, July 2–7, 2009), v. II, 293–296

[23] Bujac M., Cataranciuc S., Soltan P., “On the properties of Multidimensional Torus”, The XIV Conference on Applied and Industrial Mathematics, dedicated to the 60th anniversary of the foundation of the Faculty of Mathematics and Computer Science of Moldova State University, Chişinău, 2006, 66–70

[24] Dewdney A. K., “Degree sequences in complexes and hypergraphs”, Proc. Amer. Math. Soc., 53:2 (1975), 535–540 | DOI | MR | Zbl

[25] R. B. King (ed.), Chemical applications of topology and graph theory, ELSEVIER, 1983 | MR | Zbl

[26] Dodson C. T. J., Lok R., “Hypergraphs, homotopy and neighbourhood homology”, Ars. Com., 16-A (1983), 107–130 | MR | Zbl

[27] Dowker C. H., “Homology groups of relations”, Annals of mathematics, 56:2 (1952), 84–95 | DOI | MR | Zbl

[28] Erdos P., “On extremal problems on graphs and generalized graphs”, Israel J. Math., 2 (1964), 183–190 | DOI | MR | Zbl

[29] Fritsch R., Fritsch G., The four-color theorem, Springer-Verlag, New-York, 1998 | MR | Zbl

[30] Gantmacher F. R., Matrix theory, Moscow, 1967 (in Russian)

[31] Ghouila-Houri A., “Caracterisation des matrices totalement unimodulaires”, C. R. Acad. Sci. (Paris), 254 (1962), 1192–1194 | MR | Zbl

[32] Hilton P. J., Wylie Sh., Homology Theory: an introduction to Algebraic topology, Cambridge University Press, 1960 | MR

[33] Hoffman A. J., Kruskal J. B., “Integral boundary points of convex polyhedra”: Kuhn H. W., Tucker A. W., Linear Inequalities and Related Systems, Annals of Math. Studies, 28, Princeton Univ. Press, 1956, 223–246 | MR

[34] Holl M., Combinatorics, Mir, Moscow, 1970 (in Russian) | MR

[35] Hurewicz W., Wallman H., Dimension Theory, Princeton Univ. Press, 1941 | MR | Zbl

[36] James R. Munkres, “Topological Results in Combinatorics”, Michigan math. J., 31:1 (1984), 113–128 | DOI | MR | Zbl

[37] Kenmochi Yukiko, Atsushi Imiya, Ichikawa Akira, “Discrete combinatorial geometry”, Pattern Recognition, 30:10 (1997), 1719–1728 | DOI | Zbl

[38] Kleene St. C., Mathematical Logic, Wiley, New York, 1967 ; in Russian edition: Moscow, 1973 | MR | Zbl | Zbl

[39] Knaster B., Kuratowski C., Mazurkiewicz S., “Ein Beweis des Fixpunktsatzes fur $n$-dimensionale Simplexe”, Fundam. Math., 14 (1929), 132–137 | Zbl

[40] Kurosh A. G., Course of Higher Algebra, Moscow, 1968 (in Russian)

[41] Kurosh A. G., Lectures on general algebra, Moscow, 1962 (in Russian) | MR

[42] Lang S., Algebra, Reading, Mass., 1965 ; in Russian edition: Moscow, 1968 | MR

[43] Lovasz L., “On chromatic number of finite sets-system”, Acta Math. Acad. Sci. Hung., 19 (1968), 59–67 | DOI | MR | Zbl

[44] Martini H., Soltan P., “On the Homologies of Multi-ary Relations”, IC MCC, 45 (2003), 219–243 | MR | Zbl

[45] Melnicov O., Tyshkevich R., Yemelichev V., Sarvanov V., Lectures on graph theory, BT-Wiss-Verl., Mannheim, 1994; in Russian edition: Moscow, 1990

[46] Meyer J. C., “Quelques problemes concernant les cliques des hypergraphes $h$-complets et $q$-parti $h$-complets”, Hypergraph seminar (Columbus, 1972), Lecture Notes in Math., 411, eds. Berge and Ray-Chaudhuri, Springer-Verlag, Berlin, 1974, 127–139 | DOI | MR

[47] Moisil Grigore, Lectii despre logica rationamentului nuantat, Bucuresti, 1975 (in Romanian)

[48] Naik R. N., Rao S. B., Shrikhande S. S., Singhi N. M., “Intersection graphs on $k$-uniform linear hypergraphs”, Europ. J. Combinatorics, 3 (1982), 159–172 | DOI | MR | Zbl

[49] Oliver R., “Fixed-point sets of group actions on finite aciclic complexes”, Comment. Math. Helvetici, 50 (1975), 155–177 | DOI | MR | Zbl

[50] Padberg M. W., “Total unimodularity and the Euler subgraph problem”, Operations Research Letters, 7 (1988), 173–179 | DOI | MR | Zbl

[51] Poincare H., “Complement a l'analysis situs”, Rend. Circ. Mat. Palermo, 13 (1989), 285–343 | DOI

[52] Pontryagin L. S., Basis of Combinatorial Topology, Moscow, 1976 (in Russian) | Zbl

[53] Ringel Gerhard, Map Color Theorem, Springer-Verlag, Berlin–Heidelberg–New-York, 1974 | MR | Zbl

[54] Rishkov S., “About the combinatorial topology in Hilbert space”, Uspekhi Matematichesckikh Nauk, 13:4(82) (1958), 216–217 (in Russian)

[55] Rishkov S., “Invariant of homologies groups by infinity dimension”, All-Union Mathematics congress, v. IV, Moscow, 1958 (in Russian)

[56] Rotman J. J., An introduction to algebraic topology, Springer-Verlag, New-York, 1998 | MR | Zbl

[57] Saunders MacLane, Homology, Springer, Berlin, 1963 | MR

[58] Soltan P., “On the Homologies of Multy-ary relations and Oriented Hipergraphs”, Studii in metode de analiza numerica si optimizare, 2:1(3) (2000), 60–81 | MR

[59] Soltan P., Selected lectures in graph theory, USM, Chişinău, 2001 (in Romanian) | MR

[60] Soltan P. S, Zambitki D. K., Prisacaru K. F., Extremal problemes on graphs and algorithms of their solving, Kishinev, 1974 (in Russian)

[61] Spanier E. H., Algebraic Topology, Springer, New-York, 1966 ; in Russian edition: Moscow, 1971 | MR

[62] Teleman S., Elements of topology and differential varieties, Bucharest, 1964 (in Romanian) | Zbl

[63] The Mathematical Encyclopedia, v. 1, 1977 (in Russian)

[64] The Mathematical Encyclopedia, v. 3, 1982 (in Russian)

[65] Tomescu I., “Sur le probleme du coloriage des graphes generalises”, C. R. Acad. Sci. (Paris), 267 (1968), 250–252 | MR | Zbl

[66] Tutte W. T., “Lectures on matroids”, J. Research Nat. Bureau of Standards (USA) Sect. B, 69 (1965), 1–47 | DOI | MR | Zbl

[67] Tuza Z., “Critical hypergraphs and intersecting set-pair system”, J. Comb. Theory B, 39 (1985), 134–145 | DOI | MR | Zbl

[68] Veblen O., Alexander J. W., “Manifolds of $n$-dimensions”, Ann. of Math., 14:2 (1913), 163–178 | MR | Zbl

[69] Voloshin V. L., “On the upper chromatic number of a hypergraph”, Australian Journal of Combinatorics, 11 (1995), 25–45 | MR | Zbl

[70] Wylie S., “Intercept-finite cell complexes”, Algebraic Geometry and Topology, Symposium in honor of S. Lefschetz, Princeton Math. Series, 12, 1957, 389–399 | MR

[71] Zykov A. A., “Hypergraphs”, Uspekhi Mathematicheskikh Nauk, 29:6(180) (1974), 89–154 (in Russian) | MR | Zbl