ABSTRACT
We introduce a new notion of minors for simplicial complexes (hypergraphs), so-called homological minors. Our motivation is to propose a general approach to attack certain extremal problems for sparse simplicial complexes and the corresponding threshold problems for random complexes. In this paper, we focus on threshold problems. The basic model for random complexes is the Linial-Meshulam model Xk(n,p). By definition, such a complex has n vertices, a complete (k-1)-dimensional skeleton, and every possible k-dimensional simplex is chosen independently with probability p. We show that for every k,t ≥ 1, there is a constant C=C(k,t) such that for p ≥ C/n, the random complex Xk(n,p) asymptotically almost surely contains Kkt (the complete k-dimensional complex on t vertices) as a homological minor. As corollary, the threshold for (topological) embeddability of Xk(n,p) into R2k is at p=Θ(1/n).
The method can be extended to other models of random complexes (for which the lower skeleta are not necessarily complete) and also to more general Tverberg-type problems, where instead of continuous maps without doubly covered image points (embeddings), we consider maps without q-fold covered image points.
- M. Ajtai, V. Chvátal, M. Newborn, and E. Szemerédi. Crossing-free subgraphs. Ann. Discrete Math., 12:9--12, 1982.Google Scholar
- L. Aronshtam, N. Linial, T. Luczak, and R. Meshulam. Vanishing of the top homology of a random complex. arXiv:1010.1400v1, 2010.Google Scholar
- E. Babson, C. Hoffman, and M. Kahle. The fundamental group of random 2-complexes. J. Amer. Math. Soc., 24(1):1--28, 2011.Google ScholarCross Ref
- B. Bollobás, P. A. Catlin, and P. Erdos. Hadwiger's conjecture is true for almost every graph. European J. Combin., 1(3):195--199, 1980.Google ScholarCross Ref
- B. Bollobás and A. Thomason. Proof of a conjecture of Mader, Erdos and Hajnal on topological complete subgraphs. European J. Combin., 19(8):883--887, 1998. Google ScholarDigital Library
- W. G. Brown, P. Erdos, and V. T. Sós. Some extremal problems on r-graphs. In New directions in the theory of graphs (Proc. Third Ann Arbor Conf., Univ. Michigan, Ann Arbor, Mich, 1971), pages 53--63. Academic Press, New York, 1973.Google Scholar
- D. C. Cohen, M. Farber, and T. Kappeler. The homotopical dimension of random 2-complexes. arXiv:1005.3383v1, 2010.Google Scholar
- A. Costa, M. Farber, and T. Kappeler. Topology of random 2-complexes. arXiv:1006.4229v2, 2010.Google Scholar
- T. K. Dey. On counting triangulations in d dimensions. Comput. Geom., 3(6):315--325, 1993. Google ScholarDigital Library
- T. K. Dey and J. Pach. Extremal problems for geometric hypergraphs. Discrete Comput. Geom., 19(4):473--484, 1998.Google ScholarCross Ref
- T. K. Dey and N. R. Shah. On the number of simplicial complexes in Rd. Comput. Geom., 8(5):267--277, 1997. Google ScholarDigital Library
- P. Erdos. On extremal problems of graphs and generalized graphs. Israel J. Math., 2:183--190, 1964.Google ScholarCross Ref
- A. Flores. Über die Existenz n-dimensionaler Komplexe, die nicht in den R2n topologisch einbettbar sind. Ergeb. Math. Kolloqu., 5:17--24, 1933.Google Scholar
- J. Geelen, B. Gerards, and G. Whittle. Towards a matroid-minor structure theory. In Combinatorics, complexity, and chance, volume 34 of Oxford Lecture Ser. Math. Appl., pages 72--82. Oxford Univ. Press, Oxford, 2007.Google ScholarCross Ref
- M. Gromov. Singularities, expanders and topology of maps. Part 2: From combinatorics to topology via algebraic isoperimetry. Geom. Funct. Anal., 20(2):416--526, 2010.Google ScholarCross Ref
- A. Gundert. On the complexity of embeddable simplicial complexes. Diplomarbeit, Freie Universität Berlin, 2009.Google Scholar
- H. Hadwiger. Über eine Klassifikation der Streckenkomplexe. Vierteljschr. Naturforsch. Ges. Zürich, 88:133--142, 1943.Google Scholar
- A. Haefliger. Plongements de variétés dans le domaine stable. Sém. Bourbaki, 245, 1982.Google Scholar
- T. Kaiser. Minors of simplicial complexes. Discrete Appl. Math., 157(12):2597--2602, 2009. Google ScholarDigital Library
- G. Kalai. The diameter of graphs of convex polytopes and f-vector theory. In Applied geometry and discrete mathematics, volume 4 of DIMACS Ser. Discrete Math. Theoret. Comput. Sci., pages 387--411. Amer. Math. Soc., Providence, RI, 1991.Google Scholar
- J. Komlós and E. Szemerédi. Topological cliques in graphs. II. Combin. Probab. Comput., 5(1):79--90, 1996.Google ScholarCross Ref
- D. Kozlov. Combinatorial Algebraic Topology, volume 21 of Algorithms and Computation in Mathematics. Springer, Berlin, 2008.Google Scholar
- D. Kozlov. The threshold function for vanishing of the top homology group of random d-complexes. Proc. Amer. Math. Soc., 138(12):4517--4527, 2010.Google ScholarCross Ref
- M. Krivelevich and B. Sudakov. Minors in expanding graphs. Geom. Funct. Anal., 19(1):294--331, 2009.Google ScholarCross Ref
- K. Kuratowski. Sur le problème des courbes gauches en topologie. Fundamenta, 15:271--283, 1930.Google ScholarCross Ref
- F. T. Leighton. Complexity Issues in VLSI. MIT Press, Cambridge, MA, 1983.Google Scholar
- N. Linial and R. Meshulam. Homological connectivity of random 2-complexes. Combinatorica, 26(4):475--487, 2006. Google ScholarDigital Library
- L. Lovász. Graph minor theory. Bull. Amer. Math. Soc. (N.S.), 43(1):75--86 (electronic), 2006.Google Scholar
- T. Łuczak, B. Pittel, and J. C. Wierman. The structure of a random graph at the point of the phase transition. Trans. Amer. Math. Soc., 341(2):721--748, 1994.Google ScholarCross Ref
- W. Mader. Homomorphieeigenschaften und mittlere Kantendichte von Graphen. Math. Ann., 174:265--268, 1967.Google ScholarCross Ref
- W. Mader. 3n-5 edges do force a subdivision of $K_5$. Combinatorica, 18(4):569--595, 1998.Google ScholarCross Ref
- J. Matousek. Using the Borsuk-Ulam theorem. Springer-Verlag, Berlin, 2003.Google Scholar
- J. Matousek, M. Tancer, and U. Wagner. Hardness of embedding simplicial complexes in Rd. J. Eur. Math. Soc., 13(2):259--295, 2011.Google ScholarCross Ref
- P. McMullen. The maximum numbers of faces of a convex polytope. Mathematika, 17:179--184, 1970.Google ScholarCross Ref
- S. A. Melikhov. The van Kampen obstruction and its relatives. Proc. Steklov Inst. Math., 266(1):142--176, 2009.Google ScholarCross Ref
- R. Meshulam and N. Wallach. Homological connectivity of random k-dimensional complexes. Random Structures Algorithms, 34(3):408--417, 2009. Google ScholarDigital Library
- J. R. Munkres. Elements of Algebraic Topology. Addison-Wesley, Menlo Park, CA, 1984.Google Scholar
- E. Nevo. Higher minors and van Kampen's obstruction. Math. Scand., 101:161--176, 2007.Google ScholarCross Ref
- I. Newman and Y. Rabinovich. Finite volume spaces and sparsification. arXiv:1002.3541v3, 2010.Google Scholar
- N. Robertson, P. Seymour, and R. Thomas. Hadwiger's conjecture for K6-free graphs. Combinatorica, 13(3):279--361, 1993.Google ScholarCross Ref
- N. Robertson and P. D. Seymour. Graph minors. XX. Wagner's conjecture. J. Combin. Theory Ser. B, 92(2):325--357, 2004. Google ScholarDigital Library
- K. S. Sarkaria. Shifting and embeddability of simplicial complexes. Technical report, Max-Planck-Institut für Mathematik, Bonn, MPI/92--51, 1992.Google Scholar
- A. B. Skopenkov. Embedding and knotting of manifolds in Euclidean spaces. In Surveys in contemporary mathematics, volume 347 of London Math. Soc. Lecture Note Ser., pages 248--342. Cambridge Univ. Press, Cambridge, 2008.Google Scholar
- R. P. Stanley. The upper bound conjecture and Cohen-Macaulay rings. Studies in Appl. Math., 54(2):135--142, 1975.Google ScholarCross Ref
- A. Thomason. The extremal function for complete minors. J. Combin. Theory Ser. B, 81(2):318--338, 2001. Google ScholarDigital Library
- B. R. Ummel. Imbedding classes and n-minimal complexes. Proc. Amer. Math. Soc., 38:201--206, 1973.Google Scholar
- E. R. van Kampen. Komplexe in euklidischen Räumen. Abh. Math. Sem. Univ. Hamburg, 9:72--78, 1932.Google ScholarCross Ref
- K. Wagner. Über eine Eigenschaft der ebenen Komplexe. Math. Ann., 114(1):570--590, 1937.Google ScholarCross Ref
- C. Weber. Plongements de polyedres dans le domaine metastable. Comment. Math. Helv., 42:1--27, 1967.Google ScholarCross Ref
- J. Zaks. On minimal complexes. Pacif. J. Math., 28:721--727, 1969.Google ScholarCross Ref
Index Terms
- Minors in random and expanding hypergraphs
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