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Bounding Helly Numbers via Betti Numbers

  • Xavier Goaoc
  • Pavel Paták
  • Zuzana Patáková
  • Martin Tancer
  • Uli Wagner
Chapter

Abstract

We show that very weak topological assumptions are enough to ensure the existence of a Helly-type theorem. More precisely, we show that for any non-negative integers b and d there exists an integer h(b, d) such that the following holds. If \(\mathcal{F}\) is a finite family of subsets of \(\mathbb{R}^{d}\) such that \(\tilde{\beta }_{i}\left (\bigcap \mathcal{G}\right ) \leq b\) for any \(\mathcal{G} \subsetneq \mathcal{F}\) and every 0 ≤ i ≤ ⌈d∕2⌉− 1 then \(\mathcal{F}\) has Helly number at most h(b, d). Here \(\tilde{\beta }_{i}\) denotes the reduced \(\mathbb{Z}_{2}\)-Betti numbers (with singular homology). These topological conditions are sharp: not controlling any of these ⌈d∕2⌉ first Betti numbers allow for families with unbounded Helly number.

Our proofs combine homological non-embeddability results with a Ramsey-based approach to build, given an arbitrary simplicial complex K, some well-behaved chain map \(C_{{\ast}}(K) \rightarrow C_{{\ast}}(\mathbb{R}^{d})\).

Notes

Acknowledgements

We would like to express our immense gratitude to Jiří Matoušek, not only for raising the problem addressed in the present paper and valuable discussions about it, but, much more generally, for the privilege of having known him, as our teacher, mentor, collaborator, and friend. Through his tremendous depth and insight, and the generosity with which he shared them, he greatly influenced all of us.

We further thank Jürgen Eckhoff for helpful comments on a preliminary version of the paper, and Andreas Holmsen and Gil Kalai for providing us with useful references.

References

  1. 1.
    N. Alon, G. Kalai, Bounding the piercing number. Discrete Comput. Geom. 13, 245–256 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    N. Alon, I. Bárány, Z. Füredi, D.J. Kleitman. Point selections and weak epsilon-nets for convex hulls. Combin. Probab. Comput. 1, 189–200 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    N. Amenta, Helly-type theorems and generalized linear programming. Discrete Comput. Geom. 12, 241–261 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    N. Amenta, A short proof of an interesting Helly-type theorem. Discrete Comput. Geom. 15, 423–427 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    E.G. Bajmóczy, I. Bárány, On a common generalization of Borsuk’s and Radon’s theorem. Acta Math. Acad. Sci. Hungar. 34(3–4), 347–350 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    M. Bestvina, M. Kapovich, B. Kleiner, Van Kampen’s embedding obstruction for discrete groups. Invent. Math. 150(2), 219–235 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    A. Björner, Nerves, fibers and homotopy groups. J. Combin. Theory Ser. A 102(1), 88–93 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    K. Borsuk, On the imbedding of systems of compacta in simplicial complexes. Fundamenta Mathematicae 35, 217–234 (1948)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    G.E. Bredon, Sheaf Theory. Volume 170 of Graduate Texts in Mathematics, 2nd edn. (Springer, New York, 1997)Google Scholar
  10. 10.
    O. Cheong, X. Goaoc, A. Holmsen, S. Petitjean, Hadwiger and Helly-type theorems for disjoint unit spheres. Discrete Comput. Geom. 1–3, 194–212 (2008)CrossRefzbMATHGoogle Scholar
  11. 11.
    E. Colin de Verdiere, G. Ginot, X. Goaoc, Helly numbers of acyclic families. Adv. Mathe. 253, 163–193 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    H. Debrunner, Helly type theorems derived from basic singular homology. Am. Math. Mon. 77, 375–380 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    J. de Loera, S. Petrović, D. Stasi, Random sampling in computational algebra: Helly numbers and violator spaces. http://arxiv.org/abs/1503.08804
  14. 14.
    M. Deza, P. Frankl, A Helly type theorem for hypersurfaces. J. Combin. Theory Ser. A 45, 27–30 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    J. Dugundji, A duality property of nerves. Fundamenta Mathematicae 59, 213–219 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    J. Eckhoff, Helly, Radon and Carathéodory type theorems, in Handbook of Convex Geometry, ed. by P.M. Gruber, J.M. Wills (North Holland, Amsterdam/New York, 1993), pp. 389–448CrossRefGoogle Scholar
  17. 17.
    J. Eckhoff, K.-P. Nischke, Morris’s pigeonhole principle and the Helly theorem for unions of convex sets. Bull. Lond. Math. Soc. 41, 577–588 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    B. Farb, Group actions and Helly’s theorem. Adv. Math. 222, 1574–1588 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    A.I. Flores, Über die Existenz n-dimensionaler Komplexe, die nicht in den \(\mathbb{R}^{2n}\) topologisch einbettbar sind. Ergeb. Math. Kolloqu. 5, 17–24 (1933)Google Scholar
  20. 20.
    X. Goaoc, I. Mabillard, P. Paták, Z. Patáková, M. Tancer, U. Wagner, On generalized Heawood Inequalities for manifolds: a Van Kampen-Flores-type nonembeddability result, in 31st International Symposium on Computational Geometry (SoCG 2015), Dagstuhl, ed. by L. Arge, J. Pach. Volume 34 of Leibniz International Proceedings in Informatics (LIPIcs). Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, pp. 476–490Google Scholar
  21. 21.
    X. Goaoc, P. Paták, Z. Patáková, M. Tancer, U. Wagner, Bounding Helly numbers via Betti numbers, in 31st International Symposium on Computational Geometry (SoCG 2015), Dagstuhl, ed. by L. Arge, J. Pach. Volume 34 of Leibniz International Proceedings in Informatics (LIPIcs). Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, pp. 507–521Google Scholar
  22. 22.
    R. González-Diaz, P. Real, Simplification techniques for maps in simplicial topology. J. Symb. Comput. 40, 1208–1224 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    J.E. Goodman, A. Holmsen, R. Pollack, K. Ranestad, F. Sottile, Cremona convexity, frame convexity, and a theorem of Santaló. Adv. Geom. 6, 301–322 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    B. Grünbaum, On common transversals. Archiv der Mathematik 9, 465–469 (1958)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    A. Haefliger, Plongements différentiables dans le domaine stable. Comment. Math. Helv. 37, 155–176 (1962/1963)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    A. Hatcher, Algebraic Topology (Cambridge University Press, Cambridge, 2002)zbMATHGoogle Scholar
  27. 27.
    E. Helly, Über Mengen konvexer Körper mit gemeinschaftlichen Punkten. Jahresbericht Deutsch. Math. Verein. 32, 175–176 (1923)zbMATHGoogle Scholar
  28. 28.
    E. Helly, Über Systeme von abgeschlossenen Mengen mit gemeinschaftlichen Punkten. Monaths. Math. und Physik 37, 281–302 (1930)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    G. Kalai, Combinatorial expectations from commutative algebra, in Combinatorial Commutative Algebra, ed. by I. Peeva, V. Welker, vol. 1, no. 3 (Oberwolfach Reports, 2004), pp. 1729–1734Google Scholar
  30. 30.
    G. Kalai, R. Meshulam, Leray numbers of projections and a topological Helly-type theorem. J. Topol. 1, 551–556 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    M. Katchalski, A conjecture of Grünbaum on common transversals. Math. Scand. 59(2), 192–198 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    D. Kozlov, Combinatorial Algebraic Topology. Volume 21 of Algorithms and Computation in Mathematics (Springer, Berlin, 2008)Google Scholar
  33. 33.
    D.G. Larman, Helly type properties of unions of convex sets. Mathematika 15, 53–59 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    H. Maehara, Helly-type theorems for spheres. Discrete Comput. Geom. 4(1), 279–285 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    J. Matoušek, A Helly-type theorem for unions of convex sets. Discrete Comput. Geom. 18, 1–12 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    J. Matoušek, Using the Borsuk–Ulam Theorem (Springer, Berlin, 2003)zbMATHGoogle Scholar
  37. 37.
    S.A. Melikhov, The van Kampen obstruction and its relatives. Proc. Steklov Inst. Math. 266(1), 142–176 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    J. Milnor, Construction of universal bundles, II. Ann. Math. 63(3), 430–436 (1956)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    J. Milnor, On the Betti numbers of real varieties. Proc. Am. Math. Soc. 15, 275–280 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    L. Montejano, A new topological Helly theorem and some transversal results. Discrete Comput. Geom. 52(2), 390–398 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    T.S. Motzkin, A proof of Hilbert’s Nullstellensatz. Mathematische Zeitschrift 63, 341–344 (1955)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    J.R. Munkres, Elements of Algebraic Topology (Addison-Wesley, Menlo Park, 1984)zbMATHGoogle Scholar
  43. 43.
    C.M. Petty, Equilateral sets in Minkowski spaces. Proc. Am. Math. Soc. 29, 369–374 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    R. Rado, A theorem on general measure. J. Lond. Math. Soc. s1–21(4), 291–300 (1946)Google Scholar
  45. 45.
    F.P. Ramsey, On a problem in formal logic. Proc. Lond. Math. Soc. 30, 264–286 (1929)zbMATHGoogle Scholar
  46. 46.
    C.P. Rourke, B.J. Sanderson, Introduction to Piecewise-Linear Topology (Springer, New York, 1972). Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 69Google Scholar
  47. 47.
    M. Sharir, E. Welzl, A combinatorial bound for linear programming and related problems, in Proceedings of the 9th Symposium on Theoretical Aspects of Computer Science (1992), pp. 569–579Google Scholar
  48. 48.
    P. Shvartsman, The Whitney extension problem and Lipschitz selections of set-valued mappings in jet-spaces. Trans. Am. Math. Soc. 360, 5529–5550 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    A.B. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in Surveys in Contemporary Mathematics. Volume 347 of London Mathematical Society Lecture Note Series (Cambridge University Press, Cambridge, 2008), pp. 248–342Google Scholar
  50. 50.
    R.I. Soare, Computability theory and differential geometry. Bull. Symb. Log. 10(4), 457–486 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    J. Sosnovec, Draft of Bachelor’s thesis (2015)Google Scholar
  52. 52.
    K.J. Swanepoel, Helly-type theorems for hollow axis-aligned boxes. Proc. Am. Math. Soc. 127, 2155–2162 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    K.J. Swanepoel, Helly-type theorems for homothets of planar convex curves. Proc. Am. Math. Soc. 131, 921–932 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    M. Tancer, Intersection patterns of convex sets via simplicial complexes: a survey, in Thirty Essays on Geometric Graph Theory, ed. by J. Pach (Springer, New York, 2013), pp. 521–540CrossRefGoogle Scholar
  55. 55.
    R. Thom, Sur l’homologie des variétés algébriques réelles, in Differential and Combinatorial Topology, ed. by S.S. Cairns (Princeton University Press, Princeton, 1965), pp. 255–265Google Scholar
  56. 56.
    H. Tverberg, Proof of Grünbaum’s conjecture on common transversals for translates. Discrete Comput. Geom. 4, 191–203 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    E.R. van Kampen, Komplexe in euklidischen Räumen. Abh. Math. Sem. Univ. Hamburg 9, 72–78 (1932)CrossRefzbMATHGoogle Scholar
  58. 58.
    U. Wagner, Minors in random and expanding hypergraphs, in Proceedings of the 27th Annual Symposium on Computational Geometry (SoCG) (2011), pp. 351–360Google Scholar
  59. 59.
    C. Weber, Plongements de polyèdres dans le domaine métastable. Comment. Math. Helv. 42, 1–27 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  60. 60.
    G. Wegner, d-collapsing and nerves of families of convex sets. Archiv der Mathematik 26, 317–321 (1975)Google Scholar
  61. 61.
    R. Wenger, Helly-type theorems and geometric transversals, in Handbook of Discrete & Computational Geometry, 2nd edn., ed. by J.E. Goodman, J. O’Rourke (CRC Press LLC, Boca Raton, 2004), chapter 4, pp. 73–96Google Scholar
  62. 62.
    G.M. Ziegler, Lectures on Polytopes. Volume 152 of Graduate Texts in Mathematics (Springer, New York, 1995)Google Scholar

Copyright information

© Springer International publishing AG 2017

Authors and Affiliations

  • Xavier Goaoc
    • 1
  • Pavel Paták
    • 2
  • Zuzana Patáková
    • 3
  • Martin Tancer
    • 3
  • Uli Wagner
    • 4
  1. 1.Université Paris-Est Marne-la-ValléeMarne-la-ValléeFrance
  2. 2.Department of AlgebraCharles UniversityPragueCzech Republic
  3. 3.Department of Applied MathematicsCharles UniversityPragueCzech Republic
  4. 4.IST AustriaKlosterneuburgAustria

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