Tverberg-Type Theorems for Matroids: A Counterexample and a Proof

  • Pavle V. M. BlagojevićEmail author
  • Albert Haase
  • Günter M. Ziegler

Mathematics Subject Classification (2010)

52A35 05B35 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    I. Bárány, S. B. Shlosman and A. Szucs: On a topological generalization of a theorem of Tverberg, J. London Math. Soc. (2) 23 (1981), 158–164.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    E. G. Bajmóczy and I. Bárány: On a common generalization of Borsuk’s and Radon’s theorem, Acta Math. Acad. Sci. Hungar. 34 1979 (1980), 347–350.zbMATHGoogle Scholar
  3. [3]
    M. Özaydin: Equivariant maps for the symmetric group, Preprint 1987, Scholar
  4. [4]
    F. Frick: Counterexamples to the topological Tverberg conjecture, Oberwolfach Reports 12 (2015), 318–322.Google Scholar
  5. [5]
    P. V. M. Blagojević, F. Frick and G. M. Ziegler: Barycenters of polytope skeleta and counterexamples to the topological Tverberg conjecture, via constraints, Preprint, October 2015, arXiv:1510.07984; J. European Math. Soc., to appear.Google Scholar
  6. [6]
    P. V. M. Blagojević, F. Frick and G. M. Ziegler: Tverberg plus constraints, Bull. Lond. Math. Soc. 46 (2014), 953–967.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    I. Mabillard and U. Wagner: Eliminating higher-multiplicity intersections, I. A Whitney trick for Tverberg-type problems, Preprint, arXiv:1508.02349, August 2015.Google Scholar
  8. [8]
    I. Bárány, P. V. M. Blagojević and G. M. Ziegler: Tverberg’s theorem at 50: Extensions and counterexamples, Notices Amer. Math. Soc. 63 (2016), 732–739.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    I. Bárány, G. Kalai and R. Meshulam: A Tverberg type theorem for matroids, in: Martin Loebl, Jaroslav Nešetřil, and Robin Thomas, editors, Journey Through Discrete Mathematics. A Tribute to Jiří Matoušek, 115–121. Springer, 2017.CrossRefGoogle Scholar
  10. [10]
    J. Matoušek: Using the Borsuk-Ulam theorem, Universitext, Springer-Verlag, Berlin, 2003.Google Scholar
  11. [11]
    K. S. Sarkaria: A generalized van Kampen-Flores theorem, Proc. Amer. Math. Soc. 111 (1991), 559–565.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    A. Dold: Simple proofs of some Borsuk-Ulam results, in: H. R. Miller and S. B. Priddy, editors, Proc. Northwestern Homotopy Theory Conf., volume 19 of Contemp. Math., 65–69, 1983.CrossRefGoogle Scholar
  13. [13]
    A. Björner and M. L. Wachs: Shellable nonpure complexes and posets, I, Trans. Amer. Math. Soc. 348 (1996), 1299–1327.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    A. Björner and M. L. Wachs: Shellable nonpure complexes and posets, II, Trans. Amer. Math. Soc. 349 (1997), 3945–3975.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    P. V. M. Blagojević, A. S. D. Blagojević and J. McCleary: Equilateral triangles on a Jordan curve and a generalization of a theorem of Dold, Topology Appl. 156 (2008), 16–23.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    P. V. M. Blagojević, W. Lück and G. M. Ziegler: Equivariant topology of configuration spaces, J. Topol. 8 (2015), 414–456.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    E. R. Fadell and S. Y. Husseini: An ideal-valued cohomological index theory with applications to Borsuk-Ulam and Bourgin-Yang theorems, Ergodic Theory Dynam. Systems 8* (1988), 73–85.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    G. M. Ziegler: Shellability of chessboard complexes, Israel J. Math. 87 (1994), 97–110.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    J. Oxley: Matroid Theory, volume 21 of Oxford Graduate Texts in Mathematics, Oxford University Press, Oxford, second edition, 2011.Google Scholar
  20. [20]
    R. P. Stanley: Balanced Cohen-Macaulay complexes, Trans. Amer. Math. Soc. 249 (1979), 139–157.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    K. Baclawski: Cohen-Macaulay ordered sets, J. Algebra 63 (1980), 226–258.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    A. Björner and M. L. Wachs: On lexicographically shellable posets, Trans. Amer. Math. Soc. 277 (1983), 323–341.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    M. Goff, S. Klee and I. Novik: Balanced complexes and complexes without large missing faces, Ark. Mat. 49 (2011), 335–350.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    M. Juhnke-Kubitzke and S. Murai: Balanced generalized lower bound inequality for simplicial polytopes, Selecta Mathematica 24 (2018), 1677–1689.MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    V. B. Mnukhin and J. Siemons: Saturated simplicial complexes, J. Combin. Theory Ser. A 109 (2005), 149–179.MathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    A. Björner: Topological methods, in: Handbook of combinatorics, Vol. 2, 1819–1872, Elsevier Sci. B. V., Amsterdam, 1995.Google Scholar
  27. [27]
    A. Björner: Shellable and Cohen-Macaulay partially ordered sets, Trans. Amer. Math. Soc. 260 (1980), 159–183.MathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    J. S. Provan and L. J. Billera: Decompositions of simplicial complexes related to diameters of convex polyhedra, Math. Oper. Res. 5 (1980), 576–594.MathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    A. Y. Volovikov: On a topological generalization of Tverberg’s theorem, Mat. Zametki 59 (1996), 454–456.MathSciNetCrossRefGoogle Scholar
  30. [30]
    P. V. M. Blagojević and G. M. Ziegler: Beyond the Borsuk-Ulam theorem: The Topological Tverberg Story, in: A Journey Through Discrete Mathematics, 273–341, Springer, Cham, 2017.CrossRefGoogle Scholar
  31. [31]
    P. Paták: Tverberg type theorems for matroids, Preprint arXiv:1702.08170, February 2017.Google Scholar
  32. [32]
    X. Goaoc, I. Mabillard, P. Paták, Z. Patáková, M. Tancer and U. Wagner: On generalized Heawood inequalities for manifolds: A van Kampen-Flores-type nonembeddability result, Israel J. Math. 222 (2017), 841–866.MathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    A. Shapiro: Obstructions to the imbedding of a complex in a Euclidean space. I. The first obstruction, Ann. of Math. (2) 66 (1957), 256–269.MathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    A. Björner: The homology and shellability of matroids and geometric lattices, in: Neil White, editor, Matroid applications, chapter 7, 226–283, Cambridge University Press, Cambridge, 1992.zbMATHGoogle Scholar
  35. [35]
    G.-C. Rota: On the foundations of combinatorial theory I, Theory of Möbius functions, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 2 (1964), 340–368.CrossRefzbMATHGoogle Scholar
  36. [36]
    S. Smale: A Vietoris mapping theorem for homotopy, Proc. Amer. Math. Soc. 8 (1957), 604–610.MathSciNetCrossRefzbMATHGoogle Scholar
  37. [37]
    K. S. Sarkaria: Kuratowski complexes, Topology 30 (1991), 67–76.MathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    J. Friedman and P. Hanlon: On the Betti numbers of chessboard complexes, J. Algebraic Combin. 8 (1998), 193–203.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© János Bolyai Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Pavle V. M. Blagojević
    • 1
    • 2
    Email author
  • Albert Haase
    • 1
  • Günter M. Ziegler
    • 1
  1. 1.Inst. Math.FU BerlinBerlinGermany
  2. 2.Mat. Institut SANUBeogradSerbia

Personalised recommendations