, Volume 39, Issue 3, pp 477–500 | Cite as

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

  • Pavle V. M. Blagojević
  • Albert Haase
  • Günter M. ZieglerEmail author
Original Paper


Bárány, Kalai, and Meshulam recently obtained a topological Tverberg-type theorem for matroids, which guarantees multiple coincidences for continuous maps from a matroid complex into ℝd, if the matroid has sufficiently many disjoint bases. They make a conjecture on the connectivity of k-fold deleted joins of a matroid with many disjoint bases, which would yield a much tighter result — but we provide a counterexample already for the case of k = 2, where a tight Tverberg-type theorem would be a topological Radon theorem for matroids. Nevertheless, we prove the topological Radon theorem for the counterexample family of matroids by an index calculation, despite the failure of the connectivity-based approach.

Mathematics Subject Classification (2010)

52A35 05B35 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



We thank the referees of Combinatorica for very detailed and helpful comments, including in particular a simplification for the proof of Theorem 1.3.


  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 Berlin Heidelberg 2019

Authors and Affiliations

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

Personalised recommendations