Advertisement

The Topological Transversal Tverberg Theorem Plus Constraints

  • Pavle V. M. Blagojević
  • Aleksandra S. Dimitrijević Blagojević
  • Günter M. Ziegler
Chapter
Part of the Bolyai Society Mathematical Studies book series (BSMS, volume 27)

Abstract

In this paper we use the strength of the constraint method in combination with a generalized Borsuk–Ulam type theorem and a cohomological intersection lemma to show how one can obtain many new topological transversal theorems of Tverberg type. In particular, we derive a topological generalized transversal Van Kampen–Flores theorem and a topological transversal weak colored Tverberg theorem.

Notes

Acknowledgements

We are grateful to Florian Frick and to the referee for very good observations and useful comments.

References

  1. 1.
    A. Adem, J. R. Milgram, Cohomology of Finite Groups, vol. 309, 2nd ed., Grundlehren der Mathematischen Wissenschaften (Springer, Berlin, 2004)CrossRefGoogle Scholar
  2. 2.
    I. Bárány, S.B. Shlosman, A. Szűcs, On a topological generalization of a theorem of Tverberg. J. Lond. Math. Soc. 23, 158–164 (1981)MathSciNetCrossRefGoogle Scholar
  3. 3.
    P.V.M. Blagojević, F. Frick, G.M. Ziegler, Barycenters of Polytope Skeleta and Counterexamples to the Topological Tverberg Conjecture, Via Constraints (2015), p. 6, arXiv:1510.07984 [Preprint]
  4. 4.
    P.V.M. Blagojević, Tverberg plus constraints. Bull. Lond. Math. Soc. 46(5), 953–967 (2014)MathSciNetCrossRefGoogle Scholar
  5. 5.
    P.V.M. Blagojević, B. Matschke, G.M. Ziegler, Optimal bounds for a colorful Tverberg–Vrećica type problem. Adv. Math. 226(6), 5198–5215 (2011)MathSciNetCrossRefGoogle Scholar
  6. 6.
    P.V.M. Blagojević, B. Matschke, G.M. Ziegler, Optimal bounds for the colored Tverberg problem. J. Eur. Math. Soc 17(4), 739–754 (2015)MathSciNetCrossRefGoogle Scholar
  7. 7.
    V.L. Dol’nikov, Transversals of families of sets in \(\mathbb{R}^n\) and a relationship between Helly and Borsuk theorems. Math. Sb. 184(5), 111–132 (1993)MathSciNetzbMATHGoogle Scholar
  8. 8.
    E. Fadell, S. Husseini, An ideal-valued cohomological index theory with applications to Borsuk–Ulam and Bourgin–Yang theorems, Ergod. Theory Dynam. Syst. 8\(^*\) (1988), 73–85 [Charles Conley memorial issue]Google Scholar
  9. 9.
    F. Frick, Counterexamples to the topological Tverberg conjecture. Oberwolfach Rep. 12(1), 318–322 (2015), arXiv:1502.00947
  10. 10.
    R.N. Karasev, Tverberg’s transversal conjecture and analogues of nonembeddability theorems for transversals. Discrete Comput. Geom. 38(3), 513–525 (2007)MathSciNetCrossRefGoogle Scholar
  11. 11.
    I. Mabillard, U. Wagner, Eliminating Tverberg points, I. an analogue of the Whitney trick, in Proceedings of the 30th Annual Symposium on Computational Geometry (SoCG) (ACM, Kyoto, 2014), pp. 171–180Google Scholar
  12. 12.
    I. Mabillard, U. Wagner, Eliminating higher-multiplicity intersections, I. A Whitney trick for Tverberg-type problems (2015), p. 46, arXiv:1508.02349 [Preprint]
  13. 13.
    K.S. Sarkaria, A generalized van Kampen–Flores theorem. Proc. Am. Math. Soc. 11, 559–565 (1991)MathSciNetCrossRefGoogle Scholar
  14. 14.
    H. Tverberg, A generalization of Radon’s theorem. J. Lond. Math. Soc. 41, 123–128 (1966)MathSciNetCrossRefGoogle Scholar
  15. 15.
    H. Tverberg, S. Vrećica, On generalizations of Radon’s theorem and the ham sandwich theorem. Eur. J. Comb. 14(3), 259–264 (1993)MathSciNetCrossRefGoogle Scholar
  16. 16.
    A.Y. Volovikov, On the van Kampen–Flores theorem. Math. Notes 59(5), 477–481 (1996)MathSciNetCrossRefGoogle Scholar
  17. 17.
    S. Vrećica, Tverberg’s conjecture. Discret. Comput. Geom. 29(4), 505–510 (2003)Google Scholar
  18. 18.
    S. Vrećica, R.T. Živaljević, New cases of the colored Tverberg theorem, in Jerusalem Combinatorics ’93, ed. by H. Barcelo, G. Kalai, Contemporary Mathematics, vol. 178. (American Mathematical Society, 1994), pp. 325–334Google Scholar
  19. 19.
    R.T. Živaljević, The Tverberg–Vrećica problem and the combinatorial geometry on vector bundles. Israel J. Math. 111, 53–76 (1999)MathSciNetCrossRefGoogle Scholar
  20. 20.
    R.T. Živaljević, S. Vrećica, The colored Tverberg’s problem and complexes of injective functions. J. Combin. Theory Ser. A 61, 309–318 (1992)MathSciNetCrossRefGoogle Scholar

Copyright information

© János Bolyai Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Pavle V. M. Blagojević
    • 1
    • 2
  • Aleksandra S. Dimitrijević Blagojević
    • 2
  • Günter M. Ziegler
    • 1
  1. 1.Inst. Math.FU BerlinBerlinGermany
  2. 2.Mat. Institut SANUBeogradSerbia

Personalised recommendations