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The Tverberg-Vrećica problem and the combinatorial geometry on vector bundles

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Abstract

It is shown that many classical and many new combinatorial geometric results about finite sets of points inR d, specially the theorems of Tverberg type, can be generalized to the case of vector bundles, where they become combinatorial geometric statements about finite families of continuous cross-sections. The well known Tverberg-Vrećica conjecture is interpreted as a result of this type and its partial solution is obtained with the aid of the parametrized, ideal-valued, cohomological index theory. In the same spirit, classical “nonembeddability” and “coincidence” results like

have higher dimensional analogues. A new ingredient is that the coincidence condition is often interpreted as the existence of a common affinek-dimensional transversal, which reduces to the classical case fork=0.

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References

  1. J. F. Adams,Vector fields on spheres, Annals of Mathematics75 (1962), 603–632.

    Article  MathSciNet  Google Scholar 

  2. M. F. Atiyah and R. Bott,The moment map and equivariant cohomology, Topology23 (1984), 1–28.

    Article  MATH  MathSciNet  Google Scholar 

  3. I. Bárány and D. G. Larman,A colored version of Tverberg’s theorem, Journal of the London Mathematical Society (2)45 (1992), 314–320.

    Article  MATH  MathSciNet  Google Scholar 

  4. I. Bárány, S. B. Shlosman and A. Szücs,On a topological generalization of a theorem of Tverberg, Journal of the London Mathematical Society (2)23 (1981), 158–164.

    Article  MATH  Google Scholar 

  5. G. E. Bredon,Topology and Geometry, Graduate Texts in Mathematics 139, Springer-Verlag, Berlin, 1995.

    Google Scholar 

  6. J. Conway and C. Gordon,Knots and links in spatial graphs, Journal of Graph Theory7 (1983), 445–453.

    Article  MATH  MathSciNet  Google Scholar 

  7. A. Dold,Parametrized Borsuk-Ulam theorems, Commentarii Mathematici Helvetici63 (1988), 275–285.

    Article  MATH  MathSciNet  Google Scholar 

  8. J. Eckhoff,Helly, Radon and Carathéodory type theorems, inHandbook of Convex Geometry (P. M. Gruber and J. M. Wills, eds.), Vol. A, North-Holland, Amsterdam, 1993, pp. 389–448.

    Google Scholar 

  9. E., Fadell and S. Husseini,Relative cohomological index theories, Advances in Mathematics64 (1987), 1–31.

    Article  MATH  MathSciNet  Google Scholar 

  10. E. Fadell and S. Husseini,An ideal-valued cohomological index theory with applications to Borsuk-Ulam and Bourgin-Yang theorems, Ergodic Theory and Dynamical Systems8 * (1988), 73–85.

    Article  MathSciNet  Google Scholar 

  11. E. Fadell and S. Husseini,Index theory for G-bundle pairs with applications to Borsuk-Ulam type theorems for G-sphere bundles, inNonlinear Analysis, World Scientific, Singapore, 1987, pp. 307–337.

    Google Scholar 

  12. D. Husemoller,Fibre Bundles, McGraw-Hill, New York, 1966.

    MATH  Google Scholar 

  13. M. Izydorek and S. Rybicki,On parametrized Borsuk-Ulam theorem for free Z p -action, Proceedings of the Barcelona Conference on Algebraic Topology 1990, Lecture Notes in Mathematics1509, Springer-Verlag, Berlin, 1992, pp. 227–234.

    Google Scholar 

  14. J. Jaworowski,A continuous version of the Borsuk-Ulam theorem, Proceedings of the American Mathematical Society82 (1981), 112–114.

    Article  MATH  MathSciNet  Google Scholar 

  15. J. Matoušek,Topological methods in Combinatorics and Geometry (Lecture Notes) Kam Series (Technical Report) 94-272, Charles University, 1994.

  16. M. Milnor and J. D. Stasheff,Characteristic Classes, Princeton University Press, Princeton, 1974.

    MATH  Google Scholar 

  17. M. Nakaoka,Equivariant point theorems for fibre-preserving maps, Osaka Journal of Mathematics21 (1984), 809–815.

    MATH  MathSciNet  Google Scholar 

  18. J. Pach (ed.),New Trends in Discrete and Computational Geometry, Algorithms and Combinatorics 10, Springer-Verlag, Berlin, 1993.

    MATH  Google Scholar 

  19. R. Rado,A theorem on general measure, Journal of the London Mathematical Society26 (1946), 291–300.

    Article  MathSciNet  Google Scholar 

  20. J. Radon,Mengen konvexer Körper, die einen gemeinsamen Punkt enthalten, Mathematische Annalen83 (1921), 113–115.

    Article  MathSciNet  MATH  Google Scholar 

  21. H. Sachs,On spatial representation of finite graphs, inFinite and Infinite Sets, Colloquia Mathematica Societatis János Bolyai37 (1981).

  22. K. S. Sarkaria,A generalized Kneser conjecture, Journal of Combinatorial Theory, Series B49 (1990), 236–240.

    Article  MATH  MathSciNet  Google Scholar 

  23. K. S. Sarkaria,Kuratowski complexes, Topology30 (1991), 67–76.

    Article  MATH  MathSciNet  Google Scholar 

  24. P. Seymour,Progress on the four-color theorem, inProceedings of the International Congress of Mathematicians, Birkhäuser, Zürich, 1995.

    Google Scholar 

  25. H. Tverberg,A generalization of Radon’s theorem, Journal of the London Mathematical Society41 (1966), 123–128.

    Article  MATH  MathSciNet  Google Scholar 

  26. H. Tverberg and S. Vrećica,On generalizations of Radon’s theorem and the ham sandwich theorem, European Journal of Combinatorics14 (1993), 259–264.

    Article  MATH  MathSciNet  Google Scholar 

  27. S. Vrećica and R. Živaljević,New cases of the colored Tverberg theorem, inJerusalem Combinatorics '93 (H. Barcelo and G. Kalai, eds.), Contemporary Mathematics, American Mathematical Society, Providence, 1994.

    Google Scholar 

  28. R. Živaljević,Topological methods, inCRC Handbook of Discrete and Computational Geometry (J. E. Goodman and J. O'Rourke, eds.), CRC Press, New York, 1997.

    Google Scholar 

  29. R. Živaljević,User’s guide to equivariant methods in combinatorics, Publications of the Institute of Mathematics, Belgrade59 (73) (1996), 114–130.

    Google Scholar 

  30. R. Živaljević and S. Vrećica,An extension of the ham sandwich theorem, The Bulletin of the London Mathematical Society22 (1990), 183–186.

    Article  MathSciNet  MATH  Google Scholar 

  31. R. Živaljević and S. Vrećica,The colored Tverberg’s problem and complexes of injective functions, Journal of Combinatorial Theory, Series A61 (1992), 309–318.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Mathematics Institute SANU, Kneza Mihaila 35/1, p.f. 367, 11001, Belgrade, Serbia, Yugoslavia

    Rade T. Živaljević

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  1. Rade T. Živaljević
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Correspondence to Rade T. Živaljević.

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Supported in part by the Ministry for Science and Technology of Serbia, Grant 04M03.

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Živaljević, R.T. The Tverberg-Vrećica problem and the combinatorial geometry on vector bundles. Isr. J. Math. 111, 53–76 (1999). https://doi.org/10.1007/BF02810677

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  • DOI: https://doi.org/10.1007/BF02810677

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