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A Journey Through Discrete Mathematics pp 273–341Cite as

Beyond the Borsuk–Ulam Theorem: The Topological Tverberg Story

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Abstract

Bárány’s “topological Tverberg conjecture” from 1976 states that any continuous map of an N-simplex \(\Delta _{N}\) to \(\mathbb{R}^{d}\), for N ≥ (d + 1)(r − 1), maps points from r disjoint faces in \(\Delta _{N}\) to the same point in \(\mathbb{R}^{d}\). The proof of this result for the case when r is a prime, as well as some colored version of the same result, using the results of Borsuk–Ulam and Dold on the non-existence of equivariant maps between spaces with a free group action, were main topics of Matoušek’s 2003 book “Using the Borsuk–Ulam theorem.”

In this paper we show how advanced equivariant topology methods allow one to go beyond the prime case of the topological Tverberg conjecture.

First we explain in detail how equivariant cohomology tools (employing the Borel construction, comparison of Serre spectral sequences, Fadell–Husseini index, etc.) can be used to prove the topological Tverberg conjecture whenever r is a prime power. Our presentation includes a number of improved proofs as well as new results, such as a complete determination of the Fadell–Husseini index of chessboard complexes in the prime case.

Then, we introduce the “constraint method,” which applied to suitable “unavoidable complexes” yields a great variety of variations and corollaries to the topological Tverberg theorem, such as the “colored” and the “dimension-restricted” (Van Kampen–Flores type) versions.

Both parts have provided crucial components to the recent spectacular counter-examples in high dimensions for the case when r is not a prime power.

The research by Pavle V. M. Blagojević leading to these results has received funding from DFG via Collaborative Research Center TRR 109 “Discretization in Geometry and Dynamics.” Also supported by the grant ON 174008 of the Serbian Ministry of Education and Science.The research by Günter M. Ziegler received funding from DFG via the Research Training Group “Methods for Discrete Structures” and the Collaborative Research Center TRR 109 “Discretization in Geometry and Dynamics.”

Dedicated to the memory of Jiří Matoušek.

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References

  1. A. Adem, R.J. Milgram, Cohomology of Finite Groups, 2nd edn. Grundlehren der Mathematischen Wissenschaften, vol. 309 (Springer, Berlin, 2004)

    Google Scholar 

  2. V.I. Arnold, Experimental Mathematics. MSRI Mathematical Circles Library, vol. 16 (MSRI, Berkeley/American Mathematical Society, Providence, 2015)

    Google Scholar 

  3. S. Avvakumov, I. Mabillard, A. Skopenkov, U. Wagner, Eliminating higher-multiplicity intersections, III. Codimension 2, Preprint, 16 pages, arXiv:1511.03501. Nov 2015

    Google Scholar 

  4. E.G. Bajmóczy, I. Bárány, On a common generalization of Borsuk’s and Radon’s theorem. Acta Math. Hungar. 34, 347–350 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  5. I. Bárány, P.V.M. Blagojević, G.M. Ziegler, Tverberg’s theorem at 50: extensions and counterexamples. Not. Am. Math. Soc. 73(7), 732–739 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  6. I. Bárány, Z. Füredi, L. Lovász, On the number of halving planes. Combinatorica 10, 175–183 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  7. I. Bárány, D.G. Larman, A colored version of Tverberg’s theorem. J. Lond. Math. Soc. 2, 314–320 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  8. I. Bárány, S.B. Shlosman, A. Szűcs, On a topological generalization of a theorem of Tverberg. J. Lond. Math. Soc. 23, 158–164 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  9. B.J. Birch, On 3N points in a plane. Math. Proc. Camb. Philos. Soc. 55, 289–293 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  10. A. Björner, Topological Methods. Handbook of Combinatorics, vol. 2 (Elsevier, Amsterdam, 1995), pp. 1819–1872

    Google Scholar 

  11. A. Björner, L. Lovász, S. Vrećica, R. Živaljević, Chessboard complexes and matching complexes. J. Lond. Math. Soc. 49, 25–39 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  12. P.V.M. Blagojević, F. Frick, G.M. Ziegler, Tverberg plus constraints. Bull. Lond. Math. Soc. 46, 953–967 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. P.V.M. Blagojević, F. Frick, G.M. Ziegler, Barycenters of polytope skeleta and counterexamples to the topological Tverberg conjecture, via constraints. J. Eur. Math. Soc. (JEMS) (2015, to appear). Preprint, 6 pages, arXiv:1510.07984

    Google Scholar 

  14. P.V.M. Blagojević, W. Lück, G.M. Ziegler, Equivariant topology of configuration spaces. J. Topol. 8, 414–456 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  15. P.V.M. Blagojević, B. Matschke, G.M. Ziegler, Optimal bounds for a colorful Tverberg–Vrećica type problem. Adv. Math. 226, 5198–5215 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. P.V.M. Blagojević, B. Matschke, G.M. Ziegler, A tight colored Tverberg theorem for maps to manifolds. Topol. Appl. 158, 1445–1452 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. P.V.M. Blagojević, B. Matschke, G.M. Ziegler, Optimal bounds for the colored Tverberg problem. J. Eur. Math. Soc. (JEMS) 17, 739–754 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  18. G.E. Bredon, Equivariant Cohomology Theories. Lecture Notes in Mathematics, vol. 34 (Springer, Berlin/New York, 1967)

    Google Scholar 

  19. G.E. Bredon, Topology and Geometry. Graduate Texts in Mathematics, vol. 139 (Springer, New York, 1993)

    Google Scholar 

  20. K.S. Brown, Cohomology of Groups. Graduate Texts in Mathematics, vol. 87 (Springer, New York, 1994)

    Google Scholar 

  21. A. Dold, Simple proofs of some Borsuk–Ulam results, in Proceedings of the Northwestern Homotopy Theory Conference, ed. by H.R. Miller, S.B. Priddy. Contemporary Mathematics, vol. 19 (1983), pp. 65–69

    Google Scholar 

  22. E. Fadell, S. Husseini, An ideal-valued cohomological index theory with applications to Borsuk–Ulam and Bourgin–Yang theorems. Ergod. Theory Dynam. Syst. 8, 73–85 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  23. A. Flores, Über n-dimensionale Komplexe, die im R 2n+1absolut selbstverschlungen sind, Ergebnisse eines Math. Kolloquiums 6, 4–7 (1932/1934)

    Google Scholar 

  24. A. Fomenko, D. Fuchs, Homotopical Topology, 2nd edn. Graduate Texts in Mathematics, vol. 273 (Springer, Cham, 2016)

    Google Scholar 

  25. F. Frick, Counterexamples to the topological Tverberg conjecture. Oberwolfach Rep. 12, 318–322 (2015)

    MathSciNet  Google Scholar 

  26. M. Gromov, Singularities, expanders and topology of maps. II: from combinatorics to topology via algebraic isoperimetry. Geom. Funct. Anal. (GAFA) 20, 416–526 (2010)

    Google Scholar 

  27. P.M. Gruber, R. Schneider, Problems in geometric convexity, in Contributions to Geometry (Proceedings of the Geometry Symposium, Siegen, 1978), ed. by J. Tölke, J. Wills (Birkhäuser, Basel/Boston, 1979), pp. 255–278

    Google Scholar 

  28. B. Grünbaum, Convex Polytopes. Graduate Texts in Mathematics, vol. 221 (Springer, New York, 2003). Second edition prepared by V. Kaibel, V. Klee, G.M. Ziegler (Original edition: Interscience, London, 1967)

    Google Scholar 

  29. W.Y. Hsiang, Cohomology Theory of Topological Transformation Groups (Springer, New York/Heidelberg, 1975). Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 85

    Google Scholar 

  30. J. Jonsson, Simplicial Complexes of Graphs. Lecture Notes in Mathematics, vol. 1928 (Springer, Berlin, 2008)

    Google Scholar 

  31. I. Mabillard, U. Wagner, Eliminating Tverberg points, I. An analogue of the Whitney trick, in Proceedings of 30th Annual Symposium on Computational Geometry (SoCG), Kyoto, June 2014 (ACM, 2014), pp. 171–180

    Google Scholar 

  32. I. Mabillard, U. Wagner, Eliminating higher-multiplicity intersections, I. A Whitney trick for Tverberg-type problems, Preprint, 46 pages, Aug 2015, arXiv:1508.02349

  33. B.M. Mann, R.J. Milgram, On the Chern classes of the regular representations of some finite groups. Proc. Edinb. Math. Soc. (2) 25, 259–268 (1982)

    Google Scholar 

  34. J. Matoušek, Using the Borsuk–Ulam Theorem. Lectures on Topological Methods in Combinatorics and Geometry (Universitext, Springer, Heidelberg, 2003). Second corrected printing 2008

    Google Scholar 

  35. J. McCleary, A User’s Guide to Spectral Sequences, 2nd edn. Cambridge Studies in Advanced Mathematics, vol. 58 (Cambridge University Press, Cambridge, 2001)

    Google Scholar 

  36. M. Özaydin, Equivariant maps for the symmetric group, Preprint, 17 pages (1987), http://digital.library.wisc.edu/1793/63829

    Google Scholar 

  37. J. Radon, Mengen konvexer Körper, die einen gemeinsamen Punkt enthalten. Math. Ann. 83, 113–115 (1921)

    Article  MathSciNet  MATH  Google Scholar 

  38. K.S. Sarkaria, A generalized van Kampen–Flores theorem. Proc. Am. Math. Soc. 11, 559–565 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  39. J. Shareshian, M. Wachs, Torsion in the matching complex and chessboard complex. Adv. Math. 212, 525–570 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  40. A. Skopenkov, A user’s guide to disproof of topological Tverberg conjecture, Preprint 2016, arXiv:1605.05141

  41. S. Smale, A Vietoris mapping theorem for homotopy. Proc. Am. Math. Soc. 8, 604–610 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  42. P. Soberón, Equal coefficients and tolerance in coloured Tverberg partitions, in Proceedings of 29th Annual Symposium on Computational Geometry (SoCG), Rio de Janeiro, June 2013 (ACM, 2013), pp. 91–96

    Google Scholar 

  43. P. Soberón, Equal coefficients and tolerance in coloured Tverberg partitions. Combinatorica 35, 235–252 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  44. T. tom Dieck, Transformation Groups. Studies in Mathematics, vol. 8 (Walter de Gruyter, Berlin, 1987)

    Google Scholar 

  45. H. Tverberg, A generalization of Radon’s theorem. J. Lond. Math. Soc. 41, 123–128 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  46. E.R. Van Kampen, Komplexe in euklidischen Räumen. Abh. Math. Semin. Univ. Hamburg 9, 72–78 (1933)

    Article  MATH  Google Scholar 

  47. A.Yu. Volovikov, On a topological generalization of Tverberg’s theorem. Math. Notes 59(3), 454–456 (1996)

    Article  MathSciNet  Google Scholar 

  48. A.Yu. Volovikov, On the van Kampen-Flores theorem. Math. Notes 59(5), 477–481 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  49. S. Vrećica, R.T. Živaljević, New cases of the colored Tverberg theorem, in Jerusalem Combinatorics’93, Jerusalem, ed. by H. Barcelo, G. Kalai. Contemporary Mathematics, vol. 178 (American Mathematical Society, 1994), pp. 325–325

    Google Scholar 

  50. H. Whitney, The self-intersections of a smooth n-manifold in 2n-space. Ann. Math. 45, 220–246 (1944)

    Article  MathSciNet  MATH  Google Scholar 

  51. G.M. Ziegler, 3N colored points in a plane. Not. Am. Math. Soc. 58(4), 550–557 (2011)

    MathSciNet  MATH  Google Scholar 

  52. R.T. Živaljević, User’s guide to equivariant methods in combinatorics II. Publications de l’Institut Mathématique 64(78), 107–132 (1998)

    MathSciNet  MATH  Google Scholar 

  53. R.T. Živaljević, S. Vrećica, The colored Tverberg’s problem and complexes of injective functions. J. Combin. Theory Ser. A 61, 309–318 (1992)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

We are grateful to Alexander Engström and Florian Frick for excellent observations on drafts of this paper and many useful comments. We want to express our gratitude to Peter Landweber for his continuous help and support in improving this manuscript.

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Authors and Affiliations

  1. Institute of Mathematics, FU Berlin, Arnimallee 2, 14195, Berlin, Germany

    Pavle V. M. Blagojević

  2. Mathematical Institute SANU, Knez Mihailova 36, 11001, Beograd, Serbia

    Pavle V. M. Blagojević

  3. Institute of Mathematics, FU Berlin, Arnimallee 2, 14195, Berlin, Germany

    Günter M. Ziegler

Authors
  1. Pavle V. M. Blagojević
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Correspondence to Günter M. Ziegler .

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Editors and Affiliations

  1. Department of Applied Mathematics, Charles University, Praha, Czech Republic

    Martin Loebl

  2. Computer Science Institute of Charles University, Charles University, Praha, Czech Republic

    Jaroslav Nešetřil

  3. School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia, USA

    Robin Thomas

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Blagojević, P.V.M., Ziegler, G.M. (2017). Beyond the Borsuk–Ulam Theorem: The Topological Tverberg Story. In: Loebl, M., Nešetřil, J., Thomas, R. (eds) A Journey Through Discrete Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-44479-6_11

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