Skip to main content
SpringerLink
Log in

Eliminating higher-multiplicity intersections. III. Codimension 2

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

We study conditions under which a finite simplicial complex K can be mapped to ℝd without higher-multiplicity intersections. An almost r-embedding is a map f: K → ℝd such that the images of any r pairwise disjoint simplices of K do not have a common point. We show that if r is not a prime power and d ≥ 2r + 1, then there is a counterexample to the topological Tverberg conjecture, i.e., there is an almost r-embedding of the (d +1)(r − 1)-simplex in ℝd. This improves on previous constructions of counterexamples (for d ≥ 3r) based on a series of papers by M. Özaydin, M. Gromov, P. Blagojević, F. Frick, G. Ziegler, and the second and fourth present authors.

The counterexamples are obtained by proving the following algebraic criterion in codimension 2: If r ≥ 3 and if K is a finite 2(r − 1)-complex, then there exists an almost r-embedding K → ℝ2r if and only if there exists a general position PL map f: K → ℝ2r such that the algebraic intersection number of the f-images of any r pairwise disjoint simplices of K is zero. This result can be restated in terms of a cohomological obstruction and extends an analogous codimension 3 criterion by the second and fourth authors. As another application, we classify ornaments f: S3 ⊔ S3 ⊔ S3 → ℝ5 up to ornament concordance.

It follows from work of M. Freedman, V. Krushkal and P. Teichner that the analogous criterion for r = 2 is false. We prove a lemma on singular higher-dimensional Borromean rings, yielding an elementary proof of the counterexample.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. S. Avvakumov, I. Mabillard, A. Skopenkov and U. Wagner. Eliminating higher-multiplicity intersections. III. Codimension 2 [Extended abstract], Russian Mathematical Surveys 75 (2020), 173–174.

    Article  MATH  Google Scholar 

  2. S. Avvakumov, R. Karasev and A. Skopenkov, Stronger counterexamples to the topological Tverberg conjecture, https://arxiv.org/abs/1908.08731.

  3. D. Bar-Natan, Finite type invariants of doodles. 1, http://www.math.toronto.edu/∼drorbn/Talks/Portfolio/PortfolioPage029.pdf

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

    Article  MathSciNet  MATH  Google Scholar 

  5. I. Bárány, P. V. M. Blagojević and G. M. Ziegler, Tverberg’s Theorem at 50: Extensions and counterexamples, Notices of the American Mathematical Society 63 (2016), 732–739.

    Article  MathSciNet  MATH  Google Scholar 

  6. V. Boltyanskiy and V. Efremovich, Intuitive Combinatorial Topology, Universitext, Springer, New York, 2001

    Book  Google Scholar 

  7. P. V. M. Blagojević, F. Frick and G. M. Ziegler, Tverberg plus constraints, Bulletin of the London Mathematical Society 46 (2014), 953–967.

    Article  MathSciNet  MATH  Google Scholar 

  8. P.V.M. Blagojević, F. Frick and G. M. Ziegler, Barycenters of polytope skeleta and counterexamples to the topological Tverberg conjecture, via constraints, Journal of the European Mathematical Society 21 (2019), 2107–2116.

    Article  MathSciNet  MATH  Google Scholar 

  9. I. Báaraány and P. Soberáon, Tverberg’s theorem is 50 years old: a survey, Bulletin of the American Mathematical Society 55 (2018), 459–492.

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  11. P. V. M. Blagojevič and G. M. Ziegler, Beyond the Borsuk–Ulam theorem: The topological Tverberg story, in A Journey Through Discrete Mathematics Springer, Cham, 2017, pp. 273–341.

    Chapter  Google Scholar 

  12. M. Cencelj, D. Repovš and M. Skopenkov, Classification of embeddings of tori in the 2-metastable dimension, Sbornik Mathematics 203 (2012), 1654–1681.

    Article  MathSciNet  MATH  Google Scholar 

  13. D. Crowley and A. Skopenkov, Embeddings of non-simply-connected 4-manifolds in 7-space. III. Piecewise-linear classification.

  14. M. H. Freedman, V. S. Krushkal and P. Teichner, Van Kampen’s embedding obstruction is incomplete for 2-complexes in ℝ4, Mathematical Research Letters 1 (1994), 167–176.

    Article  MathSciNet  MATH  Google Scholar 

  15. R. Fenn and P. Taylor, Introducing doodles, in Topology of Low-Dimensional Manifolds (Proceedings of the Second Sussex Conference, Chelwood Gate, 1977), Lecture Notes in Mathematics, Vol. 722, Springer, Berlin, 1979, pp. 37–43.

    Chapter  Google Scholar 

  16. M. Gromov, Singularities, expanders and topology of maps. Part 2: From combinatorics to topology via algebraic isoperimetry, Geometric and Functional Analysis 20 (2010), 416–526.

    Article  MathSciNet  MATH  Google Scholar 

  17. P. M. Gruber and R. Schneider, Problems in geometric convexity, in Contributions to Geometry (Proc. Geom. Sympos., Siegen, 1978), Birkhäuser, Basel–Boston, MA, 1979, pp. 255–278.

    Chapter  Google Scholar 

  18. H. Chojnacki (Hanani), Über wesentlich unplättbare Kurven im dreidimensionalen Raume, Fundamenta Mathematicae 23 (1934), 135–142.

    Article  MATH  Google Scholar 

  19. A. Haefliger, Knotted (4k − 1)-spheres in 6k-space, Annals of Mathematics 75 (1962) 452–466.

    Article  MathSciNet  MATH  Google Scholar 

  20. A. Haefliger, Plongements differentiables dans le domain stable, Commentarii Mathematici Helvetici 36 (1962–63), 155–176.

  21. N. Habegger, Obstruction to embedding disks II: a proof of a conjecture by Hudson, Topology and its Applications 17 (1984), 123–130.

    Article  MathSciNet  MATH  Google Scholar 

  22. N. Habegger and U. Kaiser, Link homotopy in 2–metastable range, Topology 37 (1998) 75–94.

    Article  MathSciNet  MATH  Google Scholar 

  23. R. C. Kirby, The Topology of 4-Manifolds, Lecture Notes in Mathematics, Vol. 1374, Springer, Berlin, 1989.

    Book  MATH  Google Scholar 

  24. M. Kervaire and J. W. Milnor, On 2-spheres in 4-manifolds, Proceedings of the National Academy of Sciences of the United States of America 47 (1961), 1651–1657.

    Article  MathSciNet  MATH  Google Scholar 

  25. V. Krushkal and P. Teichner, Alexander duality, gropes and link homotopy, Geometry & Topology 1 (1997), 51–69.

    Article  MathSciNet  MATH  Google Scholar 

  26. M. de Longueville, Notes on the topological Tverberg theorem, Discrete Mathematics 247 (2002), 271–297.

    Article  MathSciNet  MATH  Google Scholar 

  27. S. Melikhov, Pseudohomotopy implies homotopy for singular links of codimension ⩾ 3, Russian Mathematical Surveys 55 (2000), 589–590.

    Article  MathSciNet  MATH  Google Scholar 

  28. S. Melikhov, A triple-point Whitney trick, Journal of Topology and Analysis 12 (2020), 1041–1046.

    Article  MathSciNet  MATH  Google Scholar 

  29. S. Melikhov, Gauss type formulas for link map invariants, preprint, https://arxiv.org/abs/1711.03530.

  30. S. Melikhov, Singular link concordance implies link homotopy in codimension ⁥ 3, preprint, https://arxiv.org/abs/1810.08299.

  31. A. Merkov, Vassiliev invariants classify plane curves and doodles, Sbornik. Mathematics 194 (2003), 1301–1330.

    Article  MathSciNet  MATH  Google Scholar 

  32. I. Mabillard and U. Wagner, Eliminating higher-multiplicity intersections, I. A Whitney trick for Tverberg-type problems, preprint, https://arxiv.org/abs/1508.02349.

  33. I. Mabillard and U. Wagner, Eliminating higher-multiplicity intersections, II. The deleted product criterion in the r-metastable range, preprint, https://arxiv.org/abs/1601.00876.

  34. I. Mabillard and U. Wagner, Eliminating Tverberg points, I. An analogue of the Whitney trick, in Computational Geometry (SoCG’14), ACM, New York, 2014, pp. 171–180.

    Google Scholar 

  35. I. Mabillard and U. Wagner, Eliminating higher-multiplicity intersections, II. The deleted product criterion in the r-metastable range, in 32nd Annual Symposium on Computational Geometry (SoCG’16), Leibniz International Proceedings in Informatics, Vol. 51, Schloss Dagstuhl, Leibniz-Zentrum für Informatik, Wadern, 2016, Article no. 51.

    Google Scholar 

  36. M. Özaydin, Equivariant maps for the symmetric group, unpublished manuscript, available online at http://minds.wisconsin.edu/handle/1793/63829.

  37. R. Porter, Milnor’s \(\bar \mu \)-invariants and Massey products, Transactions of the American Mathematical Society 257 (1980), 39–71.

    MathSciNet  MATH  Google Scholar 

  38. V. V. Prasolov and A. B. Sossinsky, Knots, Links, Braids, and 3-manifolds, Translations of Mathematical Monographs, Vol. 154, American Mathematical Society, Providence, RI, 1997.

    MATH  Google Scholar 

  39. C. P. Rourke and B. J. Sanderson, Introduction to Piecewise-Linear Topology, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 69, Springer, New York-Heidelberg, 1972.

    Book  MATH  Google Scholar 

  40. K. S. Sarkaria, A generalized van Kampen–Flores theorem, Proceedings of the American Mathematical Society 111 (1991), 559–565.

    Article  MathSciNet  MATH  Google Scholar 

  41. M. Schaefer, Toward a theory of planarity: Hanani–Tutte and planarity variants, Journal of Graph Algorithms and Applications 17 (2013), 367–440.

    Article  MathSciNet  MATH  Google Scholar 

  42. A. Shapiro, Obstructions to the imbedding of a complex in a euclidean space. I. The first obstruction, Annals of Mathematics 66 (1957), 256–269.

    Article  MathSciNet  MATH  Google Scholar 

  43. A. Skopenkov, On the generalized Massey–Rolfsen invariant for link maps, Fundamenta Mathematicae 165 (2000), 1–15.

    Article  MathSciNet  MATH  Google Scholar 

  44. A. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Mathematical Society Lecture Notes Series, Vol. 347, Cambridge University Press, Cambridge, 2008, pp. 248–342.

    MATH  Google Scholar 

  45. A. Skopenkov, Classification of smooth embeddings of 3-manifolds in 6-space, Mathematische Zeitschrift 260 (2008), 647–672.

    Article  MathSciNet  MATH  Google Scholar 

  46. A. Skopenkov, A classification of smooth embeddings of 4-manifolds in 7-space, I, Topology and its Applications 157 (2010), 2094–2110.

    Article  MathSciNet  MATH  Google Scholar 

  47. A. Skopenkov, Realizability of hypergraphs and Ramsey link theory, preprint, https://arxiv.org/abs/1402.0658.

  48. A. Skopenkov, A user’s guide to the topological Tverberg Conjecture, Russian Mathematical Surveys 73 (2018), 323–353.

    Article  MathSciNet  MATH  Google Scholar 

  49. A. Skopenkov, Eliminating higher-multiplicity intersections in the metastable dimension range, preprint, https://arxiv.org/abs/1704.00143.

  50. A. Skopenkov, Algebraic topology from algorithmic point of view, draft of a book, http://www.mccme.ru/circles/oim/algor.pdf

  51. S. Spież, Imbeddings in ℝ2mof m-dimensional compacta withdim (X ×) < 2m, Fundamenta Mathematicae 134 (1990), 105–115.

    MathSciNet  MATH  Google Scholar 

  52. J. Segal, A. Skopenkov and S. Spież, Embeddings of polyhedra in ℝmand the deleted product obstruction, Topology and its Applications 85 (1998), 225–234.

    Article  MATH  Google Scholar 

  53. H. Seifert and W. Threlfall, A Textbook of Topology, Pure and Applied Mathematics, Vol. 89, Academic Press, New York-London, 1980.

    MATH  Google Scholar 

  54. S. Spież and H. Toruńczyk, Moving compacta in ℝmapart, Topology and its Applications 41 (1991), 193–204.

    Article  MathSciNet  MATH  Google Scholar 

  55. A. Storjohann, Near optimal algorithms for computing Smith normal forms of integer matrices, in ISSAC’ 96 Proceedings of the 1996 International Symposium on Symbolic and Algebraic Computation, ACM, New York, 1996, pp. 267–274.

    Chapter  Google Scholar 

  56. R. Schneiderman and P. Teichner, Pulling apart 2-spheres in 4-manifolds, Documenta Mathematica 19 (2014), 941–992.

    Article  MathSciNet  MATH  Google Scholar 

  57. A. Skopenkov and M. Tancer, Hardness of almost embedding simplicial complexes in ℝd, Discrete & Computational Geometry 61 (2019), 452–463.

    Article  MathSciNet  MATH  Google Scholar 

  58. W. T. Tutte, Toward a theory of crossing numbers, Journal of Combinatorial Theory 8 (1970), 45–53.

    Article  MathSciNet  MATH  Google Scholar 

  59. E. R. van Kampen, Komplexe in euklidischen Räumen, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 9 (1933), 72–78; Berichtigung dazu, 152–153.

    Article  MathSciNet  MATH  Google Scholar 

  60. A. Yu. Volovikov, On a topological generalization of the Tverberg theorem, Mathematical Notes 59 (1996), 324–326.

    Article  MathSciNet  MATH  Google Scholar 

  61. A. Yu. Volovikov, On the van Kampen–Flores theorem, Mathematical Notes 59 (1996), 477–481.

    Article  MathSciNet  MATH  Google Scholar 

  62. W. T. Wu, A Theory of Embedding, Immersion and Isotopy of Polytopes in an Euclidean Space, Science Press, Peking, 1965.

    Google Scholar 

  63. C. Weber, Plongements de polyèdres dans le domain metastable, Commentarii Mathematici Helvetici 42 (1967), 1–27.

    Article  MathSciNet  MATH  Google Scholar 

  64. H. Whitney, The self-intersections of a smooth n-manifolds in 2n-space, Annals of Mathematics 45 (1944), 220–246.

    Article  MathSciNet  MATH  Google Scholar 

  65. R. T. Živaljević, Topological methods in discrete geometry, in Handbook of Discrete and Computational Geometry, CRC Press, Boca Raton, FL, 2017.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Science and Technology Austria, Am Campus 1, 3400, Klosterneuburg, Austria

    Sergey Avvakumov, Isaac Mabillard & Uli Wagner

  2. Moscow Institute of Physics and Technology, Institutskiy per., Dolgoprudnyi, 141700, Russia

    Arkadiy B. Skopenkov

  3. Independent University of Moscow, B. Vlasyevskiy, 11, Moscow, 119002, Russia

    Arkadiy B. Skopenkov

Authors
  1. Sergey Avvakumov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Isaac Mabillard
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Arkadiy B. Skopenkov
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Uli Wagner
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Uli Wagner.

Additional information

Research supported by the Swiss National Science Foundation (Project SNSF-PP00P2-138948), by the Austrian Science Fund (FWF Project P31312-N35), by the Russian Foundation for Basic Research (Grants No. 15-01-06302 and 19-01-00169), by a Simons-IUM Fellowship, and by the D. Zimin Dynasty Foundation Grant. We would like to thank E. Alkin, A. Klyachko, V. Krushkal, S. Melikhov, M. Tancer, P. Teichner and anonymous referees for helpful comments and discussions.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Avvakumov, S., Mabillard, I., Skopenkov, A.B. et al. Eliminating higher-multiplicity intersections. III. Codimension 2. Isr. J. Math. 245, 501–534 (2021). https://doi.org/10.1007/s11856-021-2216-z

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-021-2216-z

Search

Navigation