Document Open Access Logo

Eliminating Higher-Multiplicity Intersections, II. The Deleted Product Criterion in the r-Metastable Range

Authors Isaac Mabillard, Uli Wagner

PDF
Thumbnail PDF

File

LIPIcs.SoCG.2016.51.pdf
  • Filesize: 0.59 MB
  • 12 pages

Document Identifiers

Author Details

Isaac Mabillard
Uli Wagner

Cite As Get BibTex

Isaac Mabillard and Uli Wagner. Eliminating Higher-Multiplicity Intersections, II. The Deleted Product Criterion in the r-Metastable Range. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 51:1-51:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016) https://doi.org/10.4230/LIPIcs.SoCG.2016.51

Abstract

Motivated by Tverberg-type problems in topological combinatorics and by classical results about embeddings (maps without double points), we study the question whether a finite simplicial complex K can be mapped into R^d without higher-multiplicity intersections.

We focus on conditions for the existence of almost r-embeddings, i.e., maps f: K -> R^d such that the intersection of f(sigma_1), ..., f(sigma_r) is empty whenever sigma_1,...,sigma_r are pairwise disjoint simplices of K.

Generalizing the classical Haefliger-Weber embeddability criterion, we show that a well-known necessary deleted product condition for the existence of almost r-embeddings is sufficient in a suitable r-metastable range of dimensions: If r d > (r+1) dim K + 2 then there exists an almost r-embedding K-> R^d if and only if there exists an equivariant map of the r-fold deleted product of K to the sphere S^(d(r-1)-1).

This significantly extends one of the main results of our previous paper (which treated the special case where d=rk and dim K=(r-1)k, for some k> 2), and settles an open question raised there.

Subject Classification

Keywords
  • Topological Combinatorics
  • Tverberg-Type Problems
  • Simplicial Complexes
  • Piecewise-Linear Topology
  • Haefliger-Weber Theorem

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

References

  1. S. Avvakumov, I. Mabillard, A. Skopenkov, and U. Wagner. Eliminating higher-multiplicity intersections, III. Codimension 2. Preprint, http://arxiv.org/abs/1511.03501, 2015. Google Scholar
  2. E. G. Bajmóczy and I. Bárány. On a common generalization of Borsuk’s and Radon’s theorem. Acta Math. Acad. Sci. Hungar., 34(3-4):347-350 (1980), 1979. URL: http://dx.doi.org/10.1007/BF01896131.
  3. I. Bárány and D. G. Larman. A colored version of Tverberg’s theorem. J. London Math. Soc. (2), 45(2):314-320, 1992. URL: http://dx.doi.org/10.1112/jlms/s2-45.2.314.
  4. I. Bárány, S. B. Shlosman, and A. Szűcs. On a topological generalization of a theorem of Tverberg. J. London Math. Soc., II. Ser., 23:158-164, 1981. Google Scholar
  5. Imre Bárány, Zoltán Füredi, and Lászlo Lovász. On the number of halving planes. Combinatorica, 10(2):175-183, 1990. Google Scholar
  6. P. V. M. Blagojević, B. Matschke, and G. M. Ziegler. Optimal bounds for the colored Tverberg problem. J. Eur. Math. Soc., 17(4):739-754, 2015. Google Scholar
  7. Pavle V. M. Blagojević, Florian Frick, and Günter M. Ziegler. Tverberg plus constraints. Bull. Lond. Math. Soc., 46(5):953-967, 2014. URL: http://dx.doi.org/10.1112/blms/bdu049.
  8. John L. Bryant. Piecewise linear topology. In Handbook of geometric topology, pages 219-259. North-Holland, Amsterdam, 2002. Google Scholar
  9. Martin Čadek, Marek Krčál, and Lukáš Vokřínek. Algorithmic solvability of the lifting-extension problem. Preprint, arXiv:1307.6444, 2013. Google Scholar
  10. Haim Chojnacki (Hanani). Über wesentlich unplättbare Kurven im dreidimensionalen Raume. Fund. Math., 23:135-142, 1934. Google Scholar
  11. Michael H. Freedman, Vyacheslav S. Krushkal, and Peter Teichner. van Kampen’s embedding obstruction is incomplete for 2-complexes in R⁴. Math. Res. Lett., 1(2):167-176, 1994. URL: http://dx.doi.org/10.4310/MRL.1994.v1.n2.a4.
  12. Florian Frick. Counterexamples to the topological Tverberg conjecture. Preprint, arXiv:1502.00947, 2015. Google Scholar
  13. Daciberg Gonçalves and Arkadiy Skopenkov. Embeddings of homology equivalent manifolds with boundary. Topology Appl., 153(12):2026-2034, 2006. Google Scholar
  14. Mikhail Gromov. Singularities, expanders and topology of maps. part 2: From combinatorics to topology via algebraic isoperimetry. Geometric and Functional Analysis, 20(2):416-526, 2010. Google Scholar
  15. P. M. Gruber and R. Schneider. Problems in geometric convexity. In Contributions to geometry (Proc. Geom. Sympos., Siegen, 1978), pages 255-278. Birkhäuser, Basel-Boston, Mass., 1979. Google Scholar
  16. André Haefliger. Plongements différentiables dans le domaine stable. Comment. Math. Helv., 37:155-176, 1962/1963. Google Scholar
  17. Isaac Mabillard and Uli Wagner. Eliminating Tverberg points, I. An analogue of the Whitney trick. In Proc. 30th Ann. Symp. on Computational Geometry, pages 171-180, 2014. Google Scholar
  18. Isaac Mabillard and Uli Wagner. Eliminating higher-multiplicity intersections, I. a Whitney trick for Tverberg-type problems. Preprint, arXiv:1508.02349, 2015. Google Scholar
  19. Isaac Mabillard and Uli Wagner. Eliminating Higher-Multiplicity Intersections, II. The Deleted Product Criterion in the r-Metastable Range. Preprint http://arxiv.org/abs/1601.00876, 2016.
  20. S. Mardešić and J. Segal. ε-Mappings and generalized manifolds. Mich. Math. J., 14:171-182, 1967. Google Scholar
  21. Jiří Matoušek. Using the Borsuk-Ulam theorem. Springer-Verlag, Berlin, 2003. Google Scholar
  22. Jiří Matoušek, Eric Sedgwick, Martin Tancer, and Uli Wagner. Embeddability in the 3-sphere is decidable. Preprint, arXiv:1402.0815, 2014. Google Scholar
  23. Jiří Matoušek, Martin Tancer, and Uli Wagner. Hardness of embedding simplicial complexes in ℝ^d. J. Eur. Math. Soc., 13(2):259-295, 2011. Google Scholar
  24. Murat Özaydin. Equivariant maps for the symmetric group. Unpublished manuscript, available online at http://minds.wisconsin.edu/handle 63829, 1987. Google Scholar
  25. Michael J. Pelsmajer, Marcus Schaefer, and Daniel Štefankovič. Removing even crossings. J. Combin. Theory Ser. B, 97(4):489-500, 2007. URL: http://dx.doi.org/10.1016/j.jctb.2006.08.001.
  26. Michael J. Pelsmajer, Marcus Schaefer, and Daniel Štefankovič. Removing independently even crossings. SIAM J. Discrete Math., 24(2):379-393, 2010. URL: http://dx.doi.org/10.1137/090765729.
  27. Dušan Repovš and Arkadiy B. Skopenkov. New results on embeddings of polyhedra and manifolds into Euclidean spaces. Uspekhi Mat. Nauk, 54(6(330)):61-108, 1999. Google Scholar
  28. Colin Patrick Rourke and Brian Joseph Sanderson. Introduction to piecewise-linear topology. Springer Study Edition. Springer-Verlag, Berlin, 1982. Reprint. Google Scholar
  29. K. S. Sarkaria. A generalized van Kampen-Flores theorem. Proc. Amer. Math. Soc., 111(2):559-565, 1991. URL: http://dx.doi.org/10.2307/2048349.
  30. J. Segal, A. Skopenkov, and S. Spież. Embeddings of polyhedra in ℝ^m and the deleted product obstruction. Topology Appl., 85(1-3):335-344, 1998. Google Scholar
  31. J. Segal and S. Spież. Quasi embeddings and embeddings of polyhedra in ℝ^m. Topology Appl., 45(3):275-282, 1992. Google Scholar
  32. A. Skopenkov. On the deleted product criterion for embeddability in R^m. Proc. Amer. Math. Soc., 126(8):2467-2476, 1998. URL: http://dx.doi.org/10.1090/S0002-9939-98-04142-2.
  33. Arkadiy B. Skopenkov. Embedding and knotting of manifolds in Euclidean spaces. In Surveys in contemporary mathematics, volume 347 of London Math. Soc. Lecture Note Ser., pages 248-342. Cambridge Univ. Press, Cambridge, 2008. Google Scholar
  34. William T. Tutte. Toward a theory of crossing numbers. J. Combin. Theory, 8:45-53, 1970. Google Scholar
  35. H. Tverberg. A generalization of Radon’s theorem. J. London Math. Soc., 41:123-128, 1966. Google Scholar
  36. A. Yu. Volovikov. On the van Kampen-Flores theorem. Mat. Zametki, 59(5):663-670, 797, 1996. URL: http://dx.doi.org/10.1007/BF02308813.
  37. Claude Weber. Plongements de polyhèdres dans le domaine métastable. Comment. Math. Helv., 42:1-27, 1967. Google Scholar
  38. Erik Christopher Zeeman. Seminar on combinatorial topology. Institut des Hautes Études Scientifiques, 1966. Google Scholar
  39. Rade T. Živaljević. User’s guide to equivariant methods in combinatorics. Publ. Inst. Math. Beograd, 59(73):114-130, 1996. Google Scholar
  40. Rade T. Živaljević. User’s guide to equivariant methods in combinatorics. II. Publ. Inst. Math. (Beograd) (N.S.), 64(78):107-132, 1998. Google Scholar
  41. Rade T. Živaljević and Sinisa T. Vrećica. The colored Tverberg’s problem and complexes of injective functions. J. Combin. Theory Ser. A, 61(2):309-318, 1992. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact