On Codimension One Embedding of Simplicial Complexes



We study d-dimensional simplicial complexes that are PL embeddable in \(\mathbb{R}^{d+1}\). It is shown that such a complex must satisfy a certain homological condition. The existence of this obstruction allows us to provide a systematic approach to deriving upper bounds for the number of top-dimensional faces of such complexes, particularly in low dimensions.



We thank the referees for very useful remarks. Also, we thank Uli Wagner for valuable comments and for bringing Grünbaum’s paper  [8] to our attention.

This research was supported by the grant VR-2015-05308 from Vetenskapsrådet. Part of the research of the second author has been made possible by the grant KAW-stipendiet 2015.0360 from the Knut and Alice Wallenberg’s Fondation.


  1. 1.
    N.L. Biggs, E. Keith Lloyd, R.J. Wilson, Graph Theory. 1736–1936, 2nd edn. (The Clarendon Press/Oxford University Press, New York, 1986)Google Scholar
  2. 2.
    A. Björner, G. Kalai, An extended Euler-Poincaré theorem. Acta Math. 161(3–4), 279–303 (1988)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    U. Brehm, A nonpolyhedral triangulated Möbius strip. Proc. Am. Math. Soc. 89(3), 519–522 (1983)MATHGoogle Scholar
  4. 4.
    J.W. Cannon, Shrinking cell-like decompositions of manifolds. Codimension three. Ann. Math. (2) 110(1), 83–112 (1979)Google Scholar
  5. 5.
    T.K. Dey, H. Edelsbrunner, Counting triangle crossings and halving planes. Discret. Comput. Geom. 12(3), 281–289 (1994). ACM Symposium on Computational Geometry, San Diego, 1993Google Scholar
  6. 6.
    T.K. Dey, J. Pach, Extremal problems for geometric hypergraphs. Discret. Comput. Geom. 19(4), 473–484 (1998)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    C. Greene, D.J. Kleitman, Proof techniques in the theory of finite sets, in Studies in Combinatorics. MAA Studies in Mathematics, vol. 17 (Mathematical Association of America, Washington, DC, 1978), pp. 22–79Google Scholar
  8. 8.
    B. Grünbaum, Higher-dimensional analogs of the four-color problem and some inequalities for simplicial complexes. J. Comb. Theory 8, 147–153 (1970)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    A. Gundert, Extermal combinatorics II: some geometry and number theory (2008), https://gilkalai.wordpress.com/2008/07/17/extermal-combinatorics-ii-some-geometry-and-number-th eory/. Combinatorics and More
  10. 10.
    A. Gundert, On the complexity of embeddable simplicial complexes, Master’s thesis, Freie Universität Berlin, 2009Google Scholar
  11. 11.
    S. Mac Lane, A structural characterization of planar combinatorial graphs. Duke Math. J. 3(3), 460–472 (1937)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    F.H. Lutz, T. Sulanke, E. Swartz, f-vectors of 3-manifolds. Electron. J. Comb. 16(2), R13 (2009)Google Scholar
  13. 13.
    J. Matoušek, Using the Borsuk-Ulam Theorem. Lectures on Topological Methods in Combinatorics and Geometry. Universitext (Springer, Berlin, 2003)Google Scholar
  14. 14.
    J. Matoušek, M. Tancer, U. Wagner, Hardness of embedding simplicial complexes in \(\mathbb{R}^{d}\). J. Eur. Math. Soc. (JEMS) 13(2), 259–295 (2011)Google Scholar
  15. 15.
    J. Milnor, Morse Theory. Annals of Mathematics Studies AM-51 (Princeton University Press, Princeton, 1963)Google Scholar
  16. 16.
    J.R. Munkres, Elements of Algebraic Topology (Addison-Wesley Publishing Company, Menlo Park, 1984)MATHGoogle Scholar
  17. 17.
    S. Parsa, On links of vertices in simpliciald-complexes embeddable in Euclidean 2d-space (2015, preprint). arXiv:1512.05164
  18. 18.
    A. Shapiro, Obstructions to the imbedding of a complex in a Euclidean space. I. The first obstruction. Ann. Math. (2) 66, 256–269 (1957)Google Scholar
  19. 19.
    D. Sullivan, René Thom’s work on geometric homology and bordism. Bull. Am. Math. Soc. 41, 341–350 (2004)CrossRefMATHGoogle Scholar
  20. 20.
    W.T. Tutte, Toward a theory of crossing numbers. J. Comb. Theory 8, 45–53 (1970)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    E.R. van Kampen, Komplexe in euklidischen Räumen. Abh. Math. Sem. Univ. Hamburg 9(1), 72–78 (1933)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    U. Wagner, Minors in random and expanding hypergraphs, in Proceedings of the Twenty-Seventh Annual Symposium on Computational Geometry (ACM, 2011)Google Scholar
  23. 23.
    H. Whitney, Non-separable and planar graphs. Trans. Am. Math. Soc. 34(2), 339–362 (1932)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    J. Zaks, On minimal complexes. Pac. J. Math. 28 721–727 (1969)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    G.M. Ziegler, Lectures on Polytopes. Graduate Texts in Mathematics, vol. 152 (Springer, New York, 1995)Google Scholar

Copyright information

© Springer International publishing AG 2017

Authors and Affiliations

  1. 1.Department of MathematicsRoyal Institute of TechnologyStockholmSweden
  2. 2.Department of Mathematics, Discrete Geometry GroupFree University of BerlinBerlinGermany

Personalised recommendations