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On the generalized lower bound conjecture for polytopes and spheres

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Acta Mathematica

Abstract

In 1971, McMullen and Walkup posed the following conjecture, which is called the generalized lower bound conjecture: If P is a simplicial d-polytope then its h-vector (h 0, h 1, …, h d ) satisfies \( {h_0}\leq {h_1}\leq \ldots \leq {h_{{\left\lfloor {{d \left/ {2} \right.}} \right\rfloor }}} \). Moreover, if h r−1 = h r for some \( r\leq \frac{1}{2}d \) then P can be triangulated without introducing simplices of dimension ≤d − r.

The first part of the conjecture was solved by Stanley in 1980 using the hard Lefschetz theorem for projective toric varieties. In this paper, we give a proof of the remaining part of the conjecture. In addition, we generalize this result to a certain class of simplicial spheres, namely those admitting the weak Lefschetz property.

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References

  1. Bagchi, B. & Datta, B., On stellated spheres, shellable balls, lower bounds and a combinatorial criterion for tightness. Preprint, 2011. arXiv:1102.0856 [math.GT].

  2. Barnette, D.W., The minimum number of vertices of a simple polytope. Israel J. Math., 10 (1971), 121–125.

    Article  MathSciNet  MATH  Google Scholar 

  3. — A proof of the lower bound conjecture for convex polytopes. Pacific J. Math., 46 (1973), 349–354.

  4. Billera, L. J. & Lee, C. W., A proof of the sufficiency of McMullen’s conditions for f-vectors of simplicial convex polytopes. J. Combin. Theory Ser. A, 31 (1981), 237–255.

    Article  MathSciNet  MATH  Google Scholar 

  5. Bruns, W. & Herzog, J., Cohen–Macaulay Rings. Cambridge Studies in Advanced Mathematics, 39. Cambridge University Press, Cambridge, 1993.

  6. Dancis, J., Triangulated n-manifolds are determined by their \( \left( {\left\lfloor {{n \left/ {2} \right.}} \right\rfloor +1} \right) \)-skeletons. Topology Appl., 18 (1984), 17–26.

    Article  MathSciNet  MATH  Google Scholar 

  7. Flores, A., Über n-dimensionale komplexe die im \( {{\mathbb{R}}^{2n+1 }} \) absolut selbstverschlungen sind. Ergeb. Math. Kolloq. Wien, 6 (1935), 4–6.

    Google Scholar 

  8. Fulton, W., Introduction to Toric Varieties. Annals of Mathematics Studies, 131. Princeton University Press, Princeton, NJ, 1993.

  9. Goodman, J. E. & Pollack, R., Upper bounds for configurations and polytopes in R d. Discrete Comput. Geom., 1 (1986), 219–227.

    Article  MathSciNet  MATH  Google Scholar 

  10. Green, M. L., Generic initial ideals, in Six Lectures on Commutative Algebra (Bellaterra, 1996), Progr. Math., 166, pp. 119–186. Birkhäuser, Basel, 1998.

  11. Grünbaum, B., Convex Polytopes. Graduate Texts in Mathematics, 221. Springer, New York, 2003.

  12. Herzog, J. & Hibi, T., Monomial Ideals. Graduate Texts in Mathematics, 260. Springer, London, 2011.

  13. Kalai, G., Rigidity and the lower bound theorem. I. Invent. Math., 88 (1987), 125–151.

    Article  MathSciNet  MATH  Google Scholar 

  14. — Many triangulated spheres. Discrete Comput. Geom., 3 (1988), 1–14.

  15. — Some aspects of the combinatorial theory of convex polytopes, in Polytopes: Abstract, Convex and Computational (Scarborough, ON, 1993), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 440, pp. 205–229. Kluwer, Dordrecht, 1994.

  16. van Kampen, E. R., Komplexe in euklidischen Räumen. Abh. Math. Sem. Hamburg, 9 (1932), 72–78.

    Article  Google Scholar 

  17. Kleinschmidt, P. & Lee, C.W., On k-stacked polytopes. Discrete Math., 48 (1984), 125–127.

    Article  MathSciNet  MATH  Google Scholar 

  18. McMullen, P., The numbers of faces of simplicial polytopes. Israel J. Math., 9 (1971), 559–570.

    Article  MathSciNet  MATH  Google Scholar 

  19. — Triangulations of simplicial polytopes. Beitr. Algebra Geom., 45 (2004), 37–46.

  20. McMullen, P. & Walkup, D.W., A generalized lower-bound conjecture for simplicial polytopes. Mathematika, 18 (1971), 264–273.

    Article  MathSciNet  MATH  Google Scholar 

  21. Miller, E. & Sturmfels, B., Combinatorial Commutative Algebra. Graduate Texts in Mathematics, 227. Springer, New York, 2005.

  22. Munkres, J. R., Elements of Algebraic Topology. Addison–Wesley, Menlo Park, CA, 1984.

    MATH  Google Scholar 

  23. Nagel, U., Empty simplices of polytopes and graded Betti numbers. Discrete Comput. Geom., 39 (2008), 389–410.

    Article  MathSciNet  MATH  Google Scholar 

  24. Reisner, G. A., Cohen–Macaulay quotients of polynomial rings. Adv. Math., 21 (1976), 30–49.

    Article  MathSciNet  MATH  Google Scholar 

  25. Rudin, M. E., An unshellable triangulation of a tetrahedron. Bull. Amer. Math. Soc., 64 (1958), 90–91.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  27. Stanley, R. P., Cohen–Macaulay complexes, in Higher Combinatorics (Berlin, 1976), NATO Adv. Study Inst. Ser. Ser. C Math. Phys. Sci., 31, pp. 51–62. Reidel, Dordrecht, 1977.

  28. — The number of faces of a simplicial convex polytope. Adv. Math., 35 (1980), 236–238.

  29. Swartz, E., Face enumeration—from spheres to manifolds. J. Eur. Math. Soc. (JEMS), 11 (2009), 449–485.

    Article  MathSciNet  MATH  Google Scholar 

  30. Wu, W., A Theory of Imbedding, Immersion, and Isotopy of Polytopes in a Euclidean Space. Science Press, Beijing, 1965.

    MATH  Google Scholar 

  31. Ziegler, G. M., Lectures on Polytopes. Graduate Texts in Mathematics, 152. Springer, New York, 1995.

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Authors and Affiliations

  1. Department of Mathematical Science, Faculty of Science, Yamaguchi University, 1677-1 Yoshida, Yamaguchi, 753-8512, Japan

    Satoshi Murai

  2. Department of Mathematics, Ben Gurion University of the Negev, Be’er Sheva, 84105, Israel

    Eran Nevo

Authors
  1. Satoshi Murai
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  2. Eran Nevo
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Correspondence to Eran Nevo.

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Murai, S., Nevo, E. On the generalized lower bound conjecture for polytopes and spheres. Acta Math 210, 185–202 (2013). https://doi.org/10.1007/s11511-013-0093-y

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  • DOI: https://doi.org/10.1007/s11511-013-0093-y

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