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A Proof of KÜhnel’s Conjecture for n ⩾ k2 + 3k Eric Sparla

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Abstract

In this note we show that an Upper Bound Conjecture made by Kühnel for combinatorial 2k-manifolds holds for fixed k if its number of vertices is at least n ⩾ k2 + 3k. Together with known results this provides a simple proof of the conjecture for k = 1 and k = 2.

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References

  1. A. Björner and G. Kalai, On f-vectors and homology, Combinatorial Mathematics (G. J. Bloom et al., eds.), Proc. New York Acad. Sci. 555, 63–80 (1989)

  2. U. Brehm and W. Kühnel, Combinatorial manifolds with few vertices, Topology 26, 465–473 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  3. J. Eells and N. H. Kuiper, Manifolds which are like projective planes, Publ. Math. I.H.E.S. 14, 181–222(1962)

    MATH  Google Scholar 

  4. B. Grünbaum, Convex Polytopes, Interscience Publishers, New York 1967

    MATH  Google Scholar 

  5. G. Kalai, Rigidity and the Lower Bound Theorem I, Invent. Math. 88, 125–151 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  6. V. Klee, The number of vertices of a convex polytope, Can. J. Math. 16, 701–720 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  7. W. Kühnel, Manifolds in the skeletons of convex polytopes, tightness, and generalized Heawood inequalities, Polytopes: Abstract, Convex and Computational (T. Bisztriczky et al., eds.), 241–247, World Scientific, Singapore 1994

    Google Scholar 

  8. W. Kühnel, Tight Polyhedral Submanifolds and Tight Triangulations, Springer Lecture Notes in Mathematics 1612, Berlin - Heidelberg - New York 1995

  9. N. H. Kuiper, A short history of triangulation and related matters, Proc. bicenten. Congr. Wiskd. Genoot., Part I, Amsterdam 1978, Math. Cent. Tracts 100, 61–79 (1979)

    MathSciNet  Google Scholar 

  10. P. McMullen and G. C. Shephard, Convex Polytopes and the Upper Bound Conjecture, London Mathematical Society Lecture Notes Series 3, Cambridge University Press 1971

  11. R. Riordan, Combinatorial Identities, Wiley and Sons, New York 1968

    MATH  Google Scholar 

  12. R. Stanley, The upper bound conjecture and Cohen-Macaulay-rings, Studies in Applied Math. 54, 135–142 (1975)

    MathSciNet  MATH  Google Scholar 

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  1. Mathematisches Institut B4 Universität Stuttgart, D-70550, Stuttgart, Germany

    Eric Sparla

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  1. Eric Sparla
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Correspondence to Eric Sparla.

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Sparla, E. A Proof of KÜhnel’s Conjecture for n ⩾ k2 + 3k Eric Sparla. Results. Math. 31, 386–393 (1997). https://doi.org/10.1007/BF03322172

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