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ON POLYTOPAL UPPER BOUND SPHERES

  • Bhaskar Bagchi (a1) and Basudeb Datta (a2)

Abstract

Generalizing a result (the case $k= 1$ ) due to M. A. Perles, we show that any polytopal upper bound sphere of odd dimension $2k+ 1$ belongs to the generalized Walkup class ${ \mathcal{K} }_{k} (2k+ 1)$ , i.e., all its vertex links are $k$ -stacked spheres. This is surprising since it is far from obvious that the vertex links of polytopal upper bound spheres should have any special combinatorial structure. It has been conjectured that for $d\not = 2k+ 1$ , all $(k+ 1)$ -neighborly members of the class ${ \mathcal{K} }_{k} (d)$ are tight. The result of this paper shows that the hypothesis $d\not = 2k+ 1$ is essential for every value of $k\geq 1$ .

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1.Altshuler, A. and Steinberg, L., Neighborly 4-polytopes with 9 vertices. J. Combin. Theory Ser. A 15 (1973), 270287.
2.Bagchi, B. and Datta, B., Lower bound theorem for normal pseudomanifolds. Exp. Math. 26 (2008), 327351.
3.Bagchi, B. and Datta, B., On stellated spheres, shellable balls, lower bounds and a combinatorial criterion for tightness, Preprint, 2011, arXiv:1102.0856v2.
4.Bagchi, B. and Datta, B., On stellated spheres and a tightness criterion for combinatorial manifolds. Preprint, 2012, arXiv:1207.5599v1.
5.Grünbaum, B., Convex Polytopes, 2nd edn(Graduate Texts in Mathematics 221), Springer (New York, 2003).
6.McMullen, P., On simple polytopes. Invent. Math. 113 (1993), 419444.
7.McMullen, P. and Walkup, D. W., A generalized lower-bound conjecture for simplicial polytopes. Mathematika 18 (1971), 264273.
8.Murai, S. and Nevo, E., On the generalized lower bound conjecture for polytopes and spheres, Acta. Math. (to appear).
9.Novik, I., Upper bound theorems for homology manifolds. Israel J. Math. 108 (1998), 4582.
10.Stanley, R. P., The number of faces of a simplicial convex polytope. Adv. Math. 11 (1980), 236238.
11.Ziegler, G. M., Lectures on Polytopes, Springer (New York, 1995).
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ON POLYTOPAL UPPER BOUND SPHERES

  • Bhaskar Bagchi (a1) and Basudeb Datta (a2)

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