ON POLYTOPAL UPPER BOUND SPHERES. Issue 2 (28th March 2013) - Bagchi, Bhaskar; Datta, Basudeb
Abstract: Generalizing a result (the case k = 1 ) due to M. A. Perles, we show that any polytopal upper bound sphere of odd dimension 2 k + 1 belongs to the generalized Walkup class K k ( 2 k + 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 ≠ 2 k + 1, all ( k + 1 ) ‐neighborly members of the class K k ( d ) are tight. The result of this paper shows that the hypothesis d ≠ 2 k + 1 is essential for every value of k ≥ 1 .