On Polytopal Upper Bound Spheres (original) (raw)

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 K k (2k + 1), i.e., all its vertex links are k-stacked spheres. This is surprising since the k-stacked spheres minimize the face-vector (among all polytopal spheres with given f 0 ,. .. , f k−1) while the upper bound spheres maximize the face vector (among spheres with a given f 0). It has been conjectured that for d = 2k + 1, all (k + 1)-neighborly members of the class K k (d) are tight. The result of this paper shows that, for every k, the case d = 2k+1 is a true exception to this conjecture. We recall that a simplicial complex is said to be l-neighborly if each set of l vertices of the complex spans a face. As a well known consequence of the Dehn-Sommerville equations, any triangulated sphere of odd dimension d = 2k + 1 can be at most (k + 1)-neighborly (unless it is the boundary complex of a simplex). A (2k + 1)-dimensional triangulated sphere is said to be an upper bound sphere if it is (k + 1)-neighborly. This is because, by the celebrated Upper Bound Theorem, any such sphere maximizes the face vector componentwise among all (2k + 1)-dimensional triangulated closed manifolds with a given number of vertices [9]. A simplicial complex is said to be a polytopal sphere if it is isomorphic to the boundary complex of a simplicial convex polytope. For n ≥ 2k + 3, the boundary complex of an n-vertex (2k + 2)-dimensional cyclic polytope P (defined as the convex hull of any set of n points on the moment curve t → (t, t 2 ,. .. , t 2k+2)) is an example of an n-vertex polytopal upper bound sphere of dimension 2k + 1. We recall that a triangulated homology sphere S is said to be k-stacked if there is a triangulated homology ball B bounded by S all whose faces of codimension k + 1 are in the boundary S. The generalized lower bound conjecture (GLBC) due to McMullen and Walkup [7] states that a k-stacked d-sphere S minimizes the face-vector componentwise among all triangulated d-spheres T such that f i (T) = f i (S) for 0 ≤ i < k. (Here, as usual, the face-vector (f 0 (T),. .. , f d (T)) of a d-dimensional simplicial complex T is given by f i (T) = the number of i-dimensional faces of T). For polytopal spheres T , this conjecture was proved by Stanley [10] and McMullen [6]. Recently, Murai and Nevo [8] proved that a polytopal sphere (more generally, a triangulated homology sphere with the weak Lefschetz property) satisfies equality in GLBC only if it is k-stacked. A triangulated homology ball B is said to be k-stacked if all its faces of codimension k + 1 are in its boundary ∂B. Thus, a triangulated (homology) d-sphere S is k-stacked if