On Polytopal Upper Bound Spheres

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 generalizedWalkup class Kk(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 f0, . . . , fk−1) while the upper bound spheres maximize the face vector (among spheres with a given f0). It has been conjectured that for d 6= 2k + 1, all (k + 1)-neighborly members of the class Kk(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 7→ (t, t, . . . , t)) 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 fi(T ) = fi(S) for 0 ≤ i < k. (Here, as usual, the face-vector (f0(T ), . . . , fd(T )) of a d-dimensional simplicial complex T is given by fi(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 2010 Mathematics Subject Classification. Primary 52B11, 52B05; Secondary 52B22.