Centrally symmetric manifolds with few vertices

A centrally symmetric 2d-vertex combinatorial triangulation of the product of spheres Si×Sd−2−i is constructed for all pairs of nonnegative integers i and d with 0⩽i⩽d−2. For the case of i=d−2−i, the existence of such a triangulation was conjectured by Sparla. The constructed complex admits a vertex-transitive action by a group of order 4d. The crux of this construction is a definition of a certain full-dimensional subcomplex, B(i,d), of the boundary complex of the d-dimensional cross-polytope. This complex B(i,d) is a combinatorial manifold with boundary and its boundary provides a required triangulation of Si×Sd−i−2. Enumerative characteristics of B(i,d) and its boundary, and connections to another conjecture of Sparla are also discussed.