A centrally symmetric 2d-vertex combinatorial triangulation of the product of spheres S × Sd−2−i is constructed for all pairs of non-negative 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… CONTINUE READING