A classification of centrally-symmetric and cyclic 12-vertex   triangulations of $S^2 \times S^2$
  In this paper our main result states that there exist exactly three combinatorially distinct centrally-symmetric 12-vertex-triangulations of the product of two 2-spheres with a cyclic symmetry. We also compute the automorphism groups of the triangulations. These instances suggest that there is a triangulation of $S^2 \times S^2$ with 11 vertices -- the minimum number of vertices required. 
G. Lassmann, E. Sparla