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.