Enumerative properties of triangulations of spherical bundles over S1S1

We give a complete characterization of all possible pairs (f0,f1)(f0,f1), where f0f0 is the number of vertices and f1f1 is the number of edges, of any triangulation of an SkSk-bundle over S1S1. The main point is that Kühnel’s triangulations of S2k+1×S1S2k+1×S1 and the nonorientable S2kS2k-bundle over S1S1 are unique among all triangulations of (n−1)(n−1)-dimensional homology manifolds with 2n+12n+1 vertices, first Betti number nonzero, and whose orientation double cover has vanishing second Betti number.