Enumerative properties of triangulations of spherical bundles over

We give a complete characterization of all possible pairs ( f 0 , f 1 ) , where f 0 is the number of vertices and f 1 is the number of edges, of any triangulation of an S k -bundle over S 1 . The main point is that Kühnel’s triangulations of S 2 k + 1 × S 1 and the nonorientable S 2 k -bundle over S 1 are unique among all triangulations of ( n − 1 ) -dimensional homology manifolds with 2 n + 1 vertices, first Betti number nonzero, and whose orientation double cover has vanishing second Betti number.