Minimal triangulations of circle bundles

A triangulation of a circle bundle $E\underset{}{\overset{\pi }{\to }}B$ is a triangulation of the total space $E$ and the base $B$ such that the projection $\pi$ is a simplicial map. In the paper we address the following questions: Which circle bundles can be triangulated over a given triangulation of the base? What are the minimal triangulations of a bundle? A complete solution for semisimplicial triangulations was given by N. Mnëv. Our results deal with classical triangulations, that is, simplicial complexes. We give an exact answer for an infinite family of triangulated spheres (including the boundary of the $3$-simplex, the boundary of the octahedron, the suspension over an $n$-gon, the icosahedron). For the general case we present a sufficient criterion for existence of a triangulation. Some minimality results follow straightforwadly.