Fundamental Groups of Three-Dimensional Small Covers

Small covers arising from three-dimensional simple polytopes are an interesting class of 3-manifolds. The fundamental group is a rigid invariant for wide classes of 3-manifolds, particularly for orientable Haken manifolds, which include orientable small covers over flag polytopes. By using the Morse-theoretic approach, we give a procedure to get an explicit balanced presentation of the fundamental group of a closed orientable three-dimensional small cover with minimal number of generators. Our procedure is completely algorithmic and geometrical.