The fundamental group of locally standard $T$–manifolds

Abstract

We calculate the fundamental group of locally standard $T$–manifolds under the assumption that the principal $T$–bundle obtained from the free $T$–orbits is trivial. This family of manifolds contains nonsingular toric varieties which may be noncompact, quasitoric manifolds and toric origami manifolds with coörientable folding hypersurface. Although the fundamental groups of the above three kinds of manifolds are well-studied, we give a uniform and simple method to generalize the formulas of their fundamental groups.