Three-Dimensional Manifolds Defined by Coloring a Simple Polytope

Abstract

In the present paper we introduce and study a class of three-dimensional manifolds endowed with the action of the group ℤ23 whose orbit space is a simple convex polytope. These manifolds originate from three-dimensional polytopes whose faces allow a coloring into three colors with the help of the construction used for studying quasitoric manifolds. For such manifolds we prove the existence of an equivariant embedding into Euclidean space ℝ4. We also describe the action on the set of operations of the equivariant connected sum and the equivariant Dehn surgery. We prove that any such manifold can be obtained from a finitely many three-dimensional tori with the canonical action of the group ℤ23 by using these operations.