Irreducible 3-manifolds whose orientable covers are not prime

Abstract

J. L. Tollefson has asked if every closed covering space of a prime 3-manifold is prime. In the present paper, the author provides a negative answer by constructing infinitely many topologically distinct, irreducible, closed 3-manifolds with the property that none of their orientable covering spaces are prime. These 3-manifolds are distinguished by the maximum number of disjoint, nonparallel, 2-sided projective planes that they contain. The author does not know if every closed covering space of a prime, orientable 3-manifold is prime.