Surfaces, submanifolds, and aligned Fox reimbedding in non-Haken $3$-manifolds

Abstract

Understanding non-Haken 3-manifolds is central to many current endeavors in 3-manifold topology. We describe some results for closed orientable surfaces in non-Haken manifolds, and extend Fox's theorem for submanifolds of the 3-sphere to submanifolds of general non-Haken manifolds. In the case where the submanifold has connected boundary, we show also that the partial derivative-connected sum decomposition of the submanifold can be aligned with such a structure on the submanifold's complement.