The converse of the Solid Torus Theorem

Abstract

Given positive integers p and q, a ( p , q ) - solid torus is a manifold diffeomorphic to D p + 1 × S q while a ( p , q ) - torus in a closed manifold M is the image of a differentiably embedding S p × S q → M . We prove that if n = p + q + 1 with p = q = 1 or p ≠ q , then M is homeomorphic to S n whenever every ( p , q ) -torus bounds a ( p , q ) -solid torus. We also prove for p = q that every closed n-manifold for which every ( p , p ) -torus bounds an irreducible manifold is irreducible. Consequently, every closed 3-manifold for which every torus bounds an irreducible manifold is irreducible.