The converse of the Solid Torus Theorem

Given positive integers p and q, a (p,q)-solid torus is a manifold diffeomorphic to Dp+1×Sq while a (p,q)-torus in a closed manifold M is the image of a differentiably embedding Sp×Sq→M. We prove that if n=p+q+1 with p=q=1 or p≠q, then M is homeomorphic to Sn 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.