A SUFFICIENT CONDITION FOR SURFACES IN 3-MANIFOLDS TO HAVE UNIQUE PRIME DECOMPOSITIONS

In 1975, Suzuki proved that prime decompositions of closed, connected surfaces in S3 are unique up to ambient isotopy if the surface bounds a 3-manifold whose factors under the prime decomposition all have incompressible boundary. This paper extends this result to surfaces in more general 3-manifolds, when there is a prime decomposition for which every factor of the surface is incompressible on one side.