Roots of 3-manifolds and cobordisms

Given a set of simplifying moves on 3-manifolds, we apply them to a given 3-manifold M as long as possible. What we get is a root of M. For us, it makes sense to consider three types of moves: compressions along 2-spheres, proper discs and proper annuli having boundary circles in different components of the boundary of M. Our main result is that for the above moves the root of any 3-manifold exists and is unique. The same result remains true if instead of manifolds we apply the moves to 3-cobordisms.