Suppose I have a compact 3-dimensional submanifold N of S X (0,1) which has one boundary component, where S is a closed surface. Must N be a handlebody?



Are compact submanifolds of "S X (0,1)" with one boundary component handlebodies, where S is a closed surface?
No. N could be homeomorphic to the exterior of any knot in the 3-sphere (i.e. a cube with a knotted hole).


