information:filtering
Annulus and Torus theorems

The annulus theorem states that if a pair of disjoint simple closed curves on the boundary of a three manifold are freely homotopic then they cobound a properly embedded annulus. This should not be confused with the high dimensional theorem of the same name.

The torus theorem is as follows: Let M be a compact, irreducible 3-manifold with nonempty boundary. If M admits an essential map of a torus, then M admits an essential embedding of either a torus or an annulus