In mathematics, in the topology of 3-manifolds, the loop theorem is a generalization of Dehn's lemma. The loop theorem was first proven by Christos Papakyriakopoulos in 1956, along with Dehn's lemma and the Sphere theorem. A simple and useful version of the loop theorem states that if there is a map with not nullhomotopic in , then there is an embedding with the same property. The following version of the loop theorem, due to John Stallings, is given in the standard 3-manifold treatises (such as Hempel or Jaco): Let be a 3-manifold and let be a connected surface in . Let be a normal subgroup such that