Loop theorem

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 ${\displaystyle f|\partial D^{2}}$ not nullhomotopic in ${\displaystyle \partial M}$, 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 ${\displaystyle M}$ be a 3-manifold and let ${\displaystyle S}$ be a connected surface in ${\displaystyle \partial M}$. Let ${\displaystyle N\subset \pi _{1}(S)}$ be a normal subgroup such that ${\displaystyle {\mathop {\mathrm {ker} }}(\pi _{1}(S)\to \pi _{1}(M))-N\neq \emptyset }$. Let

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