Loop theorem

In the topology of 3-manifolds, the loop theorem is an ansatz about a discovery of Max Dehn (The Dehn's Lemma), who was the first to see that a continuous map from a disk to a 3-manifold whose restriction to the boundarie's disk has not singularities, implies the existence of another which is an embedding and whose restriction to the disk is equal to the restriction's original map.

The following statement called the Loop Theorem is a version due to Stalling, but written in Jaco's book:

Let $M$ be a three-manifold and let $S$ be a connected surface in $\partial M$. Let $N\subset \pi _{1}(M)$ be a normal subgroup. Let $f\colon D^{2}\to M$ a continuous map such that $f(\partial D^{2})\subset S$ and $[f|\partial D^{2}]\notin N$. Then there exists an embedding $g\colon D^{2}\to M$ such that $g(\partial D^{2})\subset S$ and $[g|\partial D^{2}]\notin N$.

The proof is a clever construction due to Papakiriakopulos about a sequence (a tower) of covering spaces. And perhaps the best detailed redaction is due to Hatcher.

But in general, accordingly, to Jaco's opinion:

...for anyone unfamiliar with the techniques of 3-manifold-topology and are here to gain a working knowledge for the study of problems in this area...., there is no better place to start.

References

W. Jaco, Lectures on 3-manifolds topology, A.M.S. regional conference series in Math 43.

J. Hempel, 3-manifolds, Princeton University Press 1976.