loop theorem

In the topology of 3-manifolds, the loop theorem is generalization of an ansatz discovered by Max Dehn (namely, Dehn’s lemma),who saw that if a continuous map from a 2-disk to a 3-manifold whose restriction to the boundary’s disk has no singularities,then there exists another embedding whose restriction to the boundary’s disk is equal to the boundary’s restriction original map.

The following statement called the loop theorem is a version from J. Stallings, but written in W. Jaco’s book.

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

The proof is a clever construction due to C. Papakyriakopoulos about a sequence (a tower) of covering spaces.Maybe the best detailed presentation is due to A. 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…, there is no better 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.

A. Hatcher, Notes on 3-manifolds, available on-line.