# 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.”

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.

Title | loop theorem |
---|---|

Canonical name | LoopTheorem |

Date of creation | 2013-03-22 15:49:13 |

Last modified on | 2013-03-22 15:49:13 |

Owner | juanman (12619) |

Last modified by | juanman (12619) |

Numerical id | 11 |

Author | juanman (12619) |

Entry type | Theorem |

Classification | msc 57M35 |

Related topic | 3Manifolds |