loop theorem

In the topologyMathworldPlanetmath of 3-manifolds, the loop theorem is generalizationPlanetmathPlanetmath 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 restrictionPlanetmathPlanetmathPlanetmath to the boundary’s disk has no singularities, then there exists another embeddingMathworldPlanetmathPlanetmathPlanetmath 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 M. Let Nπ1(M) be a normal subgroupMathworldPlanetmath. Let f:D2M be a continuous map such that f(D2)S and [f|D2]N.
Then there exists an
embedding g:D2M such that g(D2)S and [g|D2]N,

The proof is a clever construction due to C. Papakyriakopoulos about a sequence (a tower) of covering spaces. Maybe the best detailed presentationMathworldPlanetmathPlanetmathPlanetmath 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