Localising Dehn's lemma and the loop theorem in 3-manifolds

Abstract

We give a new proof of Dehn's lemma and the loop theorem. This is a fundamental tool in the topology of 3-manifolds. Dehn's lemma was originally formulated by Dehn, where an incorrect proof was given. A proof was finally given by Papakyriakopolous in his famous 1957 paper where the fundamental idea of towers of coverings was introduced. This was later extended to the loop theorem, and the version used most frequently was given by Stallings.We have shown that hierarchies for Haken 3-manifolds could be understood by a 'local version' of Dehn's lemma and the loop theorem. Developing the idea further enables us here to give a new proof of the classical theorems of Papakyriakopoulos, which do not use towers of coverings. A similar result was obtained by Johannson, with the added assumption that the 3-manifold in question was Haken. Our approach means that no extra hypotheses are necessary. Our method uses the concept of boundary patterns of hierarchies, as developed by Johannson. Marc Lackenby has independently produced a very similar proof in his lecture notes.Note that the more difficult sphere theorem [4,10,25] can then be deduced using Dehn's lemma and the loop theorem, plus the PL theory of minimal surfaces. Other applications, like the important result that a covering of an irreducible 3-manifold is irreducible, then follow also by PL minimal surface theory.

3–manifolds built from injective handlebodies

“…Then glue a handlebody to this boundary component so the surface is only incompressible in one direction. A very short hierarchy in a closed Haken manifold, as defined by I. Aitchison and H. Rubinstein in [1], can be thought of as taking a set of handlebodies, gluing each handlebody to itself so that each of the resulting manifolds has incompressible boundary. Then glue these incompressible boundaries together to produce the closed manifold.…”

“…Note there is a homeomorphism from P to some A (p,q) that sends the boundary curves of P ∩ ∂H to fibers of A (p,q) . As with the I-bundle region, we are removing the components of N that are homeomorphic to A (1,2) or A (2,2) , that is regular neighbourhoods of properly embedded annuli or Mobius bands, to get N T . This is because if there are two annuli in H − T that have a non-trivial vertical intersection then the maximal tree region can contain the regular neighbourhood of only one of the annuli.…”

“…Let A ′ ⊂ A be one of these annuli such that f 1 (A ′ ) ⊂ H − N T and let n(f 1 (A ′ )) be the regular neighbourhood of f (A ′ ) disjoint from T . Then n(f 1 (A ′ )) can be fibered as an A (1,2) fibered torus. Thus there must be an isotopy of f 1 to remove the curves A ′ ∩ Γ 1 ( there may be just one if ∂A ∩ ∂A ′ = ∅) otherwise N T ∪ n(f 1 (A ′ )) would be larger than N T , contradicting maximality.…”

An Introduction to Geometric Topology

“…Isometries of type (1), (2), and (3) are called respectively elliptic, parabolic, and hyperbolic. A hyperbolic isometry fixes two points p, q ∈ ∂H n and hence preserves the unique line l with endpoints p and q.…”

“…The triples realisable in S 2 are (2, 2, c), (2, 3, 3), (2,3,4), and (2, 3, 5): the last three tessellations are shown in Fig. 3.5 and are connected to the platonic solids.…”