Combinatorial Dehn surgery on cubed and Haken 3–manifolds

Aitchison, Iain R.; Rubinstein, J. Hyam

doi:10.2140/gtm.1999.2.1

Cited by 3 publications

(8 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…But now the conclusion follows similarly to that of [1] (c.f also [12]). The argument shows that all the surfaces in the hierarchy are algebraically incompressible and boundary algebraically incompressible, i.e.…”

Section: (M ) Is Not One-to-one Then There Is An Embedded Disksupporting

confidence: 65%

“…Suppose that M is a closed 3-manifold and S is an embedded 2-sided closed surface in M . Assume that S is not a 2-sphere nor a real projective plane and the induced map from π 1 (S) to π 1 …”

Section: Dehn's Lemma and The Loop Theoremmentioning

confidence: 99%

“…We shall establish a property which will play a similar role to the 'simplicity' condition of Waldhausen [24], for our hierarchy. Rather than following the ideas in [1] or [12], we only use the closed and spanning surfaces of the hierarchy and establish that the compressing disks for the handlebodies have to meet the boundary pattern at least 4 times, whether the disks are embedded or singular. We will go through the details, since the existence of a very short hierarchy gives an interesting, different and quite simple argument.…”

Section: (M ) Is Not One-to-one Then There Is An Embedded Diskmentioning

confidence: 99%

“…Here M i−1 is obtained by cutting M open along the first i − 1 surfaces in the hierarchy. Note the method of [1] and [12] is to study the graph Γ * formed by the preimage of the hierarchy, using the map of some singular compressing disk or boundary compressing disk for a surface such as S in the hierarchy. We look instead at the graph Γ which is the preimage of the closed and spanning surfaces of H in the domain D of the disk.…”

Section: (M ) Is Not One-to-one Then There Is An Embedded Diskmentioning

confidence: 99%

See 3 more Smart Citations

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

Aitchison

Rubinstein

2004

Math. Proc. Camb. Phil. Soc.

View full text Add to dashboard Cite

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.

show abstract

Section: (M ) Is Not One-to-one Then There Is An Embedded Disksupporting

confidence: 65%

Section: Dehn's Lemma and The Loop Theoremmentioning

confidence: 99%

Section: (M ) Is Not One-to-one Then There Is An Embedded Diskmentioning

confidence: 99%

Section: (M ) Is Not One-to-one Then There Is An Embedded Diskmentioning

confidence: 99%

See 2 more Smart Citations

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

Aitchison

Rubinstein

2004

Math. Proc. Camb. Phil. Soc.

View full text Add to dashboard Cite

show abstract

“…The situation is particularly nice in the case where the NPC cube complex X is homeomorphic to a 3-manifold. Work of Aitchison and Rubinstein (see §3 in [5]) shows that each immersed hyperplane is mapped π 1 -injectively into X. Hence if one hyperplane is embedded and 2-sided, then X is a Haken 3-manifold.…”

Section: Introductionmentioning

confidence: 96%

Unravelling the Dodecahedral Spaces

Spreer

Tillmann

2018

MATRIX Book Series

View full text Add to dashboard Cite

The hyperbolic dodecahedral space of Weber and Seifert has a natural non-positively curved cubulation obtained by subdividing the dodecahedron into cubes. We show that the hyperbolic dodecahedral space has a 6-sheeted irregular cover with the property that the canonical hypersurfaces made up of the mid-cubes give a very short hierarchy. Moreover, we describe a 60-sheeted cover in which the associated cubulation is special. We also describe the natural cubulation and covers of the spherical dodecahedral space (aka Poincaré homology sphere).In general, it is known through work of Bergeron and Wise [6] that if M is a closed hyperbolic 3-manifold, then π 1 (M) is isomorphic to the fundamental group of an NPC cube complex. However, the dimension of this cube complex may be arbitrarily large and it may not be a manifold. Agol's theorem provides a finite cover that is a special cube complex, and the π 1 -injective surfaces of Kahn and Markovic [13] are quasi-convex and hence have separable fundamental group. Thus, the above outline completes a sketch of the proof that M is virtually Haken. An embedding theorem of Haglund and Wise [11] and Agol's virtual fibring criterion [1] then imply that M is also virtually fibred.Weber and Seifert [16] described two closed 3-manifolds that are obtained by taking a regular dodecahedron in a space of constant curvature and identifying opposite sides by isometries. One is hyperbolic and known as the Weber-Seifert dodecahedral space and the other is spherical and known as the Poincaré homology sphere. Moreover, antipodal identification on the boundary of the dodecahedron yields a third closed 3-manifold which naturally fits into this family: the real projective space.The dodecahedron has a natural decomposition into 20 cubes, which is a NPC cubing in the case of the Weber-Seifert dodecahedral space. The main result of this note can be stated as follows.Theorem 2 The hyperbolic dodecahedral space WS of Weber and Seifert admits a cover of degree 60 in which the lifted natural cubulation of WS is special.In addition, we exhibit a 6-sheeted cover of WS in which the canonical immersed surface consists of six embedded surface components and thus gives a very short hierarchy of WS. The special cover from Theorem 2 is the smallest regular cover of WS that is also a cover of this 6-sheeted cover. Moreover, it is the smallest regular cover of WS that is also a cover of the 5-sheeted cover with positive first Betti number described by Hempel [12].We conclude this introduction by giving an outline of this note. The dodecahedral spaces are described in §3. Covers of the hyperbolic dodecahedral space are described in §4, and all covers of the spherical dodecahedral space and the real projective space in §5.The authors thank Daniel Groves and Alan Reid for their encouragement to write up these results, and the anonymous referee for some insightful questions and comments which triggered us to find a special cover. Cube complexes, injective surfaces and hierarchiesA cube complex is a space obtained by ...

show abstract