Dehn filling, reducible 3-manifolds, and Klein bottles

Let be a compact, connected, orientable, irreducible 3-manifold whose boundary is a torus. We announce that if two Dehn fillings create reducible manifold and manifold containing Klein bottle, then the maximal distance is three.

     

Vol3 and other exceptional hyperbolic 3-manifolds

D. Gabai, R. Meyerhoff and N. Thurston identified seven families of exceptional hyperbolic manifolds in their proof that a manifold which is homotopy equivalent to a hyperbolic manifold is hyperbolic. These families are each conjectured to consist of a single manifold. In fact, an important point in their argument depends on this conjecture holding for one particular exceptional family. In this paper, we prove the conjecture for that particular family, showing that the manifold known as in the literature covers no other manifold. We also indicate techniques likely to prove this conjecture for five of the other six families.

     

Compactifying sufficiently regular covering spaces of compact 3-manifolds

In this paper it is proven that if the group of covering translations of the covering space of a compact, connected, -irreducible 3-manifold corresponding to a non-trivial, finitely-generated subgroup of its fundamental group is infinite, then either the covering space is almost compact or the subgroup is infinite cyclic and has normalizer a non-finitely-generated subgroup of the rational numbers. In the first case additional information is obtained which is then used to relate Thurston's hyperbolization and virtual bundle conjectures to some algebraic conjectures about certain 3-manifold groups.     

Twisted face-pairing 3-manifolds

This paper is an enriched version of our introductory paper on twisted face-pairing 3-manifolds. Just as every edge-pairing of a 2-dimensional disk yields a closed 2-manifold, so also every face-pairing of a faceted 3-ball yields a closed 3-dimensional pseudomanifold. In dimension 3, the pseudomanifold may suffer from the defect that it fails to be a true 3-manifold at some of its vertices. The method of twisted face-pairing shows how to correct this defect of the quotient pseudomanifold systematically. The method describes how to modify by edge subdivision and how to modify any orientation-reversing face-pairing of by twisting, so as to yield an infinite parametrized family of face-pairings whose quotient complexes are all closed orientable 3-manifolds. The method is so efficient that, starting even with almost trivial face-pairings , it yields a rich family of highly nontrivial, yet relatively simple, 3-manifolds.

This paper solves two problems raised by the introductory paper:

(1) Replace the computational proof of the introductory paper by a conceptual geometric proof of the fact that the quotient complex of a twisted face-pairing is a closed 3-manifold. We do so by showing that the quotient complex has just one vertex and that its link is the faceted sphere dual to .

(2) The twist construction has an ambiguity which allows one to twist all faces clockwise or to twist all faces counterclockwise. The fundamental groups of the two resulting quotient complexes are not at all obviously isomorphic. Are the two manifolds the same, or are they distinct?

We prove the highly nonobvious fact that clockwise twists and counterclockwise twists yield the same manifold. The homeomorphism between them is a duality homeomorphism which reverses orientation and interchanges natural 0-handles with 3-handles, natural 1-handles with 2-handles. This duality result of (2) is central to our further studies of twisted face-pairings.

We also relate the fundamental groups and homology groups of the twisted face-pairing 3-manifolds and of the original pseudomanifold (with vertices removed).

We conclude the paper by giving examples of twisted face-pairing 3-manifolds. These examples include manifolds from five of Thurston's eight 3-dimensional geometries.

     

An inequality for polyhedra and ideal triangulations of cusped hyperbolic 3-manifolds

It is not known whether every noncompact hyperbolic 3-manifold of finite volume admits a decomposition into ideal tetrahedra. We give a partial solution to this problem: Let be a hyperbolic 3-manifold obtained by identifying the faces of convex ideal polyhedra . If the faces of are glued to , then can be decomposed into ideal tetrahedra by subdividing the 's.

     

Behavior of the extended complexity of irreducible 3-manifolds

We construct the extended complexity of irreducible 3-manifolds; unlike the usual complexity [1] it is not an integer, but an ordered tuple of five integers. The benefit of extended complexity is that it always decreases when a manifold is cut along some incompressible boundary incompressible surface.     

Incompressible Surfaces in Handlebodies and Closed 3-manifolds

in this paper we prove that for any positive integer n, 1) a handlebody of genus 2contains a separating incompressible surface of genus n, and 2) there exists a closed 3manifold of heegaard genus 2 which contains a separating incompressible surface of genus n.     

Geodesic knots in closed hyperbolic 3-manifolds

We consider the existence of simple closed geodesics or “geodesic knots” in finite volume orientable hyperbolic 3-manifolds. Every such manifold contains at least one geodesic knot by results of Adams, Hass and Scott in (Adams et al. Bull. London Math. Soc. 31: 81–86, 1999). In (Kuhlmann Algebr. Geom. Topol. 6: 2151–2162, 2006) we showed that every cusped orientable hyperbolic 3-manifold in fact contains infinitely many geodesic knots. In this paper we consider the closed manifold case, and show that if a closed orientable hyperbolic 3-manifold satisfies certain geometric and arithmetic conditions, then it contains infinitely many geodesic knots. The conditions on the manifold can be checked computationally, and have been verified for many manifolds in the Hodgson-Weeks census of closed hyperbolic 3-manifolds. Our proof is constructive, and the infinite family of geodesic knots spiral around a short simple closed geodesic in the manifold.      

Any 3-manifold 1-dominates at most finitely many 3-manifolds of -geometry

Any 3-manifold 1-dominates at most finitely many 3-manifolds supporting geometry.

     

Thin position for knots and 3-manifolds

We prove that for 2-bridge knots and 3-bridge knots in thin position the double branched cover inherits a manifold decomposition in thin position. We also argue that one should not expect this sort of correspondence to hold in general.     

The parity of the Cochran-Harvey invariants of 3-manifolds

Given a finitely presented group and an epimorphism Cochran and Harvey defined a sequence of invariants , which can be viewed as the degrees of higher-order Alexander polynomials. Cochran and Harvey showed that (up to a minor modification) this is a never decreasing sequence of numbers if is the fundamental group of a 3-manifold with empty or toroidal boundary. Furthermore they showed that these invariants give lower bounds on the Thurston norm.

Using a certain Cohn localization and the duality of Reidemeister torsion we show that for a fundamental group of a 3-manifold any jump in the sequence is necessarily even. This answers in particular a question of Cochran. Furthermore using results of Turaev we show that under a mild extra hypothesis the parity of the Cochran-Harvey invariant agrees with the parity of the Thurston norm.

     

A census of genus-two 3-manifolds up to 42 coloured tetrahedra

We improve and extend to the non-orientable case a recent result of Karábaš, Mali?ký and Nedela concerning the classification of all orientable prime 3-manifolds of Heegaard genus two, triangulated with at most 42 coloured tetrahedra.     

Hyperbolic 3-manifolds as cyclic branched coverings

There is an extensive literature on the characterization of knots in the 3-sphere which have the same 3-manifold as a common n-fold cyclic branched covering, for some integer . In the present paper, we study the following more general situation. Given two integers m and n, how are knots K ₁ and K ₂ related such that the m-fold cyclic branched covering of K ₁ coincides with the n-fold cyclic branched covering of K ₂. Or, seen from the point of view of 3-manifolds: in how many different ways can a given 3-manifold occur as a cyclic branched covering of knots in S ³. Under certain hypotheses, we solve this problem for the basic class of hyperbolic 3-manifolds and hyperbolic knots (the other basic class is that of Seifert fiber spaces resp. of torus and Montesinos knots for which the situation is well understood; the general case can then be analyzed using the equivariant sphere and torus decomposition into Seifert fiber spaces and hyperbolic manifolds). Received: December 7, 1999; revised version: May 22, 2000     

Diffeomorphism Type of Certain 3-connected Closed Smooth 12-manifolds

ＤｉｆｆｅｏｍｏｒｐｈｉｓｍＴｙｐｅｏｆＣｅｒｔａｉｎ３－ｃｏｎｎｅｃｔｅｄＣｌｏｓｅｄＳｍｏｏｔｈ１２－ｍａｎｉｆｏｌｄｓＦａｎｇＦｕｑｕａｎ（方复全）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓＮａｎｋａｉＵｎｉｖｅｒｓｉｔｙＴｉａｎｊｉｎ，...     

Invariants of piecewise-linear 3-manifolds

This paper presents an algebraic framework for constructing invariants of closed oriented 3-manifolds by taking a state sum model on a triangulation. This algebraic framework consists of a tensor category with a condition on the duals which we have called a spherical category. A significant feature is that the tensor category is not required to be braided. The main examples are constructed from the categories of representations of involutive Hopf algebras and of quantised enveloping algebras at a root of unity.

     

Parallelisability of 3-manifolds via surgery

We present two proofs that all closed, orientable 3-manifolds are parallelisable. Both are based on the Lickorish–Wallace surgery presentation; one proof uses a refinement of this presentation due to Kaplan and some basic contact geometry. This complements a recent paper by Benedetti–Lisca.     

Stiefel-Whitney surfaces and decompositions of 3-manifolds into handlebodies

Every closed nanorientable 3-manifold M can be obtained as the union of three orientable handlebodies V₁, V₂, V₃ whose interiors are pairwise disjoint. If g_i denotes the genus of V_i, g₁g₂g₃, we say that M has tri-genus (g₁, g₂, g₃), if in terms of lexicographical ordering, the triple (g₁, g₂, g₃) is minimal among all such decompositions of M into orientable handlebodies. We relate the tri-genus of M to the genus of a surface that represents the dual of the first Stiefel-Whitney class of M. This is used to determine g₁ and g₂.