non-Haken 3-manifolds Research Articles

Effective finiteness of irreducible Heegaard splittings of non-Haken 3-manifolds

The main result is a short effective proof of Tao Li’s theorem that a closed non-Haken hyperbolic 3-manifold N has at most finitely many irreducible Heegaard splittings. Along the way we show that N has finitely many branched surfaces of pinched negative sectional curvature carrying all closed index-≤1 minimal surfaces. This effective result, together with the sequel with Daniel Ketover, solves the classification problem for Heegaard splittings of non-Haken hyperbolic 3-manifolds.

Generalized handlebody sets and non-Haken 3-manifolds

In the curve complex for a surface, a handlebody set is the set of loops that bound properly embedded disks in a given handlebody bounded by the surface. A boundary set is the set of non-separating loops in the curve complex that bound two-sided, properly embedded surfaces. For a Heegaard splitting, the distance between the boundary sets of the handlebodies is zero if and only if the ambient manifold contains a non-separating, two sided incompressible surface. We show that the boundary set is 2-dense in the curve complex, i.e. every vertex is within two edges of a point in the boundary set.

Trace fields of representations and virtually Haken 3-manifolds

We introduce and study trace fields and invariant trace fields of SL2(C) and PSL2(C) representations of 3-manifold groups. We give conditions on such fields which imply that the underlying 3-manifold is virtually Haken or has virtually positive or 1 first Betti number. In particular, we define the notion of an algebraically trace-proper surface subgroup in a 3-manifold group, and show that any closed orientable irreducible 3-manifold with such a surface subgroup is virtually Haken. We give infinitely many families of closed orientable hyperbolic non-Haken 3-manifolds with algebraically trace-proper surface subgroups. In this paper a 3-manifold is always assumed to be connected and orientable, without loss of generality for the problems to be considered. A 3-manifold is said to be Haken if it is compact, irreducible, and contains a properly embedded incompressible surface. A 3-manifold is said to be virtually Haken if it has a finite cover which is Haken. The well-known virtually Haken conjecture states that every closed and irreducible 3-manifold with infinite fundamental group is virtually Haken. It is also conjectured that every closed Haken 3-manifold has virtually positive first Betti number, that is, the manifold has a finite cover which has positive first Betti number. If M is a closed hyperbolic Haken 3-manifold, then it is further conjectured that M has virtually 1 first Betti number, which means that for any positive integer n, there is a finite cover Mn of M such that the first Betti number of Mn is larger than n. These conjectures are fundamental and difficult issues in 3-manifold topology. In this paper we provide some information about these conjectures by studying traces of representations of 3-manifold groups into SL2ðCÞ or PSL2ðCÞ: For a group G and a representation r : G ! SL2ðCÞ; we define the trace field Kr of r to be the field generated by the traces of all the matrices r (g), g [ G; over the base field Q of rational numbers, that is, Kr ¼ Qðtrðr ðgÞÞ; g [ G Þ:

Surfaces, submanifolds, and aligned Fox reimbedding in non-Haken $3$-manifolds

Understanding non-Haken 3-manifolds is central to many current endeavors in 3-manifold topology. We describe some results for closed orientable surfaces in non-Haken manifolds, and extend Fox's theorem for submanifolds of the 3-sphere to submanifolds of general non-Haken manifolds. In the case where the submanifold has connected boundary, we show also that the partial derivative-connected sum decomposition of the submanifold can be aligned with such a structure on the submanifold's complement.

Small 3-Manifolds of Large Genus

We prove the existence of pure braids with arbitrarily many strands which are small, i.e. the complements contain no closed incompressible surface which is not boundary parallel. This implies the existence of irreducible non-Haken 3-manifolds of arbitrarily high Heegaard genus.

Non-Haken 3-manifolds are not large with respect to mappings of non-zero degree

Non-Haken 3-manifolds are not large with respect to mappings of non-zero degree

Comparing Heegaard splittings -the bounded case

In a recent paper we used Cerf theory to compare strongly irreducible Heegaard splittings of the same closed irreducible orientable 3-manifold. This captures all irreducible splittings of non-Haken 3-manifolds. One application is a solution to the stabilization problem for such splittings: If p ≤ q p \leq q are the genera of two splittings, then there is a common stabilization of genus 5 p + 8 q − 9 5p + 8q - 9 . Here we show how to obtain similar results even when the 3-manifold has boundary.

Comparing Heegaard splittings of non-haken 3-manifolds

Cerf theory can be used to compare two strongly irreducible Heegaard splittings of the same closed orientable 3-manifold. Any two splitting surfaces can be isotoped so that they intersect in a non-empty collection of curves, each of which is essential in both splitting surfaces. More generally, there are interesting isotopies of the splitting surfaces during which this intersection property is preserved. As sample applications we give new proofs of Waldhausen's theorem that Heegaard splittings of S 3 are standard, and of Bonahon and Otal's theorem that Heegaard splittings of lens spaces are standard. We also present a solution to the stabilization problem for irreducible non-Haken 3-manifolds: If p ⩽ q are the genera of two splittings of such a manifold, then there is a common stabilization of genus 5 p + 8 q − 9.

Full laminations in 3-manifolds

Essential laminations in 3-manifolds, introduced in [GO], are a natural generalization of both incompressible surfaces and foliations without Reeb components. One of the main reasons why essential laminations are of particular interest is that many manifolds which contain no incompressible surfaces contain essential laminations. Indeed, there is some evidence that almost all irreducible manifolds with infinite fundamental group contain essential laminations [B1;D;GM;GO;H;N;R], the only known exceptions being Seifert manifolds [B2]. So essential laminations are potentially a quite powerful tool for studying non-Haken 3-manifolds.

The word problem in a class of non-haken 3-manifolds

The word problem in a class of non-haken 3-manifolds

A Norm on the Fundamental Group of Non-Haken 3-Manifolds

A Norm on the Fundamental Group of Non-Haken 3-Manifolds

A norm on the fundamental group of non-Haken 3-manifolds

A canonical (presentation-independent) conjugacy-invariant norm is constructed on the fundamental group of any 3 3 -manifold which is orientable, irreducible, has infinite fundamental group, and contains no incompressible surface. More generally, this norm exists on any torsion-free group whose commutator quotient is finite. This norm is then computed explicitly in an example which shows that the induced metric on the group is not quasi-isometric to any word metric.

Essential laminations in non-Haken 3-manifolds

In this paper we show that an essential lamination L in a non-Haken 3-manifold M is “tightly wrapped” in the sense that any two leaves of L have intersecting closures; L therefore contains a unique minimal sublamination. We also show that these properties are inherited by any lift of L to a finite cover of M.

Alternating knots and non-Haken 3-manifolds

It is proved in this paper that there is an infinity of knot types in S 3, having essential closed embedded surfaces in their exteriors but yielding non-Hakem manifolds by Dehn surgery for infinitely many surgery coefficients.

A generalized bridge number for links in 3-manifolds

In [Sch] Schubert introduced a new invariant of knots in the 3-sphere, called the bridge number, and showed that, when reduced by 1, it is an additive invariant under the connected sum operation of knots. Several generalizations of Schubert 's theorem can be imagined. Indeed, from Schubert 's perspective the behaviour of bridge number under addition is only a corollary of a more general theorem about bridge number of satellite knots. Boileau and Lickorish have independently observed [B,L] that the following generalization of Schubert 's theorem would be useful. Bridge number as defined by Schubert begins with a projection of the knot (or link) onto a sphere in S 3. Project instead onto a higher genus unknotted surface in S 3. One advantage of this approach is that it allows bridge number to be defined for a link in a 3-manifold other than 5 '3, by projecting the link onto a Heegaard surface for the 3-manifold. Here we explore how such an invariant behaves under connected sum, using a line of argument similar to Jaco 's proof of Haken 's theorem that Heegaard genus is additive [Ja]. A crucial additional ingredient needed is Lemma4.6. Some of the arguments were motivated by [NO], in which, as an aside, a different generalization of Schubert 's theorem is attempted. They want to show that the standard bridge number behaves well under knot decomposition even when the knot is decomposed by spheres intersecting the knot more than twice (e.g. under tangle sums), l Besides recovering a new proof of Schubert 's theorem, more in the spirit of Jaco 's proof of the Haken lemma, we are also able to prove an analogous theorem for bridge number defined with respect to genus one Heegaard splittings of 5'3 or the Lens space and also with respect to irreducible Heegaard splittings of closed nonHaken 3-manifolds.

Homotopy equivalence and hemeomorphism of 3-manifolds

IN THIS paper we extend the class of 3-manifolds which are determined up to homeomorphism by their fundamental groups to the class of closed orientable irreducible 3-manifolds containing a singular surface satisfying two properties, the l-line-intersection property and the 4-plane property. A basic problem in the classification of 3-dimensional manifolds is to decide to what extent the homotopy type of a closed manifold determines the manifold up to homeomorphism. In the case of 3-manifolds with finite fundamental groups, it is known that there are homotopy equivalent manifolds which are not homeomorphic, but there are no known examples of closed orientable irreducible 3-manifolds with isomorphic infinite fundamental groups which are not homcomorphic. Waldhausen [30] and Heil Cl23 proved that if M and M’ are Haken 3-manifolds which have isomorphic fundamental groups then they are homcomorphic. If M and M’ arc hyperbolic then the Mostow rigidity theorem implies the same result [19]. but if only M is assumed to be hyperbolic then it is unknown. Boehme [3] extended Waldhausen’s theorem to certain non-Haken Seifert fiber spaces. Scott [27] showed that a closed orientable irreducible 3-manifold which is homotopy equivalent to a Seifcrt fiber space with infinite fundamental group is homeomorphic to that Seifert fiber space. Many of these Scifert fiber spaces are non-Haken. To date however, Seifert fiber spaces have provided the only examples of non-Haken 3-manifolds which are known to be determined up to homeomorphism by their fundamental groups. For manifolds with boundary there are simple examples of non-homemorphic homotopy equivalent Haken manifolds, such as the product with the circle of a thricepunctured sphere and the product with the circle of a once-punctured torus. Such examples are well understood in terms of the characteristic decomposition of the 3-manifold [17] [16]. Another possible source of troublesome examples comes by taking the connected sum of a 3-manifold with a fake homotopy 3-sphere, resulting in a homotopy equivalent non-homeomorphic 3-manifold. This possibility would be ruled out by a successful solution to the Poincari conjecture. To get around this potential problem we work with irreducible 3-manifolds, in which any 2-sphere bounds a bail. Since non-orientable P2-irreducible 3-manifolds are always Haken, we restrict our attention to the orientable case. For simplicity of notation, we sometimes follow the convention of not distinguishing between a map of a surface into a manifold M and the image of the map. When it is

Nonorientable Surfaces in Some Non-Haken 3-Manifolds

Nonorientable Surfaces in Some Non-Haken 3-Manifolds

Nonorientable surfaces in some non-Haken 3-manifolds

If a closed, irreducible, orientable 3 3 -manifold M M does not possess any 2 2 -sided incompressible surfaces, then it can be very useful to investigate embedded one-sided surfaces in M M of minimal genus. In this paper such 3 3 -manifolds M M are studied which admit embeddings of the nonorientable surface K K of genus 3 3 . We prove that a 3 3 -manifold M M of the above type has at most 3 3 different isotopy classes of embeddings of K K representing a fixed element of H 2 ( M , Z 2 ) {H_2}(M,\,{Z_2}) . If M M is either a binary octahedral space, an appropriate lens space or Seifert manifold, or if M M has a particular type of fibered knot, then it is shown that the embedding of K K in M M realizing a specific homology class is unique up to isotopy.