Finite maximal orientation reversing group actions on handlebodies and 3-manifolds

We study arbitrary (that is not necessarily orientation preserving) finite group actions on 3-dimensional orientable or nonorientable handlebodies of genus g. For g>1, the maximal possible order is 24(g−1); we characterize the corresponding groups of this order and also the occuring quotient orbifolds. Then we use this to study finite group actions of large order (with respect to the equivariant Heegaard genus g) on closed 3-manifolds, again concentrating on the maximal case of order 24(g−1). Our results extend corresponding results in the orientation preserving setting. Whereas for arbitrary finite group actions on handlebodies much more types of quotient orbifolds occur than in the orientation preserving case, it turns out that for closed 3-manifolds the situation is quite rigid, in contrast to the orientation preserving case where one has many possibilities to construct manifolds with large group actions.     

Three-manifolds with Heegaard genus at most two represented by crystallisations with at most 42 vertices

It is known that every closed compact orientable 3-manifold M can be represented by a 4-edge-coloured 4-valent graph called a crystallisation of M. Casali and Grasselli proved that 3-manifolds of Heegaard genus g can be represented by crystallisations with a very simple structure which can be described by a 2(g+1)-tuple of non-negative integers. The sum of first g+1 integers is called complexity of the admissible 2(g+1)-tuple. If c is the complexity then the number of vertices of the associated graph is 2c. In the present paper we describe all prime 3-manifolds of Heegaard genus two described by 6-tuples of complexity at most 21.     

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.     

Non degenerating Dehn fillings on genus two Heegaard splittings of knots′ complements

It is Thurston's result that for a hyperbolic knot K in S~3, almost all Dehn fillings on its complement result in hyperbolic 3-manifolds except some exceptional cases. So almost all produced 3-manifolds have the same geometry. It is known that its complement in S~3, denoted by E(K), admits a Heegaard splitting. Then it is expected that there is a similar result on Heegaard distance for Dehn fillings. In this paper, Dehn fillings on genus two Heegaard splittings are studied. More precisely, we prove that if the distance of a given genus two Heegaard splitting of E(K) is at least 3, then for any two degenerating slopes on ?E(K), there is a universal bound of their distance in the curve complex of ?E(K).     

Heegaard splittings of twisted torus knots

Little is known on the classification of Heegaard splittings for hyperbolic 3-manifolds. Although Kobayashi gave a complete classification of Heegaard splittings for the exteriors of 2-bridge knots, our knowledge of other classes is extremely limited. In particular, there are very few hyperbolic manifolds that are known to have a unique minimal genus splitting. Here we demonstrate that an infinite class of hyperbolic knot exteriors, namely exteriors of certain “twisted torus knots” originally studied by Morimoto, Sakuma and Yokota, have a unique minimal genus Heegaard splitting of genus two. We also conjecture that these manifolds possess irreducible yet weakly reducible splittings of genus three. There are no known examples of such Heegaard splittings.     

On the Heegaard genus of contact 3-manifolds

It is well-known that the Heegaard genus is additive under connected sum of 3-manifolds. We show that the Heegaard genus of contact 3-manifolds is not necessarily additive under contact connected sum. We also prove some basic properties of the contact genus (a.k.a. open book genus [Rubinstein J.H., Comparing open book and Heegaard decompositions of 3-manifolds, Turkish J. Math., 2003, 27(1), 189–196]) of 3-manifolds, and compute this invariant for some 3-manifolds.     

Nowhere-zero flows in low genus graphs

A nowhere-zero k-flow is an assignment of edge directions and integer weights in the range 1,…, k ? 1 to the edges of an undirected graph such that at every vertex the flow in is equal to the flow out. Tutte has conjectured that every bridgeless graph has a nowhere-zero 5-flow. We show that a counterexample to this conjecture, minimal in the class of graphs embedded in a surface of fixed genus, has no face-boundary of length <7. Moreover, in order to prove or disprove Tutte's conjecture for graphs of fixed genus γ, one has to check graphs of order at most 28(γ ? 1) in the orientable case and 14(γ ? 2) in the nonorientable case. So, in particular, it follows immediately that every bridgeless graph of orientable genus ?1 or nonorientable genus ?2 has a nowhere-zero 5-flow. Using a computer, we checked that all graphs of orientable genus ?2 or nonorientable genus ?4 have a nowhere-zero 5-flow.     

Covering homotopy 3-spheres

We prove the following theorem: for any closed orientable 3-manifoldM and any homotopy 3-sphere Σ, there exists a simple 3-fold branched coveringp:M→Σ. We also propose the conjecture that, for any primitive branched coveringp:M→N between orientable 3-manifolds,g(M)≥g(N), whereg denotes the Heegaard genus. By the above mentioned result, the genus 0 case of such conjecture is equivalent to the Poincaré conjecture.     

A Note on Heegaard Splittings of Amalgamated 3-Manifolds

Let M be a compact orientable irreducible 3-manifold, and F be an essential connected closed surface in M which cuts M into two manifolds M1 and M2. If Mi has a minimal Heegaard splitting Mi = Vi ∪Hi Wi with d(H1) + d(H2) ≥ 2(g(M1) + g(M2) -g(F)) + 1, then g(M) = g(M1) + g(M2) - g(F).     

An alternative proof of Lickorish–Wallace theorem

We introduce the concept of s-distance of an unstabilized Heegaard splitting. We prove if a 3-manifold admits an unstabilized genus g Heegaard splitting with s-distance m  , then surgery on some (m−1)$(m-1)$ components link may produce a 3-manifold which admits a stabilized genus g Heegaard splitting. We also give an alternative proof of the fundamental theorem of surgery theory, which states that every closed orientable 3-manifold is obtained by surgery on some link in 3-sphere.     

Heegaard Gradient of Seifert Fibered 3-Manifolds

The infimal Heegaard gradient of a 3-manifold was defined andstudied by Marc Lackenby in an approach towards proving thewell-known virtually Haken conjecture. As instructive examples,Seifert fibered 3-manifolds are considered in this paper. Theauthor shows that a compact orientable Seifert fibered 3-manifoldhas zero infimal Heegaard gradient if and only if it virtuallyfibers over either the circle or a surface other than the 2-sphereor, equivalently, if it has infinite fundamental group. 2000Mathematics Subject Classification 57M10 (primary), 57N10, 57M50(secondary).     

High distance Heegaard splittings of 3-manifolds

J. Hempel [J. Hempel, 3-manifolds as viewed from the curve complex, Topology 40 (3) (2001) 631-657] used the curve complex associated to the Heegaard surface of a splitting of a 3-manifold to study its complexity. He introduced the distance of a Heegaard splitting as the distance between two subsets of the curve complex associated to the handlebodies. Inspired by a construction of T. Kobayashi [T. Kobayashi, Casson-Gordon's rectangle condition of Heegaard diagrams and incompressible tori in 3-manifolds, Osaka J. Math. 25 (3) (1988) 553-573], J. Hempel [J. Hempel, 3-manifolds as viewed from the curve complex, Topology 40 (3) (2001) 631-657] proved the existence of arbitrarily high distance Heegaard splittings.In this work we explicitly define an infinite sequence of 3-manifolds {Mn} via their representative Heegaard diagrams by iterating a 2-fold Dehn twist operator. Using purely combinatorial techniques we are able to prove that the distance of the Heegaard splitting of Mn is at least n.Moreover, we show that π₁(Mn) surjects onto π₁(Mn−1). Hence, if we assume that M⁰ has nontrivial boundary then it follows that the first Betti number β₁(Mn)>0 for all n?1. Therefore, the sequence {Mn} consists of Haken 3-manifolds for n?1 and hyperbolizable 3-manifolds for n?3.     

Unstabilized and uncritical self-amalgamation along essential subsurfaces

Suppose V ∪_S W is a strongly irreducible Heegaard splitting of a compact connected orientable 3-manifold M and F_1 and F_2 are pairwise disjoint homeomorphic essential subsurfaces in ?_V. In this paper,we give a sufficient condition such that the self-amalgamation of V ∪_S W along F_1 and F_2 is unstabilized and uncritical.     

Free actions on handlebodies

The equivalence (or weak equivalence) classes of orientation-preserving free actions of a finite group G on an orientable three-dimensional handlebody of genus g?1 can be enumerated in terms of sets of generators of G. They correspond to the equivalence classes of generating n-vectors of elements of G, where n=1+(g−1)/|G|, under Nielsen equivalence (or weak Nielsen equivalence). For Abelian and dihedral G, this allows a complete determination of the equivalence and weak equivalence classes of actions for all genera. Additional information is obtained for other classes of groups. For all G, there is only one equivalence class of actions on the genus g handlebody if g is at least 1+?(G)|G|, where ?(G) is the maximal length of a chain of subgroups of G. There is a stabilization process that sends an equivalence class of actions to an equivalence class of actions on a higher genus, and some results about its effects are obtained.     

Heegaaed分解有不交曲线性质的一个充分条件

In the paper,we give two conditions that the Heegaard splitting admits the disjoint cnrve property.The main result is that for a genus g(g≥2)strongly irreducible Heegaard splitting(C1,C2;F),let Di be an essential disk in Ci,i=1,2,satisfying(1)at least one of (の)D4 and (の)D2 is separating in F and |(の)D1 (∩)(の)D2|≤ 2g-1;or(2)both (の)D1 and (の)D2 are non-separating in F and |(の)D1 (∩)(の)D2|≤ 2g-2,then(C1,C2;F)has the disjoint curve property.     

The amalgamation of high distance Heegaard splittings is always efficient

Let M be a compact orientable manifold, and F be an essential closed surface which cuts M into two 3-manifolds M ₁ and M ₂. Let be a Heegaard splitting for i = 1, 2. We denote by d(S _i ) the distance of . If d(S ₁), d(S ₂) ≥ 2(g(M ₁) + g(M ₂) − g(F)), then M has a unique minimal Heegaard splitting up to isotopy, i.e. the amalgamation of and . Ruifeng Qiu is supported by NSFC(10625102).     

沿着不连通曲面的Heegaard分解的不可稳定化的融合积

Let M be a 3-manifold, F= {F1 , F2 , . . . , Fn } be a collection of essential closed surfaces in M (for any i, j ∈ {1, ..., n}, ifi≠j, Fi is not parallel to Fj and Fi ∩Fj = φ) and0 M be a collection of components of M. Suppose M-UFi ∈FFi×(-1, 1) contains k components M1 , M2 , . . . , Mk . If each M i has a Heegaard splitting ViUSiWi with d(Si) > 4(g(M1 ) + ··· + g(Mk )), then any minimal Heegaard splitting of M relative to 0M is obtained by doing amalgamations and self-amalgamations from minimal Heegaard splittings or -stabilization of minimal Heegaard splittings of M1 , M2 , . . . , Mk .     

字定理在Heegaard分解方面的一个应用

The word theorem states that x can be denoted as a rotation inserting word of A if x is in the normal closure of A in F(X). As an application of the theorem, in this note a condition that guarantees reducing the genus of Heegaard splitting of 3-manifolds is given. This leads Poincare conjecture to a new formulation.     

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 are the genera of two splittings, then there is a common stabilization of genus . Here we show how to obtain similar results even when the 3-manifold has boundary.

     

Heegaard Genus Formula for Haken Manifolds

Suppose M is a compact orientable 3-manifold and a properly embedded orientable boundary incompressible essential surface. Denote the completions of the components of M – Q with respect to the path metric by M ¹, ...,M ^k . Denote the smallest possible genus of a Heegaard splitting of M, or M ^j respectively, for which ∂M, or ∂M ^j respectively, is contained in one compression body by g(M, ∂M), or g(M ^j , ∂M ^j ) respectively. Denote the maximal number of non-parallel essential annuli that can be simultaneously embedded in M ^j by n _j . Then