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.     

The Heegaard genera of surface sums

Let M be a compact orientable 3-manifold, and let F be a separating (resp. non-separating) incompressible surface in M which cuts M into two 3-manifolds M₁ and M₂ (resp. a manifold M₁). Then M is called the surface sum (resp. self surface sum) of M₁ and M₂ (resp. M₁) along F, denoted by M=M₁F_∪M₂ (resp. M=M₁F_∪). In this paper, we will study how g(M) is related to χ(F), g(M₁) and g(M₂) when both M₁ and M₂ have high distance Heegaard splittings.     

Isotoping Heegaard surfaces in neat positions with respect to critical distance Heegaard surfaces

Suppose a closed orientable 3-manifold M has a genus g Heegaard surface P with distance d(P)=2g. Let Q be another genus g Heegaard surface which is strongly irreducible. Then we show that there is a height function f:M→I induced from P such that by isotopy, Q is deformed into a position satisfying the following; (1) fQ_| has 2g+2 critical points p₀,p₁,…,p2g+1 with f(p₀)<f(p₁)<?<f(p2g+1) where p₀ is a minimum and p2g+1 is a maximum, and p₁,…,p2g are saddles, (2) if we take regular values ri (i=1,…,2g+1) such that f(pi−1)<ri<f(pi), then f−1(ri)∩Q consists of a circle if i is odd, and f−1(ri)∩Q consists of two circles if i is even.     

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.     

Branched spines and Heegaard genus of 3-manifolds

We prove that an invariant of closed 3-manifolds, called the block number, which is defined via flow-spines, equals the Heegaard genus, except for S ³ and S ² × S ¹. We also show that the underlying 3-manifold is uniquely determined by a neighborhood of the singularity of a flow-spine. This allows us to encode a closed 3-manifold by a sequence of signed labeled symbols. The behavior of the encoding under the connected sum and a criterion for reducibility are studied.     

Stabilizing and destabilizing Heegaard splittings of sufficiently complicated 3-manifolds

Let M ₁ and M ₂ be compact, orientable 3-manifolds with incompressible boundary, and M the manifold obtained by gluing with a homeomorphism ${\phi : {\partial}M_1 \to {\partial}M_2}$ . We analyze the relationship between the sets of low genus Heegaard splittings of M ₁, M ₂, and M, assuming the map ${\phi}$ is “sufficiently complicated”. This analysis yields counter-examples to the Stabilization Conjecture, a resolution of the higher genus analogue of a conjecture of Gordon, and a result about the uniqueness of expressions of Heegaard splittings as amalgamations.     

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.     

Heegaard genera of high distance are additive under annulus sum

Let Mi be a compact orientable 3-manifold, and Ai a non-separating incompressible annulus on ∂Mi, i=1,2. Let h:A₁→A₂ be a homeomorphism, and M=M₁h_∪M₂ the annulus sum of M₁ and M₂ along A₁ and A₂. In the present paper, we show that if Mi has a Heegaard splitting ViSi_∪Wi with distance d(Si)?2g(Mi)+3 for i=1,2, then g(M)=g(M₁)+g(M₂). Moreover, if g(Fi)?2, i=1,2, then the minimal Heegaard splitting of M is unique.     

Hyperbolic spatial graphs in 3-manifolds

For any closed connected orientable 3-manifold M, we present a method for constructing infinitely many hyperbolic spatial embeddings of a given finite graph with no vertex of degree less than two from hyperbolic spatial graphs in S³ via the Heegaard splitting theory. These spatial embeddings are adjustable so as to take cycle subgraphs into specified homotopy classes of loops in M.     

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.     

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).     

Cohomogeneity one Riemannian manifolds of non-positive curvature

We study a G-manifold M which admits a G-invariant Riemannian metric g of non-positive curvature. We describe all such Riemannian G-manifolds (M,g) of non-positive curvature with a semisimple Lie group G which acts on M regularly and classify cohomogeneity one G-manifolds M of a semisimple Lie group G which admit an invariant metric of non-positive curvature. Some results on non-existence of invariant metric of negative curvature on cohomogeneity one G-manifolds of a semisimple Lie group G are given.     

The Heegaard Genus of Amalgamated 3-Manifolds

Let M and M′ be simple 3-manifolds, each with connected boundary of genus at least two. Suppose that Mand M′ are glued via a homeomorphism between their boundaries. Then we show that, provided the gluing homeomorphism is ‘sufficiently complicated’, the Heegaard genus of the amalgamated manifold is completely determined by the Heegaard genus of Mand M′ and the genus of their common boundary. Here, a homeomorphism is ‘sufficiently complicated’ if it is the composition of a homeomorphism from the boundary ofM to some surface S, followed by a sufficiently high power of a pseudo-Anosov onS, followed by a homeomorphism to the boundary of M′. The proof uses the hyperbolic geometry of the amalgamated manifold, generalised Heegaard splittings and minimal surfaces.     

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.     

The distance between two separating,reducing slopes is at most 4

Let M be a simple 3-manifold such that one component of ∂M, say F, has genus at least two. For a slope α on F, we denote by M(α) the manifold obtained by attaching a 2-handle to M along a regular neighborhood of α on F. If M(α) is reducible, then α is called a reducing slope. In this paper, we shall prove that the distance between two separating, reducing slopes on F is at most 4. This work is supported by NSFC (10625102).     

The self-amalgamation of high distance Heegaard splittings is always efficient

Let M be a compact orientable irreducible 3-manifold, F be an essential non-separating closed surface in M. We denote by η(F) the open regular neighborhood of F. If M−η(F) has a high distance Heegaard splitting, then M has a unique minimal Heegaard splitting up to isotopy.     

Stiefel–Whitney surfaces and the tri-genus of non-orientable 3-manifolds

Every non-orientable 3-manifold M can be expressed as a union of three orientable handlebodies V ₁,V ₂,V ₃ whose interiors are pairwise disjoint. If g _i denotes the genus of ∂V _i and g ₃≤g ₂≤g ₃, then the tri-genus of M is the minimum triple (g ₁,g ₂,g ₃), ordered lexicographically. If the Bockstein of the first Stiefel–Whitney class βw ₁(M)=0, then M has tri-genus (0,2g,g ₃), where g is the minimal genus of a 2-sided Stiefel Whitney surface of M. In this paper it is shown that, if βw ₁(M)&\ne;0, then M has tri-genus (1,2g−1,g ₃), where g is the minimal genus of a (1-sided) Stiefel–Whitney surface. As an application the tri-genus of certain graph manifolds is computed. Received: 28 April 1999     

Estimating Matveev's complexity via crystallization theory

In [M.R. Casali, Computing Matveev's complexity of non-orientable 3-manifolds via crystallization theory, Topology Appl. 144(1-3) (2004) 201-209], a graph-theoretical approach to Matveev's complexity computation is introduced, yielding the complete classification of closed non-orientable 3-manifolds up to complexity six. The present paper follows the same point-of view, making use of crystallization theory and related results (see [M. Ferri, Crystallisations of 2-fold branched coverings of S³, Proc. Amer. Math. Soc. 73 (1979) 271-276; M.R. Casali, Coloured knots and coloured graphs representing 3-fold simple coverings of S³, Discrete Math. 137 (1995) 87-98; M.R. Casali, From framed links to crystallizations of bounded 4-manifolds, J. Knot Theory Ramifications 9(4) (2000) 443-458]) in order to significantly improve existing estimations for complexity of both 2-fold and three-fold simple branched coverings (see [O.M. Davydov, The complexity of 2-fold branched coverings of a 3-sphere, Acta Appl. Math. 75 (2003) 51-54] and [O.M. Davydov, Estimating complexity of 3-manifolds as of branched coverings, talk-abstract, Second Russian-German Geometry Meeting dedicated to 90-anniversary of A.D.Alexandrov, Saint-Petersburg, Russia, June 2002]) and 3-manifolds seen as Dehn surgery (see [G. Amendola, An algorithm producing a standard spine of a 3-manifold presented by surgery along a link, Rend. Circ. Mat. Palermo 51 (2002) 179-198]).     

On the Heegaard Floer homology of a surface times a circle

We make a detailed study of the Heegaard Floer homology of the product of a closed surface Σg of genus g with S¹. We determine HF⁺(Σg×S¹,s;C) completely in the case c₁(s)=0, which for g?3 was previously unknown. We show that in this case HF^∞ is closely related to the cohomology of the total space of a certain circle bundle over the Jacobian torus of Σg, and furthermore that HF⁺(Σg×S¹,s;Z) contains nontrivial 2-torsion whenever g?3 and c₁(s)=0. This is the first example known to the authors of torsion in Z-coefficient Heegaard Floer homology. Our methods also give new information on the action of H₁(Σg×S¹) on HF⁺(Σg×S¹,s) when c₁(s) is nonzero.