A Sufficient Condition for the Genus of an Annulus Sum of Two 3-manifolds to Be Non-degenerate

Let Mi be a compact orientable 3-manifold, and Ai a non-separating incompressible annulus on a component of δMi, say Fi, i = 1, 2. Let h ： A1 → A2 be a homeomorphism, and M→M1 ∪h M2, the annulus sum of Mi and M2 along A1 and A2. Suppose that Mi has a Heegaard splitting Vi ∪Si Wi with distance d（Si） ≥ 2g（Mi） ＋ 2g（F3-i） ＋ 1, i = 1, 2. Then g（M） = g（M1） ＋ g（M2）, and the minimal Heegaard splitting of M is unique, which is the natural Heegaard splitting of M induced from Vi∪S1 Wi and V2 ∪S2 W2.     

A Note on Heegaard Genus of Self-amalgamated 3-Manifold

Let M be a connected orientable compact irreducible 3-manifold. Suppose that αM consists of two homeomorphic surfaces F1 and F2, and both F1 and F2 are compressible in M. Suppose furthermore that g(M, F1) = g(M) + g(F1), where g(M, F1)is the Heegaard genus of M relative to F1. Let Mfbe the closed orientable 3-manifold obtained by identifying F1 and F2 using a homeomorphism f : F1 → F2. The authors show that if f is sufficiently complicated, then g(Mf) = g(M, αM) + 1.     

A Lower Bound for the Sectional Genus of Quasi-Polarized Surfaces

Let (X,L) be a quasi-polarized variety, i.e. X is a smooth projective variety over the complex numbers and L is a nef and big divisor on X. Then we conjecture that g(L) = q(X), whereg(L) is the sectional genus of L and . In this paper, we treat the case . First we prove that this conjecture is true for , and we classify (X,L) withg(L)=q(X), where is the Kodaira dimension of X. Next we study some special cases of .     

Incompressible surfaces in handlebodies and closed 3-manifolds of Heegaard genus 2

In this paper, we shall prove that for any integer 0$">, 1) a handlebody of genus 2 contains a separating incompressible surface of genus , 2) there exists a closed 3-manifold of Heegaard genus which contains a separating incompressible surface of genus .

     

平环和的Heegaard亏格的一个下界

Let Mi, i = 1,2, be a compact orientable 3-manifold, and Ai an incompressible annulus on a component Fi of OMi. Suppose A1 is separating on F1 and A2 is non-separating on F2. Let M be the annulus sum of M1 and M2 along A1 and A2. In the present paper, we give a lower bound for the genus of the annulus sum M in the condition of the Heegaard distances of the submanifolds M1 and M2     

Classifying Genus Two 3-Manifolds up to 34 Tetrahedra

In this paper, we present a catalogue of all genus two 3-manifolds admitting a contracted triangulation with at most 34 simplexes. Then we give a complete classification of the above manifolds. Mathematics Subject Classifications (2000) 57M05, 57N10, 57M15.Work performed under the auspicies of the G.N.S.A.G.A. of the C.N.R. (National Research Council of Italy) and financially supported by M.I.U.R. of Italy (project Strutture geometriche delle varietà reali e complesse).     

Bounded 3-manifolds with Distance n Heegaard Splittings

We prove that for any integer n≥2 and g ≥ 2, there are bounded 3-manifolds admitting distance n, genus g Heegaard splittings with any given bound-aries.     

Unstabilized Self-amalgamation of a Heegaard Splitting along Disks

In this paper, we prove that a self-amalgamation of a strongly irreducible Heegaard splitting along disks is unstabilized.     

大距离Heegaard分解的三穿孔球面和

设$M_i~(i=1,2)$是一个紧致可定向的三维流形, $F_i$是$M_i$边界上的一个不可压缩曲面, $M=M_{1}\cup_{f}M_{2}$, 其中$f$是$F_1$到$F_2$一个同胚,对于具有特定条件的相粘曲面$F_i$, 如果$M_i$具有一个Heegaard距离至少是$2(g(M_1)+g(M_2))+1$的Heegaard分解,则$g(M)=g(M_1)+g(M_2)$.     

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.     

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

Heegaard spines of 3-manifolds

Summary We describe a procedure to construct a 4-coloured graph representing a closed, connected 3-manifold M starting from a Heegaard diagram of M. As a consequence, we prove that, to each Heegaard diagram of a (closed) 3-manifold M, canonically corresponds a spine (Heegaard spine) of M.     

根点自粘合后图的亏格分布

利用删点、加边原理,多种乘法法则,自粘合定理给出了一个双根图在其中一个根点的度为任意大的情形下根点自粘合后图的亏格分布,推广了Gross在文[Genus distribution of graph amalgamations:self-pasting at root-vertices,Aust.J.Comb.,2011,49:19-38]中两个根点度均为2的类似结果.     

与直径和围长有关的最大亏格的下界的一个注记

改正了文章"与直径和围长有关的最大亏格的下界(数学学报2004,47(6):1201-1204)"中的一个错误结论,并得到了如下结果:设G是直径为d(G)的简单图,若G的围长g(G)■d(G),则ξ(G)■2,从而γM(G)■(1/2)β(G)-1.     

Spanning Annuli in 3—manifolds

Al3-manifOldsandsurfacesconsideredinthisPaperareassumedtobecompactandori-entable,andallconcePtsandnotationsnotdefinedinthepaperareStandardfsee,forexample[2,3j.AcompressionbodyHisconstructedbyadding2-handlestoSXIalongacollectionofpairwisdisjointsimpleclosedcurvesonSX{o},andcaPpingoffanyresulting2-spherebound-arycomponentSwith3-balls,whereSisaconnectedclosedorientablesurface.ThecomponentSX{1}Of8Hisdenoted8 Handthesurface8H-8 H,whichmayormaynotbeconnect-ed,isdenoted8H.If8H=gi,Hisahandleb…     

A formula on Heegaard genus of self-amalgamated 3-manifolds

Let M be an orientable compact irreducible and ∂-irreducible 3-manifold, and suppose ∂M consists of two boundary components F₁ and F₂ with g(F₁)=g(F₂)>1. Let Mf be the closed orientable 3-manifold obtained by identifying F₁ and F₂ via a homeomorphism f:F₁→F₂. With the assumption that M is small or g(M,F₁)=g(M)+g(F₁), we show that if f is sufficiently complicated, then g(Mf)=g(M,∂M)+1.     

The pants sum of high distance Heegaard splittings简

Let Mi be a compact orientable 3-manifold, and Fi be an incompressible surface on δMi, i -= 1,2. Let f ： F1 →F2 be a homeomorphism, and M = M1 UI M2. In this paper, under certain assumptions for the attaching surface Fi, we show that if both M1 and M2 have Heegaavd splittings with distance at least 2（g（M1）＋ g（M2））＋ 1, then g（M） = g（M1）＋g（M2）.     

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.     

Heegaard surfaces and measured laminations, II: Non-Haken 3-manifolds

A famous example of Casson and Gordon shows that a Haken 3-manifold can have an infinite family of irreducible Heegaard splittings with different genera. In this paper, we prove that a closed non-Haken 3-manifold has only finitely many irreducible Heegaard splittings, up to isotopy. This is much stronger than the generalized Waldhausen conjecture. Another immediate corollary is that for any irreducible non-Haken 3-manifold , there is a number such that any two Heegaard splittings of are equivalent after at most stabilizations.

     

On(e)-reducible 3-manifolds Which Admit Complete Surface Systems

In the present paper, we consider a class of compact orientable 3-manifolds with one boundary component, and suppose that the manifolds are ?-reducible and admit complete surface systems. One of our main results says that for a compact orientable, irreducible and ?-reducible 3-manifold M with one boundary component F of genus n > 0 which admits a complete surface system S′, if D is a collection of pairwise disjoint compression disks for ?M , then there exists a complete surface system S for M , which is equivalent to S′, such that D is disjoint from S . We also obtain some properties of such 3-manifolds which can be embedded in S3.