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.     

Local detection of strongly irreducible Heegaard splittings

Let S be a Heegaard splitting surface of a compact orientable 3-manifold M. If S is strongly irreducible, the manner in which it can intersect a ball or a solid torus in M is very constrained and the allowable configurations are simple and useful. Splitting surfaces not conforming to these simple local pictures must be weakly reducible.     

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).     

Heegaard分解的重新分块

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How a strongly irreducible Heegaard splitting intersects a handlebody

In [Topology Appl. 90 (1998) 135] Scharleman showed that a strongly irreducible Heegaard splitting surface Q of a 3-manifold M can, under reasonable side conditions, intersect a ball or a solid torus in M in only a few possible ways. Here we extend those results to describe how Q can intersect a handlebody in M.     

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.     

Heegaard splittings 的曲面连通和

Suppose Mi = Vi ∪ Wi （i = 1,2） are Heegaard splittings. A homeomorphism f ： F1 → F2 produces an attached manifold M = M1 ∪F1=F2 M2, where Fi ∪→ δ_Wi. In this paper we define a surface sum of Heegaard splittings induced from the Heegaard splittings of M1 and M2, and give a sufficient condition when the surface sum of Heegaard splitting is stabilized. We also give examples showing that the surface sum of Heegaard splittings can be unstabilized. This indicates that the surface sum of Heegaard splittings and the amalgamation of Heegaard splittings can give different Heegaard structures.     

Thin Position of Heegaard Splittings

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A Note on Surfaces Bounding Hyperbolic 3-Manifolds

We consider the problem of whether a given hyperbolic surface occurs as the totally geodesic boundary of a compact hyperbolic 3-manifold (as some or as the only boundary component). We discuss some explicit examples of hyperbolic surfaces, in particular the surface associated to the small stellated dodecahedron (one of the four Kepler-Poinsot polyhedra) which is the boundary of a hyperbolic icosahedral 3-manifold.     

A Note on Surfaces Bounding Hyperbolic 3-Manifolds

We consider the problem of whether a given hyperbolic surface occurs as the totally geodesic boundary of a compact hyperbolic 3-manifold (as some or as the only boundary component). We discuss some explicit examples of hyperbolic surfaces, in particular the surface associated to the small stellated dodecahedron (one of the four Kepler-Poinsot polyhedra) which is the boundary of a hyperbolic icosahedral 3-manifold.     

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.     

On Separation Index of Heegaard Splittings

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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.     

有亏格为1的Heegaard分解的三维流形中的环面纽结

将存在亏格为1的Heegaard分解T’1∪F T’2的三维流形记为M=L(p,q),其中p和q是互素整数,q/p为T’2的纬线在T’1上的斜率.若环面F上的简单闭曲线γ在M中非平凡,则称γ是M中的环面纽结.本文对在M中沿环面纽结作m/n-Dehn手术所得流形进行了分类,并给出了两个实心环体沿边界上平环作融合所得流形是L(p,q)中环面纽结补的特征描述.     

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 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）.     

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 .     

具有有限组不相交压缩圆片的Heegaard分解

设$V\cup_SW$是一个闭的三维流形亏格为$g$的, 弱可约的Heegaard分解, 并且在合痕意义下只有有限组位于曲面不同侧的不相交的压缩圆片, 则它存在一个广义的Heegaard分解: $V\cup_SW=(V_1\cup_{S_1}W_1)\cup_F(W_2\cup_{S_2}V_2)$, 并且满足对于每个$i=1,2$, 压缩体$W_i$都只有一个分离的压缩圆片且$d(S_i)\geq 2$. 进一步的, 如果有有限且多于1组不相交的压缩圆片, 则至少一个$d(S_i)$等于2, 并且Heegaard曲面满足临界性质.     

关于有界线性算子非负分裂的比较定理

Marek和Szyld建立了有界线性算子非负分裂的比较定理.他们还提出了严格不等式成立的条件,但没有进行详细证明.本注记首先用几个反例说明那里的保证严格不等式成立的条件是不充分的,然后给出正确的关于严格不等式的比较定理.     

A Note on a Class of Strongly Coupled Semilinear Reaction-diffusion Equations

A class of strongly coupled semilinear reaction-diffusion equations is discussed. The FitzHugh-Nagumo equation in [1] is only an example. The results that strongly coupled term generates an analytic semigroup extended the results in [4]. The results of the existence and uniqueness of solution and the stability of zero solution improved that in [1]. This paper also discusses the existence of periodic solution.