On reducible Heegaard splittings

Let V∪S W be a reducible Heegaard splitting of genus g = g(S)≥2.For a maximal prime connected sum decomposition of V∪S W,let q denote the number of the genus 1 Heegaard splittings of S2×S1 in the decomposition,and p the number of all other prime factors in the decomposition.The main result of the present paper is to describe the relation of p,q and dim(C V∩CW).     

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.     

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 Lower Bound of the Genus of a Self-amalgamated 3-manifolds

Let M be a compact connected oriented 3-manifold with boundary, Q1, Q2 C 0M be two disjoint homeomorphic subsurfaces of cgM, and h ： Q1 → Q2 be an orientation-reversing homeomorphism. Denote by Mh or MQ1=Q2 the 3-manifold obtained from M by gluing Q1 and Q2 together via h. Mh is called a self-amalgamation of M along Q1 and Q2. Suppose Q1 and Q2 lie on the same component F1 of δM1, and F1 - Q1 ∪ Q2 is connected. We give a lower bound to the Heegaard genus of M when M＇ has a Heegaard splitting with sufficiently high distance.     

组合群论中的一个定理

Let F= F(X) be a free group of rand n, A be a finite subset of F(X) and x∈X be a generator. The 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). Finally, an application of the theorem in Heegaard splitting of 3manifolds is given.     

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 .     

Equivalent Separating Curves on Closed Surface of Genus 2

Let {A,B} be a complete system of the closed orientable surface F of genus 2. A simple closed curve C on F is separating with respect to (A, B) if it is disjoint from A∪B and it cuts F into two once-punctured tori X, Y with A(?)X, B(?)Y. Letγbe a simple closed curve on F which is disjoint from A∪B and intersects C essentially in two points. In this paper, we show that up to isotopy, {hnγ(C):n∈Z} is the set containing all the simple closed curves on F which is separating with respect to (A,B), where hγis the Dehn twist alongγon F. This also shows how two simple closed curves on F which are separating with respect to (A,B) are related. The result can be applied to yield all Haken spheres of a Heegaard splitting V∪F W which are weakly equivalent to a given Heken sphere of the splitting.     

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.     

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.     

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.     

关于内亏格1的弱可约SD-分解(英)

设(M；H_1,H_2；F_0)为带边3-流形M的一个SD-分解．称该分解为可约的(或弱可约的)若存在本质圆片D_1■H)_1,D_2■H_2使得■D_1,■D_2■F_0并且■D_1=■D_2(或■D_1∩■D_2=■)．称(M；H_1,H_2；F_0)为内亏格1若F_0为穿孔环面．本文主要结果：一个弱可约的内亏格1的SD-分解或是可约的或是双经的．     

关于图中存在不含给定k-因子的[a,b]-因子的度条件

设$1\leq a<b, 0\leq k$是整数. 设$G$是一个含有$k$-因子$Q$且阶为$|G|$的图. 设\delta(G)$表示$G$的最小度, 且$\delta(G)\geq a+k$. 如果$Q$连通, 设$\varepsilon=k$, 否则设$\varepsilon=k+1$.证明:当$b\geq a+\varepsilon-1$时, 如果对$G$的任意两个不相邻的点$x$和$y$都有max$\{d_G(x),d_G(y)\}\geq {\rm max}\{{{a|G|} \over {a+b}},{{(|G|+(a-1)(2a+b+\varepsilon-2))} \over {b+1}}\}+k$, 那么$G$有一个$[a, b]$-因子$F$ 使得 $E(F)\cap E(Q)=\emptyset$. 这个度条件是最佳的, 条件$b\geqa+\varepsilon-1$不能去掉. 进一步,得到图存在含给定$k$-因子的$[a, b]$-因子的度条件.     

Pairs of Disjoint Dominating Sets and the Minimum Degree of Graphs

For a connected graph G of order n and minimum degree δ we prove the existence of two disjoint dominating sets D ₁ and D ₂ such that, if δ ≥ 2, then ${|D_1\cup D_2|\leq \frac{6}{7}n}$ unless G = C ₄, and, if δ ≥ 5, then ${|D_1\cup D_2|\leq 2\frac{1+\ln(\delta+1)}{\delta+1}n}$ . While for the first estimate there are exactly six extremal graphs which are all of order 7, the second estimate is asymptotically best-possible.     

On 3-Submanifolds of S^3 Which Admit Complete Spanning Curve Systems

Let M be a compact connected 3-submanifold of the 3-sphere S~3 with one boundary component F such that there exists a collection of n pairwise disjoint connected orientable surfaces S = {S_1, ···, S_n} properly embedded in M, ?S = {?S_1, ···, ?S_n}is a complete curve system on F. We call S a complete surface system for M, and ?S a complete spanning curve system for M. In the present paper, the authors show that the equivalent classes of complete spanning curve systems for M are unique, that is, any complete spanning curve system for M is equivalent to ?S. As an application of the result,it is shown that the image of the natural homomorphism from the mapping class group M(M) to M(F) is a subgroup of the handlebody subgroup Hn.     

循环群$Z_{2}$的模向量不变式

Let G be the finite cyclic group Z_2 and V be a vector space of dimension 2n with basis x_1,...,x_n,y_1,...,y_n over the field F with characteristic 2.If σ denotes a generator of G,we may assume that σ（x_i）= ayi,σ（y_i）= a～-1x_i,where a ∈ F.In this paper,we describe the explicit generator of the ring of modular vector invariants of F[V]～G.We prove that F[V]～G = F[l_i = x_i ＋ ay_i,q_i = x_iy_i,1 ≤ i ≤ n,M_I = X_I ＋ a～-I-Y_I],where I∈An = {1,2,...,n},2 ≤-I-≤ n.     

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.     

Aizenman's Theorem for Orthogonal Polynomials on the Unit Circle

For suitable classes of random Verblunsky coefficients, including independent, identically distributed, rotationally invariant ones, we prove that if $$ \bbE \biggl( \int\f{d\theta}{2\pi} \biggl|\biggl( \f{\calC + e^{i\theta}}{\calC-e^{i\theta}} \biggr)_{k\ell}\biggr|^p \biggr) \leq C_1 e^{-\kappa_1 \abs{k-\ell}} $$ for some $\kappa_1 < 0$ and $p < 1$, then for suitable $C_2$ and $\kappa_2 >0$, $$ \bbE \Bigl( \sup_n \abs{(\calC^n)_{k\ell}}\Bigr) \leq C_2 e^{-\kappa_2 \abs{k-\ell}}. $$ Here $\calC$ is the CMV matrix.     

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.     

关于超越亚纯系数微分方程亚纯解的零点

本文研究了非齐次线性微分方程的复振荡问题，其中，D0，D1，…，D(k-1)，是超越亚纯函数．当存在某个Ds（1≤s≤k-1）比其它Dj（j≠s）有较快增长的意义下起支配作用时，得到了微分方程（I）亚纯解的零点收敛指数的精确估计式．