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$g(M)=g(M_1)+g(M_2)$.     

两个压缩体融合为一个压缩体的充分必要条件北大核心CSCD

本文给出了两个压缩体沿紧致连通曲面(带边曲面或闭曲面)融合仍是一个压缩体(有非空负边界)的充分必要条件,还给出了两个3维流形沿着边界上的紧致连通带边曲面融合中的融合曲面为边界不可压缩的一个特征描述,同时还证明了压缩体的每个Heegaard分解是标准的.     

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.     

紧带边流形到欧氏空间的嵌入

本文研究了紧带边流形到欧氏空间的嵌入问题,证明了K-连通、边界为(K-1)-连通的紧带边流形能嵌入和整齐嵌入到某些欧氏空间的一个充分必要条件;作为应用,给出了多个边界分支的K-连通紧带边流形,在每个边界分支为(K-1)-连通的情况下,到某些欧氏空间的嵌入结果。     

流形的双连通和分解

所论流形均假设为无边微分或PL紧致流形。设M_1和M_2为两个n维流形,从M_1,M_2各挖去两个不相交的n维闭胞腔之内部,而得M_1,M_2。然后将M_1的边缘与M_2的边缘用一个同胚来粘合,而得一新的n维流形,称为M_1和M_2的双连通和,记作M_1§M_2。一个流形若可表示为两个连通流形的双连通和,则称为可约化的;否则称为不可约化的。     

小3-流形中的双侧可压缩非Heegaard曲面

研究小3-流形中的双侧可压缩非Heegaard 曲面, 给出了这类曲面所具有的若干特殊的性质.     

3维流形中的双侧可压缩曲面与薄形分解

对一个3维流形的任何一组彼此不相交的非过剩的双侧可压缩曲面集所含有的元素个数证明了是有上界的,记为No(M).记L0(M)为M的薄形分解的长度,则有L0(M)≤N0(M) |(e)M|,记L0(V ∪ W)为M的Heegaard分解M=V ∪ W的薄形分解的长度,则对M的任意不可稳定化的Heegaard分解都有L0(V ∪ W)≤N0(W) |(e)M|.     

紧扰动下的谱连通算子是强不可约的 *

回答了D .A .Herrero于1988年提出的一个猜测 :可分复Hilbert空间上每个谱连通算子在紧扰动下是强不可约的 .     

强连通竞赛图的圈分解

设T为n阶强连通竞赛图．本文通过详细刻画不能进行圈分解的强连通竞赛图的特征，证明了满足max{^ ，δ^-}≥5k-5和k≥2的强连通竞赛图T,能够分解为k个圈．     

流形上的k-Hessian方程的Neumann边值问题

本文考虑紧致具有全脐边界的Riemann流形上k-Hessian方程的Neumann边值问题;通过对k-Hessian方程的解做零阶、一阶、二阶估计和使用连续性方法,得到流形上k-Hessian方程的Neumann边值问题的存在性结果.     

On Separation Index of Heegaard Splittings

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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.     

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).     

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 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）.     

Infinite class of new sign ambiguities

In 1934, two kinds of multiplicative relations, the norm and the Davenport-Hasse relations, between Gauss sums, were known. In 1964, H. Hasse conjectured that the norm and the Davenport-Hasse relations were the only multiplicative relations connecting Gauss sums over Fp. However, in 1966, K. Yamamoto provided a simple counterexample disproving the conjecture. This counterexample was a new type of multiplicative relation, called a sign ambiguity, involving a ± sign not connected to elementary properties of Gauss sums. In this paper, we give an explicit product formula involving Gauss sums which generates an infinite class of new sign ambiguities, and we resolve the ambiguous sign by using Stickelberger?s theorem.     

Spanning 2‐trails from degree sum conditions

Suppose G is a simple connected n‐vertex graph. Let σ₃(G) denote the minimum degree sum of three independent vertices in G (which is ∞ if G has no set of three independent vertices). A 2‐trail is a trail that uses every vertex at most twice. Spanning 2‐trails generalize hamilton paths and cycles. We prove three main results. First, if σ₃G)≥ n ‐ 1, then G has a spanning 2‐trail, unless G ? K_1,3. Second, if σ₃(G) ≥ n, then G has either a hamilton path or a closed spanning 2‐trail. Third, if G is 2‐edge‐connected and σ₃(G) ≥ n, then G has a closed spanning 2‐trail, unless G ? K_2,3 or K (the 6‐vertex graph obtained from K_2,3 by subdividing one edge). All three results are sharp. These results are related to the study of connected and 2‐edge‐connected factors, spanning k‐walks, even factors, and supereulerian graphs. In particular, a closed spanning 2‐trail may be regarded as a connected (and 2‐edge‐connected) even [2,4]‐factor. © 2004 Wiley Periodicals, Inc. J Graph Theory 45: 298–319, 2004     

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…