CW复形的乘积的伦型

<正> §1.主要结果 CW复形的乘积不一定是一个CW复形,在文[2]中,Milnor证明了两个CW复形的乘积一定和一个CW复形具有相同的伦型.由此可以很容易地证明,有限多个CW复形的乘积一定和一个CW复形具有相同的伦型.我们研究了任意多个(有限或无穷)的情况.我们用表示与CW复形伦型相同的所有空间所成的集合,用表示可压缩为一点的所有空间所成的集合,我们有以下结果:     

CW复形可乘的一个充要条件

<正> 作为单纯复形的推广,J.H.C.Whitehead引入了重要的CW复形的概念并给出CW复形一系列性质.关于CW复形的乘积,其结果是:设K,L是两个CW复形,若L是局部有限的,则乘积复形K×L也是CW复形.这里L的局部有限性假定是否必需,他于脚注中提出这个问题.Dowker C.H.[2;页563]举例说明这个假定一般说是不能取消的.过后,Milnor J.[3:引理2.1]指出另一结果,即当K与L皆为可数CW复形     

Gorenstein投射、内射和平坦复形

证明了在任意结合环R上,复形C是Gorenstein投射复形当且仅当每个层次的模C~m是Gorenstein投射模,由此给出了复形Gorenstein投射维数的性质刻画.并证明了对于正合复形C,若对于任意投射模Q,函子Hom(-,Q)作用复形C后仍然得到正合复形,则C是Gorenstein投射复形当且仅当对于所有的m∈Z,有Ker(δ_C~m)是Gorenstein投射模.类似地,本文也讨论了关于Gorenstein内射和Gorenstein平坦复形的相应结果.     

正则可折叠方体复形中广义长方体的一些性质

在正则可折叠方体复形的研究中,广义长方体的概念起着重要的作用。本文叙述了方体复形中的广义长方体(GC)的定义,证明了正则可折叠方体复形K中的广义长方体有好些良好的性质。例如,K中任一组两两相交的广义长方体的交集仍是个非空的广义长方体;存在着从K的多面体|K|到K中任一个非空广义长方体的不增加距离的收缩映射,等等。     

有关重复排列的一类计数问题

利用生成函数解决了从n个元素的集合中任意重复选取r个元素且这r个元素中含有不同元素的个数一定时，所构成的不同r-序列的方法数．     

桥数为3的纽结上的Dehn手术与透镜空间L(3,1)

对于桥教最多为3的纽结,证明了Gordon的L(3,1)猜测,即如果K是三维球面中桥数不超过3的非平凡纽结,那么K上的Dehn手术不能产生透镜空间L(3,1).     

关于Poincaré定理的注记

§1.引言命p,q,n是三个正整数,p+q=n,通常,从考虑n维定向组合同调流形K及其对偶复形K~＊的定向元素之相交指数出发,可以证明(见[2],467-483页).定理1.复形K的p维上同调群~PH~G(K)与复形K~＊的q维同调群~qH_G(K~＊)彼此同构.由于K和K~＊具有同一的重心重分K′,而同调群是重心重分的不变量,所以,从定理     

論多面体映入n維球的分类問題

<正> 設K为n+r維有限多面体,S~n为n維球.把連續映像f:K~(n+r)→S~n的全体看成一个集合,在这个集合中二个映像f_0,f_1之間,如果存在同伦关系,即有連續映像F:K~(n+r)×I→S~n存在,使得     

（M,N）树的几个判定定理

(m,n)树是图论中树的概念在n维复形上的推广。关于n维复形和它的(m,n)回路,(m,n)连通,(m,n)简单,纯n维复形,(m,n)树等概念见文献[3],[4]。 用a_k(K)表复形K中的k维单形个数。又记     

R~n中构形和单纯复形的一个最优度量不等式

对于R~n中一般位置的点构形,定义了第r个极小凸包距离的概念,证明了极小凸包距离和极小点-超平面距离之间的一个最优不等式.该不等式的一个直接推论是:对于R~n中一个k-维单纯复形K,我们能用其顶点集的极小点-超平面距离下估计K的Gromov-Guth厚度.进一步,在每一个维数k,构造了例子说明该下界几乎是最优的.     

有限群在特征为零的任意域上的表示的一些注记

Let G be a finite group and K a field of characteristic zero.It is well-known that if K is a splitting field for G,then G is abelian if and only if any irreducible representation of G has degree 1.In this paper,we generalize this result to the case that K is an arbitrary field of characteristic zero（that is,K need not be a splitting field for G）,and we also obtain the orthogonality relations of irreducible K-characters of G in this case.Our results generalize some well-known theorems.     

On the Alexander polynomials of knots with Gordian distance one

We consider a condition on a pair of the Alexander polynomials of knots which are realizable by a pair of knots with Gordian distance one. We show that there are infinitely many mutually disjoint infinite subsets in the set of the Alexander polynomials of knots such that every pair of distinct elements in each subset is not realizable by any pair of knots with Gordian distance one. As one of the subsets, we have an infinite set containing the Alexander polynomials of the trefoil knot and the figure eight knot. We also show that every pair of distinct Alexander polynomials such that one is the Alexander polynomial of a slice knot is realizable by a pair of knots of Gordian distance one, so that every pair of distinct elements in the infinite subset consisting of the Alexander polynomials of slice knots is realizable by a pair of knots with Gordian distance one. These results solve problems given by Y. Nakanishi and by I. Jong.     

Buchstaber Invariants of Universal Complexes

Davis and Januszkiewicz introduced (real and complex) universal complexes to give an equivalent definition of characteristic maps of simple polytopes,which now can be seen as "colorings".The author derives an equivalent definition of Buchstaber invariants of a simplicial complex K,then interprets the difference of the real and complex Buchstaber invariants of K as the obstruction to liftings of nondegenerate simplicial maps from K to the real universal complex or the complex universal complex.It was proved by Ayzenberg that real universal complexes can not be nondegenerately mapped into complex universal complexes when dimension is 3.This paper presents that there is a nondegenerate map from 3-dimensional real universal complex to 4-dimensional complex universal complex.     

$H(2)$-Unknotting Number of a Knot

An $H(2)$-move is a local move of a knot which is performed by adding a half-twisted band. It is known an $H(2)$-move is an unknotting operation. We define the $H(2)$-unknotting number of a knot $K$ to be the minimum number of $H(2)$-moves needed to transform K into a trivial knot. We give several methods to estimate the $H(2)$-unknotting number of a knot. Then we give tables of $H(2)$-unknotting numbers of knots with up to 9 crossings.     

Homology Groups of Simplicial Complements

This paper deals with homology groups induced by the exterior algebra generated by the simplicial compliment of a simplicial complex K. By using ech homology and Alexander duality, the authors prove that there is a duality between these homology groups and the simplicial homology groups of K.     

Thin position and essential planar surfaces

Abby Thompson proved that if a link is in thin position but not in bridge position, then the knot complement contains an essential meridional planar surface, and she asked whether some thin level surface must be essential. This note is to give a positive answer to this question, showing that if a link is in thin position but not bridge position, then a thinnest level surface is essential. A theorem of Rieck and Sedgwick follows as a consequence, which says that thin position of a connected sum of small knots comes in the obvious way.     

P2r,6图的优美性

Kathiresan KM证实$P_{2r,2m-1}$($r,m$皆为任意正整数)是优美的且猜想:除了$(a,b)=(2r-1,4m-2)$外,所有的$P_{a,b}$都是优美的.杨元生证实$P_{2r+1,2m+1}$是优美的,并且证实了当$r\leq7$, $r=9$时的$P_{2r,2m}$是优美的.严谦泰证实$r$为奇数时$P_{2r,2m}$是优美的.采用回溯和分支限界算法搜索到了一个适合于所有$P_{2r,b}$图($r,b$皆为任意正整数)的优美标号,用函数构造法提取其规律并从数学的严格性进行了证明,使得所有的$P_{2r,b}$图($r,b$ 皆为任意正整数)的优美性得到了证实.     

复环面情形的Suita猜想

对任意复环面的情形证明了推广的Suita猜想,即απK≥c~2(α∈R),其中c是修正后的对数容度,K是对角线上的Bergman核.还阐明了对任意亏格≥2的紧Riemann面情形的公开问题.文中结果的证明部分地依赖于椭圆函数理论.     

On the semisimplicity of polyhedral isometries

If a polyhedral complex has only finitely many isometry types of cells, then all of its cellular isometries are semisimple. If is 1-connected and non-positively curved, then any solvable group that acts freely by cellular isometries on is finitely generated and contains an abelian subgroup of finite index.

     

Relationship Between the Stiffly Weighted Pseudoinverse and Multi-Level Constrained Rseudoinverse

It is known that for a given matrix A of rank r, and a set D of positive diagonal matrices, supw∈D‖（W^1/2A) W^1/2‖ = (miniσ (A^(i))^-1, in which (A^(i) is a submatrix of A formed with r = (rank(A)) rows of A, such that (A^(i) has full row rank r. In many practical applications this value is too large to be used. In this paper we consider the case that both A and W(∈D) are fixed with W severely stiff. We show that in this case the weighted pseudoinverse (W^1/2‖A) W^1/2‖ is close to a multilevel constrained weighted pseudoinverse therefore ‖(W^1/2A) W^1/‖2 is uniformly bounded.We also prove that in this case the solution set the stiffly weighted least squares problem is close to that of corresponding multi-level constrained least squares problem.