拓扑偶的自同伦等价群

研究拓扑偶范畴中的自同伦等价群.对于同伦可结合与同伦可逆的Co-H空间X与Y,我们将在一定的条件下得到一条可裂的短正合序列:1→(?)(X)→(?)(X(?)Y)→(?)(Y)→1.     

同伦带的长度与宽度

<正> 若X、Y是拓扑空间,f:X→Y是连续映射,S_q~i是Steenrod运算,则有交换图式:(q=0,1,2…) 又若(1)式中各群都是有限生成的,则(1)表为一系列矩阵等式(本文所提到的矩阵都是二元域Z_2上的矩阵).进而,给定自然数n,只考虑q=n,n+1;j=2及f为同伦等价,则(1)化为:     

关于co-H-空间上的映射(Ⅱ)

对于 co-H-空间 X 和 Y,本文对所有 X 到 Y 的 co-H-映射的同伦等价类所成的群[X,Y]~H([X,Y]的子群)的秧进行估值,得到如下结论:当 X 满足一定条件时ρ[X,Y]~H≤(?)β_K(X)β_k(Y)以及当取同纬映象结构时ρ[SX,SY]~H=(?)β_K(X)β_k(Y).另外本文还讨论了 co-H-空间和回路空间上的 F-等价问题,并获得一些结果.     

关于CO-H-空间上的映射(Ⅰ)

对CO-H-空间X和Y,本文得到[X,Y]上的元素为有限阶元素的一个充分条件,并讨论了全体从X到Y的CO-H-映射的同伦等价类所组成的集合[X,Y]_(co-H)对[X,Y]上的加法运算封闭的条件.     

论附属于某种连续映像的不等式及其在纤维空间中的应用

<正> §1. 设 X,Y 为拓扑空间,又设 f:X→Y 为连续映像.J.H.C.Whitehead 证明 X,Y 为 CW 丛而 f 能导出基本群及上同调群间的同模对应时,f 为同伦对等映像.映像 f 是否存在,不仅与 X,Y 的基本群及上同调群的构造有关,而与 X,Y 内在的几何结构有密切的关系.连续照像 f 导出 X,Y 之间上同调群的准同模对应 f,那末 f 能与某些准同模对应相交换,由此 J.H.C.Whitehead 指出正则准同模的观念.由[4]可知正则同模论供应我们许多同伦不变量,它们是直接可以计算的东西,并且对于 X,Y 间连续映像的分类问题,应该有密切的关系.     

关于链同伦前伸的注记

首先,本文在链复形和链映射范畴d中证明了各种同伦扩张性质(HEP,WHEP,RWHEP与VWHEP)是等价的.这不同于拓扑空间和连续映射范畴Top.其次,在范踌d中也有Puppe序列,本文证明了约化奇异链函子保持Puppe序列的右正合性.最后,与范畴Top相类似,在链同伦前伸方块中一个链同伦等价诱导出另一个链同伦等价.     

关于链同伦前伸的注记

首先,本文在链复形和链映射范畴d中证明了各种同伦扩张性质(HEP,WHEP,RWHEP与VWHEP)是等价的.这不同于拓扑空间和连续映射范畴Top.其次,在范踌d中也有Puppe序列,本文证明了约化奇异链函子保持Puppe序列的右正合性.最后,与范畴Top相类似,在链同伦前伸方块中一个链同伦等价诱导出另一个链同伦等价.     

Locale同伦理论

本文引入了locale连续映射同伦的概念,建立了locale同伦范畴,构造性地证明了任一locale连续映射都同伦等价于一个locale包含映射。通过引入locale H群的概念(它是locale群概念的自然推广),建立了locale同伦范畴到群同态范畴的一个反变函子。特别地,我们建立了locale同伦群范畴上的基本群函子,证明了locale L上以p为基点的基本群同构于L的谱空间pt(L)上以p为基点的基本群。因此,基本群函子是locale范畴中的一个同伦不变量。     

拓扑群作用下度量空间中链回归点集的研究

仿造度量空间中链回归点的定义,给出了拓扑群作用下度量空间中G-链回归点的概念,并将度量空间中链回归点的一些结论,推广到拓扑群作用下度量空间中,得到如下结果:1)同胚伪等价映射f的G-链回点集等于它的逆映射f~(-1)的G-链回归点集;2)伪等价映射f的G-链回点集和G-链等价集对G强不变;3)同胚等价映射f的G-链回点集f对强不变.4)等价映射f限制在它的G-链回归点集上形成的G-链回归点集就是等价映射f在度量G-空间X上形成的G-链回归点集.这些结果丰富了拓扑群作用下度量空间中G-链回归点的理论.     

关于同伦正则态射

该文在点标拓扑空间的范畴中引进了同伦正则态射的概念,研究了它存在的条件,性质以及它与同伦单（满）态,同伦正则单（满）态和同伦等价之间的密切关系。     

Some Properties of Aut（X） and the Subgroup Aut∑（X）

Let Aut. （X） denote the group of homotopy classes of self-homotopy equivalences of X, which induce identity automorphisms of homology group. We describe a decomposition of Aut. （X1 V…VXn） as a product of its simpler subgroups. We consider the subgroup Aut∑（X） of all self homotopy classes α of X such that ∑α=1∑X： ∑X → ∑X, and also give some properties of Aut∑（X）.     

6p阶可解对称图的分类

设ｘ是简单无向图，Ｇ是Ａｕｔ（Ｘ）的一个于群，Ｘ称为Ｇ－对称的，如果Ｇ在ｘ的１－孤（即两相邻顶点构成的有序偶）集合上的作用是传递的；ｘ称为对称图，如果Ｘ是Ａｕｔ（ｘ）－对称的；ｘ称为可解对称的，如果Ａｕｔ（Ｘ）包含可解子群Ｇ，使Ｘ是Ｇ－对称的．本文给出了具有６Ｐ个顶点的可解对称图的一个分类，这里ｐ≥５是素数．     

Some Properties of Aut$_∗(X)$ and the Subgroup Aut$_Σ(X)$

Let Aut$_∗(X)$ denote the group of homotopy classes of self-homotopy equivalences of $X$, which induce identity automorphisms of homology group. We describe a decomposition of Aut$_∗(X_1∨· · ·∨X_n)$ as a product of its simpler subgroups. We consider the subgroup Aut$_Σ(X)$ of all self homotopy classes α of $X$ such that $Σα = 1_{ΣX} : ΣX → ΣX$, and also give some properties of Aut$_Σ(X)$.     

Smooth automorphisms and path-connectedness in Borel dynamics

Let Aut(X, B) be the group of all Borel automorphisms of a standard Borel space (X, B). We study topological properties of Aut(X, B) with respect to the uniform and weak topologies, τ and p, defined in [Bezuglyi S., Dooley A.H., Kwiatkowski J., Topologies on the group of Borel automorphisms of a standard Borel space, Preprint 2003]. It is proved that the class of smooth automorphisms is dense in (Aut(X, B), p). Let Ctbl(X) denote the group of Borel automorphisms with countable support. It is shown that the topological group Aut₀(X, B) = Aut(X, B)/Ctbl(X) is path-connected with respect to the quotient topology τ₀. It is also proved that Aut₀(X, B) has the Rokhlin property in the quotient topology p₀, i.e., the action of Aut₀(X,B) on itself by conjugation is topologically transitive.     

AUTOMORPHISM GROUPS OF 4-VALENT CONNECTED CAYLEY GRAPHS OF p-GROUPS

61. IntroductionLet G be a trite grouP and S a subs6t of G such thst 1' S and S = S--1. The Cayleygraph X = Cay(G, S) Of G with respect to S is defined to have vertex set V(X) = G and edgeset E(X) = {(g, ag) I g E G, s E' S}. ~ the defection the following two faCts are obvious:(1) the automorphism group Ant(X) of X contains GR, the right regular representation ofG, as a subgroup, and (2) X is cormected if and only if S generates the group G.FOr a Cayley graph X = Cay(G, S) Of …     

ON NONLINEAR DIFFERENTIAL GALOIS THEORY

This Is an accom ofa work In course ofprogress．The aim Is the following:     

The derivation algebra of the associative algebra C q [X,Y,X -1,Y -1] *

The derivation algebra Lof the associative algebra C_q[X,Y,X ^-1,Y ^-1] (q ⁿ :≠1,n∈ N)and the automorphism group Aut Lof L are given.     

Bispaces admitting only bicomplete or only totally bounded quasi-metrics

We characterize quasi-metrizable bispaces that admit only bicomplete quasimetrics by means of doubly primitive sequences, and deduce that if (X, S, T) is a quasi-metrizable bispace admitting only bicomplete quasi-metrics and either (X, S) or (X, T) is hereditarily Lindelöf, then (X, S ∨ T) is compact. We also give an example which shows that hereditary Lindelöfness cannot be omitted in the above result. Finally, we show that a quasi-pseudometrizable bispace (X, S, T) admits only totally bounded quasi-pseudometrics if and only if (X, S ∨ T) is compact, and deduce that a quasi-pseudometrizable topological space admits only totally bounded quasi-pseudometrics if and only if it is hereditarily compact and quasi-sober (equivalently, if and only if it admits a unique quasi-uniformity).