拓扑和的自同伦等价群

如果拓扑空间X,Y的拓扑和X∨Y的自同伦等价可以对角化,则X∨Y的自同伦等价群Aut(X∨Y)可表示为它的两个子群Autx(X∨Y)与AutY(X∨y)的乘积.     

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.     

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.     

夹心半群S(X,Y,θ)上的α-同余

本文讨论了夹心半群T(X,Y,θ)上的α-同余与集合Y上Tθ-等价关系之间的联系,证明了对于每个Tθ-等价关系E,夹心半群同余格C(T(X,Y,θ))的完全子格γ-1(E)中的最大元是一个α-同余,并判明Symons同余l,d都是α-同余.对于某些拓扑空间X,Y,确定了夹心半群S(X,Y,θ)上的最小(最大)真α-同余.     

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VWHEP与WHEP的比较

<正> 在文[1]与[2]中,我们对经典的同伦扩张性质(HEP)与弱同伦扩张性质(WHEP)进行了讨论,它们之间有许多相似之处.文[8]的推论3指出:若(X,A)有WHEP,且有映射ω:X→I使得ω~(-1)(0)=A,则(X,A)也有HEP(即成为上纤维偶).比HEP与WHEP更广泛的一个概念是VWHEP,它在同伦论中的重要性在于:刻划了扩张问题仅依赖于映射的同伦类这种特殊情况.文[4]、[5]对VWHEP已有不少讨论.我们的目的是进一步比较WHEP与VWHEP.尽管VWHEP与WHEP也有不少相似之处,但还有相当的差别.似乎这种差别比WHEP与HEP间的差别要大些.     

向量场的积分曲线分类和链群与有向同伦不变性

利用积分曲线的极限集,对紧流形M上向量场X所确定的积分曲线进行了分类,在分类的基础上定义了向量场X的链群、极限集闭链群和极限集边缘链群,以及两个同类群,最后还引入了向量场正向同伦和负向同伦的概念,并证明了极限集是正向同伦和负向同伦不变的,链群、极限集闭链群和极限集边缘链群以及两个同类群在向量场双向同伦的情况下是同构的.     

向量场的积分曲线分类和链群与有向同伦不变性

利用积分曲线的极限集,对紧流形M上向量场X所确定的积分曲线进行了分类,在分类的基础上定义了向量场X的链群、极限集闭链群和极限集边缘链群,以及两个同类群,最后还引入了向量场正向同伦和负向同伦的概念,并证明了极限集是正向同伦和负向同伦不变的,链群、极限集闭链群和极限集边缘链群以及两个同类群在向量场双向同伦的情况下是同构的.     

夹心半群T(X,Y,θ)上的一个同余链

设T(X, Y,θ)是两个非空集合X, Y上的夹心半群,其中θ是从Y到X的一个映射.本文刻画了这个半群T(X, Y,θ)上的一些同余,得到了半群上的同余链.     

关于同伦满态与覆叠空间

本文在点标道路连通ＣＷ空间的同伦范畴中，利用同伦推出示性了同伦满态，得出了若ｆ：Ｘ－Ｙ是同伦满态，则对π１Ｙ的任一正规子群Ｈ，升腾映射ｆ：Ｘ（ｆ－１＃（Ｈ））→■（Ｈ）也是同伦满态．     

Algebraic cocycles on normal, quasi-projective varieties

This paper extends to quasi-projective varieties earlier work by the author and H. Blaine Lawson concerning spaces of algebraic cocycles on projective varieties. The topological monoid C_r(Y) (U) of effective cocycles on a normal, quasi-projective variety U with values in a projective variety Y consists of algebraic cycles on U×Y equi-dimensional of relative dimension r over U. A careful choice of topology enables the establishment of various good properties: the definition is essentially algebraic, the group completion Z_r (Y) (U) has 'sensible' homotopy groups, the construction is contravariant with respect to U, convariant with respect to Y, and there is a natural 'quality map" to the topological group of cycles on U×Y. The fundamental theorem presented here is the extension of Friedlander-Lawson duality to this context: the duality map Z_r (Y) (U) to Z_{_r+m} (U × Y) is a homotopy equivalence provided that both U and Y are smooth (where m=dim U). Various application are given, especially the determination of the homotopy types of certain topological groups of algeb raic morphisms.     

關於Borsuk的絕對同倫擴充性質的討論

<正> 序言.在同倫論中,常常需要考慮滿足這種性質的拓撲空間X設Y為任意的一個正規空間,B為Y的任何一個非空閉集,任何一個由B×(0,1)+Y×(0)到X的映像都可以扩充為一個由Y×(0,1)到X的映像,我們稱這種性質為絕對同倫扩充性質,具有這種性質的空間以及用AHE表示.Borsuk曾經介紹這樣一個重要的定理:     

保持谱半径的乘法映射

设X和Y为无限维Banach空间,Φ：B（X）→B（Y）是保持谱半径的满射,且秩为1算子,则Φ具有形式Φ（T）=ATA∧-1,这里A：X→Y或是线性拓扑同构映射或是线性拓扑同构映射的共轭。     

Topological continuous convergence

Let X be a limit space, Y a topological space. We show that c(X,Y), the limitierung of continuous convergence on LIM(X,Y), is topological whenever X is basic locally compact. For regular Y, local compactness of X is sufficient. In both cases, c(X,Y) coincides with the compact-open topology. If X satisfies a certain regularity condition, the fact that c(X,Y) is topological implies, conversely, that X is (basic) locally compact.The author would like to thank S. Weck for some inspiring discussions.     

A Survey of the Homotopy Properties of Inclusion of Certain Types of Configuration Spaces into the Cartesian Product

Let X be a topological space.In this survey the authors consider several types of configuration spaces,namely,the classical(usual)configuration spaces F_n(X)and D_n(X),the orbit configuration spaces F_n~G(X)and F_n~G(X)/S_nwith respect to a free action of a group G on X,and the graph configuration spaces F_n~Γ(X)and F_n~Γ(X)/H,whereΓis a graph and H is a suitable subgroup of the symmetric group S_n.The ordered configuration spaces F_n(X),F_n~G(X),F_n~Γ(X)are all subsets of the n-fold Cartesian product ∏_1~nX of X with itself,and satisfy F_n~G(X)?F_n(X)?F_n~Γ(X)?∏_1~nX.If A denotes one of these configuration spaces,the authors analyse the difference between A and ∏_1~nXfrom a topological and homotopical point of view.The principal results known in the literature concern the usual configuration spaces.The authors are particularly interested in the homomorphism on the level of the homotopy groups of the spaces induced by the inclusionι:A-→∏_1~nX,the homotopy type of the homotopy fibre I_ιof the mapιvia certain constructions on various spaces that depend on X,and the long exact sequence in homotopy of the fibration involving I_ιand arising from the inclusionι.In this respect,if X is either a surface without boundary,in particular if X is the 2-sphere or the real projective plane,or a space whose universal covering is contractible,or an orbit space S~k/Gof the k-dimensional sphere by a free action of a Lie group G,the authors present recent results obtained by themselves for the first case,and in collaboration with Golasi′nski for the second and third cases.The authors also briefly indicate some older results relative to the homotopy of these spaces that are related to the problems of interest.In order to motivate various questions,for the remaining types of configuration spaces,a few of their basic properties are described and proved.A list of open questions and problems is given at the end of the paper.     

Characterization of the Function Classes C(Y)|X

Let X be a topological space and let Φ ? C(X). Then there exists a topological space Y containing X as a subspace and such that Φ = C(Y)¦X, if and only if Φ is weakly composition closed, i.e., for any index set I, any fi ∈ Φ (i ∈ I) and any continuous map k : RI → Rwe have k ° 〈fi〉 ∈ Φ, where 〈fi〉 : X → RI is the map with i-th coordinate fi. The analogous statement is valid for functions to any T1 space, rather than to R, and even we can consider functions to any set of T1 spaces, and then a generalization of the above statement is valid, with a suitably defined weak composition closedness property. We also show that some earlier results on characterization of function classes Φ ? C(X) of the form C(Y)¦X, with Y some extension of a given topological space X, and on the characterization of function classes C(〈X, T〉), with T some topology on a given set X, respectively, can be generalized in an analogous way as above, by means of composition properties analogous to the above one or by filter closedness (for functions to any set of T3 spaces, or to any set of topological spaces, respectively).     

復式乘積和球面乘積的關係

<正> 1.設X為一拓撲空間,以a_i表示X的n_i次同偷羣∏_(ni)(X)的一個元素.在球面乘積S~(n1)×…×S~(nr)中,最高次元的腔胞記為e~(n1+…+nr).在S~(n1)×…×S~(nr)中除去e~(n1+…+nr)而得一空間Y.設S~(nl)為nl次元球,為了印刷方便起見,把S~(nl)記作S(nl).在S(ni)中取一參考點x_i~o.設i_1相似文献   

Spaces of Densely Continuous Forms

The set C(X,Y) of continuous functions from a topological space X into a topological space Y is extended to the set D(X,Y) of densely continuous forms from X to Y, such form being a kind of multifunction from X to Y. The topologies of pointwise convergence, uniform convergence, and uniform convergence on compact sets are defined for D(X,Y), for locally compact spaces X and metric spaces Y having a metric satisfying the Heine–Borel property. Under these assumptions, D(X,Y) with the uniform topology is shown to be completely metrizable. In addition, if X is compact, D(X,Y) is completely metrizable under the topology of uniform convergence on compact sets. For this latter topology, an Ascoli theorem is established giving necessary and sufficient conditions for a subset of D(X,Y) to be compact.