ON DIFFERENTIAL HOPF ALGEBRAS
ON DIFFERENTIAL HOPF ALGEBRAS
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关于微分HOPF代数
DOI：
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发表时间：
1963
期刊：
影响因子：
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通讯作者：
W. Browder
中科院分区：
文献类型：
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作者：
W. Browder
微分霍普夫代数出现在代数拓扑学的几个背景下。一个//-空间的Bockstein谱序列是许多作者研究过的一个例子[3; 1; 7; 8]。Borel [3]和Araki [1]证明了关于特殊类型微分Hopf代数结构的代数定理。这些特殊的定理使他们能够确定奇扭转的上同调例外李群。若X和T是//-空间且/：X-> T是一个乘积纤维映射，则/的谱序列是Hopf代数的谱序列.这种情况首先由J. C.摩尔[17]，后来由作者[5; 6]。[5]的技巧后来被作者（在未发表的工作中）扩展到证明关于同调和上同调悬置的定理，即，当X是//-空间Y的路径空间时。证明依赖于一个关于这个谱序列结构的一般定理。摩尔用一个不同的谱序列证明了其中的一些悬挂定理[12]。本文研究了微分Hopf代数，证明了其同调结构的一般定理。这些定理推广了Borel和Araki的结果。应用到乘法纤维映射的情况下，我们得到了一个关于谱序列的结构的一般定理（即使在非循环的情况下），特别是，产生简单的证明上述悬浮定理。应用于Bockstein谱序列，我们得到了关于//-空间中挠率的信息.微分Hopf代数的研究依赖于两个谱序列，这两个谱序列可以在不同的情况下定义。如果其中一个被定义，那么该谱序列的项满足另一个被定义的必要条件。这样我们就得到了另一个谱序列（2）的项的谱序列，并且这个谱序列有一个非常简单的形式，这使得计算它的同调的形式很容易。因此极限的结构
Differential Hopf algebras arise in several contexts in algebraic topology. The Bockstein spectral sequence of an //-space is one example that has been investigated by many authors [3; 1; 7; 8]. Borel [3] and Araki [1] proved algebraic theorems about the structure of differential Hopf algebras of special kinds. These special theorems enabled them to determine the odd torsion in the cohomology of the exceptional Lie groups. If X and Tare //-spaces and/:X-> Tis a fibre map which is multiplicative, then the spectral sequence of / is a spectral sequence of Hopf algebras. This situation was first discussed by J. C. Moore [17], and later by the author [5; 6]. The techniques of [5] were later extended by the author (in unpublished work) to prove theorems about the homology and cohomology suspensions, i.e., when X is the space of paths of the //-space Y. The proofs rested upon a general theorem about the structure of this spectral sequence. Some of these suspension theorems had been proved by Moore using a different spectral sequence of Hopf algebras [12]. In this paper we make a study of differential Hopf algebras, and prove general theorems on the structure of their homology. These theorems generalize the results of Borel and Araki. Applied to the case of multiplicative fibre maps, we obtain a general theorem about the structure of the spectral sequence (even in the nonacyclic case), which, in particular, yields simple proofs of the suspension theorems mentioned above. Applied to the Bockstein spectral sequence, we get information on torsion in //-spaces. This study of differential Hopf algebras depends on two spectral sequences which may be defined in different circumstances. If one of them is defined, then the terms of that spectral sequence satisfy the conditions necessary for the other to be defined. Thus we get a spectral sequence for the term of the other spectral sequence(2), and this spectral sequence has a very simple form which makes it easy to calculate the form of its homology. Thus the structure of the limit