Matric Massey products
Matric Massey products
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DOI：
10.1016/0021-8693(69)90027-1
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发表时间：
1969-08
影响因子：
0.9
通讯作者：
Jon P. May
中科院分区：
文献类型：
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作者：
Jon P. May
人们很早就认识到，微分代数和微分代数上的微分模的同调不仅有积，而且有高阶运算，即梅西积。这些操作在很大程度上被忽视了，因为计算它们很困难，而且它们似乎缺乏概念上的兴趣。我们将在这里介绍和研究这些操作的概括；新的运算将在 n 元组矩阵上定义，而不是在 n 元组元素上。当然，推广对简化计算没有任何作用，尽管本文中谱序列的结果会产生这种效果。然而，较大类别的操作确实具有重要的概念意义。在代数拓扑中有多种情况，几何自然地给出了可分解性的概念。新的运算正是用代数方式描述这些概念所需要的。例如，如果 G 是连通拓扑幺半群，则庞特里亚金环 H,(G) 中可分解元素的几何概念是同调悬浮 cr*: H,(G)+ H,(BG) 的核元素，而 f: act ker cr.+ 正是所有可分解为矩阵 Massey 乘积的元素的集合。如果 B 是单连通空间，则对偶命题成立；如果 cr*: H”(B)-* H*(, QB) 是上同调悬置，则 ker CT* 是可分解为矩阵 Massey 积的所有元素的集合。其他此类情况将在 [lo], nhcre 中给出，上述陈述已得到证明。此外，这些运算将在 [10] 中用于计算各种齐次空间和主丛的上同调，并开发一种计算 mod 2 的算法任何简单连接的两阶段空间的上同调可以在[9]中找到。我们在本文中的程序如下，我们将在第 1 节中定义矩阵 3Iassey 积并证明它们的自然性。我们将在第 2 节中证明我们的操作满足某些线性关系并研究它们的不确定性。本节的主要目的是证明，至少在合理的技术假设下，矩阵 Massey 积是值得尊敬的。
It has long been recognized that the homologies of differential algebras and of differential modules over differential algebras have not only products but also higher order operations, namely Massey products. These operations have largely been ignored because of the difficulty in computing them and because of their seeming lack of conceptual interest. We shall here introduce and study generalizations of these operations; the new operations will be defined on n-tuples of matrices rather than on n-tuples of elements. Of course, the generalization does nothing to simplify the computations, although the results on spectral sequences in this paper will have this effect. The larger class of operations does, however, have essential conceptual interest. There are a variety of situations in algebraic topology where the geometry naturally gives a notion of decomposability. The new operations arc precisely what is required to dcscribc these notions algebraically. For example, if G is a connected topological monoid, then the geometric notion of a decomposable element in the Pontryagin ring H,(G) is an element of the kernel of the homology suspension cr*: H,(G)+ H,(BG), and in f: act ker cr.+ is exactly the set of all elements decomposable as matric Massey products. If B is a simply connected space, the dual statement is true; if cr*: H”(B)-* H*(, QB) is the cohomology suspension, then ker CT* is the set of all elements which are decomposable as matric Massey products. Other such situations will be given in [lo], nhcre the statements above are proven. Moreover, these operations wil1 be used in [10] to compute the cohomologies of a wide variety of homogeneous spaces and principal bundles and to develop an algorithm for the computation of the mod 2 cohomology of any simply connected two-stage space. Statements of the results in question may be found in [9].Our program in this paper is as follows. We shall define matric 3Iassey products and prove their naturality in Section 1. We shall prove certain linearity relations satisfied by our operations and study their indetermina. cy in Section 2. The main purpose of this section is to show that, at least under reasonable technical assumptions, matric Massey products are respectable 533