A general algebraic approach to steenrod operations
A general algebraic approach to steenrod operations
复制标题
杆操作的通用代数方法
DOI：
10.1007/bfb0058524
复制
发表时间：
1970
影响因子：
1.3
通讯作者：
J. May
中科院分区：
文献类型：
--
作者：
J. May
我的介绍。自从在拓扑空间的上同调中引入Steenrod运算以来，类似的运算在各种其他情况下的存在已经变得很清楚。例如，在简单受限李代数的上同调中，在协交换Hopf代数的上同调中，在无限环空间的上同调中（其中它们由Araki和Kudo[3]引入mod 2，由Dyer和Lashof[6]引入mod p， p> 2）存在Steenrod运算。这篇说明性论文的目的是发展一个一般的代数设置，其中所有这些操作可以同时研究。这种方法允许对操作的基本属性（包括Adem关系）进行单一证明，适用于上述所有示例。与Steenrod运算的分类处理相反，Steenrod[25 30]开发的优雅证明实际上在我们的代数设置中有所简化。此外，即使是现有最一般的Steenrod运算的范畴研究，即Epstein[7]的范畴研究，也不能应用于迭代循环空间。
I. Introduction. Since the introduction of the Steenrod operations in the cohomology of topological spaces, it has become clear that similar operations exist in a variety of other situations. For example, there are Steenrod operations in the cohomology of simplicial restricted Lie algebras, in the cohomology of cocommutative Hopf algebras, and in the homology of infinite loop spaces (where they were introduced mod 2 by Araki and Kudo [3] and mod p, p> 2, by Dyer and Lashof [6]).The purpose of this expository paper is to develop a general algebraic setting in which all such operations can be studied simultaneously. This approach allows a single proof, applicable to all of the above examples, of the basic properties of the operations, including the Adem relations. In contrast to categorical treatments of Steenrod operations, the elegant proofs developed by Steenrod [25 30] actually simplify somewhat in our algebraic setting. Further, even the most general existing categorical study of Steenrod operations, that of Epstein [7], cannot be applied to iterated loop spaces.