Steenrod Squares in Spectral Sequences谱序列中的斯廷罗德正方形
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作者： William M. Singer著
出 版 社：
出版时间： 2006-8-1字数：版次： 1页数： 152印刷时间： 2006/08/01开本： 16开印次： 1纸张： 胶版纸I S B N ： 9780821841419包装： 精装内容简介
This book develops a general theory of Steenrod operations in spectral sequences. It gives special attention to the change-of-rings spectral sequence for the cohomology of an extension of Hopf algebras and to the Eilenberg-Moore spectral sequence for the cohomology of classifying spaces and homotopy orbit spaces. In treating the change-of-rings spectral sequence, the book develops from scratch the necessary properties of extensions of Hopf algebras and constructs the spectral sequence in a form particularly suited to the introduction of Steenrod squares. The resulting theory can be used effectively for the computation of the cohomology rings of groups and Hopf algebras, and of the Steenrod algebra in particular, and so should play a useful role in stable homotopy theory. Similarly the book offers a self-contained construction of the Eilenberg-Moore spectral sequence, in a form suitable for the introduction of Steenrod operations. The corresponding theory is an effective tool for the computation of the cohomology rings of the classifying spaces of the exceptional Lie groups, and it promises to be equally useful for the computation of the cohomology rings of homotopy orbit spaces and of the classifying spaces of loop groups.
目录
Preface
Chapter 1.Conventions
1.Vector Spaces
2.Algebras, Coalgebras, and Modules
3.Dual Modules
4.Bialgebras and Hopf Algebras
5.O－Modules and Relative Homological Algebra
6.Algebras with Coproducts
7.Chains and Cochains
8.Differential Algebras and Coalgebras
9.Simplicial O-Modules
10.Homology of Simplicial Sets and Simplicial Groups
11.Cotriples, Simplicial Objects, and Projective Resolutions
12.Simplicial O-Coalgebras and Steenrod Operations
13.Steenrod Operations on the Cohomology of Simplicial Sets
14.Steenrod Operations on the Cohomology of Hopf Algebras
15.Bisimplicial Objects
16.The Spectral Sequence of a Bisimplicial @-Module
17.Cup-k Products and Bisimplicial O-Modules
Chapter 2.The Spectral Sequence of a Bisimplicial Coalgebra
1.Bisimplicial O-Coalgebras
2.Filtrations
3.The Spectral Sequence
4.Bisimplicial Sets with Group Action
5.Application to the Serre Spectral Sequence
6.Application to Andre-Quillen Cohomology
Chapter 3.Bialgebra Actions on the Cohomology of Algebras
1.Left Action by a Bialgebra
2.Left Action by an Algebra with Coproducts
3.Right Action by a Hopf algebra
Chapter 4.Extensions of Hopf Algebras
1.Convolutions and Conjugations
2.Some Properties of Extensions
3.Adjunction Isomorphism and Change-of-Rings Spectral Sequence
Chapter 5.Steenrod Operations in the Change-of-Rings Spectral Sequence
1.The Spectral Sequence with its Products and Steenrod Squares
2.Steenrod Operations on Ext
3.Central Extensions
4.The Operations at the E2-1evel
5.A Simple Example
6.Application to the Cohomology of the Steenrod Algebra
7.Application to Finite sub-Hopf Algebras of the Steenrod Algebra
8.Applications to the Cohomology of Groups
Chapter 6.The Eilenberg-Moore Spectral Sequence
1.Kan Fibrations and Twisted Cartesian Products
2.Bisimplicial Models for Fiber Bundles
3.Construction of the Spectral Sequence
4.Calculation of the E2-Term
Chapter 7.Steenrod Operations in the Eilenberg-Moore Spectral Sequence
1.The Spectral Sequence with its Products and Steenrod Squares
2. Steenrod Operations on Ext
3.The Operations at the E2-1evel
4.Applications
Bibliography
Index
