Steenrod Squares in Spectral Sequences譜序列中的斯廷羅德正方形

內(nèi)容概要

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.

書籍目錄

PrefaceChapter 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-ModulesChapter 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 CohomologyChapter 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 algebraChapter 4.Extensions of Hopf Algebras　1.Convolutions and Conjugations　2.Some Properties of Extensions　3.Adjunction Isomorphism and Change-of-Rings Spectral SequenceChapter 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 GroupsChapter 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-TermChapter 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.ApplicationsBibliographyIndex

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