By William M. Singer
This booklet develops a normal idea of Steenrod operations in spectral sequences. It offers detailed recognition to the change-of-rings spectral series for the cohomology of an extension of Hopf algebras and to the Eilenberg-Moore spectral series for the cohomology of classifying areas and homotopy orbit areas. In treating the change-of-rings spectral series, the publication develops from scratch the mandatory homes of extensions of Hopf algebras and constructs the spectral series in a sort rather fitted to the creation of Steenrod squares. The ensuing conception can be utilized successfully for the computation of the cohomology earrings of teams and Hopf algebras, and of the Steenrod algebra particularly, and so may still play an invaluable position in solid homotopy conception. equally the publication bargains a self-contained development of the Eilenberg-Moore spectral series, in a kind appropriate for the advent of Steenrod operations. The corresponding idea is an efficient software for the computation of the cohomology earrings of the classifying areas of the phenomenal Lie teams, and it grants to be both worthy for the computation of the cohomology earrings of homotopy orbit areas and of the classifying areas of loop groups.
Readership: Graduate scholars and examine mathematicians drawn to algebraic topology.
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Extra resources for Steenrod Squares in Spectral Sequences
Example text
82) CE®CC C(E x C) CE. 7 is a left c-space then CF becomes a differential left module over CC. 87 that the dual chain complex Hom(CF, 1) becomes a differential right module over CC. 84) (CF, 1)0 CC -* CF, 1) 1. 85) £ x E is a mapping of right c-spaces, where c acts on E x ( through the diagonal d: —p x Q. As a consequence of this, and Clearly the diagonal d : —* of the naturality of the shuffle, we have: be a simplicial group, and £ a right c-space. 98. Let CIIE : CE —+ C(E x E) is a homomorphism of differential right CC-modules, where CC acts on C(E x E) through the homomorphism of differential algebras Suppose now a simplicial group, E a right c-space and F a left c-space.
Simplicial ê-Modules The material in this section is central to the book. We review properties of the "shuffle" mapping of Eilenberg and Mac Lane, and of its homotopy inverse, the Alexander-Whitney map. We review properties of Steenrod's "cup-k products" which measure the deviation of the Alexander-Whitney map from commutativity. We single out a class of cup-k products that we call "special". These have filtration- preserving properties that will play a key role when, in the next chapter, we put Steenrod operations into spectral sequences.
This formula will play an important role when, in Chapter 2, we put Steenrod operations into spectral sequences. We will review the Adem relations, and observe that the Steenrod operations on cohomology, subject to the Adem relations, form an "algebra with coproducts" in the sense of Section 6. The operations Sq" were first defined by Steenrod [83] on the mod-2 cohomology of a topological space. Later, Dold [16] defined them in a more general context, replacing the singular chains on a topological space by an arbitrary simplicial coalgebra over Z/2.