By William M. Singer
This e-book develops a basic conception of Steenrod operations in spectral sequences. It provides certain consciousness 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 required homes of extensions of Hopf algebras and constructs the spectral series in a kind relatively suited for the advent of Steenrod squares. The ensuing concept can be utilized successfully for the computation of the cohomology earrings of teams and Hopf algebras, and of the Steenrod algebra specifically, and so should still play an invaluable function in good homotopy thought. equally the e-book bargains a self-contained development of the Eilenberg-Moore spectral series, in a kind appropriate for the advent of Steenrod operations. The corresponding thought is a good software for the computation of the cohomology jewelry of the classifying areas of the outstanding Lie teams, and it provides to be both important 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 attracted to algebraic topology.
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Additional resources for Steenrod Squares in Spectral Sequences
Suppose a simplicial group and 2: a left c-space. 91) of differential algebras. PROOF. 75) is a homomorphism of right modules over CC ® CC. Therefore itis also a homomorphism of right modules over CC, where CC is acting on both Hom(CF, l)®Hom(CF, I) and Hom(CF®CF, 1) through . 91). 101. 104. Suppose g a simplicial group and F a left c-space. 85), the action of algebra. 104. 31. 27 below). 100 remains true if the simplicial sets e and E' on which ç and ç' act from the right, are replaced by simplicial sets F, F' on which 1.
Homology of Simplicial Sets and Simplicial Groups The material in this section is needed only for our treatment of the EilenbergMoore spectral sequence, in Chapters 6 and 7. The chains and cochains on simplicial sets and simplicial groups inherit certain products and coproducts from the shuffle and Alexander-Whitney mappings. Similarly, the chains on a simplicial set that We discuss these has an action of a simplicial group inherit an action of structures in this section. None of these results are new, and some are well known.
2. 57) Do(a®r)=a®r. 58) 3. 59) ODk + DkO = Dk_1 + TDk_1T. The existence of a cup-k product, and its uniqueness up to a suitable notion of homotopy, is proved by Dold  using the method of acyclic models . The next three propositions give some useful properties of cup-k products. If V is a vector space over Z/2, there is associated a "constant" simplicial vector space, which is isomorphic in each dimension to V, and for which the faces and degeneracies are all identity maps. We denote this simplicial vector space also byV.