By William M. Singer
This publication develops a common concept of Steenrod operations in spectral sequences. It provides precise 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 booklet develops from scratch the mandatory homes of extensions of Hopf algebras and constructs the spectral series in a sort quite suited for the advent of Steenrod squares. The ensuing idea 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 sturdy homotopy idea. equally the publication bargains a self-contained building of the Eilenberg-Moore spectral series, in a kind compatible for the creation of Steenrod operations. The corresponding concept is an efficient software for the computation of the cohomology jewelry of the classifying areas of the phenomenal Lie teams, and it can provide to be both worthwhile 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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Sample text
122) H*7Z ® Sqk : H*R. HER. The product is the cup-product as originally defined by Whitney on the cohomology of a topological space. 78) is the negative of this one. The squaring operations are those originally defined by Steenrod. 122) is the identity operation. 114, together with the facts that the Verschiebung on R is the identity mapping, and a2 = a for all a in the coefficient ring Z/2. 115) are to be interpreted with Sq° = 1. 116), with Sq° = 1. , known from the work of Milnor [52] to be a Hopf algebra, with coproduct 14.
90 and the naturality of f. x is a mapping If c is a simplicial group then clearly the diagonal d: : CC —* C(C x C) is a homomorphism of differential of simplicial groups, so algebras. 81) is also a homomorphism of differential algebras. 'g 0 becomes is an algebra homomorphism, and so, as is classical, conclude that for the mapping defined by v(g) = a bialgebra. 26) for an antipode. So we have the Zig —p Eg, lig}. 1)g, classical result that is a Hopf algebra with structure maps acts Suppose now that c is a simplicial group and S a simplicial set on which from the right (a "right c-space"), with action a : S x —* S.
S 0. We can also regard N as graded on Z x 5, with N —s = N and = 0 if n > 0. 126) are given by Sqk(u®v)= Sqku®v2 for all u E E 15. Bisimplicial Objects This section will give our conventions concerning bisimplicial objects, their augmentations, and associated diagonal objects. We will be working with bisimplicial objects X = {Xp,q i° 2 0, q 2 0} over a category C, throughout this book. We write for the "horizontal" face and : Xpq and degeneracy operators: d," : Xp,q s," for the Xp_i,q; "vertical" operators.
