**Mathematical Surveys and Monographs**

Volume: 129;
2006;
155 pp;
Hardcover

MSC: Primary 16; 18; 55;

**Print ISBN: 978-0-8218-4141-9
Product Code: SURV/129**

List Price: $64.00

AMS Member Price: $51.20

MAA Member Price: $57.60

**Electronic ISBN: 978-1-4704-1356-9
Product Code: SURV/129.E**

List Price: $60.00

AMS Member Price: $48.00

MAA Member Price: $54.00

#### Supplemental Materials

# Steenrod Squares in Spectral Sequences

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*William M. Singer*

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.

#### Readership

Graduate students and research mathematicians interested in algebraic topology.

#### Reviews & Endorsements

...this book gives a definitive reference on Steenrod operations in first quadrant spectral sequences addressed to experts or experienced mathematicians interested in applications of the theory.

-- Zentralblatt MATH

#### Table of Contents

# Table of Contents

## Steenrod Squares in Spectral Sequences

- Contents iii4 free
- Preface v6 free
- Chapter 1. Conventions 114 free
- 1. Vector Spaces 114
- 2. Algebras, Coalgebras, and Modules 417
- 3. Dual Modules 821
- 4. Bialgebras and Hopf Algebras 821
- 5. Θ … ΞModules and Relative Homological Algebra 1124
- 6. Algebras with Coproducts 1528
- 7. Chains and Cochains 2033
- 8. Differential Algebras and Coalgebras 2336
- 9. Simplicial Θ-Modules 2538
- 10. Homology of Simplicial Sets and Simplicial Groups 2942
- 11. Cotriples, Simplicial Objects, and Projective Resolutions 3548
- 12. Simplicial Θ-Coalgebras and Steenrod Operations 3750
- 13. Steenrod Operations on the Cohomology of Simplicial Sets 4053
- 14. Steenrod Operations on the Cohomology of Hopf Algebras 4154
- 15. Bisimplicial Objects 4457
- 16. The Spectral Sequence of a Bisimplicial Θ-Module 4558
- 17. Cup-K Products and Bisimplicial Θ-Modules 4760

- Chapter 2. The Spectral Sequence of a Bisimplicial Coalgebra 4962
- Chapter 3. Bialgebra Actions on the Cohomology of Algebras 7588
- Chapter 4. Extensions of Hopf Algebras 95108
- Chapter 5. Steenrod Operations in the Change-of-Rings Spectral Sequence 113126
- 1. The Spectral Sequence with its Products and Steenrod Squares 113126
- 2. Steenrod Operations on Ext*,*[sub(Ω)](P,Ext[sup(τ)][sub(*)](Q,N)) 115128
- 3. Central Extensions 120133
- 4. The Operations at the E[sub(2)]-level 120133
- 5. A Simple Example 124137
- 6. Application to the Cohomology of the Steenrod Algebra 126139
- 7. Application to Finite sub-Hopf Algebras of the Steenrod Algebra 127140
- 8. Applications to the Cohomology of Groups 129142

- Chapter 6. The Eilenberg-Moore Spectral Sequence 131144
- Chapter 7. Steenrod Operations in the Eilenberg-Moore Spectral Sequence 141154
- Bibliography 149162
- Index 153166