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Steenrod Squares in Spectral Sequences http://dx.doi.org/10.1090/surv/129 Mathematical Surveys and Monographs Volume 129 Steenrod Squares in Spectral Sequences William M. Singer ,£^>c American Mathematical Society ^VDED EDITORIAL COMMITTEE Jerry L. Bona Peter S. Landweber Michael G. Eastwood Michael P. Loss J. T. Stafford, Chair 2000 Mathematics Subject Classification. Primary 16E40, 18G25, 18G30, 18G40, 55R20, 55R40, 55S10, 55T05, 55T15, 55T20. For additional information and updates on this book, visit www. ams.org/bookpages/surv-129 Library of Congress Cataloging-in-Publication Data Singer, William M., 1942- Steenrod squares in spectral sequences / William M. Singer. p. cm. — (Mathematical surveys and monographs, ISSN 0076-5376 ; v. 129) Includes bibliographical references and index. ISBN 0-8218-4141-6 (alk. paper) 1. Spectral sequences (Mathematics) 2. Steenrod algebra. I. Singer, William M., 1942- II. Series: Mathematical surveys and monographs ; v. 129. QA612.8.S56 2006 515/.24—dc22 2006045953 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. © 2006 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. @ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1 11 10 09 08 07 06 Contents Preface v Chapter 1. Conventions 1 1. Vector Spaces 1 2. Algebras, Coalgebras, and Modules 4 3. Dual Modules 8 4. Bialgebras and Hopf Algebras 8 5. 0 — H Modules and Relative Homological Algebra 11 6. Algebras with Coproducts 15 7. Chains and Cochains 20 8. Differential Algebras and Coalgebras 23 9. Simplicial 6-Modules 25 10. Homology of Simplicial Sets and Simplicial Groups 29 11. Cotriples, Simplicial Objects, and Projective Resolutions 35 12. Simplicial ©-Coalgebras and Steenrod Operations 37 13. Steenrod Operations on the Cohomology of Simplicial Sets 40 14. Steenrod Operations on the Cohomology of Hopf Algebras 41 15. Bisimplicial Objects 44 16. The Spectral Sequence of a Bisimplicial 9-Module 45 17. Cup-A; Products and Bisimplicial 6-Modules 47 Chapter 2. The Spectral Sequence of a Bisimplicial Coalgebra 49 1. Bisimplicial 6-Coalgebras 50 2. Filtrations 53 3. The Spectral Sequence 55 4. Bisimplicial Sets with Group Action 65 5. Application to the Serre Spectral Sequence 71 6. Application to Andre-Quillen Cohomology 73 Chapter 3. Bialgebra Actions on the Cohomology of Algebras 75 1. Left Action by a Bialgebra 75 2. Left Action by an Algebra with Coproducts 84 3. Right Action by a Hopf algebra 89 Chapter 4. Extensions of Hopf Algebras 95 1. Convolutions and Conjugations 96 2. Some Properties of Extensions 102 3. Adjunction Isomorphism and Change-of-Rings Spectral Sequence 107 iv CONTENTS Chapter 5. Steenrod Operations in the Change-of-Rings Spectral Sequence 113 1. The Spectral Sequence with its Products and Steenrod Squares 113 2. Steenrod Operations on Ext£*(P, Ext* (Q, N)) 115 3. Central Extensions 120 4. The Operations at the P2-level 120 5. A Simple Example 124 6. Application to the Cohomology of the Steenrod Algebra 126 7. Application to Finite sub-Hopf Algebras of the Steenrod Algebra 127 8. Applications to the Cohomology of Groups 129 Chapter 6. The Eilenberg-Moore Spectral Sequence 131 1. Kan Fibrations and Twisted Cartesian Products 132 2. Bisimplicial Models for Fiber Bundles 133 3. Construction of the Spectral Sequence 137 4. Calculation of the P2-Term 138 Chapter 7. Steenrod Operations in the Eilenberg-Moore Spectral Sequence 141 1. The Spectral Sequence with its Products and Steenrod Squares 141 2. Steenrod Operations on Ext^gtH+E^K+J7) 142 3. The Operations at the P2-level 144 4. Applications 148 Bibliography 149 Index 153 Preface In his thesis [74], J.-P. Serre initiated the use of spectral sequences in alge­ braic topology. Serre equipped his spectral sequence with a theory of products that greatly enhanced its effectiveness in computing the cohomology of a fiber space. Then it was natural to ask whether Steenrod's squaring operations could be intro­ duced into spectral sequences as well. For example, in a collection of problems in algebraic topology [46] published in 1955, William Massey wrote: "Problem 6. Is it possible to introduce Steenrod squares and reduced p'th powers into the spectral sequence of a fibre mapping so that, for the term E2, they behave according to the usual rules for squares and reduced p'th powers in a product space?" The problem was first addressed by S. Araki [6] and R. Vazquez-Garcia [87], who both developed a theory of Steenrod operations in the Serre spectral sequence. L. Kristensen [34] also wrote down such a theory, and used it to calculate the cohomology rings of some two-stage Postnikov systems. Later, in the papers [77, 78], the present writer developed a theory of Steenrod operations for a class of first- quadrant cohomology spectral sequences. The theory applied to the Serre spectral sequence, but also 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 a classifying space. The present work offers a more detailed exposition of the theory originally set out in [77] and [78], and generalizes its results. To introduce the context for our theory, we recall the context in which Steenrod operations were originally defined, first by Steenrod [83], and then more generally by Dold [16]. One starts with a commutative, simplicial coalgebra R over the ground ring Z/2. Associated to R is a chain complex CR, defined by (CR)n = Rn, with differential the sum of the face operators of R. Steenrod and Dold showed how to use the "cup-z" products, D{ : C(R x R) —> CR 0 CR, together with the coproduct on R, to define the Steenrod squaring operations on the homology of the cochain complex dual to CR: (0.1) Sqk : tfn(Hom(C#,Z/2)) -• Hn+k(Rom(CR,Z/2)). To get a general theory of Steenrod operations in spectral sequences we gener­ alize this construction in several directions. We begin by changing the ground ring from Z/2 to any Z/2-bialgebra 6 with commutative coproduct. This generaliza­ tion is useful in dealing with Steenrod operations on the cohomology rings of Hopf algebras. Because we wish to discuss spectral sequences, we replace the simplicial coalgebra it! by a bisimplicial object X over the category of 6-coalgebras. This bisimplicial object determines a "total" chain complex CX of O-modules by the rule (CX)n = (&p+q==n(Xp^q). The differential is defined as a sum of the horizontal and vertical face operators in the usual way. We define Steenrod operations on the vi PREFACE homology of the "dual" cochain complex, replacing the coefficient ring Z/2 of (0.1) by an arbitrary 6-algebra N: k n n+/c (0.2) Sq : H (RomQ(CX,N)) -> iJ (Home(CX, TV)). This construction is completely analogous to (0.1). But the bisimplicial object X also determines a double chain complex CX, with (CX)Ptq = Xp^q, and vertical and horizontal differentials defined as sums, respectively, of vertical and horizontal face operators. So a naturally defined first-quadrant spectral sequence ("the spectral sequence of a bisimplicial coalgebra") converges to the groups H*(HOIRQ(CX^N)) that appear in (0.2). We show how to put Steenrod operations into this spectral sequence. That is, for all r > 2 and all p, q, k > 0 we define Steenrod squares: (0.3a) Sqk : E™ -• E™+k if 0 < k < q, (0.3b) Sqk : E™ -• E\p+k — q,2q if q<k. Here t is an integer that must in general be chosen larger than r, in order that Sqk be well-defined (the exact value of t is given in Theorem 2.15). However, if r = 2 then t = 2. All Steenrod squares defined on £2 take values in E2, and they carry a typical class a G E™ to positions on the P — Q plane as shown by the diagram: Q Sqqa. Sqq+1a. Sq •p+q, Sq2a. Sq1a. PREFACE vn The squaring operations on Er are determined by the operations on E2, by passage to the subquotient (Theorem 2.16). For reasons made obvious by the diagram, we refer to the family (0.3a) as "vertical" operations, and the family (0.3b) as "diagonal". The Steenrod operations commute with the differentials of the spectral sequence (Theorem 2.17 and the diagram that follows it). The operations (0.3) are defined when r = t = 00, and agree with those defined on the graded vector space associated to the filtration on the target groups i7*(Home(CX, TV)) of the spectral sequence (Theorem 2.22). We apply the general theory in three cases. A. Dress observed [17] that the Serre spectral sequence can be obtained as the spectral sequence of a bisimplicial coalgebra.
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