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Complex Cobordism and Stable Homotopy Groups of Spheres COMPLEX COBORDISM AND STABLE HOMOTOPY GROUPS OF SPHERES SECOND EDITION DOUGLAS C. RAVENEL AMS CHELSEA PUBLISHING !MERICAN-ATHEMATICAL3OCIETYs0ROVIDENCE 2HODE)SLAND http://dx.doi.org/10.1090/chel/347.H COMPLEX COBORDISM AND STABLE HOMOTOPY GROUPS OF SPHERES SECOND EDITION DOUGLAS C. RAVENEL AMS CHELSEA PUBLISHING American Mathematical Society • Providence, Rhode Island 2000 Mathematics Subject Classification. Primary 55-02; Secondary 55N22, 55Q40, 55Q45, 55Q50, 55Q51, 55T15, 55T25. For additional information and updates on this book, visit www.ams.org/bookpages/chel-347 Library of Congress Cataloging-in-Publication Data Ravenel, Douglas C. Complex cobordism and stable homotopy groups of spheres / Douglas C. Ravenel.—2nd ed. p. cm. Includes bibliographical references and index. ISBN 0-8218-2967-X (alk. paper) 1. Homotopy groups. 2. Sphere. 3. Spectral sequences (Mathematics) 4. Cobordism theory. I. Title. QA612.78 .R39 2003 514.24—dc22 2003062646 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]. c 2004 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 18 17 16 15 14 13 To my wife, Michelle Contents List of Figures xi List of Tables xiii Preface to the Second Edition xv Preface to the First Edition xvii Commonly Used Notations xix Chapter 1. An Introduction to the Homotopy Groups of Spheres 1 1. Classical Theorems Old and New 2 Homotopy groups. The Hurewicz and Freudenthal theorems. Stable stems. The Hopf map. Serre’s finiteness theorem. Nishida’s nilpotence theorem. Cohen, Moore and Neisendorfer’s exponent theorem. Bott per- iodicity. The J-homomorphism. n 2. Methods of Computing π∗(S )5 Eilenberg–Mac Lane spaces and Serre’s method. The Adams spectral sequence. Hopf invariant one theorems. The Adams–Novikov spectral sequence. Tables in low dimensions for p =3. 3. The Adams–Novikov E2-term, Formal Group Laws, and the Greek Letter Construction 12 Formal group laws and Quillen’s theorem. The Adams–Novikov E2- term as group cohomology. Alphas, betas and gammas. The Morava– Landweber theorem and higher Greek letters. Generalized Greek letter elements. 4. More Formal Group Law Theory, Morava’s Point of View, and the Chromatic Spectral Sequence 19 The Brown–Peterson spectrum. Classification of formal group laws. Morava’s group action, its orbits and stabilizers. The chromatic resolution and the chromatic spectral sequence. Bockstein spectral sequences. Use of cyclic subgroups to detect Arf invariant elements. Morava’s vanishing theorem. Greek letter elements in the chromatic spectral sequence. 5. Unstable Homotopy Groups and the EHP Spectral Sequence 24 The EHP sequences. The EHP spectral sequence. The stable zone. The inductive method. The stable EHP spectral sequence. The Adams vector field theorem. James periodicity. The J-spectrum. The spectral v vi CONTENTS ∞ sequence for J∗(RP )andJ∗(BΣp). Relation to the Segal conjecture. The Mahowald root invariant. Chapter 2. Setting up the Adams Spectral Sequence 41 1. The Classical Adams Spectral Sequence 41 Mod (p) Eilenberg–Mac Lane spectra. Mod (p) Adams resolutions. Differentials. Homotopy inverse limits. Convergence. The extension prob- lem. Examples: integral and mod (pi) Eilenberg–Mac Lane spectra. 2. The Adams Spectral Sequence Based on a Generalized Homology Theory 49 E∗-Adams resolutions. E-completions. The E∗-Adams spectral se- quence. Assumptions on the spectrum E. E∗(E) is a Hopf algebroid. The canonical Adams resolution. Convergence. The Adams filtration. 3. The Smash Product Pairing and the Generalized Connecting Homomorphism 53 The smash product induces a pairing in the Adams spectral sequence. A map that is trivial in homology raises Adams filtration. The connecting homomorphism in Ext and the geometric boundary map. Chapter 3. The Classical Adams Spectral Sequence 59 1. The Steenrod Algebra and Some Easy Calculations 59 Milnor’s structure theorem for A∗. The cobar complex. Multiplica- tion by p in the E∞-term. The Adams spectral sequence for π∗(MU). Computations for MO, bu and bo. 2. The May Spectral Sequence 67 May’s filtration of A∗. Nonassociativity of May’s E1-term and a way to avoid it. Computations at p = 2 in low dimensions. Computations with the subalgebra A(2) at p =2. 3. The Lambda Algebra 74 ΛasanAdamsE1-term. The algebraic EHP spectral sequence. Serial numbers. The Curtis algorithm. Computations below dimension 14. James periodicity. The Adams vanishing line. d1 is multiplication by 3 λ−1. Illustration for S . 4. Some General Properties of Ext 85 Exts for s ≤ 3. Behavior of elements in Ext2. Adams’ vanishing line of slope 1/2forp = 2. Periodicity above a line of slope 1/5forp =2. Elements not annihilated by any periodicity operators and their relation to im J. An elementary proof that most of these elements are nontrivial. 5. Survey and Further Reading 94 Exotic cobordism theories. Decreasing filtrations of A∗ and the result- ing spectral sequences. Application to MSp. Mahowald’s generalizations CONTENTS vii of Λ. vn-periodicity in the Adams spectral sequence. Selected references to related work. Chapter 4. BP-Theory and the Adams–Novikov Spectral Sequence 103 1. Quillen’s Theorem and the Structure of BP∗(BP) 103 Complex cobordism. Complex orientation of a ring spectrum. The formal group law for a complex oriented homology theory. Quillen’s the- orem equating the Lazard and complex cobordism rings. Landweber and Novikov’s theorem on the structure of MU∗(MU). The Brown–Peterson spectrum BP. Quillen’s idempotent operation and p-typical formal group laws. The structure of BP∗(BP). 2. A Survey of BP-Theory 111 Bordism groups of spaces. The Sullivan–Baas construction. The Johnson–Wilson spectrum BPn.TheMoravaK-theories K(n). The Landweber filtration and exact functor theorems. The Conner–Floyd iso- morphism. K-theory as a functor of complex cobordism. Johnson and Yosimura’s work on invariant regular ideals. Infinite loop spaces associ- ated with MU and BP; the Ravenel–Wilson Hopf ring. The unstable Adams–Novikov spectral sequence of Bendersky, Curtis and Miller. 3. Some Calculations in BP∗(BP) 117 The Morava-Landweber invariant prime ideal theorem. Some invari- ant regular ideals. A generalization of Witt’s lemma. A formula for the universal p-typical formal group law. Formulas for the coproduct and con- jugation in BP∗(BP). A filtration of BP∗(BP)/In. 4. Beginning Calculations with the Adams–Novikov Spectral Sequence 130 The Adams–Novikov spectral sequence and sparseness. The alge- braic Novikov spectral sequence of Novikov and Miller. Low dimensional Ext of the algebra of Steenrod reduced powers. Bockstein spectral se- quences leading to the Adams–Novikov E2-term. Calculations at odd primes. Toda’s theorem on the first nontrivial odd primary Novikov dif- ferential. Chart for p = 5. Calculations and charts for p =2.Comparison with the Adams spectral sequence. Chapter 5. The Chromatic Spectral Sequence 147 1. The Algebraic Construction 148 Greek letter elements and generalizations. The chromatic resolu- tion, spectral sequence, and cobar complex. The Morava stabilizer alge- bra Σ(n). The change-of-rings theorem. The Morava vanishing theorem. Signs of Greek letter elements. Computations with βt. Decomposability of γ1. Chromatic differentials at p = 2. Divisibility of α1βp. 1 2. Ext (BP∗/In) and Hopf Invariant One 158 0 0 0 1 Ext (BP∗). Ext (M1 ). Ext (BP∗). Hopf invariant one elements. 1 The Miller-Wilson calculation of Ext (BP∗/In). viii CONTENTS 3. Ext(M 1)andtheJ-Homomorphism 165 Ext(M 1). Relation to im J. Patterns of differentials at p =2.Comp- utations with the mod (2) Moore spectrum. 4. Ext2 and the Thom Reduction 172 Results of Miller, Ravenel and Wilson (p>2) and Shimomura (p =2) 2 on Ext (BP∗). Behavior of the Thom reduction map. Arf invariant differ- entials at p>2. Mahowald’s counterexample to the doomsday conjecture. 5. Periodic Families in Ext2 178 Smith’s construction of βt. Obstructions at p =3.ResultsofDavis, Mahowald, Oka, Smith and Zahler on permanent cycles in Ext2. Decom- posables in Ext2. 6. Elements in Ext3 and Beyond 183 Products of alphas and betas in Ext3. Products of betas in Ext4.A possible obstruction to the existence of V (4). Chapter 6. Morava Stabilizer Algebras 187 1. The Change-of-Rings Isomorphism 188 Theorems of Ravenel and Miller. Theorems of Morava. General nonsense about Hopf algebroids. Formal group laws of Artin local rings. Morava’s proof. Miller and Ravenel’s proof. 2. The Structure of Σ(n) 192 Relation to the group ring for Sn. Recovering the grading via an eigenspace decomposition. A matrix representation of Sn. A splitting of Sn when p n. Poincar´e duality and periodic cohomology of Sn. 3. The Cohomology of Σ(n) 198 A May filtration of Σ(n) and the May spectral sequence. The open subgroup theorem. Cohomology of some associated Lie algebras. H1 and H2. H∗(S(n)) for n =1, 2, 3.
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