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To my children,Christian Rene,Heidi and Anna,Preface xi. Preface xi,Introduction xiii,Introduction xiii,Chapter 1 The main theorems 1. 1 1 Homotopy 1,1 2 Functors 2,1 3 Suspension 4,1 4 Self maps and the nilpotence theorem 6. 1 5 Morava K theories and the periodicity theorem 7. Chapter 2 Homotopy groups and the chromatic filtration 11. 2 1 The definition of homotopy groups 11,2 2 Classical theorems 12. 2 3 Cofibres 14,2 4 Motivating examples 16,2 5 The chromatic filtration 20.

Chapter 3 M U theory and formal group laws 27,3 1 Complex bordism 27. 3 2 Formal group laws 28,3 3 The category C 31,3 4 Thick subcategories 36. Chapter 4 Morava s orbit picture and Morava stabilizer. 4 1 The action of on L 39,4 2 Morava stabilizer groups 40. 4 3 Cohomological properties of Sn 43,Chapter 5 The thick subcategory theorem 47. 5 1 Spectra 47,5 2 Spanier Whitehead duality 50,5 3 The proof of the thick subcategory theorem 53.

Chapter 6 The periodicity theorem 55,6 1 Properties of vn maps 56. 6 2 The Steenrod algebra and Margolis homology groups 60. 6 3 The Adams spectral sequence and the vn map on Y 64. 6 4 The Smith construction 69, Chapter 7 Bousfield localization and equivalence 73. 7 1 Basic definitions and examples 73,7 2 Bousfield equivalence 75. 7 3 The structure of hM U i 79,7 4 Some classes bigger than hM U i 80. 7 5 E n localization and the chromatic filtration 82. Chapter 8 The proofs of the localization smash product. and chromatic convergence theorems 87,8 1 Ln BP and the localization theorem 87.

8 2 Reducing the smash product theorem to a special. example 90, 8 3 Constructing a finite torsion free prenilpotent. spectrum 92, 8 4 Some cohomological properties of profinite groups 96. 8 5 The action of Sm on F K m CP 98,8 6 Chromatic convergence 101. Chapter 9 The proof of the nilpotence theorem 105,9 1 The spectra X n 106. 9 2 The proofs of the first two lemmas 108,9 3 A paradigm for proving the third lemma 112.

9 4 The Snaith splitting of 2 S 2m 1 114,9 5 The proof of the third lemma 118. 9 6 Historical note theorems of Nishida and Toda 121. Appendix A Some tools from homotopy theory 125,A 1 CW complexes 125. A 2 Loop spaces and spectra 127, A 3 Generalized homology and cohomology theories 132. A 4 Brown representability 136,A 5 Limits in the stable homotopy category 137. A 6 The Adams spectral sequence 145,Appendix B Complex bordism and BP theory 151.

B 1 Vector bundles and Thom spectra 151,B 2 The Pontrjagin Thom construction 158. B 3 Hopf algebroids 160,B 4 The structure of M U M U 166. B 5 BP theory 173,B 6 The Landweber exact functor theorem 180. B 7 Morava K theories 182, B 8 The change of rings isomorphism and the chromatic. spectral sequence 186,Appendix C Some idempotents associated with the.

symmetric group 191,C 1 Constructing the idempotents 191. C 2 Idempotents for graded vector spaces 195, C 3 Getting strongly type n spectra from partially type. n spectra 198,Appendix Bibliography 203,Bibliography 203. Appendix Index 211, This research leading to this book began in Princeton in 1974 75. when Haynes Miller Steve Wilson and I joined forces with the. goal of understanding what the ideas of Jack Morava meant for. the stable homotopy groups of spheres Due to widely differing. personal schedules our efforts spanned nearly 24 hours of each. day we met during the brief afternoon intervals when all three of. us were awake Our collaboration led to MRW77 and Morava. eventually published his work in Mor85 and I gave a broader. account of it in my first book Rav86 but that was not the end. of the story, I suspected that there was some deeper structure in the stable.

homotopy category itself that was reflected in the pleasing alge. braic patterns described in the two papers cited above I first. aired these suspicions in a lecture at the homotopy theory confer. ence at Northwestern University in 1977 and later published them. in Rav84 which ended with a list of seven conjectures Their. formulation was greatly helped by the notions of localization and. equivalence defined by Bousfield in Bou79b and Bou79a. I had some vague ideas about how to approach the conjec. tures but in 1982 when Waldhausen asked me if I expected to see. them settled before the end of the century I could offer him no. assurances It was therefore very gratifying to see all but one of. them proved by the end of 1986 due largely to the seminal work of. Devinatz Hopkins and Smith DHS88 The mathematics sur. rounding these conjectures and their proofs is the subject of this. The one conjecture of Rav84 not proved here is the telescope. conjecture 7 5 5 I disproved a representative special case of it in. 1990 an outline of the argument can be found in Rav92 I find. this development equally satisfying If the telescope conjecture. had been proved the subject might have died Its failure leads. to interesting questions for future work On the other hand had. I not believed it in 1977 I would not have had the heart to go. through with Rav84, This book has two goals to make this material accessible to a. general mathematical audience and to provide algebraic topolo. gists with a coherent and reasonably self contained account of this. material The nine chapters of the book are directed toward the. first goal The technicalities are suppressed as much as possible. at least in the earlier chapters The three appendices give descrip. tions of the tools needed to perform the necessary computations. In essence almost all of the material of this book can be found. in previously published papers The major exceptions are Chap. ter 8 excluding the first section which hopefully will appear in. more detailed form in joint work with Mike Hopkins HR and. Appendix C which was recently written up by Jeff Smith Smi. In both cases the results were known to their authors by 1986. This book itself began as a series of twelve lectures given at. Northwestern University in 1988 then repeated at the University. of Rochester and MSRI Berkeley in 1989 at New Mexico State. University in 1990 and again at Rochester and Northwestern in. 1991 I want to thank all of my listeners for the encouragement. that their patience and enthusiasm gave me Special thanks are. due to Sam Gitler and Hal Sadofsky for their careful attention to. certain parts of the manuscript, I am also grateful to all four institutions and to the National. Science Foundation for helpful financial support,D C Ravenel. Introduction, In Chapter 1 we will give the elementary definitions in ho. motopy theory needed to state the main results the nilpotence. theorem 1 4 2 and the periodicity theorem 1 5 4 The latter. implies the existence of a global structure in the homotopy groups. of many spaces called the chromatic filtration This is the subject. of Chapter 2 which begins with a review of some classical results. about homotopy groups, The nilpotence theorem says that the complex bordism functor.

reveals a great deal about the homotopy category This functor. and the algebraic category C defined in 3 3 2 in which it takes. its values are the subject of Chapters 3 and 4 This discussion is. of necessity quite algebraic with the theory of formal group laws. playing a major role, In C it is easy to enumerate all the thick subcategories de. fined in 3 4 1 The thick subcategory theorem 3 4 3 says that. there is a similar enumeration in the homotopy category itself. This result is extremely useful it means that certain statements. about a large class of spaces can be proved by verifying them. only for very carefully chosen examples The thick subcategory. theorem is derived from the nilpotence theorem in Chapter 5. In Chapter 6 we prove the periodicity theorem using the thick. subcategory theorem First we prove that the set of spaces satis. fying the periodicity theorem forms a thick subcategory this re. quires some computations in certain noncommutative rings This. thickness statement reduces the proof of the theorem to the con. struction of a few examples this requires some modular represen. tation theory due to Jeff Smith, In Chapter 7 we introduce the concepts of Bousfield localiza. tion 7 1 1 and 7 1 3 and Bousfield equivalence 7 2 1 These are. useful both for understanding the structure of the homotopy cat. egory and for proving the nilpotence theorem The proof of the. nilpotence theorem itself is given in Chapter 9 modulo certain. details for which the reader must consult DHS88, There are three appendices which give more technical back. ground for many of the ideas discussed in the text Appendix. A recalls relevant facts known to most homotopy theorists while. Appendix B gives more specialized information related to com. plex bordism theory and BP theory Appendix C which is still. more technical describes some results about representations of. the symmetric group due to Jeff Smith Smi, The appendices are intended to enable a sufficiently moti. vated nonspecialist to follow the proofs of the text in detail. However as an introduction to homotopy theory they are very. unbalanced By no means should they be regarded as a substitute. for a more thorough study of the subject, We will now spell out the relation between the conjectures.

stated in the last section and listed on the last page of Rav84. and the theorems proved here in the order in which they were. stated there Part a of the nilpotence conjecture is the self map. form of the nilpotence theorem 1 4 2 and part b is essentially. the smash product form 5 1 4 Part c is the periodicity theo. rem 1 5 4 of which the realizability conjecture is an immediate. consequence This is not quite true since we do not prove that. the self map can be chosen so that its cofibre is a ring spectrum. This has been proved recently by Devinatz Dev The class in. variance conjecture is Theorem 7 2 7 The telescope conjecture. is stated here as 7 5 5 but is likely to be false in general The. smashing conjecture is the smash product theorem 7 5 6 and the. localization conjecture is Theorem 7 5 2 Finally the Boolean al. gebra conjecture slightly modified to avoid problems with the. telescope conjecture is Theorem 7 2 9, Two major results proved here that were not conjectured in. Rav84 are the thick subcategory theorem 3 4 3 and the chro. matic convergence theorem 7 5 7,The main theorems, The aim of this chapter is to state the nilpotence and period. icity theorems 1 4 2 and 1 5 4 with as little technical fussing as. possible Readers familiar with homotopy theory can skip the first. three sections which contain some very elementary definitions. 1 1 Homotopy, A basic problem in homotopy theory is to classify continuous. maps up to homotopy Two continuous maps from a topological. space X to Y are homotopic if one can be continuously deformed. into the other A more precise definition is the following. Definition 1 1 1 Two continuous maps f0 and f1 from X to. Y are homotopic if there is a continuous map called a homo. such that for t 0 or 1 the restriction of h to X t is ft If f1. is a constant map i e one that sends all of X to a single point. in Y then we say that f0 is null homotopic and that h is a null. homotopy A map which is not homotopic to a constant map is. essential The set of homotopy classes of maps from X to Y is. denoted by X Y, For technical reasons it is often convenient to consider maps. which send a specified point x0 X called the base point to a. given point y0 Y and to require that homotopies between such. maps send all of x0 0 1 to y0 Such maps and homotopies. are said to be base point preserving The set of equivalence. classes of such maps under base point preserving homotopies is. denoted by X x0 Y y0,2 1 THE MAIN THEOREMS, Under mild hypotheses needed to exclude pathological cases.

if X and Y are both path connected and Y is simply connected. the sets X Y and X x0 Y y0 are naturally isomorphic. In many cases e g when X and Y are compact manifolds. or algebraic varieties over the real or complex numbers this set is. countable In certain cases such as when Y is a topological group. it has a natural group structure This is also the case when X is. a suspension 1 3 1 and 2 1 2, In topology two spaces are considered identical if there is a. homeomorphism a continuous map which is one to one and onto. and which has a continuous inverse between them A homo. topy theorist is less discriminating than a point set topologist. two spaces are identical in his eyes if they satisfy a much weaker. equivalence relation defined as follows, Definition 1 1 2 Two spaces X and Y are homotopy equiv. alent if there are continuous maps f X Y and g Y X such. that gf and f g are homotopic to the identity maps on X and Y. The maps f and g are homotopy equivalences A space that. is homotopy equivalent to a single point is contractible Spaces. which are homotopy equivalent have the same homotopy type. For example every real vector space is contractible and a solid. torus is homotopy equivalent to a circle,1 2 Functors. In algebraic topology one devises ways to associate various al. gebraic structures groups rings modules etc with topological. spaces and homomorphisms of the appropriate sort with continu. Definition 1 2 1 A covariant functor F from the category. NILPOTENCE AND PERIODICITY IN STABLE HOMOTOPY THEORY BY DOUGLAS C RAVENEL To my children Christian Ren e Heidi and Anna Contents Preface xi Preface xi Introduction xiii Introduction xiii Chapter 1 The main theorems 1 1 1 Homotopy 1 1 2 Functors 2 1 3 Suspension 4 1 4 Self maps and the nilpotence theorem 6 1 5 Morava K theories and the periodicity theorem 7 Chapter 2 Homotopy

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