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Nilpotence and Periodicity in Stable Homotopy Theory 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 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. MU-theory and formal group laws 27 3.1. Complex bordism 27 3.2. Formal group laws 28 3.3. The category CΓ 31 3.4. Thick subcategories 36 Chapter 4. Morava's orbit picture and Morava stabilizer groups 39 4.1. The action of Γ on L 39 4.2. Morava stabilizer groups 40 4.3. Cohomological properties of Sn 43 vii 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 hMUi 79 7.4. Some classes bigger than hMUi 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. LnBP 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 1 8.5. The action of Sm on FK(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 Ω2S2m+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 viii 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 MU∗(MU) 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 Index 211 ix Preface 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 book. 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 xi 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 June, 1992 xii 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 (CΓ, 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 CΓ 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 xiii 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.
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