Springer Monographs in Mathematics Yuli B. Rudyak On Thom Spectra, Orientability, and Cobordism 123 YuliB.Rudyak DepartmentofMathematics UniversityofFlorida 358LittleHall POBox118105 Gainesville,FL32611–8105 USA e-mail:[email protected]fl.edu Corrected2ndprinting2008 ISBN978-3-540-62043-3 SpringerMongraphsinMathematicsISSN1439-7382 LibraryofCongressControlNumber:97032730 MathematicsSubjectClassification(1991):55Nxx,55Rxx,55Sxx,57Nxx,57Qxx,57Rxx ©1998Springer-VerlagBerlinHeidelberg Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerial isconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broad- casting,reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationof thispublicationorpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLaw ofSeptember9,1965,initscurrentversion,andpermissionforusemustalwaysbeobtainedfrom Springer.ViolationsareliableforprosecutionundertheGermanCopyrightLaw. Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnot imply,even intheabsenceofaspecificstatement,thatsuchnames areexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Coverdesign:WMXDesignGmbH,Heidelberg Printedonacid-freepaper 987654321 springer.com Dedicated to my parents Foreword For many years, Algebraic Topology rests on three legs: “ordinary” Coho- mology,K-theory,andCobordism.An introductionto the firstlegandsome of its applications constitute the curriculum of a typical first year graduate course. There have been all too few books addressed to students who have completedsuchanintroduction,andthepresentvolumeisthefirstsuchguide inthesubjectofCobordismsinceRobertStong’sencyclopedicandinfluential notes of a generation ago. The pioneering work of Pontryagin and Thom forged a deep connection between certain geometric problems (such as the classification of manifolds) andhomotopytheory,throughthemediumoftheThomspace.Computations become possible upon stabilization, and this provided some of the first and most compelling examples of “spectra.” Since its inception the subject has thus represented a merger of the Rus- sian and Western mathematical schools. This international tradition was continued with the more or less simultaneous work by Novikov and Milnor on complex cobordism, and later by Quillen. More recently Dennis Sullivan openedthe wayto the study of“manifoldswith singularities,”a study taken up most forcefully by the Russian school, notably by Vershinin, Botvinnik, and Rudyak. Attention to pedagogy is another Russian tradition which you will find amply fulfilled in this book. There is a fine introduction to the stable homo- topycategory.The subtle andincreasinglyimportantissueofphantommaps isaddressedherewithcare.Equallycarefulisthetreatmentoforientability,a subjecttowhichthe authorhascontributedgreatly.Andthe variousaspects ofthetheoryofCobordism,especiallythecentralcaseofcomplexcobordism, are naturally given a detailed and ample telling. ProfessorRudyakhasalsoperformedaservicetothehistoryofsciencein this book, giving detailed and informed attributions. This same care makes the book easy to use by the student, for when proofs are not given here specific references are. It is to be hoped that this book is the first in a new generation of text- books, reflecting the current vigor of the subject. Haynes Miller Cambridge, MA April, 1997 Preface IstartedtowritethisbookinMoscowandfinishedinHeidelberg.Iamgrate- fultotheChairofHigherMathematicsoftheMoscowCivilEngineeringInsti- tute (the Chairman is S.Ja.Havinson)and to the Forschergruppe “Topologie und nichtkommutative Geometrie” (sponsored by Deutsche Forschungsge- meinschaft) at the Mathematical Institute of the University of Heidelberg which partially supported me during the writing of the book. I express my especial gratitude to my Ph.D. advisor Michael Postnikov, andIamgladtothankalltheparticipantsoftheAlgebraicTopologySeminar (supervisedbyM.Postnikov)atMoscowState University,wheremytopolog- ical tastes and preferences were formed and developed. I am also very grateful to Albrecht Dold, Hans–Werner Henn and Dieter Puppe for a lot of fruitful discussions at Heidelberg. Among the others whose suggestions have served me well, I note Nils Baas, Boris Botvinnik, Victor Buchstaber, Alexei Chernavski, Werner End, Oswald Gschnitzer, Edward Fadell, Andrey Khokhlov, Karl-Heinz Knapp, Matthias Kreck, Peter Landweber, Ran Levi, Mark Mahowald, Peter May, Haynes Miller, Alexandr Mishchenko, Sergey Novikov, Erich Ossa, Andrey Pazhitnov, Yury Solov’ev, Michael Stan’ko, Robert Stong, Paul Turner, and Vladimir Vershinin. I am grateful to the mathematics editorial of Springer-Verlag (Heidel- berg), especially to Ruth Allewelt, Martin Peters and Catriona Byrne, for theirkindassistanceandhelp.Thanksarealsoduetothe productioneditors Leonie Kunz and Karl-Friedrich Koch as well as to the TEX editor Thomas Rudolf. IwouldliketothankSamuelMaltbyforproof-readingthemanuscriptfor publication. Finally, Gregor Hoffleit, Waldemar Klemm and Hannes Reinecke helped me with UNIX and TEX. Toallofthese andothers whohavehelped me,I expressmy bestthanks. Yuli B. Rudyak Heidelberg Februar, 1998 Table of Contents Foreword ................................................... VII Preface ..................................................... IX Introduction ................................................ 1 Chapter I. Notation, Conventions and Other Preliminaries . 9 §1. Generalities .......................................... 9 §2. Algebra ............................................. 12 §3. Topology ............................................ 14 Chapter II. Spectra and (Co)homology Theories ............ 33 §1. Preliminaries on Spectra ............................... 33 §2. The Smash product of Spectra, Duality, Ring and Module Spectra .............................................. 45 §3. (Co)homology Theories and Their Connection with Spectra .............................................. 53 §4. Homotopy Properties of Spectra ........................ 79 §5. Localization .......................................... 97 §6. Algebras, Coalgebras and Hopf Algebras ................. 107 §7. Graded Eilenberg–Mac Lane Spectra .................... 120 Chapter III. Phantoms ..................................... 135 §1. Phantoms and the Inverse Limit Functor ................ 135 §2. Derived Functors of the Inverse Limit Functor ............ 143 §3. Representability Theorems ............................. 152 §4. A Spectral Sequence .................................. 162 §5. A Sufficient Condition for the Absence of Phantoms ....... 171 §6. Almost Equivalent Spectra (Spaces) ..................... 174 §7. Multiplications and Quasi-multiplications ................ 180 Chapter IV. Thom Spectra ................................. 185 §1. Fibrations and Their Classifying Spaces ................. 185 §2. Structures on Fibrations ............................... 222 §3. A Glance at Locally Trivial Bundles .................... 228 §4. Rn-Bundles and Spherical Fibrations .................... 232 §5. Thom Spaces and Thom Spectra ....................... 250 §6. Homotopy Properties of Certain Thom Spectra ........... 268 §7. Manifolds and (Co)bordism ............................ 276 XII Table of Contents Chapter V. Orientability and Orientations .................. 299 §1. Orientations of Bundles and Fibrations .................. 300 §2. Orientations of Manifolds .............................. 316 §3. Orientability and Integrality ........................... 323 §4. Obstructions to Orientability ........................... 327 §5. Realizability of the Obstructions to Orientability ......... 335 Chapter VI. K and KO-Orientability ....................... 339 §1. Some Secondary Operations on Thom Classes ............ 339 §2. Some Calculations with Classifying Spaces ............... 351 §3. k-Orientability ....................................... 359 §4. kO-Orientability ...................................... 374 §5. A Few Geometric Observations ......................... 379 Chapter VII. Complex (Co)bordism ........................ 383 §1. Homotopy and Homology Properties of the Spectrum MU . 383 §2. C-oriented Spectra ................................... 393 §3. Operations on MU. Idempotents. The Brown–Peterson Spectrum BP ........................................ 402 §4. Invariant Prime Ideals. The Filtration Theorem .......... 419 §5. Formal Groups ....................................... 428 §6. Formal Groups Input .................................. 433 §7. The Steenrod–tom Dieck Operations .................... 446 Chapter VIII. (Co)bordism with Singularities .............. 457 §1. Definitions and Basic Properties ........................ 457 §2. Multiplicative Structures .............................. 466 §3. Obstructions and the Steenrod–tom Dieck Operations ..... 474 §4. A Universality Theorem for MU with Singularities ........ 481 §5. Realization of Homology Classes by PL Manifolds with Singularities ......................................... 487 Chapter IX. Complex (Co)bordism with Singularities ....... 495 §1. Brown–Peterson(Co)homology with Singularities ......... 495 §2. The Spectra P(n) ..................................... 498 §3. Homological Properties of the Spectra P(n) .............. 505 §4. The Exactness Theorem ............................... 514 §5. Commutative Ring Spectra of Characteristic 2 ........... 521 §6. The Spectra BP(cid:2)n(cid:3) and Homological Dimension .......... 528 §7. Morava K-Theories ................................... 538 References .................................................. 553 List of Notations ........................................... 573 Subject Index .............................................. 579 Introduction First, tell what you are going to talk about, then tell this, and then tell what you havetalked about. Manuals of a senior country priest for beginners The contents of this book are concentrated around Thom spaces (spec- tra),orientabilitytheoryand(co)bordismtheory(including(co)bordismwith singularities), framed by (co)homology theories and spectra. These matters haveformedone of the main lines ofdevelopmentfor the last50yearsinthe areaofalgebraicandgeometrictopology.Inthe bookI considersomeresults obtained in this field in the last 20–30 years, settled enough in order to be exposedinamonographandclosetomyresearchinterests.AsfarasI know, there are no books which cover substantial parts of the presented material. InthebookItriedtoprovethosereferencedresultswhichwerenotproved in any monograph (unfortunately, there are a few exceptions there). More- over, when I quote a result which I do not prove here, I quote the original paper and a monograph where this result is treated as well. There are also occasional remarks containing historical and bibliographical comments, ad- ditional results not included in the text, exercises, etc. A reference to Theorem III.4.5 is to Theorem 4.5 of Ch. III (which is in §4 ofthe chapter);if the chapternumber is omitted, itis to a theoremofthe chapter at hand. The scheme of interconnections of chapters is very simple: I⇒II⇒III⇒IV⇒V⇒VII⇒VIII⇒IX ⇓ VI Iwillnotoverviewthecontents,butIwilldiscussthe subjectofthebook and the place which it occupies in algebraic topology.