On Thom Spectra, Orientability, and Cobordism

592 Pages·1998·5.294 MB·English

by  Yuli B. Rudyak (auth.)

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
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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.
2 Introduction
Conceptional foundations
From the conceptual point of view, we consider the (inter)connections be-
tween geometry and homotopy theory, since Thom spectra and related mat-
ters are now the main tools for this interplay. Here I say a few words about
this.
Algebraic topology studies topological spaces via their algebraic invari-
ants. Evidently, these algebraic invariants should be simple enough in order
to be computable and deep (and so complicated) enough in order to keep
some essential information about a space. How does algebraic topology suc-
ceed in slipping between these two dangers: non-computable informativity
andnon-informativecomputability?The answeris thathomotopy providesa
desired balance between informativity and computability. Therefore, a rea-
sonable way from topology to algebra passes through homotopy theory. (If
you like artistic expressions, I can say that homotopy theory works like a
camera when we make an algebraic photograph of the topological world.)
In other words, one should reduce a geometric problem to a homotopic one
andthencomputethecorrespondinghomotopyinvariants.Thus,interconnec-
tionsbetweengeometryandhomotopytheoryplayapivotalroleinalgebraic
topology.
One of the first results in this area was the Gauss–Bonnet formula, re-
lating a geometrical invariant (the curvature) to a homotopical one (the
Euler characteristic). Proceeding, we can recall the Riemann–Roch Theo-
rem, the Poincar´e integrality theory, relationships between critical points of
a smooth function on a smooth manifold and its homotopy type (Lusternik–
Schnirelmann, Morse), the de Rham Theorem, etc. Hence, the geometry–
homotopy interconnections are very classical things, with a noble genealogy.
On the other hand, we’ll see below that this instrument works very success-
fully in the present as well.
The Characters
Here I discuss (briefly and roughly, because the body of the book contains
the details) the main concepts which appear in the book.
(Co)homology theories. We reserve the term “classical cohomology” or
“ordinary cohomology” for the functors H∗(−;G). The term “cohomology
theory” is used for what was previously called “generalized” or “extraordi-
nary” cohomology theory, i.e., for functors which satisfy all the Eilenberg–
Steenrod axioms except the dimension axioms. Similarly for homology theo-
ries.
Every homology theory h∗(−) yields a so-called dual cohomology theory
h∗(−),andviceversa.They areconnectedviathe equality(cid:2)hi(X)=(cid:2)hn−i(Y)
where Y is n-dual to X (and tilde denotes the reduced (co)homology).