Modern Classical Homotopy Theory · 2019. 2. 12. · Modern Classical Homotopy Theory Graduate Studies in Mathematics Volume 127. Modern Classical ... The core of the subject I’m

American Mathematical Society

Jeffrey Strom

Modern Classical Homotopy Theory

Graduate Studies in Mathematics

Volume 127


http://dx.doi.org/10.1090/gsm/127


Jeffrey Strom

American Mathematical SocietyProvidence, Rhode Island

Graduate Studies in Mathematics

Volume 127

EDITORIAL COMMITTEE

David Cox (Chair)Rafe Mazzeo

Martin ScharlemannGigliola Staffilani

2010 Mathematics Subject Classification. Primary 55Nxx, 55Pxx, 55Qxx, 55Sxx, 55Uxx.

For additional information and updates on this book, visitwww.ams.org/bookpages/gsm-127

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Strom, Jeffrey, 1969–Modern classical homotopy theory / Jeffrey Strom.

p. cm. — (Graduate studies in mathematics ; v. 127)Includes bibliographical references and index.ISBN 978-0-8218-5286-6 (alk. paper)1. Homotopy theory. I. Title.

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Dedicated to my mom and dad

Contents

Preface xvii

Part 1. The Language of Categories

Chapter 1. Categories and Functors 3

§1.1. Diagrams 3

§1.2. Categories 5

§1.3. Functors 7

§1.4. Natural Transformations 11

§1.5. Duality 14

§1.6. Products and Sums 15

§1.7. Initial and Terminal Objects 18

§1.8. Group and Cogroup Objects 21

§1.9. Homomorphisms 24

§1.10. Abelian Groups and Cogroups 25

§1.11. Adjoint Functors 26

Chapter 2. Limits and Colimits 29

§2.1. Diagrams and Their Shapes 29

§2.2. Limits and Colimits 31

§2.3. Naturality of Limits and Colimits 34

§2.4. Special Kinds of Limits and Colimits 35

§2.5. Formal Properties of Pushout and Pullback Squares 40

vii

viii Contents

Part 2. Semi-Formal Homotopy Theory

Chapter 3. Categories of Spaces 45

§3.1. Spheres and Disks 45

§3.2. CW Complexes 46

§3.3. Example: Projective Spaces 51

§3.4. Topological Spaces 53

§3.5. The Category of Pairs 58

§3.6. Pointed Spaces 60

§3.7. Relating the Categories of Pointed and Unpointed Spaces 63

§3.8. Suspension and Loop 66

§3.9. Additional Problems and Projects 68

Chapter 4. Homotopy 69

§4.1. Homotopy of Maps 69

§4.2. Constructing Homotopies 74

§4.3. Homotopy Theory 80

§4.4. Groups and Cogroups in the Homotopy Category 84

§4.5. Homotopy Groups 87

§4.6. Homotopy and Duality 89

§4.7. Homotopy in Mapping Categories 91

§4.8. Additional Problems 98

Chapter 5. Cofibrations and Fibrations 99

§5.1. Cofibrations 100

§5.2. Special Properties of Cofibrations of Spaces 104

§5.3. Fibrations 107

§5.4. Factoring through Cofibrations and Fibrations 110

§5.5. More Homotopy Theory in Categories of Maps 115

§5.6. The Fundamental Lifting Property 118

§5.7. Pointed Cofibrations and Fibrations 122

§5.8. Well-Pointed Spaces 124

§5.9. Exact Sequences, Cofibers and Fibers 129

§5.10. Mapping Spaces 133

§5.11. Additional Topics, Problems and Projects 136

Chapter 6. Homotopy Limits and Colimits 143

§6.1. Homotopy Equivalence in Diagram Categories 144

Contents ix

§6.2. Cofibrant Diagrams 146

§6.3. Homotopy Colimits of Diagrams 151

§6.4. Constructing Cofibrant Replacements 155

§6.5. Examples: Pushouts, 3× 3s and Telescopes 160

§6.6. Homotopy Limits 167

§6.7. Functors Applied to Homotopy Limits and Colimits 173

§6.8. Homotopy Colimits of More General Diagrams 176


Chapter 7. Homotopy Pushout and Pullback Squares 181

§7.1. Homotopy Pushout Squares 181

§7.2. Recognition and Completion 185

§7.3. Homotopy Pullback Squares 188

§7.4. Manipulating Squares 190

§7.5. Characterizing Homotopy Pushout and Pullback Squares 195


Chapter 8. Tools and Techniques 199

§8.1. Long Cofiber and Fiber Sequences 199

§8.2. The Action of Paths in Fibrations 203

§8.3. Every Action Has an Equal and Opposite Coaction 205

§8.4. Mayer-Vietoris Sequences 209

§8.5. The Operation of Paths 211

§8.6. Fubini Theorems 212

§8.7. Iterated Fibers and Cofibers 214

§8.8. Group Actions 216

Chapter 9. Topics and Examples 221

§9.1. Homotopy Type of Joins and Products 221

§9.2. H-Spaces and co-H-Spaces 225

§9.3. Unitary Groups and Their Quotients 230

§9.4. Cone Decompositions 237

§9.5. Introduction to Phantom Maps 245

§9.6. G. W. Whitehead’s Homotopy Pullback Square 249

§9.7. Lusternik-Schnirelmann Category 250


x Contents

Chapter 10. Model Categories 261

§10.1. Model Categories 262

§10.2. Left and Right Homotopy 266

§10.3. The Homotopy Category of a Model Category 268

§10.4. Derived Functors and Quillen Equivalence 268

§10.5. Homotopy Limits and Colimits 270

Part 3. Four Topological Inputs

Chapter 11. The Concept of Dimension in Homotopy Theory 275

§11.1. Induction Principles for CW Complexes 276

§11.2. n-Equivalences and Connectivity of Spaces 277

§11.3. Reformulations of n-Equivalences 280

§11.4. The J. H. C. Whitehead Theorem 286


Chapter 12. Subdivision of Disks 289

§12.1. The Seifert-Van Kampen Theorem 289

§12.2. Simplices and Subdivision 295

§12.3. The Connectivity of Xn ↪→ X 298

§12.4. Cellular Approximation of Maps 299

§12.5. Homotopy Colimits and n-Equivalences 300


Chapter 13. The Local Nature of Fibrations 305

§13.1. Maps Homotopy Equivalent to Fibrations 306

§13.2. Local Fibrations Are Fibrations 308

§13.3. Gluing Weak Fibrations 310

§13.4. The First Cube Theorem 313

Chapter 14. Pullbacks of Cofibrations 317

§14.1. Pullbacks of Cofibrations 317

§14.2. Pullbacks of Well-Pointed Spaces 319

§14.3. The Second Cube Theorem 320

Chapter 15. Related Topics 323

§15.1. Locally Trivial Bundles 323

§15.2. Covering Spaces 326

§15.3. Bundles Built from Group Actions 330

Contents xi

§15.4. Some Theory of Fiber Bundles 333

§15.5. Serre Fibrations and Model Structures 336

§15.6. The Simplicial Approach to Homotopy Theory 341

§15.7. Quasifibrations 346


Part 4. Targets as Domains, Domains as Targets

Chapter 16. Constructions of Spaces and Maps 353

§16.1. Skeleta of Spaces 354

§16.2. Connectivity and CW Structure 357

§16.3. Basic Obstruction Theory 359

§16.4. Postnikov Sections 361

§16.5. Classifying Spaces and Universal Bundles 363


Chapter 17. Understanding Suspension 373

§17.1. Moore Paths and Loops 373

§17.2. The Free Monoid on a Topological Space 376

§17.3. Identifying the Suspension Map 379

§17.4. The Freudenthal Suspension Theorem 382

§17.5. Homotopy Groups of Spheres and Wedges of Spheres 383

§17.6. Eilenberg-MacLane Spaces 384

§17.7. Suspension in Dimension 1 387

§17.8. Additional Topics and Problems 389

Chapter 18. Comparing Pushouts and Pullbacks 393

§18.1. Pullbacks and Pushouts 393

§18.2. Comparing the Fiber of f to Its Cofiber 396

§18.3. The Blakers-Massey Theorem 398

§18.4. The Delooping of Maps 402

§18.5. The n-Dimensional Blakers-Massey Theorem 405


Chapter 19. Some Computations in Homotopy Theory 413

§19.1. The Degree of a Map Sn → Sn 414

§19.2. Some Applications of Degree 417

§19.3. Maps Between Wedges of Spheres 421

xii Contents

§19.4. Moore Spaces 424

§19.5. Homotopy Groups of a Smash Product 427

§19.6. Smash Products of Eilenberg-MacLane Spaces 429

§19.7. An Additional Topic and Some Problems 432

Chapter 20. Further Topics 435

§20.1. The Homotopy Category Is Not Complete 435

§20.2. Cone Decompositions with Respect to Moore Spaces 436

§20.3. First p-Torsion Is a Stable Invariant 438

§20.4. Hopf Invariants and Lusternik-Schnirelmann Category 445

§20.5. Infinite Symmetric Products 448


Part 5. Cohomology and Homology

Chapter 21. Cohomology 459

§21.1. Cohomology 459

§21.2. Basic Computations 467

§21.3. The External Cohomology Product 473

§21.4. Cohomology Rings 475

§21.5. Computing Algebra Structures 479

§21.6. Variation of Coefficients 485

§21.7. A Simple Kunneth Theorem 487

§21.8. The Brown Representability Theorem 489

§21.9. The Singular Extension of Cohomology 494

§21.10. An Additional Topic and Some Problems and Projects 495

Chapter 22. Homology 499

§22.1. Homology Theories 499

§22.2. Examples of Homology Theories 505

§22.3. Exterior Products and the Kunneth Theorem for Homology 508

§22.4. Coalgebra Structure for Homology 509

§22.5. Relating Homology to Cohomology 510

§22.6. H-Spaces and Hopf Algebras 512

Chapter 23. Cohomology Operations 515

§23.1. Cohomology Operations 516

§23.2. Stable Cohomology Operations 518

Contents xiii

§23.3. Using the Diagonal Map to Construct CohomologyOperations 521

§23.4. The Steenrod Reduced Powers 525

§23.5. The Adem Relations 528

§23.6. The Algebra of the Steenrod Algebra 533

§23.7. Wrap-Up 538

Chapter 24. Chain Complexes 541

§24.1. The Cellular Complex 542

§24.2. Applying Algebraic Universal Coefficients Theorems 547

§24.3. The General Kunneth Theorem 548

§24.4. Algebra Structures on C∗(X) and C∗(X) 550

§24.5. The Singular Chain Complex 551

Chapter 25. Topics, Problems and Projects 553

§25.1. Algebra Structures on Rn and Cn 553

§25.2. Relative Cup Products 554

§25.3. Hopf Invariants and Hopf Maps 556

§25.4. Some Homotopy Groups of Spheres 563

§25.5. The Borsuk-Ulam Theorem 565

§25.6. Moore Spaces and Homology Decompositions 567

§25.7. Finite Generation of π∗(X) and H∗(X) 570

§25.8. Surfaces 572

§25.9. Euler Characteristic 573

§25.10. The Kunneth Theorem via Symmetric Products 576

§25.11. The Homology Algebra of ΩΣX 576

§25.12. The Adjoint λX of idΩX 577

§25.13. Some Algebraic Topology of Fibrations 579

§25.14. A Glimpse of Spectra 580

§25.15. A Variety of Topics 581


Part 6. Cohomology, Homology and Fibrations

Chapter 26. The Wang Sequence 591

§26.1. Trivialization of Fibrations 591

§26.2. Orientable Fibrations 592

§26.3. The Wang Cofiber Sequence 593

xiv Contents

§26.4. Some Algebraic Topology of Unitary Groups 597

§26.5. The Serre Filtration 600


Chapter 27. Cohomology of Filtered Spaces 605

§27.1. Filtered Spaces and Filtered Groups 606

§27.2. Cohomology and Cone Filtrations 612

§27.3. Approximations for General Filtered Spaces 615

§27.4. Products in E∗,∗1 (X) 618

§27.5. Pointed and Unpointed Filtered Spaces 620

§27.6. The Homology of Filtered Spaces 620

§27.7. Additional Projects 621

Chapter 28. The Serre Filtration of a Fibration 623

§28.1. Identification of E2 for the Serre Filtration 623

§28.2. Proof of Theorem 28.1 625

§28.3. External and Internal Products 631

§28.4. Homology and the Serre Filtration 633


Chapter 29. Application: Incompressibility 635

§29.1. Homology of Eilenberg-MacLane Spaces 636

§29.2. Reduction to Theorem 29.1 636


§29.4. Consequences of Theorem 29.1 641


Chapter 30. The Spectral Sequence of a Filtered Space 645

§30.1. Approximating Grs Hn(X) by Es,nr (X) 646

§30.2. Some Algebra of Spectral Sequences 651

§30.3. The Spectral Sequences of Filtered Spaces 654

Chapter 31. The Leray-Serre Spectral Sequence 659

§31.1. The Leray-Serre Spectral Sequence 659

§31.2. Edge Phenomena 663

§31.3. Simple Computations 671

§31.4. Simplifying the Leray-Serre Spectral Sequence 673


Contents xv

Chapter 32. Application: Bott Periodicity 681

§32.1. The Cohomology Algebra of BU(n) 682

§32.2. The Torus and the Symmetric Group 682

§32.3. The Homology Algebra of BU 685

§32.4. The Homology Algebra of ΩSU(n) 689

§32.5. Generating Complexes for ΩSU and BU 690

§32.6. The Bott Periodicity Theorem 692

§32.7. K-Theory 695


Chapter 33. Using the Leray-Serre Spectral Sequence 699

§33.1. The Zeeman Comparison Theorem 699

§33.2. A Rational Borel-Type Theorem 702

§33.3. Mod 2 Cohomology of K(G,n) 703

§33.4. Mod p Cohomology of K(G,n) 706

§33.5. Steenrod Operations Generate Ap 710

§33.6. Homotopy Groups of Spheres 711

§33.7. Spaces Not Satisfying the Ganea Condition 713

§33.8. Spectral Sequences and Serre Classes 714


Part 7. Vistas

Chapter 34. Localization and Completion 721

§34.1. Localization and Idempotent Functors 722


§34.3. Homotopy Theory of P-Local Spaces 729

§34.4. Localization with Respect to Homology 734

§34.5. Rational Homotopy Theory 737

§34.6. Further Topics 742

Chapter 35. Exponents for Homotopy Groups 745

§35.1. Construction of α 747

§35.2. Spectral Sequence Computations 751

§35.3. The Map γ 754


§35.5. Nearly Trivial Maps 756

xvi Contents

Chapter 36. Classes of Spaces 759

§36.1. A Galois Correspondence in Homotopy Theory 760

§36.2. Strong Resolving Classes 761

§36.3. Closed Classes and Fibrations 764

§36.4. The Calculus of Closed Classes 767

Chapter 37. Miller’s Theorem 773

§37.1. Reduction to Odd Spheres 774

§37.2. Modules over the Steenrod Algebra 777

§37.3. Massey-Peterson Towers 780

§37.4. Extensions and Consequences of Miller’s Theorem 785

Appendix A. Some Algebra 789

§A.1. Modules, Algebras and Tensor Products 789

§A.2. Exact Sequences 794

§A.3. Graded Algebra 795

§A.4. Chain Complexes and Algebraic Homology 798

§A.5. Some Homological Algebra 799

§A.6. Hopf Algebras 803

§A.7. Symmetric Polynomials 806

§A.8. Sums, Products and Maps of Finite Type 807

§A.9. Ordinal Numbers 808

Bibliography 811

Index of Notation 821

Index 823

Preface

The subject of topology can be described as the study of the category Topof all topological spaces and the continuous maps between them. But manytopological problems, and their solutions, do not change if the maps involvedare replaced with ‘continuous deformations’ of themselves. The equivalencerelation—called homotopy—generated by continuous deformations of mapsrespects composition, so that there is a ‘quotient’ homotopy category hTopand a functor Top → hTop. Homotopy theory is the study of this functor.Thus homotopy theory is not entirely confined to the category hTop: it isfrequently necessary, or at least useful, to use constructions available onlyin Top in order to prove statements that are entirely internal to hTop; andthe homotopy category hTop can shed light even on questions in Top thatare not homotopy invariant.

History. The core of the subject I’m calling ‘classical homotopy theory’is a body of ideas and theorems that emerged in the 1950s and was laterlargely codified in the notion of a model category. This includes the notionsof fibration and cofibration, CW complexes, long fiber and cofiber sequences,loop space, suspension, and so on. Brown’s representability theorems showthat homology and cohomology are also contained in classical homotopytheory.

One of the main complications in homotopy theory is that many, if notmost, diagrams in the category hTop do not have limits or colimits. Thusmany theorems were proved using occasionally ingenious and generally adhoc constructions performed in the category Top. Eventually many of theseconstructions were codified in the dual concepts of homotopy colimit and

xvii

xviii Preface

homotopy limit, and a powerful calculus for working with them was devel-oped. The language of homotopy limits and colimits and the techniques formanipulating them made it possible to easily state and conceptually provemany results that had previously seemed quite difficult and inscrutable.

Once the basic theory has been laid down, the most interesting and usefultheorems are those that break the categorical barrier between domain andtarget. The basic example of such a theorem is the Blakers-Massey theorem,which compares homotopy pushout squares to homotopy pullback squares.Other excellent examples of duality-breaking theorems are the Hilton-Milnortheorem on the loop space of a wedge and Ganea’s theorem (which is dualto the most important special case of the Blakers-Massey theorem). All ofthese results were first proved with a great deal of technical finesse but cannow be established easily using homotopy pushouts and pullbacks.

The Aim of This Book. The aim of this book is to develop classical ho-motopy theory and some important developments that flow from it using themore modern techniques of homotopy limits and colimits. Thus homotopypushouts and homotopy pullbacks play a central role.

The book has been written with the theory of model categories firmly inmind. As is probably already evident, we make consistent and unapologeticuse of the language of categories, functors, limits and colimits. But we aregenuinely interested in the homotopy theory of spaces so, with the exceptionof a brief account of the abstract theory of model categories, we work withspaces throughout and happily make use of results that are special for spaces.Indeed, the third part of the book is devoted to the development of fourbasic properties that set the category of spaces apart from generic modelcategories.

I have generally used topological or homotopy-theoretical argumentsrather than algebraic ones. This almost always leads to simpler statementsand simpler arguments. Thus my book attempts to upset the balance (ob-served in many algebraic topology texts) between algebra and topology, infavor of topology. Algebra is just one of many tools by which we understandtopology. This is not an anti-algebra crusade. Rather, I set out hoping tofind homotopy-theoretical arguments wherever possible, with the expecta-tion that at certain points, the simplicity or clarity afforded by the standardalgebraic approach would outweigh the philosophical cleanliness of avoidingit. But I ended up being surprised: at no point did I find that ‘extra’ algebramade any contribution to clarity or simplicity.

Omissions. This is a very long book, and many topics that were in myearliest plans have had to be (regretfully) left out. I had planned threechapters on stable homotopy, extraordinary cohomology and nilpotence and

Preface xix

another on Goodwillie calculus. But in the book that emerged it seemedthematically appropriate to draw the line at stable homotopy theory, sospace and thematic consistency drove these chapters to the cutting roomfloor.

Problems and Exercises. Many authors of textbooks assert that theonly way to learn the subject is to do the exercises. I have taken this toheart, and so there are no outright proofs in the book. Instead, theorems arefollowed by multi-part problems that guide the readers to find the proofs forthemselves. To the expert, these problems will read as terse proofs, perhapssuitable for exposition in a journal article. Reading this text, then, is apreparation for the experience of reading research articles. There are alsoa great many other problems incorporated into the main flow of the text,problems that develop interesting tangential results, explore applications, orcarry out explicit calculations.

In addition, there are numerous exercises. These are intended to helpthe student develop some habits of mind that are extremely useful whenreading mathematics. After definitions, the reader is asked to find examplesand nonexamples, to explore how the new concept fits in with previousideas, etc. Other exercises ask the reader to compare theorems with previousresults, to test whether hypotheses are needed, or can be weakened, and soon.

Audience. This book was written with the idea that it would be used bystudents in their first year or two of graduate school. It is assumed that thereader is familiar with basic algebraic concepts such as groups and rings. Itis also assumed that the student has had an introductory course in topology.It would be nice if that course included some mention of the fundamentalgroup, but that is not necessary.

Teaching from This Book. This book covers more topics, in greaterdepth, than can be covered in detail in a typical two-semester homotopytheory or algebraic topology sequence. That being said, a good goal for atwo-semester course would be to cover the high points of Parts 1 – 4 in thefirst semester and Parts 5 – 6 in the second semester, followed by some orall of Part 7 if time permits.

Here’s some more detail.

The first semester would start with a brief (one day) introduction to thelanguage of category theory before heading on to Part 2 to develop the basictheory of cofibrations, fibrations, and homotopy limits and colimits. Part1 is an overview of the basics of category theory and shouldn’t be coveredin its own right at all; refer back to it as needed to bring in more advancedcategory-theoretical topics. Chapters 3 and 4, in which the category of

xx Preface

spaces is established and the concept of homotopy is developed should becovered fairly thoroughly. Chapter 5 is on cofibrations and fibrations. Thebasic properties should be explored, and the mapping cylinder and its dualshould be studied carefully; it’s probably best to gloss over the distinctionbetween the pointed and unpointed cases. State the Fundamental Lift-ing Property and the basic factorization theorems without belaboring theirproofs. The fact that fiber and cofiber sequences lead to exact sequencesof homotopy sets should be explored in detail. Chapter 6 is on homotopycolimits and limits. Cover homotopy pushouts in detail, appealing to du-ality for homotopy pullbacks, and give a brief discussion of the issues formore general diagrams. Chapter 7 is on homotopy pullback and pushoutsquares and should be covered in some detail. Chapters 8 and 9 offer a hugecollection of topics. For the moment, only Section 8.1 (Long cofiber andfiber sequences) and perhaps Section 9.2 (on H-Spaces and co-H-spaces) arereally mandatory. Other sections can be covered as needed or assigned tostudents as homework. Chapter 10 is a brief account of abstract model cat-egories. It is included for ‘cultural completeness’ and, since it does not enterinto the main flow of the text, it can be skipped in its entirety. Part 3 coversthe four major special features of the homotopy theory of spaces. Chapters11 through 14 should be covered in detail. Chapter 15 is a combination oftopics and cultural knowledge. Sections 15.1 and 15.2 are crucial, but therest can be glossed over if need be. Part 4 is where the four basic topologicalinputs are developed into effective tools for studying homotopy-theoreticalproblems. Chapters 16 through 19 should all be covered in detail. Chapter20 contains topics which can be assigned to students as homework.

The second semester should pick up with Part 5 where we develop coho-mology (and homology). Chapters 21 through 24 should be covered prettythoroughly. Chapter 25 is a vast collection of topics, which can be coveredat the instructor’s discretion or assigned as homework. Part 6 is about thecohomology of fiber sequences, leading ultimately to the Leray-Serre spec-tral sequence, which is notoriously forbidding when first encountered. Theexposition here is broken into small pieces with a consistent emphasis onthe topological content. Many of the basic ideas and a nice application arecovered in Chapters 26 through 29; this would be a fine place to stop iftime runs out. Otherwise, Chapters 30 and 31 get to the full power of theLeray-Serre spectral sequence. This power is used in Chapter 32 to provethe Bott Periodicity Theorem. Chapter 33 is another topics chapter, whichincludes the cohomology of Eilenberg-MacLane spaces and some computa-tions involving the homotopy groups of spheres. Finally, Part 7 covers somevery fun and interesting topics: localization and completion, a discussion ofthe exponents of homotopy groups of spheres including a proof of Selick’stheorem on the exponent of π∗(S3); the theory of closed classes and a dual

Preface xxi

concept known as strong resolving classes; and a proof of Miller’s theoremon the space of maps from BZ/p to a simply-connected finite complex.

Acknowledgements. A book such as this, I have come to realize, is essen-tially an attempt to set down the author’s point of view on his subject. Mypoint of view has been shaped by many people, beginning with Ed Fadell,Sufian Husseini and Steve Hutt, who were my first teachers in the subject.Early in my career, my horizons were greatly expanded by conversation andcollaboration with Bob Bruner and Chuck McGibbon, and even more soduring my long, pleasant and fruitful collaboration with Martin Arkowitz.

At various points during the writing of this book I have turned to othersfor clarification or advice on certain points that escaped me. Thanks aredue to Peter May, whose kind responses to my emailed questions greatlyimproved a chapter that is, unfortunately, no longer included in the book.The community at the website MathOverflow offered useful advice on manyquestions.

My thanks are also due to the students who were guinea pigs for earlyversions of this text. Specifically, the enthusiasm of David Arnold, JimClarkson, Julie Houck, Rob Nendorf, Nick Scoville, and Jason Trowbridgewas inspirational. I must also thank John Martino and Jay Wood for teach-ing the algebraic topology sequence at Western Michigan University usingearly drafts of this text.

Finally, I must gratefully acknowledge the support of my family duringthe long writing process. Dolores was exceedingly—albeit decreasingly—patient with my nearly endless string of pronouncements that I was ‘almostdone’, and my sanity was preserved by my son Brandon, who unknowinglyand innocently forced me every day to stop working and have fun.

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[9] Jose Adem, The iteration of the Steenrod squares in algebraic topology, Proc. Nat. Acad.Sci. U. S. A. 38 (1952), 720–726. MR0050278 (14,306e)

[10] , Relations on iterated reduced powers, Proc. Nat. Acad. Sci. U. S. A. 39 (1953),636–638. MR0056293 (15,53c)

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[12] Marcelo Aguilar, Samuel Gitler, and Carlos Prieto, Algebraic topology from a homotopicalviewpoint, Universitext, Springer-Verlag, New York, 2002. Translated from the Spanish byStephen Bruce Sontz. MR1908260 (2003c:55001)

[13] David J. Anick, A counterexample to a conjecture of Serre, Ann. of Math. (2) 115 (1982),no. 1, 1–33, DOI 10.2307/1971338. MR644015 (86i:55011a)

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Index of Notation

X ∗ Y , 221

X � Y , 64

X+, 64

X−, 64

X(s)/(t), 611

[X,Y ], 70

α ∗[s] β, 109α � β, 374

�, 59

∇, 17

〈X,Y 〉, 70←−α , 73

u • v, 474

A(f), 421

Bn(C∗), 544BI, 177

...C, 18

Cf , 130

Cn(X;G), 542

Cop, 10

cat(X), 251

Celln(X), 542

Chain, 543

χ(X), 573

χ(X), 573

clA(X), 239

colim, 56

connP (X), 439

conn(X), 279

CW∗, 462CX, 67

Cyl, 90

Dn, 45

Δn, 259

Δn, 295

ΔQ, 32

deg(f), 414

Δ, 16

Ef , 112

η, 556

ExtnR( ? , ? ), 802

F∗, 462[f ], 70

f , 74

〈f〉, 70f∗, 10f∗, 11

G(V ), 230

Grk(Fn+k), 231

h, 502

H(α), 446

H (α), 256

Hn, 467

Hn, 460

Hn, 506

hn, 461

hn, 500

hn, 467

hn, 500

Hn(C∗), 544Hσ(α), 446

hT , 80

in, 18

821

822 Index of Notation

Jn(X), 377J(X), 377

K(G,n), 385

LA(f), 238LA(f), 238Λ(X), 259lim, 56lim1, 247

Mf , 111, 125Mn×n(F), 230map∗(X,Y ), 61map◦(X,Y ), 55

nil(I), 555

ΩM , 374Ωp, 109ΩX, 67

P d, 527PM , 374Ph(X,Y ), 245π0, 71πn, 87pr, 15, 18

Sn, 45sn, 502ΣX, 66sd(K), 298Sets∗, 19Σ◦, 46SP∞(X), 449SPn(X), 449

Sqd, 527

T , 53, 54T∗, 53, 54, 60T◦, 53, 54T(2), 58T k(X), 251Tz,V , 233ΘX , 246

Vk(Fn+k), 232

W∗, 171

x , 73

Zn(C∗), 544

Index

∞-connected, 279

∞-equivalence, 278

3× 3 diagram, 166, 173

homotopy colimit, 166

homotopy limit, 173

A-cone decomposition, 238

A-module, 796

A-morphism, 94

abelianization, 388

abstract simplicial complex, 296, 342

action

naturality, 206

acyclic, 700, 799

acyclic cofibration, 121, 263

acyclic fibration, 121, 263

Adams operations, 697

Adem relations, 520, 533

adjoint, 26, 54

loop space, 207

suspension, 207

adjoint functors

and cofibrant diagrams, 151

cofibrations, 122

fibrations, 122

adjunction

counit, 28

unit, 28

admissible, 537, 538, 592, 705

for h∗, 592admissible map, 204

admissible path

for p, 204

admissible trivialization, 592

algebra

graded, 796

algebraic closure, 4

algebraic Kunneth map, 803

algebraic loop, 287

amalgamated free product, 291

antipodal map, 415, 472

antipode, 52

associated graded, 610

associative, 227, 475

augmentation, 780, 797

augmentation ideal, 780, 797

augmented algebra, 797

barycenter, 297

barycentric subsimplex, 297

base, 108

based maps, 60

basepoint, 19, 60

change, 211

of I, 67, 70

bidegree, 559, 587, 612, 653

bigraded, 612

bilinear, 790

Blakers-Massey theorem, 398

n-dimensional, 406

cellular, 769

geometric version, 401

mod Q, 445

Bockstein, 520, 567

map, 517

operation, 517

Borel construction, 220, 335

Borsuk-Ulam theorem, 566

Bott map, 692

Bott-Samelson theorem, 577

boundaries, 544, 798

823

824 Index

boundary, 295boundary map, 241, 543, 798

product, 243bounded below, 795Bousfield equivalent, 725Brown Representability Theorem, 490

bundle isomorphism, 335bundle map, 324, 334bundle of groups, 625

Cartan formula, 525, 533category, 5

discrete, 33Lusternik-Schnirelmann, 250, 555pointed, 18

category of maps, 91cell

closed, 47

open, 47cellular action, 217Cellular Approximation Theorem, 299

1-dimensional, 74, 78cellular chain complex

product space, 549cellular cochain complex, 543

cellular inequality, 759, 764cellular map, 48, 78, 299cellular replacement, 340

existence, 355uniqueness, 354

chain algebra, 550, 653chain complex, 543, 798

chain map, 544, 798change of fiber, 335characteristic map, 47characterization

acyclic cofibrations, 121, 129acyclic fibrations, 121, 129

cofibrations, 121, 129fibrations, 121, 129

Clark, 18class, 5classified, 370classifying space, 177, 370closed

under homotopy limits, 409under smash, 762under weak homotopy equivalence, 409under wedge, 762

closed class, 759, 764closed cofibration, 104closed under suspension, 238, 762

closed under wedges, 238clutching, 603

map, 603co-H-space, 228

retract, 228

suspension, 359coagumentation, 722coalgebra, 803

homomorphism, 804coboundary map, 543cocartesian, 196cochain complex, 543, 798cocommutative, 25, 804cocomplete, 262cocycles, 544coefficient group, 462, 500coefficient transformations, 516coefficients, 427, 461coend, 346coequalizer, 38cofiber, 130

homotopy invariance, 163homotopy pushout, 184

of a composite, 216of contractible subspace, 132of projection map, 223of trivial map, 194standard, 130, 131unpointed, 131

cofiber sequence, 131, 460exactness, 131long sequence, 201Mayer-Vietoris, 210

cofibrant, 101, 147, 148, 264cofibrant diagram

characterization, 158cofibrant replacement, 150

prepushout diagrams, 148cofibration, 99, 100, 262

acyclic, 121adjoint pair, 129closed, 104composition, 100converting to, 112exact sequence, 201, 202exactness, 130

mapping space, 105, 134pointed, 122product, 106pushout, 103retract, 121trivial, 121

cofibrationsinclusion maps, 101pushouts, 103

cofinalsubdiagram, 40

cofinality, 809cogroup object, 23

cocommutative, 25Cohen-Moore-Neisendorfer lemma, 215coherent homotopies, 92

Index 825

cohomology, 460BZ/p, 520algebraic, 544coefficients, 460exact sequence of fibration, 583James construction, 480of a product, 488of Moore space, 470projective space, 470reduced, 467telescope, 465unitary group, 597unreduced, 467with local coefficients, 625

cohomology algebraprojective space, 480

cohomology class, 460cohomology group, 544

cohomology operation, 516Moore space, 567

cohomology theory, 459, 461extraordinary, 462multiplictative, 586ordinary, 462

colimit, 15, 343× 3 diagram, 41I × J diagram, 42commute with adjoint, 35naturality, 34pushout, 37

collapse, 654commutative, 4, 25, 227, 475, 792

on the nose, 82graded, 419, 796homotopy, 81strictly, 82

commutative differential graded algebra,739

commutativitygraded, 418

commutator, 388

commutator subgroup, 388compact, 97compact subspace

of CW complex, 50compact-open topology, 54compactly generated, 55compactness

and CW complexes, 49comparison

cofiber to suspension of fiber, 397cofiber to base, 395homotopy pullback to homotopy

pushout, 394homotopy pushout to homotopy

pullback, 398comparison map, 51, 394, 397, 398, 463

compatible, 136, 137complement, 231complete, 262completion, 735component, 795compressible, 635comultiplication, 804concatenate, 374concatenation, 74

infinite, 277of homotopies, 73of paths, 72rigid, 374

concentrated, 462, 795cone, 67

reduced, 67cone decomposition, 237

cellular, 470

cone filtration, 608, 613cone length

A-cone length, 238Lusternik-Schnirelmann category, 251of a map, 238of a space, 239

connected, 797connected cover, 362connected sum, 572connective, 737connectivity, 279

cohomology, 464homology, 502mod Q, 443mod C, 716of a space, 279of half-smash product, 358of join, 358of product, 358of smash product, 358of smash product of maps, 359of suspension, 358, 383

constant diagram, 159

constant homotopy, 74continuity

of map from a CW complex, 49contractible, 82

smash product, 581contraficients, 427contravariant functor, 9converges, 654converting to

fibration, 111, 113convex, 75convex combination, 75coordinate map, 421, 807coproduct, 17, 38covariant functor, 8covering, 326

826 Index

covering homotopy extension property, 120criticize this argument, 48cube diagram, 405Cube Theorem

First, 314Second, 321

cup length, 555cup product, 476currying, 55CW n-ad, 97CW complex, 47

compact, 49

finite, 49finite-dimensional, 47induction, 276infinite-dimensional, 47of finite type, 637pointed, 61

CW decompositions, 48

CW induction, 276CW pair, 60CW product, 51CW replacement, 353

uniqueness, 354CW structures, 48

cycles, 544, 798cylinder, 67, 92, 95

external, 95internal, 95reduced, 67standard, 90

cylinder object, 90, 266

de Polignac’s formula, 640, 755deck transformation, 329decomposables, 577, 797deformation retract, 76degeneracy maps, 345

degenerate, 345degree, 414, 452, 497, 516, 518, 537, 653,

795of twist map, 418

derivation, 595derived, 652deviation, 747

diagonal action, 219diagonal functor, 32diagonal indexing, 612diagonal map, 16, 476

reduced, 476diagram, 3, 30diagram category, 30

diagram homotopy equivalence, 145differential, 543, 798dimension, 47, 296, 653, 795

of a cohomology class, 460dimension zero, 47

direct category, 176direct limit, 39direct sum

of graded modules, 796discrete category, 33disk, 45

disk bundle, 677distinguished, 348divided polynomial algebra, 794

graded, 796divisible groups, 802domain, 4, 34

domain type, 15double factorization, 113double mapping cylinder, 161dual, 37duality, 15, 17, 107, 188, 202, 209, 223, 228,

248, 402and homotopy, 89

E-equivalence, 722E-local, 722Eckmann-Hilton argument, 229Eckmann-Hilton duality, 265edge homomorphism, 665

bottom, 665EHP sequence, 742, 743Eilenberg-MacLane space, 384, 459, 506,

516existence, 386maps between, 385of type (G,n), 384

symmetric product, 451uniqueness, 386

Eilenberg-MacLane spacegeneralized, 451

Eilenberg-Steenrod axioms, 460, 461elementary symmetric function, 535elementary symmetric polynomial, 806

endpoint, 374enough projectives, 799equalizer, 37equivalence, 6equivalence of categories, 14equivalent maps, 27

equivariant, 216essential, 82Euler characteristic, 573

reduced, 573Euler class, 675evaluation map, 54, 374evenly covered, 326

exact, 130exact couple, 616, 617, 650

derived, 652exact functor, 795exact sequence, 794

Index 827

split, 517exactness

unreduced homology, 500excess, 537, 538, 705Excision Axiom, 461exhaustive, 612

exponent, 636, 745exponential law, 54, 55, 57, 65, 81, 84, 791

for pairs, 59pointed, 62

Ext-p-complete, 735extend

to cone, 82extension, 100

functor, 156extension problem, 14exterior algebra, 793, 797

graded, 796exterior product

homology, 508with respect to a pairing, 632

external cohomology productfor cartesian products, 475

external product, 427, 473cohomology, 474

F-algebra, 553f -equivalence, 724f -local, 724face, 342face map, 342factors through, 4

fat wedge, 251fiber, 108, 131, 337

homotopy, 131of a composite, 216of trivial map, 194suspension, 453

fiber bundle, 334fiber sequence, 132

exactness, 132homotopy pullback, 189long sequence, 202Mayer-Vietoris, 210

fiber-cofiber construction, 395

fibrant, 110, 168, 264pointed diagram, 171

fibrant replacement, 168, 171fibration, 99, 108, 262, 325

adjoint pair, 129converting to, 113evaluation, 109

mapping space, 133, 135orientable, 203path space, 109pointed, 123product, 110

pullback, 110retract, 121sequence, 132trivial, 110unpointed, 123

filtered R-algebra, 609

filtered map, 608filtered space, 607filtration, 605, 607, 613

ascending, 607, 609descending, 609finite, 612

of an element, 613filtration degree, 612filtration quotients, 605, 610finite, 30finite type, 807finite-type

map, 422

map of wedges, 469wedge, 762

Five Lemma, 795fixed point, 417fixed points

space of, 217folding map, 17

forgetful functor, 10free, 218free abelian group, 27free graded commutative algebra, 797free loop space, 141, 259free module, 790

free product, 290free resolution, 386, 425Freudenthal Suspension Theorem, 383

dual, 403in dimension 1, 388

functor, 8contravariant, 9

covariant, 8extension, 156forgetful, 10homotopy, 80lifting, 156preserve homotopy colimit, 174preserve homotopy limit, 174

represented, 11fundamental class, 490fundamental group, 9fundamental groupoid, 205Fundamental Lifting Property

pointed, 128

unpointed, 118Fundamental Theorem of Algebra, 419, 554

G-CW complex, 217G-CW replacement, 219

828 Index

G-maps, 216G-space, 216Ganea condition, 257, 713, 741Ganea construction, 254, 395, 444, 665, 666Ganea’s conjecture, 257GEM, 451

general position, 295generalized CW complex, 244generating complex, 599, 690generating function, 575generator, 294

Hn(Sn;Z), 502

generic, 734geometric realization, 297, 342graded, 795

R-algebra, 796R-module, 795algebra, 796divided polynomial algebra, 796

exterior algebra, 796module, 796polynomial algebra, 796tensor algebra, 796

graded R-algebra, 475graded abelian group, 460graded commutative, 419, 796

algebra, 796algebra, free, 797

graded commutativity, 428graded module

suspension, 795graded Serre class, 716

graded twist map, 796Gram-Schmidt, 332graph, 133, 216Grassmannian, 231group

as a category, 6group object, 21

commutative, 25Gysin sequence, 675, 682

h∗-equivalence, 734h∗-local, 734H-map, 226

fiber, 227H-space, 226

characterization, 226product, 227projective space, 553retract, 226strict unit, 226

Hahn-Mazurkowicz, 75half-smash product, 64

with suspension, 140halo, 319Hawaiian earring, 371

HELP, 282Hilton-Milnor theorem, 390holonomy, 205homogeneous, 460homogeneous coordinates, 52homology, 384, 544

algebraic, 544, 798cellular structure, 504connectivity, 502Moore space, 567spheres, 502unitary group, 598

homology decompositions, 569homology group, 544homology module, 798homology suspension, 577, 578homology theory, 500homomorphism, 24, 226, 454

of modules, 789homotopic, 70, 72, 74, 267homotopy, 70, 72, 92, 95, 145, 267

cell-by-cell construction, 56constant, 74free, 70, 72Moore, 376of diagram morphisms, 145of homotopies, 74of paths, 72pointed, 70reverse, 74straight-line, 75under A, 95unpointed, 70

homotopy category, 261, 268pointed, 80unpointed, 80

homotopy classes, 70homotopy cocartesian cube, 405homotopy cocartesian square, 405homotopy colimit, 151, 176, 271

pointed vs. unpointed, 214

Quillen-Serre, 341standard, 155

homotopy commutative, 81homotopy equivalence, 92, 96

of diagrams, 144, 145pointwise, 93, 144

homotopy equivalent, 81, 267maps, 93

Homotopy Extension Lifting Property, 282homotopy extension property, 100homotopy fiber, 131homotopy fixed point, 218homotopy functor, 71homotopy groups, 87

exact sequence of cofibration, 409finite, 570

Index 829

finitely generated, 570of a pair, 401of a triad, 402smash product, 427spheres, 383, 384wedges of spheres, 383

homotopy idempotent functors, 724homotopy invariant, 267homotopy lifting

relative, 119homotopy lifting property, 107, 108, 203

pointed, 123relative, 120weak, 308

homotopy limit, 169Quillen-Serre, 341

homotopy orbit, 218homotopy orbit space, 220

homotopy pullbackcommutes with mapping spaces, 195completion, 190composition of squares, 191fiber sequence, 189homotopy equivalence, 190loop space, 189Quillen-Serre, 341recognition, 190trivial map, 194

homotopy pullback square, 189characterization, 195composition, 191product, 193Quillen-Serre, 341

homotopy pushout, 161, 435cofiber, 184commutes with products, 194composition of squares, 191homotopy equivalence, 185iterated, 192Quillen-Serre, 341suspension, 184

trivial map, 194homotopy pushout square, 184

characterization, 195completion, 187composition, 191Quillen-Serre, 341Serre, 405wedge, 193

homotopy theory, 80homotopy type, 81, 82

of a graph, 133homotopy under, 92Hopf algebra, 512, 513, 805

Steenrod algebra, 584Hopf construction, 558Hopf fibration

real, 330Hopf invariant, 256, 446, 557, 561, 713

and reduced diagonal, 560and linking number, 556

Berstein-Hilton, 446James Hopf, 562Lusternik-Schnirelmann category, 445

Hopf map, 326, 556

Hopf set, 446, 561Hopf’s theorem, 587Hurewicz model structure, 264Hurewicz map, 502, 569, 635Hurewicz theorem, 503, 507

hypoabelian group, 388

idempotent, 196idempotent functor, 722inclusion map, 4incompressible, 635

indecomposables, 797induced, 35induced map, 56infinite loop space, 493, 507, 580

infinite symmetric product, 448infinite unitary group, 686initial object, 18initial point, 374injective, 801

injective resolution, 801interval

basepoint, 67, 70inverse limit, 39

isomorphism, 488iterated cofiber, 215iterated fiber, 215, 406

J. H. C. Whitehead theorem, 286James construction, 377, 379, 445

cohomology, 480models loops on ΣX, 379multiplication, 378universal properties, 378weak equivalence, 378

James Hopf invariant, 562James splitting, 389, 562join, 221

pointed, 221

unpointed, 221

k-cartesian, 406k-frame, 232Kunneth theorem

symmetric product, 576

topological, 549Klein bottle, 423, 593Kudo Transgression Theorem, 707Kunneth map, 474, 513

830 Index

algebraic, 803homology, 508

Kunneth theorem, 487, 488algebraic, 803homology, 509

Lebesgue Number Lemma, 77left adjoint, 26left derived functor, 269left homotopic, 266left homotopy, 90

left lifting property, 121Leibniz rule, 653length, 237Leray-Hirsch theorem, 674Leray-Serre spectral sequence, 74

relative, 662lifting

functor, 156lifting function, 109

dual, 103uniqueness, 140

lifting problem, 14lim1

cohomology, 466

limit, 15, 31, 323× 3 diagram, 42commute with adjoint, 35naturality, 34pullback, 36

limit ordinal, 809

linear, 75, 295, 343local section, 330localization, 268localized at, 723localized away, 723locally finite, 370locally trivial bundle, 324

long A-cone decomposition, 239long cofiber sequence, 200long fiber sequence, 202loop map

homotopy pullback, 249loop operation, 578loop space, 67, 84, 577, 738

free, 141group object in T∗, 87homotopy pullback, 189

Lusternik-Schnirelmann category, 251, 252,555, 713

cone length, 251Ganea criterion, 254

homotopy invariance, 251homotopy pushout, 253Hopf invariant, 445Lusternik-Schnirelmann cover, 250, 555mapping cone, 256

of a map, 252of CW complexes, 253product, 256

manifold, 497

mapfinite type, 807pointed, 60

map under, 94mapping cone, 130

unpointed, 130mapping cylinder, 111

in map(T◦), 115pointed, 125

mapping space, 54and colimits, 57and limits, 57cofibration, 105, 134

fibration, 133, 135pointed, 61well-pointed, 126

mapping torus, 179Mather cube, 305, 395, 594, 627matrix, 20, 421Mayer-Vietoris

cofiber sequence, 210exact sequence, 465, 501fiber sequence, 210

measured path, 374Milnor sign convention, 413, 418minimal, 739, 741minimal model, 740

Mislin genus, 734Mittag-Leffler, 775model category, 89, 99, 261, 262module, 789

graded, 796moment, 537

monoid object, 21, 227monster, 423, 499, 578Moore homotopy, 376Moore loops, 374Moore path, 374

based, 374Moore space, 425

cohomology of, 470cohomology operation, 567homology, 567maps between, 568

Moore suspension, 379morphism, 5

in pointed category, 20

matrix representation, 807trivial, 19

multiplication, 226multiplicative, 476, 608multiplicative cohomology theory, 586

Index 831

multiplicity, 415

n-ad, 97n-connected, 279

map, 279n-cube diagram, 196

n-equivalence, 277at P, 443mod Q, 443pointwise, 301

n-skeleton, 354natural, 12

isomorphism, 12natural grading, 612natural transformation, 11

of cohomology theories, 462of represented functors, 12

NDR pair, 104negative, 506

neighborhood deformation retract, 104strong, 104

Neisendorfer localization, 786nerve, 176

Newton polynomial, 697, 807nilpotency, 555nonzero, 417norm, 553normed algebra, 553

north pole, 46northern hemisphere, 46nose

on the, 82

nullhomotopic, 82nullhomotopy, 82numerable, 309numerable bundles, 324

objects, 5

obstruction, 360, 642obstruction theory, 360, 385odd map, 565on the nose, 82open cells, 47

opposite category, 10, 265opposite-simple category, 169orbit space, 217ordinary cohomology theory, 462

ordinary homology theory, 500existence, 507

orientable, 592fibration, 203

orthogonal group, 231, 603over B, 94

p-adic integers, 735p-completion, 735P-connectivity, 438, 439

P-group, 438P-injective, 443P-local, 730P-localization, 724P-surjective, 443pair, 58, 401, 461

‘product’, 59mapping space, 59

pairing, 427homology with cohomology, 510

parametrized concatenation, 109partition of unity, 309path, 70

reverse, 73path homotopy, 72path object, 89, 266

standard, 90path space, 83, 89

perfect, 388phantom map, 245, 582

existence, 582nonexistence, 582

πn(Sn), 383PID, 487, 801piecewise linear, 75, 297Poincare series, 574pointed category, 18pointed homotopy, 72pointed map, 60

composition, 20pointed model category, 263, 264pointed set, 19, 71pointed space, 60pointwise cofibration, 167pointwise equivalence

in hT , 93pointwise equivalent, 146pointwise fibration, 147, 270pointwise homotopy equivalence, 93, 94, 96

in hT , 146pointwise weak equivalence, 270

polyhedron, 296polynomial algebra, 793

graded, 796Pontrjagin algebra, 512Postnikov approximation, 361Postnikov section

cellular structure, 361existence, 361uniqueness, 361

prepullback, 36prepushout, 37presentation, 294prespectra, 580primitive, 510, 804principal bundle

associated, 335

832 Index

principal cofibration, 237principal ideal domain, 487, 801product, 15, 17, 18

as functor, 16

flat, 223infinite, 807infinite weak product, 376of CW complexes, 50of group objects, 26

of groups, 17of many objects, 18of mapping cones, 224of maps, 16of sets, 17

pointed spaces, 62product rule, 653projections, 15projective, 790, 799projective resolution, 800

projective space, 51cohomology, 470cohomology algebra, 480CW decomposition, 53CW structure, 53

diagonal map, 482fiber bundle, 325infinite-dimensional, 52truncated, 586

pullback, 36

composition of, 41fibration, 110product, 65square, 36

punctured, 196

punctured n-cube, 405pure quaternion, 420pushout, 37

composition of, 40of equivalence, 40

square, 37wedge, 65

quadratic part, 740quasi-isomorphism, 799quasifibration, 347quasiregular, 743

quaternions, 419Quillen adjunction, 270Quillen equivalence, 268, 270quotient map, 49

R-algebra, 792graded, 796

R-modulegraded, 795

rational homotopy theory, 797rationalization, 733

rationally elliptic, 741rationally hyperbolic, 741reduced p powers, 527reduced diagonal, 260, 476reduced product type, 454regular prime, 743

regular value, 415, 497relation, 294relative CW complex, 60relative Sullivan algebra, 739reparametrization, 73represented cohomology, 460

represented functor, 11represents, 614resolving class, 409respects homotopy, 71retract, 7

of a map, 7reverse, 74

reverse path, 73Riemannian metric, 677right adjoint, 26right homotopic, 266right homotopy, 90right lifting property, 121rigid concatenation, 374

rigidification, 178root, 156RPT complex, 454RPT model, 455

Schubert symbol, 237

Second Cube Theorem, 398section, 203

Ωf , 207Serre class, 714Serre cofibrations, 336Serre exact sequence, 583

Serre fibration, 336, 337sequence, 337

Serre filtration, 600, 601, 623Serre model structure, 264shape, 29, 30shape diagram, 29shear map, 229, 287

shift map, 39, 246, 465, 501short exact sequence, 794shuffle product, 688simple category, 155simple system, 704, 708

p-simple, 708simplex, 295, 296

boundary, 295standard, 295

simplex category, 348simplicial complex, 296, 342

abstract, finite, 296

Index 833

structure, 296simplicial map, 343simplicial object, 346simplicial set, 345simplicial structure, 342simply-connected, 88, 279, 739singular chain complex, 551singular extension, 494, 501singular simplex, 342singular simplical set, 346skeleta, 47

n-equivalence, 358naturality, 354

skeleton, 47, 572small diagrams, 30small object argument, 165smash product, 61

Eilenberg-MacLane spaces, 429

bilinear, 427contractible, 581homotopy groups, 427of CW complexes, 62

smash product pairing, 427southern hemisphere, 46space, 54

pointed, 60, 62spaces over, 94spaces under, 94spanning tree, 133special orthogonal group, 231special unitary group, 231, 598spectra, 580spectral sequence, 645, 647

convergence, 657first quadrant, 660free monogenic, 703homomorphism, 651of algebras, 653

spectrum, 580sphere, 45

abelian cogroup in T∗, 86as pushout, 46cellular decompositions, 49homotopy groups, 383infinite-dimensional, 52

sphere bundle, 677spherical fibrations, 675spherically resolvable, 761split, 223, 795split exact sequence, 517, 795stabilizes, 505stable cohomology operation, 518

additive, 519stable homotopy groups, 389

homology theory, 505stable homotopy sets

exact sequence, 410

stable operationrecognition, 518

stable phenomena, 389stable range, 389stably equivalent, 696standard filtration, 608, 609

of a smash product, 609standard form, 292Steenrod algebra, 519, 777

basis, 537Hopf algebra, 584

Steenrod operations, 527main properties, 533

Steenrod reduced powers, 525Steenrod square, 527

indecomposable, 536stereographic projection, 46Stiefel manifold, 232, 599

strictly commutative, 82Strøm model structure, 264strong deformation retract, 76, 104strong homotopy pullback square, 189, 307strong homotopy pushout square, 186

recognition, 186strong resolving classes, 759, 760strongly closed classes, 759, 760strongly cocartesian, 196strongly homotopy cocartesian, 406structure, 296, 385

simplicial complex, 296structure group, 334subcomplex, 47, 297subquotient, 606subsimplex, 296, 342successor ordinal, 809Sullivan algebra, 739Sullivan conjecture, 785sum, 17, 18, 38

infinite, 807support, 309surface, 572

suspension, 46, 66, 84, 461, 487, 738, 796as natural transformation, 379cogroup object in T∗, 86connectivity, 383fiber, 453functor, 66graded module, 795homotopy pushout, 184of a sphere, 67reduced, 66unreduced, 46

suspension map, 379symmetric polynomial, 535, 683, 806symmetric product, 576

Eilenberg-MacLane space, 451homology, 576

834 Index

Kunneth theorem, 576symmetric square, 259symplectic group, 231, 603system of local coefficients, 625

tab-and-glue, 311

target, 4, 34target type, 15telescope

cohomology, 465telescope diagram, 39, 163

tensor algebra, 793graded, 796universal property, 793

tensor product, 427, 790graded, 474

of chain complexes, 799terminal object, 18Thom class, 678Thom Isomorphism Theorem, 678Thom space, 676, 677

topological space, 54pointed, 62

total degree, 612, 653total dimension, 612total left derived functor, 269

total space, 108total square, 536tower, 39, 173track, 138transgression, 667, 668

transgressive, 667transgressive pair, 667transition functions, 333tree, 133tree-like, 159

triad, 402trivial, 82, 263trivial bundle, 323trivial cofibration, 121trivial fibration, 121, 591

trivial homotopy, 74trivial morphism, 19trivialization, 591truncated polynomial algebras, 793truncated projective space, 586

twist map, 25, 418, 792two-cones, 602type, 743

unique, 53unital, 475

unitary group, 231universal bundle

existence, 366universal coefficients decomposition, 634,

700

Universal Coefficients Theorem, 486, 660universal cover, 328universal example, 102, 124, 225, 523, 584,

790, 797universal phantom map, 246

universal problem, 4universal property, 15unreduced homology

exactness, 500

unreduced homology theory, 500unstable algebra, 534unstable conditions, 533, 534unstable module, 534

unstable relations, 525

vanishes, 360vector field, 417versal, 246vertices, 295

very nice, 780

W-local, 722Wang cofiber sequence, 594Wang exact sequence, 594Wang sequence, 593

cohomology, 594, 595

homology, 595weak category, 260, 561weak equivalence, 262, 278

and cohomology, 464

Weak Equivalence Axiom, 461, 463homology, 501

weak fibration, 306strong homotopy pullback, 307

weak Hausdorff, 55weak homotopy equivalence, 278weak lifting function, 308weak product, 421

infinite, 376

weakly contractible, 279weakly equivalent, 263wedge, 19, 61, 463Wedge Axiom, 463

homology, 501well-pointed, 124, 188

cofibration, 126fibration, 126

mapping space, 126Whitehead exact sequence, 583Whitehead product

generalized, 225H-space, 226

Whitney sum, 695wreath product, 531

X -cellular, 764X-null, 785

Index 835

Y-resolvable, 761Yoneda lemma, 13, 379

Zabrodsky lemma, 411, 765Zabrodsky mixing, 733

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GSM/127

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The core of classical homotopy theory is a body of ideas and theorems that emerged in the 1950s and was later largely codifi ed in the notion of a model category. This core includes the notions of fi bration and cofi bration; CW complexes; long fi ber and cofi ber sequences; loop spaces and suspensions; and so on. Brown’s representability theorems show that homology and cohomology are also contained in classical homotopy theory.

This text develops classical homotopy theory from a modern point of view, meaning that the exposition is informed by the theory of model categories and that homotopy limits and colimits play central roles. The exposition is guided by the principle that it is generally preferable to prove topo-logical results using topology (rather than algebra). The language and basic theory of homotopy limits and colimits make it possible to penetrate deep into the subject with just the rudiments of algebra. The text does reach advanced territory, including the Steenrod algebra, Bott periodicity, localization, the Exponent Theorem of Cohen, Moore, and Neisendorfer, and Miller’s Theorem on the Sullivan Conjecture. Thus the reader is given the tools needed to understand and participate in research at (part of) the current frontier of homotopy theory. Proofs are not provided outright. Rather, they are presented in the form of directed problem sets. To the expert, these read as terse proofs; to novices they are challenges that draw them in and help them to thoroughly understand the arguments.

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Modern Classical Homotopy Theory · 2019. 2. 12. · Modern Classical Homotopy Theory Graduate Studies in Mathematics Volume 127. Modern Classical ... The core of the subject I’m