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Modern Classical Homotopy Theory Modern Classical Homotopy Theory Jeffrey Strom Graduate Studies in Mathematics Volume 127 American Mathematical Society http://dx.doi.org/10.1090/gsm/127 Modern Classical Homotopy Theory Modern Classical Homotopy Theory Jeffrey Strom Graduate Studies in Mathematics Volume 127 American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE David Cox (Chair) Rafe Mazzeo Martin Scharlemann Gigliola Staffilani 2010 Mathematics Subject Classification. Primary 55Nxx, 55Pxx, 55Qxx, 55Sxx, 55Uxx. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-127 Library of Congress Cataloging-in-Publication Data 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. QA612.7.S77 2011 514.24—dc23 2011019677 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. c 2011 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 161514131211 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 §6.9. Additional Topics, Problems and Projects 178 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 §7.6. Additional Topics, Problems and Projects 196 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 §9.8. Additional Problems and Projects 258 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 §11.5. Additional Problems 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 §12.6. Additional Problems and Projects 303 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 §15.8. Additional Problems and Projects 348 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 §16.6. Additional Problems and Projects 371 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-Mac Lane 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 §18.6. Additional Topics, Problems and Projects 409 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-Mac Lane 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 §20.6. Additional Topics, Problems and Projects 452 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 K¨unneth 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 K¨unneth 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.
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