Modern classical homotopy theory

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1. Verfasser:	Strom, Jeffrey 1969- (VerfasserIn)
Format:	Buch
Sprache:	English
Veröffentlicht:	Providence, R.I. American Mathematical Society 2011
Schriftenreihe:	Graduate studies in mathematics 127
Schlagworte:	Homotopy theory Algebraic topology / Homology and cohomology theories / Homology and cohomology theories Algebraic topology / Homotopy theory / Homotopy theory Algebraic topology / Homotopy groups / Homotopy groups Algebraic topology / Operations and obstructions / Operations and obstructions Algebraic topology / Applied homological algebra and category theory / Applied homological algebra and category theory Homotopietheorie
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adam_text	Titel: Modern classical homotopy theory Autor: Strom, Jeffrey Jahr: 2011 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 vu Contents vin ____________________-------------------— 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 x 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 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 ƒ 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 5 - Sn 414 §19.2. Some Applications of Degree 417 §19.3. Maps Between Wedges of Spheres 421 Contents xa_______________________________ §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 Runneth 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 Kiinneth 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 Cohomology Operations 521 §23.4. The Steenrod Reduced Powers 525 §23.5. The Ádem 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 Kiinneth 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 tt(X) and H(X) 570 §25.8. Surfaces 572 §25.9. Euler Characteristic 573 §25.10. The Kiinneth Theorem via Symmetric Products 576 §25.11. The Homology Algebra of Q.Y,X 576 §25.12. The Adjoint x of idnx 577 §25.13. Some Algebraic Topology of Fibrations 579 §25.14. A Glimpse of Spectra 580 §25.15. A Variety of Topics 581 §25.16. Additional Problems and Projects 585 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 Contents §26.4. Some Algebraic Topology of Unitary Groups 597 §26.5. The Serre Filtration 600 §26.6. Additional Topics, Problems and Projects 603 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{*{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 §28.5. Additional Problems 633 Chapter 29. Application: Incompressibility 635 §29.1. Homology of Eilenberg-Mac Lane Spaces 636 §29.2. Reduction to Theorem 29.1 636 §29.3. Proof of Theorem 29.2 638 §29.4. Consequences of Theorem 29.1 641 §29.5. Additional Problems and Projects 642 Chapter 30. The Spectral Sequence of a Filtered Space 645 §30.1. Approximating Gts Hn{X) by Epn{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 §31.5. Additional Problems and Projects 679 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 QSU(n) 689 §32.5. Generating Complexes for ÜSU and BU 690 §32.6. The Bott Periodicity Theorem 692 §32.7. if-Theory 695 §32.8. Additional Problems and Projects 698 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 §33.9. Additional Problems and Projects 716 Part 7. Vistas Chapter 34. Localization and Completion 721 §34.1. Localization and Idempotent Functors 722 §34.2. Proof of Theorem 34.5 726 §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 a 747 §35.2. Spectral Sequence Computations 751 §35.3. The Map 7 754 §35.4. Proof of Theorem 35.3 754 §35.5. Nearly Trivial Maps 756 xvi Contents Chapter 36. Classes of Spaces §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.l. 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 gH Index of Notation g2i Index 823
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spellingShingle	Strom, Jeffrey 1969- Modern classical homotopy theory Graduate studies in mathematics Homotopy theory Algebraic topology / Homology and cohomology theories / Homology and cohomology theories msc Algebraic topology / Homotopy theory / Homotopy theory msc Algebraic topology / Homotopy groups / Homotopy groups msc Algebraic topology / Operations and obstructions / Operations and obstructions msc Algebraic topology / Applied homological algebra and category theory / Applied homological algebra and category theory msc Homotopietheorie (DE-588)4128142-1 gnd
subject_GND	(DE-588)4128142-1
title	Modern classical homotopy theory
title_auth	Modern classical homotopy theory
title_exact_search	Modern classical homotopy theory
title_full	Modern classical homotopy theory Jeffrey Strom
title_fullStr	Modern classical homotopy theory Jeffrey Strom
title_full_unstemmed	Modern classical homotopy theory Jeffrey Strom
title_short	Modern classical homotopy theory
title_sort	modern classical homotopy theory
topic	Homotopy theory Algebraic topology / Homology and cohomology theories / Homology and cohomology theories msc Algebraic topology / Homotopy theory / Homotopy theory msc Algebraic topology / Homotopy groups / Homotopy groups msc Algebraic topology / Operations and obstructions / Operations and obstructions msc Algebraic topology / Applied homological algebra and category theory / Applied homological algebra and category theory msc Homotopietheorie (DE-588)4128142-1 gnd
topic_facet	Homotopy theory Algebraic topology / Homology and cohomology theories / Homology and cohomology theories Algebraic topology / Homotopy theory / Homotopy theory Algebraic topology / Homotopy groups / Homotopy groups Algebraic topology / Operations and obstructions / Operations and obstructions Algebraic topology / Applied homological algebra and category theory / Applied homological algebra and category theory Homotopietheorie
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work_keys_str_mv	AT stromjeffrey modernclassicalhomotopytheory

Modern classical homotopy theory

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