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A Concise Course in Algebraic Topology

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A Concise Course in Algebraic Topology J. P. May Contents Introduction 1 Chapter 1. The fundamental group and some of its applications 5 1. What is algebraic topology? 5 2. The fundamental group 6 3. Dependence on the basepoint 7 4. Homotopy invariance 7 1 5. Calculations: π1(R) = 0 and π1(S )= Z 8 6. The Brouwer fixed point theorem 10 7. The fundamental theorem of algebra 10 Chapter2. CategoricallanguageandthevanKampentheorem 13 1. Categories 13 2. Functors 13 3. Natural transformations 14 4. Homotopy categories and homotopy equivalences 14 5. The fundamental groupoid 15 6. Limits and colimits 16 7. The van Kampen theorem 17 8. ExamplesofthevanKampentheorem 19 Chapter 3. Covering spaces 21 1. The definition of covering spaces 21 2. The unique path lifting property 22 3. Coverings of groupoids 22 4. Group actions and orbit categories 24 5. Theclassificationofcoveringsofgroupoids 25 6. Theconstructionofcoveringsofgroupoids 27 7. Theclassificationofcoveringsofspaces 28 8. Theconstructionofcoveringsofspaces 30 Chapter 4. Graphs 35 1. The definition of graphs 35 2. Edge paths and trees 35 3. The homotopy types of graphs 36 4. CoversofgraphsandEulercharacteristics 37 5. Applications to groups 37 Chapter 5. Compactly generated spaces 39 1. The definition of compactly generated spaces 39 2. Thecategoryofcompactlygeneratedspaces 40 v vi CONTENTS Chapter 6. Cofibrations 43 1. The definition of cofibrations 43 2. Mapping cylinders and cofibrations 44 3. Replacing maps by cofibrations 45 4. Acriterionforamaptobeacofibration 45 5. Cofiber homotopy equivalence 46 Chapter 7. Fibrations 49 1. The definition of fibrations 49 2. Path lifting functions and fibrations 49 3. Replacing maps by fibrations 50 4. Acriterionforamaptobeafibration 51 5. Fiber homotopy equivalence 52 6. Change of fiber 53 Chapter8. Basedcofiberandfibersequences 57 1. Based homotopy classes of maps 57 2. Cones, suspensions, paths, loops 57 3. Based cofibrations 58 4. Cofiber sequences 59 5. Based fibrations 61 6. Fiber sequences 61 7. Connectionsbetweencofiberandfibersequences 63 Chapter 9. Higher homotopy groups 65 1. The definition of homotopy groups 65 2. Longexactsequencesassociatedtopairs 65 3. Longexactsequencesassociatedtofibrations 66 4. A few calculations 66 5. Change of basepoint 68 6. n-Equivalences, weak equivalences, and a technical lemma 69 Chapter 10. CW complexes 73 1. The definition and some examples of CW complexes 73 2. SomeconstructionsonCWcomplexes 74 3. HELP and the Whitehead theorem 75 4. The cellular approximation theorem 76 5. Approximation of spaces by CW complexes 77 6. Approximation of pairs by CW pairs 78 7. Approximation of excisive triads by CW triads 79 Chapter 11. The homotopy excision and suspension theorems 83 1. Statement of the homotopy excision theorem 83 2. The Freudenthal suspension theorem 85 3. Proofofthehomotopyexcisiontheorem 86 Chapter 12. A little homological algebra 91 1. Chain complexes 91 2. Maps and homotopies of maps of chain complexes 91 3. Tensor products of chain complexes 92 CONTENTS vii 4. Short and long exact sequences 93 Chapter 13. Axiomatic and cellular homology theory 95 1. Axioms for homology 95 2. Cellular homology 97 3. Verification of the axioms 100 4. The cellular chains of products 101 5. Some examples: T , K, and RP n 103 Chapter 14. Derivations of properties from the axioms 107 1. Reducedhomology;basedversusunbasedspaces 107 2. Cofibrations and the homology of pairs 108 3. Suspensionandthelongexactsequenceofpairs 109 4. Axioms for reduced homology 110 5. Mayer-Vietoris sequences 112 6. The homology of colimits 114 Chapter 15. The Hurewicz and uniqueness theorems 117 1. The Hurewicz theorem 117 2. TheuniquenessofthehomologyofCWcomplexes 119 Chapter 16. Singular homology theory 123 1. The singular chain complex 123 2. Geometric realization 124 3. Proofs of the theorems 125 4. Simplicial objects in algebraic topology 126 5. Classifying spaces and K(π,n)s 128 Chapter17. Somemorehomologicalalgebra 131 1. Universal coefficients in homology 131 2. The K¨unneth theorem 132 3. Hom functors and universal coefficients in cohomology 133 4. Proofoftheuniversalcoefficienttheorem 135 5. Relations between ⊗ and Hom 136 Chapter 18. Axiomatic and cellular cohomology theory 137 1. Axioms for cohomology 137 2. Cellular and singular cohomology 138 3. Cup products in cohomology 139 4. An example: RP n andtheBorsuk-Ulamtheorem 140 5. Obstruction theory 142 Chapter 19. Derivations of properties from the axioms 145 1. Reducedcohomologygroupsandtheirproperties 145 2. Axioms for reduced cohomology 146 3. Mayer-Vietorissequencesincohomology 147 4. Lim1 and the cohomology of colimits 148 5. TheuniquenessofthecohomologyofCWcomplexes 149 Chapter 20. The Poincar´eduality theorem 151 1. Statement of the theorem 151 viii CONTENTS 2. The definition of the cap product 153 3. Orientations and fundamental classes 155 4. The proof of the vanishing theorem 158 5. The proof of the Poincar´eduality theorem 160 6. The orientation cover 163 Chapter 21. The index of manifolds; manifolds with boundary 165 1. The Euler characteristic of compact manifolds 165 2. The index of compact oriented manifolds 166 3. Manifolds with boundary 168 4. Poincar´eduality for manifolds with boundary 169 5. The index of manifolds that are boundaries 171 Chapter 22. Homology, cohomology, and K(π,n)s 175 1. K(π,n)s and homology 175 2. K(π,n)s and cohomology 177 3. Cup and cap products 179 4. Postnikov systems 182 5. Cohomology operations 184 Chapter23. Characteristicclassesofvectorbundles 187 1. The classification of vector bundles 187 2. Characteristic classes for vector bundles 189 3. Stiefel-Whitney classes of manifolds 191 4. Characteristic numbers of manifolds 193 5. ThomspacesandtheThomisomorphismtheorem 194 6. The construction of the Stiefel-Whitney classes 196 7. Chern,Pontryagin,andEulerclasses 197 8. A glimpse at the general theory 200 Chapter 24. An introduction to K-theory 203 1. The definition of K-theory 203 2. The Bott periodicity theorem 206 3. The splitting principle and the Thom isomorphism 208 4. The Chern character; almost complex structures on spheres 211 5. The Adams operations 213 6. The Hopf invariant one problem and its applications 215 Chapter 25. An introduction to cobordism 219 1. Thecobordismgroupsofsmoothclosedmanifolds 219 2. Sketch proof that N∗ is isomorphic to π∗(T O) 220 3. Prespectra and the algebra H∗(T O; Z2) 223 4. The Steenrod algebra and its coaction on H∗(T O) 226 5. The relationship to Stiefel-Whitney numbers 228 6. Spectra and the computation of π∗(T O)= π∗(MO) 230 7. An introduction to the stable category 232 Suggestions for further reading 235 1. Aclassicbookandhistoricalreferences 235 2. Textbooksinalgebraictopologyandhomotopytheory 235 CONTENTS ix 3. Books on CW complexes 236 4. Differential forms and Morse theory 236 5. Equivariant algebraic topology 237 6. Categorytheoryandhomologicalalgebra 237 7. Simplicial sets in algebraic topology 237 8. TheSerrespectralsequenceandSerreclasstheory 237 9. TheEilenberg-Moorespectralsequence 237 10. Cohomology operations 238 11. Vector bundles 238 12. Characteristic classes 238 13. K-theory 239 14. Hopf algebras; the Steenrod algebra, Adams spectral sequence 239 15. Cobordism 240 16. Generalized homology theory and stable homotopy theory 240 17. Quillen model categories 240 18. Localization and completion; rational homotopy theory 241 19. Infinite loop space theory 241 20. Complex cobordism and stable homotopy theory 242 21. Follow-ups to this book 242 Introduction The first year graduate program in mathematics at the University of Chicago consists of three three-quarter courses, in analysis, algebra, and topology. The first two quarters of the topology sequence focus on manifold theory and differential geometry, including differential forms and, usually, a glimpse of de Rham cohomol- ogy. The third quarter focuses on algebraic topology. I have been teaching the third quarter off and on since around 1970. Before that, the topologists, including me, thought that it would be impossible to squeeze a serious introduction to al- gebraic topology into a one quarter course, but we were overruled by the analysts and algebraists, who felt that it was unacceptable for graduate students to obtain their PhDs without having some contact with algebraic topology. This raises a conundrum. A large number of students at Chicago go into topol- ogy, algebraic and geometric. The introductory course should lay the foundations for their later work, but it should also be viable as an introduction to the subject suitable for those going into other branches of mathematics. These notes reflect my efforts to organize the foundations of algebraic topology in a way that caters to both pedagogical goals. There are evident defects from both points of view. A treatment more closely attuned to the needs of algebraic geometers and analysts would include Cechˇ cohomology on the one hand and de Rham cohomology and perhaps Morse homology on the other. A treatment more closely attuned to the needs of algebraic topologists would include spectral sequences and an array of calculations with them. In the end, the overriding pedagogical goal has been the introduction of basic ideas and methods of thought. Our understanding of the foundations of algebraic topology has undergone sub- tle but serious changes since I began teaching this course. These changes reflect in part an enormous internal development of algebraic topology over this period, one which is largely unknown to most other mathematicians, even those working in such closely related fields as geometric topology
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