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 5. Calculations: π 1 (R) = 0 and π 1 (S 1 ) = Z 8 6. The Brouwer fixed point theorem 10 7. The fundamental theorem of algebra 10 Chapter 2. Categorical language and the van Kampen theorem 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. Examples of the van Kampen theorem 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. The classification of coverings of groupoids 25 6. The construction of coverings of groupoids 27 7. The classification of coverings of spaces 28 8. The construction of coverings of spaces 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. Covers of graphs and Euler characteristics 37 5. Applications to groups 37 Chapter 5. Compactly generated spaces 39 1. The definition of compactly generated spaces 39 2. The category of compactly generated spaces 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. A criterion for a map to be a cofibration 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. A criterion for a map to be a fibration 51 5. Fiber homotopy equivalence 52 6. Change of fiber 53 Chapter 8. Based cofiber and fiber sequences 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. Connections between cofiber and fiber sequences 63 Chapter 9. Higher homotopy groups 65 1. The definition of homotopy groups 65 2. Long exact sequences associated to pairs 65 3. Long exact sequences associated to fibrations 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. Some constructions on CW complexes 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. Proof of the homotopy excision theorem 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. Reduced homology; based versus unbased spaces 107 2. Cofibrations and the homology of pairs 108 3. Suspension and the long exact sequence of pairs 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. The uniqueness of the homology of CW complexes 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 Chapter 17. Some more homological algebra 131 1. Universal coefficients in homology 131 2. The K¨unneth theorem 132 3. Hom functors and universal coefficients in cohomology 133 4. Proof of the universal coefficient theorem 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 and the Borsuk-Ulam theorem 140 5. Obstruction theory 142 Chapter 19. Derivations of properties from the axioms 145 1. Reduced cohomology groups and their prop erties 145 2. Axioms for reduced cohomology 146 3. Mayer-Vietoris sequences in cohomology 147 4. Lim 1 and the cohomology of colimits 148 5. The uniqueness of the cohomology of CW complexes 149 Chapter 20. The Poincar´e duality 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´e duality 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 b oundary 168 4. Poincar´e duality 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 Chapter 23. Characteristic classes of vector bundles 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. Thom spaces and the Thom isomorphism theorem 194 6. The construction of the Stiefel-Whitney classes 196 7. Chern, Pontryagin, and Euler classes 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. The cobordism groups of smooth closed manifolds 219 2. Sketch proof that N ∗ is isomorphic to π ∗ (T O) 220 3. Prespectra and the algebra H ∗ (T O; Z 2 ) 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. A classic book and historical references 235 2. Textbooks in algebraic topology and homotopy theory 235 CONTENTS ix 3. Books on CW complexes 236 4. Differential forms and Morse theory 236 5. Equivariant algebraic topology 237 6. Category theory and homological algebra 237 7. Simplicial sets in algebraic topology 237 8. The Serre spectral sequence and Serre class theory 237 9. The Eilenberg-Moore spectral sequence 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 c ourses, 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 ˇ Cech 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 and algebraic geometry. Moreover, this development is poorly reflected in the textbooks that have appeared over this period. Let me give a small but technically important example. The study of gen- eralized homology and cohomology theories pervades modern algebraic topology. These theories satisfy the e xcision axiom. One constructs most such theories ho- motopically, by constructing representing objects called spectra, and one must then prove that excision holds. There is a way to do this in general that is no more dif- ficult than the standard verification for singular homology and cohomology. I find this proof far more conceptual and illuminating than the standard one even when specialized to singular homology and cohomology. (It is based on the approxima- tion of excisive triads by weakly equivalent CW triads.) This should by now be a 1 2 INTRODUCTION standard approach. However, to the best of my knowledge, there exists no rigorous exposition of this approach in the literature, at any level. More centrally, there now exist axiomatic treatments of large swaths of homo- topy theory based on Quillen’s theory of closed model categories. While I do not think that a first course should introduce such abstractions, I do think that the ex- position should give emphasis to those features that the axiomatic approach shows to be fundamental. For example, this is one of the reasons, although by no means the only one, that I have dealt with cofibrations, fibrations, and weak equivalences much more thoroughly than is usual in an introductory course. Some parts of the theory are dealt with quite classically. The theory of fun- damental groups and covering spaces is one of the few parts of algebraic topology that has probably reached definitive form, and it is well treated in many sources. Nevertheless, this material is far too important to all branches of mathematics to be omitted from a first course. For variety, I have made more use of the funda- mental groupoid than in standard treatments, 1 and my use of it has some novel features. For conceptual interest, I have emphasized different categorical ways of modeling the topological situation algebraically, and I have taken the opportunity to introduce some ideas that are central to equivariant algebraic topology. Poincar´e duality is also too fundamental to omit. There are more elegant ways to treat this topic than the classical one given here, but I have preferred to give the theory in a quick and standard fashion that reaches the desired conclusions in an economical way. Thus here I have not presented the truly modern approach that applies to generalized homology and cohomology theories. 2 The reader is warned that this book is not des igned as a textbook, although it could be used as one in exceptionally strong graduate programs. Even then, it would be impossible to cover all of the material in detail in a quarter, or even in a year. There are sections that should be omitted on a first reading and others that are intended to whet the student’s appetite for further developments. In practice, when teaching, my lectures are regularly interrupted by (purposeful) digressions, most often directly prompted by the questions of students. These introduce more advanced topics that are not part of the formal introductory course: cohomology operations, characteristic classes, K-theory, cobordism, etc., are often first intro- duced earlier in the lectures than a linear development of the subject would dictate. These digressions have been expanded and written up here as sketches without complete proofs, in a logically coherent order, in the last four chapters. These are topics that I feel must be introduced in some fashion in any serious graduate level introduction to algebraic topology. A defect of nearly all existing texts is that they do not go far enough into the subject to give a feel for really substantial applications: the reader sees spheres and projective spaces, maybe lens spaces, and applications accessible with knowledge of the homology and cohomology of such spaces. That is not enough to give a real feeling for the subject. I am aware that this treatment suffers the same defect, at least before its sketchy last chapters. Most chapters end with a set of problems. Most of these ask for computa- tions and applications based on the material in the text, some extend the theory and introduce further concepts, some ask the reader to furnish or complete proofs 1 But see R. Brown’s book cited in §2 of the suggestions for further reading. 2 That approach derives Poincar´e duality as a consequence of Spanier-Whitehead and Atiyah duality, via the Thom isomorphism for oriented vector bundles. [...]... literature for the reader interested in going further in algebraic topology These notes have evolved over many years, and I claim no originality for most of the material In particular, many of the problems, especially in the more classical chapters, are the same as, or are variants of, problems that appear in other texts Perhaps this is unavoidable: interesting problems that are doable at an early stage... (X, x) as a category with a single object x, and it is a skeleton of Π(X) 16 CATEGORICAL LANGUAGE AND THE VAN KAMPEN THEOREM 6 Limits and colimits Let D be a small category and let C be any category A D-shaped diagram in C is a functor F : D −→ C A morphism F −→ F of D-shaped diagrams is a natural transformation, and we have the category D[C ] of D-shaped diagrams in C Any object C of C determines... composition of paths β · α Deduce that π1 (G, e) is Abelian CHAPTER 2 Categorical language and the van Kampen theorem We introduce categorical language and ideas and use them to prove the van Kampen theorem This method of computing fundamental groups illustrates the general principle that calculations in algebraic topology usually work by piecing together a few pivotal examples by means of general constructions... to introduce some ideas that are central to equivariant algebraic topology, the study of spaces with group actions In any case, this material is far too important to all branches of mathematics to omit 1 The definition of covering spaces While the reader is free to think about locally contractible spaces, weaker conditions are appropriate for the full generality of the theory of covering spaces A space... sets and from Abelian groups to sets, and we have the free Abelian group functor from sets to Abelian groups 13 14 CATEGORICAL LANGUAGE AND THE VAN KAMPEN THEOREM 3 Natural transformations A natural transformation α : F −→ G between functors C −→ D is a map of functors It consists of a morphism A : F (A) −→ G (A) for each object A of C such that the following diagram commutes for each morphism f : A −→... then a (left) action of G on a set S is the same thing as a covariant functor G −→ S (A right action is the same thing as a contravariant functor.) If B is a small groupoid, it is therefore natural to think of a covariant functor T : B −→ S as a generalization of a group action For each object b of B, T restricts to an action of π(B, b) on T (b) We say that the functor T is transitive if this group action... in the category of transitive G-sets is an equivalence of categories because O(G) is a full subcategory that contains a skeleton We could shrink O(G) to a skeleton by choosing one H in each conjugacy class of subgroups of G, but the resulting equivalent subcategory is a less natural mathematical object 7 The classification of coverings of spaces In this section and the next, we shall classify covering... in the literature in full generality The proof well illustrates how to manipulate colimits formally We have used the van Kampen theorem as an excuse to introduce some basic categorical language, and we shall use that language heavily in our treatment of covering spaces in the next chapter Theorem (van Kampen) Let O = {U } be a cover of a space X by path connected open subsets such that the intersection... < ε} A function p : X −→ Y is continuous if it takes nearby points to nearby points Precisely, p−1 (U ) is open if U is open If X and Y are metric spaces, this means that, for any x ∈ X and ε > 0, there exists δ > 0 such that p(Uδ (x)) ⊂ Uε (p(x)) Algebraic topology assigns discrete algebraic invariants to topological spaces and continuous maps More narrowly, one wants the algebra to be invariant with... called proof by categorical nonsense 18 CATEGORICAL LANGUAGE AND THE VAN KAMPEN THEOREM Theorem (van Kampen) Let X be path connected and choose a basepoint x ∈ X Let O be a cover of X by path connected open subsets such that the intersection of finitely many subsets in O is again in O and x is in each U ∈ O Regard O as a category whose morphisms are the inclusions of subsets and observe that the functor