mathematics - algebraic topology

Allen Hatcher Copyright c 2000 by Allen Hatcher Paper or electronic copies for noncommercial use may be made freely without explicit permission from the author. All other rights reserved. Table of Contents Chapter 0. Some Underlying Geometric Notions 1 Homotopy and Homotopy Type 1. Cell Complexes 5. Operations on Spaces 8. Two Criteria for Homotopy Equivalence 11. The Homotopy Extension Property 14. Chapter 1. The Fundamental Group 21 1. Basic Constructions 25 Paths and Homotopy 25. The Fundamental Group of the Circle 28. Induced Homomorphisms 34. 2. Van Kampen’s Theorem 39 Free Products of Groups 39. The van Kampen Theorem 41. Applications to Cell Complexes 49. 3. Covering Spaces 55 Lifting Properties 59. The Classification of Covering Spaces 62. Deck Transformations and Group Actions 69. Additional Topics A. Graphs and Free Groups 81. B. K(G,1) Spaces and Graphs of Groups 86. Chapter 2. Homology 97 1. Simplicial and Singular Homology 102 ∆ Complexes 102. Simplicial Homology 104. Singular Homology 108. Homotopy Invariance 110. Exact Sequences and Excision 113. The Equivalence of Simplicial and Singular Homology 128. 2. Computations and Applications 134 Degree 134. Cellular Homology 137. Mayer-Vietoris Sequences 149. Homology with Coefficients 153. 3. The Formal Viewpoint 160 Axioms for Homology 160. Categories and Functors 162. Additional Topics A. Homology and Fundamental Group 166. B. Classical Applications 168. C. Simplicial Approximation 175. Chapter 3. Cohomology 183 1. Cohomology Groups 188 The Universal Coefficient Theorem 188. Cohomology of Spaces 195. 2. Cup Product 204 The Cohomology Ring 209. A K ¨ unneth Formula 215. Spaces with Polynomial Cohomology 221. 3. Poincar ´ e Duality 228 Orientations and Homology 231. The Duality Theorem 237. Connection with Cup Product 247. Other Forms of Duality 250. Additional Topics A. The Universal Coefficient Theorem for Homology 259. B. The General K ¨ unneth Formula 266. C. H–Spaces and Hopf Algebras 279. D. The Cohomology of SO(n) 291. E. Bockstein Homomorphisms 301. F. Limits 309. G. More About Ext 316. H. Transfer Homomorphisms 320. I. Local Coefficients 327. Chapter 4. Homotopy Theory 337 1. Homotopy Groups 339 Definitions and Basic Constructions 340. Whitehead’s Theorem 346. Cellular Approximation 348. CW Approximation 351. 2. Elementary Methods of Calculation 359 Excision for Homotopy Groups 359. The Hurewicz Theorem 366. Fiber Bundles 374. Stable Homotopy Groups 383. 3. Connections with Cohomology 392 The Homotopy Construction of Cohomology 393. Fibrations 404. Postnikov Towers 409. Obstruction Theory 415. Additional Topics A. Basepoints and Homotopy 421. B. The Hopf Invariant 427. C. Minimal Cell Structures 429. D. Cohomology of Fiber Bundles 432. E. The Brown Representability Theorem 448. F. Spectra and Homology Theories 453. G. Gluing Constructions 456. H. Eckmann-Hilton Duality 461. I. Stable Splittings of Spaces 468. J. The Loopspace of a Suspension 471. K. Symmetric Products and the Dold-Thom Theorem 477. L. Steenrod Squares and Powers 489. Appendix 521 Topology of Cell Complexes 521. The Compact-Open Topology 531. Bibliography 535 Index 540 Preface This book was written to be a readable introduction to Algebraic Topology with rather broad coverage of the subject. Our viewpoint is quite classical in spirit, and stays largely within the confines of pure Algebraic Topology. In a sense, the book could have been written thirty years ago since virtually all its content is at least that old. However, the passage of the intervening years has helped clarify what the most important results and techniques are. For example, CW complexes have proved over time to be the most natural class of spaces for Algebraic Topology, so they are em- phasized here much more than in the books of an earlier generation. This emphasis also illustrates the book’s general slant towards geometric, rather than algebraic, as- pects of the subject. The geometry of Algebraic Topology is so pretty, it would seem a pity to slight it and to miss all the intuition that it provides. At deeper levels, alge- bra becomes increasingly important, so for the sake of balance it seems only fair to emphasize geometry at the beginning. Let us say something about the organization of the book. At the elementary level, Algebraic Topology divides naturally into two channels, with the broad topic of Ho- motopy on the one side and Homology on the other. We have divided this material into four chapters, roughly according to increasing sophistication, with Homotopy split between Chapters 1 and 4, and Homology and its mirror variant Cohomology in Chapters 2 and 3. These four chapters do not have to be read in this order, how- ever. One could begin with Homology and perhaps continue on with Cohomology before turning to Homotopy. In the other direction, one could postpone Homology and Cohomology until after parts of Chapter 4. However, we have not pushed this latter approach to its natural limit, in which Homology and Cohomology arise just as branches of Homotopy Theory. Appealing as this approach is from a strictly logical point of view, it places more demands on the reader, and since readability is one of our first priorities, we have delayed introducing this unifying viewpoint until later in the book. There is also a preliminary Chapter 0 introducing some of the basic geometric concepts and constructions that play a central role in both the homological and ho- motopical sides of the subject. Each of the four main chapters concludes with a selection of Additional Topics that the reader can sample at will, independent of the basic core of the book contained in the earlier parts of the chapters. Many of these extra topics are in fact rather important in the overall scheme of Algebraic Topology, though they might not fit into the time constraints of a first course. Altogether, these Additional Topics amount to nearly half the book, and we have included them both to make the book more comprehensive and to give the reader who takes the time to delve into them a more substantial sample of the true richness and beauty of the subject. Not included in this book is the important but somewhat more sophisticated topic of spectral sequences. It was very tempting to include something about this marvelous tool here, but spectral sequences are such a big topic that it seemed best to start with them afresh in a new volume. This is tentatively titled ‘Spectral Sequences in Algebraic Topology’ and referred to herein as [SSAT]. There is also a third book in progress, on vector bundles, characteristic classes, and K–theory, which will be largely independent of [SSAT] and also of much of the present book. This is referred to as [VBKT], its provisional title being ‘Vector Bundles and K–Theory.’ In terms of prerequisites, the present book assumes the reader has some famil- iarity with the content of the standard undergraduate courses in algebra and point-set topology. One topic that is not always a part of a first point-set topology course but which is quite important for Algebraic Topology is quotient spaces, or identification spaces as they are sometimes called. Good sources for this are the textbooks by Arm- strong and J ¨ anich listed in the Bibliography. A book such as this one, whose aim is to present classical material from a fairly classical viewpoint, is not the place to indulge in wild innovation. Nevertheless there is one new feature of the exposition that may be worth commenting upon, even though in the book as a whole it plays a relatively minor role. This is a modest extension of the classical notion of simplicial complexes, which we call ∆ complexes. These have made brief appearances in the literature previously, without a standard name emerging. The idea is to weaken the condition that each simplex be embedded, to require only that the interiors of simplices are embedded. (In addition, an ordering of the vertices of each simplex is also part of the structure of a ∆ complex.) For example, if one takes the standard picture of the torus as a square with opposite edges identified and divides the square into two triangles by cutting along a diagonal, then the result is a ∆ complex structure on the torus having 2 triangles, 3 edges, and 1 vertex. By contrast, it is known that a simplicial complex structure on the torus must have at least 14 triangles, 21 edges, and 7 vertices. So ∆ complexes provide a significant improvement in efficiency, which is nice from a pedagogical viewpoint since it cuts down on tedious calculations in examples. A more fundamental reason for considering ∆ complexes is that they just seem to be very natural objects from the viewpoint of Algebraic Topology. They are the natural domain of definition for simplicial homology, and a number of standard constructions produce ∆ complexes rather than simplicial complexes, for instance the singular complex of a space, or the classifying space of a discrete group or category. It is the author’s intention to keep this book available online permanently, as well as publish it in the traditional manner for those who want the convenience of a bound copy. With the electronic version it will be possible to continue making revisions and additions, so comments and suggestions from readers will always be welcome. The web address is: http://www.math.cornell.edu/˜hatcher One can also find here the parts of the other two books that are currently available. Standard Notations R n : n dimensional Euclidean space, with real coordinates C n : complex n space I = [0, 1]: the unit interval S n : the unit sphere in R n+1 , all vectors of length 1 D n : the unit disk or ball in R n , all vectors of length ≤ 1 ∂D n = S n−1 : the boundary of the n disk 11: the identity function from a set to itself : disjoint union ≈: isomorphism Z n : the integers modn A ⊂ B or B ⊃ A: set-theoretic containment, not necessarily proper The aim of this short preliminary chapter is to introduce a few of the most com- mon geometric concepts and constructions in algebraic topology. The exposition is somewhat informal, with no theorems or proofs until the last couple pages, and it should be read in this informal spirit, skipping bits here and there. In fact, this whole chapter could be skipped now, to be referred back to later for basic definitions. To avoid overusing the word ‘continuous’ we adopt the convention that maps be- tween spaces are always assumed to be continuous unless otherwise stated. Homotopy and Homotopy Type One of the main ideas of algebraic topology is to consider two spaces to be equiv- alent if they have ‘the same shape’ in a sense that is much broader than homeo- morphism. To take an everyday example, the letters of the alphabet can be written either as unions of finitely many straight and curved line segments, or in thickened forms that are compact subsurfaces of the plane bounded by simple closed curves. In each case the thin letter is a subspace of the thick letter, and we can continuously shrink the thick letter to the thin one. A nice way to do this is to decompose a thick letter, call it X, into line segments connecting each point on the outer boundary of X to a unique point of the thin subletter X, as indicated in the figure. Then we can shrink 2 Chapter 0. Some Underlying Geometric Notions X to X by sliding each point of X − X into X along the line segment that contains it. Points that are already in X do not move. We can think of this shrinking process as taking place during a time interval 0 ≤ t ≤ 1, and then it defines a family of functions f t : X → X parametrized by t ∈ I = [0, 1], where f t (x) is the point to which a given point x ∈ X has moved at time t . Naturally we would like f t (x) to depend continuously on both t and x , and this will be true if we have each x ∈ X − X move along its line segment at constant speed so as to reach its image point in X at time t = 1, while points x ∈ X are stationary, as remarked earlier. These examples lead to the following general definition. A deformation retrac- tion of a space X onto a subspace A is a family of maps f t : X → X , t ∈ I , such that f 0 = 11 (the identity map), f 1 (X) = A, and f t | | A = 11 for all t. The family f t should be continuous in the sense that the associated map X ×I → X , (x, t)  f t (x), is continuous. It is easy to produce many more examples similar to the letter examples, with the deformation retraction f t obtained by sliding along line segments. The first figure below shows such a deformation retraction of a M ¨ obius band onto its core circle. The other three figures show deformation retractions in which a disk with two smaller open subdisks removed shrinks to three different subspaces. In all these examples the structure that gives rise to the deformation retraction can be described by means of the following definition. For a map f : X → Y , the map- ping cylinder M f is the quotient space of the disjoint union (X ×I) Y obtained by identifying each (x, 1) ∈ X ×I with f(x)∈ Y. X×I X Y Y M f f X () In the letter examples, the space X is the outer boundary of the thick letter, Y is the thin letter, and the map f : X → Y sends the outer endpoint of each line segment to its inner endpoint. A similar description applies to the other examples. Then it is a general fact that a mapping cylinder M f deformation retracts to the subspace Y by sliding each point (x, t) along the segment {x}×I ⊂ M f to the endpoint f(x)∈ Y. Not all deformation retractions arise in this way from mapping cylinders, how- ever. For example, the thick X deformation retracts to the thin X, which in turn [...]... path ft (x) for the extended ft One of the main techniques of algebraic topology is to study topological spaces by forming algebraic images of them Most often these algebraic images are groups, but more elaborate structures such as rings, modules, and algebras also arise The mechanisms which create these images — the ‘lanterns’ of algebraic topology, one might say — are known formally as functors and... point To begin, let X be the union of an infinite sequence of cones on the Cantor set arranged end-to-end, as in the figure Next, form the one-point compactification of X × R This embeds i0.8 n R3 as a closed disk with curved ‘fins’ attached along circular arcs, and with the one-point compactification of X as a cross-sectional slice Finally, Y is obtained from this by wrapping one more cone on the Cantor set... complication, however: The topology on X × Y as a cell complex is sometimes slightly weaker than the product topology, with more open sets than the product topology has, though the two topologies coincide if either X or Y has only finitely many cells, or if both X and Y have countably many cells This is explained in the Appendix In practice this subtle point of point-set topology rarely causes problems... g and hf are homotopy equivalences 12 Show that a homotopy equivalence f : X →Y induces a bijection between the set of path-components of X and the set of path-components of Y , and that f restricts to a homotopy equivalence from each path-component of X to the corresponding path-component of Y 13 Show that any two deformation retractions rt0 and rt1 of a space X onto a subspace A can be joined by... projected onto homomorphisms between their algebraic images, so topologically related spaces have algebraically related images With suitably constructed lanterns one might hope to be able to form images with enough detail to reconstruct accurately the shapes of all spaces, or at least of large and interesting classes of spaces This is one of the main goals of algebraic topology, and to a surprising extent... could go further and write just π1 X In general, a space is called simply-connected if it is path-connected and has trivial fundamental group The following result is probably the reason for this term Proposition 1.6 A space X is simply-connected iff there is a unique homotopy class of paths connecting any two points in X Proof: Path-connectedness is the existence of paths connecting every pair of points,... and Y Two Criteria for Homotopy Equivalence 11 The smash product X ∧ Y is a cell complex if X and Y are cell complexes with x0 and y0 0 cells, assuming that we give X × Y the cell-complex topology rather than the product topology in cases when these two topologies differ For example, S m ∧S n has a cell structure with just two cells, of dimensions 0 and m+n , hence S m ∧S n = S m+n In particular,... a subcomplex, by regarding each S k as being obtained inductively from the equatorial S k−1 by attaching two k cells, the components of S k −S k−1 The infinite-dimensional sphere S ∞ = then becomes a cell complex as well Note that the two-to-one quotient map S ∞ n Sn →RP∞ that identifies antipodal points of S ∞ identifies the two n cells of S ∞ to the single n cell of RP∞ In the examples of cell complexes... complicated pieces of machinery But this machinery also has a certain intrinsic beauty This first chapter introduces one of the simplest and most important of the functors of algebraic topology, the fundamental group, which creates an algebraic image of a space from the loops in the space, that is, paths starting and ending at the same point The Idea of the Fundamental Group To get a feeling for what the... other parts of mathematics Not all retractions come from deformation retractions For example, every space X retracts onto any point x0 ∈ X via the map sending all of X to x0 But a space that deformation retracts onto a point must certainly be path-connected, since a deformation retraction of X to a point x0 gives a path joining each x ∈ X to x0 It is less trivial to show that there are path-connected