Algebraic topology

Allen Hatcher Copyright c 2001 by Allen Hatcher Paper or electronic copies for noncommercial use may be made freely without explicit permission from the author All other rights reserved Preface ix Standard Notations xii Chapter Some Underlying Geometric Notions Homotopy and Homotopy Type Cell Complexes Operations on Spaces Two Criteria for Homotopy Equivalence 10 The Homotopy Extension Property 14 Chapter The Fundamental Group 1.1 Basic Constructions 21 25 Paths and Homotopy 25 The Fundamental Group of the Circle 28 Induced Homomorphisms 33 1.2 Van Kampen’s Theorem 38 Free Products of Groups 39 The van Kampen Theorem 41 Applications to Cell Complexes 48 1.3 Covering Spaces Lifting Properties 58 The Classification of Covering Spaces 61 Deck Transformations and Group Actions 68 Additional Topics 1.A Graphs and Free Groups 81 1.B K(G,1) Spaces and Graphs of Groups 85 54 Table of Contents vi Chapter Homology 2.1 Simplicial and Singular Homology 95 100 ∆ Complexes 100 Simplicial Homology 102 Singular Homology 106 Homotopy Invariance 108 Exact Sequences and Excision 111 The Equivalence of Simplicial and Singular Homology 126 2.2 Computations and Applications 132 Degree 132 Cellular Homology 135 Mayer-Vietoris Sequences 147 Homology with Coefficients 151 2.3 The Formal Viewpoint 158 Axioms for Homology 158 Categories and Functors 160 Additional Topics 2.A Homology and Fundamental Group 164 2.B Classical Applications 167 2.C Simplicial Approximation 175 Chapter Cohomology 183 3.1 Cohomology Groups 188 The Universal Coefficient Theorem 188 Cohomology of Spaces 195 3.2 Cup Product 204 The Cohomology Ring 209 A Kă unneth Formula 216 Spaces with Polynomial Cohomology 222 3.3 Poincar´ e Duality 228 Orientations and Homology 231 The Duality Theorem 237 Connection with Cup Product 247 Other Forms of Duality 250 Additional Topics 3.A Universal Coefficients for Homology 259 3.B The General Kă unneth Formula 266 3.C H–Spaces and Hopf Algebras 279 3.D The Cohomology of SO(n) 290 3.E Bockstein Homomorphisms 301 3.F Limits and Ext 309 3.G Transfer Homomorphisms 319 3.H Local Coefficients 325 Table of Contents Chapter Homotopy Theory 4.1 Homotopy Groups vii 335 337 Definitions and Basic Constructions 338 Whitehead’s Theorem 344 Cellular Approximation 346 CW Approximation 350 4.2 Elementary Methods of Calculation 358 Excision for Homotopy Groups 358 The Hurewicz Theorem 364 Fiber Bundles 373 Stable Homotopy Groups 382 4.3 Connections with Cohomology 391 The Homotopy Construction of Cohomology 391 Fibrations 403 Postnikov Towers 408 Obstruction Theory 413 Additional Topics 4.A Basepoints and Homotopy 419 4.B The Hopf Invariant 425 4.C Minimal Cell Structures 427 4.D Cohomology of Fiber Bundles 429 4.E The Brown Representability Theorem 446 4.F Spectra and Homology Theories 450 4.G Gluing Constructions 454 4.H Eckmann-Hilton Duality 458 4.I Stable Splittings of Spaces 464 4.J The Loopspace of a Suspension 468 4.K The Dold-Thom Theorem 473 4.L Steenrod Squares and Powers 485 Appendix 517 Topology of Cell Complexes 517 The Compact-Open Topology 527 Bibliography Index 531 537 This book was written to be a readable introduction to algebraic topology with rather broad coverage of the subject The viewpoint is quite classical in spirit, and stays well within the confines of pure algebraic topology In a sense, the book could have been written thirty or forty years ago since virtually everything in it is at least that old However, the passage of the intervening years has helped clarify what are the most important results and techniques For example, CW complexes have proved over time to be the most natural class of spaces for algebraic topology, so they are emphasized 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, aspects 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 it provides At the elementary level, algebraic topology separates naturally into the two broad channels of homology and homotopy This material is here divided into four chapters, roughly according to increasing sophistication, with homotopy split between Chapters and 4, and homology and its mirror variant cohomology in Chapters and These four chapters not have to be read in this order, however One could begin with homology and perhaps continue with cohomology before turning to homotopy In the other direction, one could postpone homology and cohomology until after parts of Chapter If this latter strategy is pushed to its natural limit, homology and cohomology can be developed 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 the first priorities of the book, this homotopic interpretation of homology and cohomology is described only after the latter theories have been developed independently of homotopy theory Preceding the four main chapters there is a preliminary Chapter introducing some of the basic geometric concepts and constructions that play a central role in both the homological and homotopical sides of the subject This can either be read before the other chapters or skipped and referred back to later for specific topics as they become needed in the subsequent chapters 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 x Preface constraints of a first course Altogether, these additional topics amount to nearly half the book, and they are included here 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 is 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 familiarity with the content of the standard undergraduate courses in algebra and point-set topology In particular, the reader should know about quotient spaces, or identification spaces as they are sometimes called, which are quite important for algebraic topology Good sources for this concept are the textbooks [Armstrong 1983] and [Jă anich 1984] listed in the Bibliography A book such as this one, whose aim is to present classical material from a rather 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 a simplicial complex that goes under the name of a ∆ complex in this book The idea is to allow different faces of a simplex to coincide, so only the interiors of simplices are embedded and simplices are no longer uniquely determined by their vertices (As a technical point, 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 triangles, edges, and vertex By contrast, it is known that a simplicial complex structure on the torus must have at least 14 triangles, 21 edges, and 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 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 In spite of this naturality, ∆ complexes have appeared explicitly in the literature only rarely, and no standard name for the notion has emerged Preface xi This book will remain available online in electronic form after it has been printed in the traditional fashion The web address is http://www.math.cornell.edu/˜hatcher One can also find here the parts of the other two books in the sequence that are currently available Although the present book has gone through countless revisions already, including corrections of many small errors both typographical and mathematical that were found by careful readers of earlier versions, it is a virtual certainty that some errors remain, so the web page will contain also a list of corrections Readers are encouraged to submit their candidates for entries on this list to the email address posted on the web page With the electronic version of the book it will be possible to continue making revisions and additions as well as corrections, so comments and suggestions from readers will always be welcome xii Standard Notations Z , Q , R , C , H , O : the integers, rationals, reals, complexes, quaternions, and Cayley octonions Zn : the integers mod n Rn : n dimensional Euclidean space Cn : complex n space I = [0, 1] : the unit interval S n : the unit sphere in Rn+1 , all vectors of length D n : the unit disk or ball in Rn , all vectors of length ≤ ∂D n = S n−1 : the boundary of the n disk 11 : the identity function from a set to itself : disjoint union of sets or spaces ×, : product of sets, groups, or spaces ≈ : isomorphism A ⊂ B or B ⊃ A : set-theoretic containment, not necessarily proper iff : if and only if Appendix 528 The Compact-Open Topology that finer and finer compact covers {Ki } of Y are taken to smaller and smaller open covers {Ui } of f (Y ) One of the main cases of interest in homotopy theory is when Y = I , so X I is the space of paths in X In this case one can check that a system of basic neighborhoods of a path f : I →X consists of the open sets i M(Ki , Ui ) where the Ki ’s are a partition of I into nonoverlapping closed intervals and Ui is an open neighborhood of f (Ki ) The compact-open topology is the same as the topology of uniform convergence in many cases: Proposition A.13 topology on X Y If X is a metric space and Y is compact, then the compact-open is the same as the metric topology defined by the metric d(f , g) = supy∈Y d(f (y), g(y)) Proof: First we show that every open ε ball Bε (f ) about f ∈ X Y contains a neigh- borhood of f in the compact-open topology Since f (Y ) is compact, it is covered by finitely many balls Bε/3 f (yi ) Let Ki ⊂ Y be the closure of f −1 Bε/3 (f (yi )) , so Ki is compact, Y = To show that i i Ki , and f (Ki ) ⊂ Bε/2 f (yi ) = Ui , hence f ∈ M(Ki , Ui ) ⊂ Bε (f ) , suppose that g ∈ i i M(Ki , Ui ) M(Ki , Ui ) For any y ∈ Y , say y ∈ Ki , we have d g(y), f (yi ) < ε/2 since g(Ki ) ⊂ Ui Likewise we have d f (y), f (yi ) < ε/2 , so d f (y), g(y) ≤ d f (y), f (yi ) + d g(y), f (yi ) < ε Since y was arbitrary, this shows g ∈ Bε (f ) Conversely, we show that for each open set M(K, U ) and each f ∈ M(K, U ) there is a ball Bε (f ) ⊂ M(K, U) Since f (K) is compact, it has a distance ε > from the complement of U Then d(f , g) < ε/2 implies g(K) ⊂ U since g(K) is contained in an ε/2 neighborhood of f (K) So Bε/2 (f ) ⊂ M(K, U ) The next proposition contains the essential properties of the compact-open topology from the viewpoint of algebraic topology Proposition A.14 If Y is locally compact, then : (a) The evaluation map e : X Y × Y →X , e(f , y) = f (y) , is continuous (b) A map f : Y × Z →X is continuous iff the map f : Z →X Y , f (z)(y) = f (y, z) , is continuous In particular, part (b) provides the point-set topology justifying the adjoint relation ΣX, Y = X, ΩY in §4.3, since it implies that a map ΣX →Y is continuous iff the associated map X →ΩY is continuous, and similarly for homotopies of such maps Namely, think of a basepoint-preserving map ΣX →Y as a map f : I × X →Y taking ∂I × X ∪ {x0 }× I to the basepoint of Y , so the associated map f : X →Y I has image in the subspace ΩY ⊂ Y I A homotopy ft : ΣX →Y gives a map F : I × X × I →Y taking ∂I × X × I ∪I × {x0 }× I to the basepoint, with F a map X × I →ΩY ⊂ Y I defining a basepoint-preserving homotopy ft The Compact-Open Topology Proof: Appendix 529 (a) For (f , y) ∈ X Y × Y let U ⊂ X be an open neighborhood of f (y) Since Y is locally compact, continuity of f implies there is a compact neighborhood K ⊂ Y of y such that f (K) ⊂ U Then M(K, U )× K is a neighborhood of (f , y) in X Y × Y taken to U by e , so e is continuous at (f , y) (b) Suppose f : Y × Z →X is continuous To show continuity of f it suffices to show that for a subbasic set M(K, U) ⊂ X Y , the set f −1 M(K, U ) = { z ∈ Z | f (K, z) ⊂ U } is open in Z Let z ∈ f −1 M(K, U ) Since f −1 (U ) is an open neighborhood of the compact set K × {z} , there exist open sets V ⊂ Y and W ⊂ Z whose product V × W satisfies K × {z} ⊂ V × W ⊂ f −1 (U ) So W is a neighborhood of z in f −1 M(K, U ) (The hypothesis that Y is locally compact is not needed here.) For the converse half of (b) note that f is the composition Y × Z →Y × X Y →X of 11× f and the evaluation map, so part (a) gives the result Proposition A.15 If X is a compactly generated Hausdorff space and Y is locally compact, then the product topology on X × Y is compactly generated Proof: First a preliminary observation: A function f : X × Y →Z is continuous iff its restrictions f : C × Y →Z are continuous for all compact C ⊂ X For, using (b) of the previous proposition, the first statement is equivalent to f : X →Z Y being continuous and the second statement is equivalent to f : C →Z Y being continuous for all compact C ⊂ X Since X is compactly generated, the latter two statements are equivalent To prove the proposition we just need to show the identity map X × Y →(X × Y )c is continuous By the previous paragraph, this is equivalent to continuity of the inclusion maps C × Y →(X × Y )c for all compact C ⊂ X Since Y is locally compact, it is compactly generated, and C is compact Hausdorff hence locally compact, so the same reasoning shows that continuity of C × Y →(X × Y )c is equivalent to continuity of C × C →(X × Y )c for all compact C ⊂ Y But on the compact set C × C , the two topologies on X × Y agree, so we are done (This proof is from [Dugundji 1966].) Proposition A.16 The map X Y × Z →(X Y )Z , f f , is a homeomorphism if Y is locally compact Hausdorff and Z is Hausdorff Proof: First we show that a subbasis for X Y × Z is formed by the sets M(A× B, U ) as A and B range over compact sets in Y and Z respectively and U ranges over open sets in X Given a compact K ⊂ Y × Z and f ∈ M(K, U ) , let KY and KZ be the projections of K onto Y and Z Then KY × KZ is compact Hausdorff, hence normal, so for each point k ∈ K there are compact sets Ak ⊂ Y and Bk ⊂ Z such that Ak × Bk is a compact neighborhood of k in f −1 (U ) ∩ (KY × KZ ) By compactness of K a finite number of the products Ak × Bk cover K Discarding the others, we then have f ∈ k M(Ak × Bk , U) ⊂ M(K, U ) , which shows that the sets M(A× B, U ) form a subbasis Under the bijection X Y × Z →(X Y )Z these sets M(A× B, U ) correspond to the sets M(B, M(A, U)) , so it will suffice to show the latter sets form a subbasis for (X Y )Z We 530 Appendix The Compact-Open Topology show more generally that X Y has as a subbasis the sets M(K, V ) as V ranges over a subbasis for X and K ranges over compact sets in Y , assuming that Y is Hausdorff Given f ∈ M(K, U) , write U as a union of basic sets Uα with each Uα an intersection of finitely many sets Vα,j of the given subbasis The cover of K by the open sets f −1 (Uα ) has a finite subcover, say by the open sets f −1 (Ui ) Since K is compact Hausdorff, hence normal, we can write K as a union of compact subsets Ki with Ki ⊂ f −1 (Ui ) Then f lies in M(Ki , Ui ) = M(Ki , i Hence f lies in i,j M(Ki , Vij ) = j Vij ) = j i M(Ki , Ui ) ⊂ M(K, U ) Since M(Ki , Vij ) for each i,j M(Ki , Vij ) is a finite intersection, this shows that the sets M(K, V ) form a subbasis for (X Y )Z Books J F Adams, Algebraic Topology: a Student’s Guide, Cambridge Univ Press, 1972 J F Adams, Stable Homotopy and Generalised Homology, Univ of Chicago Press, 1974 J F Adams, Infinite Loop Spaces, Ann of Math Studies 90, 1978 A Adem and R J Milgram, Cohomology of Finite Groups, Springer-Verlag, 1994 M Aigner and G Ziegler, Proofs from THE BOOK, Springer-Verlag, 1999 P Alexandroff and H Hopf, Topologie, Chelsea, 1972 (reprint of original 1935 edition) M A Armstrong, Basic Topology, Springer-Verlag, 1983 M F Atiyah, K–Theory, W A Benjamin, 1967 H J Baues, Homotopy Type and Homology, Oxford Univ Press, 1996 D J Benson, Representations and Cohomology, Volume II: Cohomology of Groups and Modules, Cambridge Univ Press, 1992 D J Benson, Polynomial Invariants of Finite Groups, Cambridge Univ Press, 1993 R Bott and L Tu, Differential Forms in Algebraic Topology, Springer-Verlag GTM 82, 1982 G Bredon, Topology and Geometry, Springer-Verlag GTM 139, 1993 K Brown, Cohomology of Groups, Springer-Verlag GTM 87, 1982 R Brown, The Lefschetz Fixed Point Theorem, Scott Foresman, 1971 M Cohen, A Course in Simple-Homotopy Theory, Springer-Verlag GTM 10, 1973 J Dieudonn´ e, A History of Algebraic and Differential Topology 1900-1960, Birkhă auser, 1989 A Dold, Lectures on Algebraic Topology, Springer-Verlag, 1980 J Dugundji, Topology, Allyn & Bacon, 1966 H.-D Ebbinghaus et al., Numbers, Springer-Verlag GTM 123, 1991 S Eilenberg and N Steenrod, Foundations of Algebraic Topology, Princeton Univ Press, 1952 Y F´ elix, S Halperin, and J.-C Thomas, Rational Homotopy Theory, Springer-Verlag GTM 205, 2001 R Fenn, Techniques of Geometric Topology, Cambridge Univ Press, 1983 A T Fomenko and D B Fuks, A Course in Homotopic Topology, Izd Nauka, 1989 (In Russian; an English translation of an earlier version was published by Akad´ emiai Kiad´ o, Budapest, 1986.) M Freedman and F Quinn, Topology of 4-Manifolds, Princeton Univ Press, 1990 532 Bibliography W Fulton, Algebraic Topology: A First Course, Springer-Verlag, 1995 B Gray, Homotopy Theory, Academic Press, 1975 M Greenberg and J Harper, Algebraic Topology: A First Course, Addison-Wesley, 1981 P Griffiths and J Morgan, Rational Homotopy Theory and Differential Forms, Birkhă auser, 1981 P J Hilton, An Introduction to Homotopy Theory, Cambridge University Press, 1953 P J Hilton, Homotopy Theory and Duality, Gordon and Breach, 1966 P J Hilton and S Wylie, Homology Theory, Cambridge Univ Press, 1967 P J Hilton and U Stammbach, A Course in Homological Algebra, Springer-Verlag, 1970 P J Hilton, G Mislin, and J Roitberg, Localization of Nilpotent Groups and Spaces, NorthHolland, 1975 D Husemoller, Fibre Bundles, McGraw-Hill, 1966 (later editions by Springer-Verlag) I M James, ed., Handbook of Algebraic Topology, North-Holland, 1995 I M James, ed., History of Topology, North-Holland, 1999 K Jă anich, Topology, Springer-Verlag, 1984 R M Kane, The Homology of Hopf Spaces, North-Holland, 1988 R Kirby and L Siebenmann, Foundational Essays on Topological Manifolds, Smoothings, and Triangulations, Ann of Math Studies 88, 1977 S Kochman, Stable Homotopy Groups of Spheres, Springer Lecture Notes 1423, 1990 S Kochman, Bordism, Stable Homotopy and Adams Spectral Sequences, Fields Institute Monographs 7, A.M.S., 1996 D Kă onig, Theory of Finite and Infinite Graphs, Birkhă auser, 1990 A Lundell and S Weingram, The Topology of CW Complexes, Van Nostrand Reinhold, 1969 S MacLane, Homology, Springer-Verlag, 1963 S MacLane, Categories for the Working Mathematician, Springer-Verlag GTM 5, 1971 I Madsen and R Milgram, The Classifying Spaces for Surgery and Cobordism of Manifolds, Ann of Math Studies 92, 1979 W Massey, Algebraic Topology: An Introduction, Harcourt, Brace & World, 1967 (reprinted by Springer-Verlag) W Massey, A Basic Course in Algebraic Topology, Springer-Verlag, 1993 C R F Maunder, Algebraic Topology, Cambridge Univ Press, 1980 (reprinted by Dover Publications) J P May, Simplicial Objects in Algebraic Topology, Van Nostrand, 1967 (reprinted by Univ Chicago Press) J P May, A Concise Course in Algebraic Topology, Univ Chicago Press, 1999 J Milnor, Topology from the Differentiable Viewpoint, Univ Press of Virginia, 1965 Bibliography 533 J Milnor and J Stasheff, Characteristic Classes, Ann of Math Studies 76, 1974 R Mosher and M Tangora, Cohomology Operations and Applications in Homotopy Theory, Harper and Row, 1968 D Ravenel, Complex Cobordism and Stable Homotopy Groups of Spheres, Academic Press, 1986 D Ravenel, Nilpotence and Periodicity in Stable Homotopy Theory, Ann of Math Studies 128, 1992 E Rees and J D S Jones, eds., Homotopy Theory: Proceeding of the Durham Symposium 1985, Cambridge Univ Press, 1987 D Rolfsen, Knots and Links, Publish or Perish, 1976 H Seifert and W Threlfall, Lehrbuch der Topologie, Teubner, 1934 P Selick, Introduction to Homotopy Theory, Fields Institute Monographs 9, A.M.S., 1997 J.-P Serre, A Course in Arithmetic, Springer-Verlag, 1973 E Spanier, Algebraic Topology, McGraw-Hill, 1966 (reprinted by Springer-Verlag) N Steenrod, The Topology of Fibre Bundles, Princeton Univ Press, 1951 N Steenrod and D Epstein, Cohomology Operations, Ann of Math Studies 50, 1962 R Stong, Notes on Cobordism Theory, Princeton Univ Press, 1968 D Sullivan, Geometric Topology, xeroxed notes from MIT, 1970 R Switzer, Algebraic Topology, Springer-Verlag, 1975 H Toda, Composition Methods in Homotopy Groups of Spheres, Ann of Math Studies 49, 1962 K Varadarajan, The Finiteness Obstruction of C T C Wall, Wiley, 1989 C Weibel, An Introduction to Homological Algebra, Cambridge Univ Press, 1994 G Whitehead, Elements of Homotopy Theory, Springer-Verlag GTM 62, 1978 J Wolf, Spaces of Constant Curvature, Publish or Perish, 1984 Papers J F Adams, On the non-existence of elements of Hopf invariant one, Ann of Math 72 (1960), 20–104 J F Adams, Vector fields on spheres, Ann of Math 75 (1962), 603–632 J F Adams, On the groups J(X) IV, Topology (1966), 21–71 J F Adams and M F Atiyah, K–theory and the Hopf invariant, Quart J Math 17 (1966), 31–38 J F Adams, A variant of E H Brown’s representability theorem, Topology 10 (1971), 185–198 J F Adams and C Wilkerson, Finite H–spaces and algebras over the Steenrod algebra, Ann of Math 111 (1980), 95–143 534 Bibliography M Barratt and J Milnor, An example of anomalous singular homology, Proc A.M.S 13 (1962), 293–297 M Bestvina and N Brady, Morse theory and finiteness properties of groups, Invent Math 129 (1997), 445–470 A Borel, Sur la cohomologie des espaces fibr´ es principaux, Ann of Math 57 (1953), 115–207 R Bott and H Samelson, On the Pontryagin product in spaces of paths, Comment Math Helv 27 (1953), 320–337 R Bott and J Milnor, On the parallelizability of spheres, Bull A.M.S 64 (1958), 87–89 R Bott, The stable homotopy of the classical groups, Ann of Math 70 (1959), 313–337 E Brown and A Copeland, An homology analog of Postnikov systems, Michigan Math J (1959), 313–330 E Brown, Cohomology theories, Ann of Math 75 (1962), 467–484 M Brown, A proof of the generalized Schoenflies theorem, Bull A.M.S 66 (1960), 74–76 J Cannon and G Conner, The combinatorial structure of the Hawaiian earring group, Top and its Appl 106 (2000), 225–271 G Carlsson and R J Milgram, Stable homotopy and iterated loopspaces, pp 505–583 in Handbook of Algebraic Topology, ed I M James, Elsevier 1995 J Davis and R J Milgram, A survey of the spherical space form problem, Math Reports (1985), 223–283, Harwood Acad Pub A Dold and R Thom, Quasifaserungen und unendliche symmetrischen Produkte, Ann of Math 67 (1958), 239–281 C Dowker, Topology of metric complexes, Am J Math 74 (1952), 555–577 W Dwyer, H Miller, and C Wilkerson, Homotopical uniqueness of classifying spaces, Topology 31 (1992), 29–45 W Dwyer and C Wilkerson, A new finite loop space at the prime two, Journal A.M.S (1993), 37–63 E Dyer and A Vasquez, Some small aspherical spaces, J Austral Math Soc 16 (1973), 332– 352 S Eilenberg, Singular homology theory, Ann of Math 45 (1944), 407-447 R Edwards, A contractible, nowhere locally connected compactum, Abstracts A.M.S 20 (1999), 494 S Feder and S Gitler, Mappings of quaternionic projective spaces, Bol Soc Mat Mexicana 18 (1978), 33–37 J Harris and N Kuhn, Stable decompositions of classifying spaces of finite abelian p groups, Math Proc Camb Phil Soc 103 (1988), 427–449 P Hoffman and G Porter, Cohomology realizations of Q [x], Ox Q 24 (1973), 251–255 R Holzsager, Stable splitting of K(G,1), Proc A.M.S 31 (1972), 305–306 Bibliography 535 ¨ ber die Abbildungen der dreidimensionalen Sph¨ are auf die Kugelflă ache, Math Ann H Hopf, U 104 (1931), 637665 I M James, Reduced product spaces, Ann of Math 62 (1955), 170–197 M Kervaire, Non-parallelizability of the n sphere for n > 7, Proc N.A.S 44 (1958), 280–283 S Kochman and M Mahowald, On the computation of stable stems, Contemp Math 181 (1995), 299–316 I Madsen, C B Thomas, and C T C Wall, The topological spherical space form problem, II: existence of free actions, Topology 15 (1976), 375–382 J P May, A general approach to Steenrod operations, Springer Lecture Notes 168 (1970), 153– 231 J P May, Weak equivalences and quasifibrations, Springer Lecture Notes 1425 (1990), 91–101 C Miller, The topology of rotation groups, Ann of Math 57 (1953), 95–110 J Milnor, Construction of universal bundles I, II, Ann of Math 63 (1956), 272–284, 430–436 J Milnor, Groups which act on S n without fixed points, Am J Math 79 (1957), 623–630 J Milnor, Some consequences of a theorem of Bott, Ann of Math 68 (1958), 444–449 J Milnor, On spaces having the homotopy type of a CW complex, Trans A.M.S 90 (1959), 272–280 J Milnor, On axiomatic homology theory, Pac J Math 12 (1962), 337–341 J Milnor and J Moore, On the structure of Hopf algebras, Ann of Math 81 (1965), 211–264 J Milnor, On the Steenrod homology theory, pp 79–96 in Novikov Conjectures, Index Theorems, and Rigidity, ed S Ferry, A Ranicki, and J Rosenberg, Cambridge Univ Press, 1995 J Neisendorfer, Primary homotopy theory, Mem A.M.S 232 (1980) D Notbohm, Spaces with polynomial mod–p cohomology, Math Proc Camb Phil Soc., (1999), 277–292 D Quillen, Rational homotopy theory, Ann of Math 90 (1969), 205–295 D Rector, Loop structures on the homotopy type of S 3, Springer Lecture Notes 249 (1971), 99–105 C Robinson, Moore-Postnikov systems for non-simple fibrations, Ill J Math 16 (1972), 234– 242 P Scott and T Wall, Topological methods in group theory, London Math Soc Lecture Notes 36 (1979), 137–203 J.-P Serre, Homologie singuli` ere des espaces fibr´ es, Ann of Math 54 (1951), 425–505 J.-P Serre, Groupes d’homotopie et classes de groupes ab´ eliens, Ann of Math 58 (1953), 258–294 536 Bibliography A Sieradski, Algebraic topology for two dimensional complexes, pp 51–96 in Two-Dimensional Homotopy and Combinatorial Group Theory, ed C Hog-Angeloni, W Metzler, and A Sieradski., Cambridge Univ Press, 1993 S Shelah, Can the fundamental group of a space be the rationals?, Proc A.M.S 103 (1988), 627–632 J Stallings, A finitely presented group whose dimensional integral homology is not finitely generated, Am J Math 85 (1963), 541–543 D Sullivan, Genetics of homotopy theory and the Adams conjecture, Ann of Math 100 (1974), 1–79 H Toda, Note on the cohomology ring of certain spaces, Proc A.M.S 14 (1963), 89–95 E van Kampen, On the connection between the fundamental groups of some related spaces, Am J Math 55 (1933), 261–267 C T C Wall, Finiteness conditions for CW complexes, Ann of Math 81 (1965), 56–69 G Whitehead, Generalized homology theories, Trans A.M.S 102 (1962), 227–283 J H C Whitehead, Combinatorial homotopy II, Bull A.M.S 55 (1949), 453–496 abelian space 340, 415 boundary 104, 251 action of π1 on πn 340, 343, 419 boundary homomorphism 103, 106, 114 action of π1 on the fiber of a covering space Brouwer 31, 112, 124, 132, 171, 175 67 Brown representability 446 action of a group 69, 455 BSO(n) 438 acyclic space 140 BSU(n) 438 Adams 425 bundle of groups 328 adjoint 393, 460 Burnside problem 78 admissible monomial 497 Ad´ em relations 494, 499 cap product 237 Alexander 129, 175 Cartan formula 487, 488 Alexander duality 252 category 160 Alexander horned sphere 167, 168 Cayley graph, complex 75 amalgamation 454 ˇ Cech cohomology 255 aspherical space 341 ˇ Cech homology 255 attaching cells cell attaching spaces 13, 454 cell complex augmented chain complex 108 cellular approximation theorem 347 cellular chain complex 137 Barratt-Kahn-Priddy theorem 372 cellular cohomology 200 barycenter 117 cellular homology 137, 151 barycentric coordinates 101 cellular map 346 barycentric subdivision 117 chain 103, 106 base space 375 chain complex 104 basepoint 26, 27 chain homotopy 111 basepoint-preserving homotopy 35, 355, 419 chain map 109 basis 40 change of basepoint 27, 339 Betti number 128 characteristic map 7, 517 binomial coefficient 285, 489 circle 28 Bockstein homomorphism 301, 486 classifying space 163 Borel construction 321, 456, 501 closed manifold 229 Borel theorem 283 closure-finite 519 Borsuk–Ulam theorem 32, 38, 174 coboundary 196 Bott periodicity 382, 395 coboundary map 189, 195 538 Index cochain 189, 195 deck transformation 68 cocycle 196 decomposable operation 495 coefficients 151, 159, 196, 460 deformation retraction 2, 35, 344, 521 cofiber 459 deformation retraction, weak 18 cofibration 458 degree 132, 256 cofibration sequence 396, 460 Delta-complex 101 Cohen–Macauley ring 226 cohomology group 189, 196 cohomology operation 486 cohomology ring 209 cohomology theory 200, 312, 446, 452 cohomology with compact supports 240 cohomotopy groups 452 colimit 458, 460 diagonal 281 diagram of spaces 454, 460 dihedral group 73 dimension 6, 124, 229 direct limit 241, 309, 453, 458, 460 directed set 241 divided polynomial algebra 222, 284, 288 collar 251 division algebra 171, 220, 426 commutative diagram 109 dodecahedral group 140 commutative graded ring 213 Dold–Thom theorem 481 commutativity of cup product 213 dominated 526 compact supports 240, 332 dual Hopf algebra 287 compact-open topology 527 compactly generated topology 521, 529 Eckmann–Hilton duality 458 complex of spaces 455, 460, 464 edge 81 compression lemma 344 edgepath 84 cone EHP sequence 472 connected graded algebra 281 connected sum 255 contractible 4, 155 contravariant 161, 199 coproduct 281, 459 covariant 161 covering homotopy property 58 covering space 54, 319, 340, 375 covering space action 70 covering transformation 68 Eilenberg 129 Eilenberg–MacLane space 85, 363, 391, 408, 451, 473 ENR, Euclidean neighborhood retract 525 Euler characteristic 6, 84, 144 Euler class 436, 442 exact sequence 111 excess 497 excision 117, 199, 358 cross product 208, 216, 221, 266, 275, 276 excisive triad 474 cup product 247 Ext 193, 314, 315 CW approximation 350 extension lemma 346 CW complex 5, 517 extension problem 413 CW pair exterior algebra 215, 282 cycle 104 external cup product 208, 216 Index 539 face 101 holim 460 fiber 373 homologous cycles 104 fiber bundle 374, 429 homology 104 fiber homotopy equivalence 404 homology decomposition 463 fiber-preserving map 404 homology of groups 146, 421 fibration 373 homology theory 158, 312, 452 fibration sequence 407, 460 homotopy 3, 25 finitely generated homology 421, 525 homotopy equivalence 3, 10, 36, 344 finitely generated homotopy 362, 390, 421 homotopy extension property 14 five-lemma 127 homotopy fiber 405, 459, 477 fixed point 31, 71, 112, 177, 227, 491 homotopy group 338 flag 434, 445 homotopy group with coefficients 460 frame 299, 379 homotopy lifting property 58, 373, 377 free action 71 homotopy of attaching maps 13, 16 free algebra 225 homotopy type free group 40, 75, 83 Hopf 132, 171, 220, 279, 283 free product 39 Hopf algebra 281 free product with amalgamation 90 Hopf bundle 359, 373, 375, 376, 390 free resolution 191, 261 Hopf invariant 425, 445, 487, 488 Freudenthal suspension theorem 358 Hopf map 377, 378, 383, 425, 428, 472, function space 527 473, 496 functor 161 Hurewicz homomorphism 367, 484 fundamental class 234, 392 Hurewicz theorem 368, 370, 388 fundamental group 26 fundamental theorem of algebra 30 induced fibration 404 induced homomorphism 33, 108, 109, 116, Galois correspondence 61 199, 213 general linear group GLn 291 infinite loopspace 395 good pair 112 invariance of dimension 124 Gram-Schmidt orthogonalization 291, 380 invariance of domain 170 graph 6, 11, 81 inverse limit 310, 408, 460 graph of groups 90 inverse path 27 graph product of groups 90 isomorphism of actions 68 Grassmann manifold 224, 379, 433, 437, isomorphism of covering spaces 65 443 iterated mapping cylinder 455, 464 groups acting on spheres 73, 133, 389 Gysin sequence 436, 442 J(X) , James reduced product 222, 280, 286, 287, 465, 468 H–space 279, 417, 418, 420, 426 J homomorphism 385 HNN extension 91 join 9, 20, 455 hocolim 458, 460 Jordan curve theorem 167 Index 540 K(G, 1) space 85 mapping torus 52, 149, 455 k invariant 410, 473 maximal tree 82 Klein bottle 50, 72, 91, 100 MayerVietoris axiom 447 Kă unneth formula 217, 266, 272, 273, 355, Mayer–Vietoris sequence 147, 157, 159, 201 430 Milnor 406, 407 minimal chain complex 303 Lefschetz 129, 177, 227 Mittag–Leffler condition 318 Lefschetz duality 252 monoid 161 Lefschetz number 177 Moore space 141, 274, 310, 318, 389, 460, lens space 73, 86, 142, 249, 280, 302, 308, 389 Leray–Hirsch theorem 430 463, 473 Moore–Postnikov tower 412 morphism 160 Lie group 280 lift 28, 58 natural transformation 163 lifting criterion 59 naturality 125 lifting problem 413 n connected cover 413 limit 458, 460 n connected space, pair 344 lim-one 311, 409 nerve 255, 456 linking 45 nonsingular pairing 248 local coefficients: cohomology 326, 331 normal covering space 68 local coefficients: homology 326 nullhomotopic local degree 134 local homology 124, 254 object 160 local orientation 232 obstruction 415 local trivialization 375 obstruction theory 414 locally contractible 521, 523 octonion 171, 279, 292, 376, 496 locally finite homology 334 Ω spectrum 394 locally path-connected 60 open cover 457 long exact sequence: cohomology 198 orbit, orbit space 70, 455 long exact sequence: fibration 374 orientable manifold 232 long exact sequence: homology 112, 114, orientable sphere bundle 440 116 orientation 103, 232, 233 long exact sequence: homotopy 342 orientation class 234 loop 26 orthogonal group O(n) 290, 306, 433 loopspace 393, 406, 468 p adic integers 311 manifold 229, 525, 527 path 25 manifold with boundary 250 path lifting property 58 mapping cone 13, 180 pathspace 405 mapping cylinder 2, 180, 345, 455, 459 permutation 66 mapping telescope 136, 310, 455, 526 plus construction 372, 418 Index Poincar´ e 128 relative cycle 113 Poincar´ e conjecture 388 relative homology 113 Poincar´ e duality 239, 243, 251, 333 relative homotopy group 341 Poincar´ e series 228, 435 reparametrization 26 Pontryagin product 285, 296 retraction 3, 35, 112, 146, 523 541 Postnikov tower 352, 408 primary obstruction 417 Schoenflies theorem 167 primitive element 282, 296 semilocally simply-connected 61 principal fibration 410, 418 sheet 59 prism 110 short exact sequence 112, 114 product of CW complexes 8, 522 shrinking wedge 48 product of ∆ complexes 276 shuffle 276 product of paths 26 simplex 9, 100 product of simplices 276 simplicial approximation theorem 175 product space 33, 266, 341, 529 simplicial cohomology 200 projective plane 50, 100, 104, 210, 377 simplicial complex 105 projective space: complex 6, 138, 210, 224, simplicial homology 104, 126 227, 248, 280, 320, 378, 437, 489 projective space: quaternion 212, 224, 228, 248, 320, 376, 378, 437, 489, 490 projective space: real 6, 72, 86, 142, 152, 178, 210, 227, 248, 320, 437, 489 simplicial map 175 simply-connected 28 simply-connected manifold 428 singular complex 106 singular homology 106 properly discontinuous 70 singular simplex 106 pullback 404, 431, 459 skeleton 5, 517 Puppe sequence 396 slant product 278 pushout 459, 464 smash product 10, 221, 268 quasi-circle 77 quasifibration 477 quaternion 73, 171, 279, 292, 445 Quillen 372 quotient CW complex spectrum 452 sphere bundle 440, 442 Spin(n) 289 split exact sequence 145 stable homotopy group 382, 450 stable splitting 489 rank 40, 144 stable stem 382 realization 455 star 176 reduced cohomology 197 Steenrod algebra 494 reduced homology 108 Steenrod homology 255 reduced suspension 12, 393 Steenrod squares, powers 485 rel 3, 16 Stiefel manifold 299, 379, 434, 445, 491 relative boundary 113 subcomplex 7, 518 relative cohomology 197 subgraph 82 542 Index surface 49, 86, 91, 100, 139, 205, 239, 388 transfer homomorphism 173, 319 suspension 8, 135, 221, 464, 471 tree 82 suspension spectrum 452 triple 116, 342 symmetric polynomials 433 truncated polynomial algebra 282 symmetric product 280, 363, 479 symplectic group Sp(n) 224, 380, 432 unique lifting property 60 unitary group U (n) 224, 380, 432 tensor algebra 286, 469 universal coefficient theorem 193, 262, 461 tensor product 216, 326 universal cover 57, 66 tensor product of chain complexes 271 Thom class 439, 508 van Kampen 41 Thom isomorphism 439 vector field 133, 491 Thom space 439, 508 vertex 81, 101 Toda bracket 385 topological group 279 weak homotopy equivalence 350 Tor 261, 265 weak topology 5, 81, 519 torsion coefficient 128 wedge sum 10, 42, 124, 158, 200, 378, 464 torus 33, 72, 100, 104, 225 Whitehead product 379, 428 torus knot 46 Whitehead tower 354 total space 375 Whitehead’s theorem 344, 365, 416 ... introduction to algebraic topology with rather broad coverage of the subject The viewpoint is quite classical in spirit, and stays well within the confines of pure algebraic topology In a sense,... subject The geometry of algebraic topology is so pretty, it would seem a pity to slight it and to miss all the intuition it provides At the elementary level, algebraic topology separates naturally... ft (x) for the extended ft Algebraic topology can be roughly defined as the study of techniques for forming algebraic images of topological spaces Most often these algebraic images are groups,