Algebraic tropology allen hatcher

Allen Hatcher Copyright c 2002 by Cambridge University Press Single paper or electronic copies for noncommercial use may be made freely without explicit permission from the author or publisher 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 29 Induced Homomorphisms 34 1.2 Van Kampen’s Theorem 40 Free Products of Groups 41 The van Kampen Theorem 43 Applications to Cell Complexes 50 1.3 Covering Spaces Lifting Properties 60 The Classification of Covering Spaces 63 Deck Transformations and Group Actions 70 Additional Topics 1.A Graphs and Free Groups 83 1.B K(G,1) Spaces and Graphs of Groups 87 56 Chapter Homology 2.1 Simplicial and Singular Homology 97 102 ∆ Complexes 102 Simplicial Homology 104 Singular Homology 108 Homotopy Invariance 110 Exact Sequences and Excision 113 The Equivalence of Simplicial and Singular Homology 128 2.2 Computations and Applications 134 Degree 134 Cellular Homology 137 Mayer-Vietoris Sequences 149 Homology with Coefficients 153 2.3 The Formal Viewpoint 160 Axioms for Homology 160 Categories and Functors 162 Additional Topics 2.A Homology and Fundamental Group 166 2.B Classical Applications 169 2.C Simplicial Approximation 177 Chapter Cohomology 185 3.1 Cohomology Groups 190 The Universal Coefficient Theorem 190 Cohomology of Spaces 197 3.2 Cup Product 206 The Cohomology Ring 211 A Kă unneth Formula 218 Spaces with Polynomial Cohomology 224 3.3 Poincar´ e Duality 230 Orientations and Homology 233 The Duality Theorem 239 Connection with Cup Product 249 Other Forms of Duality 252 Additional Topics 3.A Universal Coefficients for Homology 261 3.B The General Kă unneth Formula 268 3.C H–Spaces and Hopf Algebras 281 3.D The Cohomology of SO(n) 292 3.E Bockstein Homomorphisms 303 3.F Limits and Ext 311 3.G Transfer Homomorphisms 321 3.H Local Coefficients 327 Chapter Homotopy Theory 4.1 Homotopy Groups 337 339 Definitions and Basic Constructions 340 Whitehead’s Theorem 346 Cellular Approximation 348 CW Approximation 352 4.2 Elementary Methods of Calculation 360 Excision for Homotopy Groups 360 The Hurewicz Theorem 366 Fiber Bundles 375 Stable Homotopy Groups 384 4.3 Connections with Cohomology 393 The Homotopy Construction of Cohomology 393 Fibrations 405 Postnikov Towers 410 Obstruction Theory 415 Additional Topics 4.A Basepoints and Homotopy 421 4.B The Hopf Invariant 427 4.C Minimal Cell Structures 429 4.D Cohomology of Fiber Bundles 431 4.E The Brown Representability Theorem 448 4.F Spectra and Homology Theories 452 4.G Gluing Constructions 456 4.H Eckmann-Hilton Duality 460 4.I Stable Splittings of Spaces 466 4.J The Loopspace of a Suspension 470 4.K The Dold-Thom Theorem 475 4.L Steenrod Squares and Powers 487 Appendix 519 Topology of Cell Complexes 519 The Compact-Open Topology 529 Bibliography Index 533 539 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 There is, however, one small novelty in the exposition that may be worth commenting upon, even though in the book as a whole it plays a relatively minor role This is the use of what we call ∆ complexes, which are a mild generalization of the classical notion of a simplicial complex The idea is to decompose a space into simplices allowing different faces of a simplex to coincide and dropping the requirement that simplices are uniquely determined by their vertices 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, 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 Historically, ∆ complexes were first introduced by Eilenberg and Zilber in 1950 under the name of semisimplicial complexes This term later came to mean something different, however, and the original notion seems to have been largely ignored since 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, including the correction of many small errors both typographical and mathematical found by careful readers of earlier versions, it is inevitable that some errors remain, so the web page will include a list of corrections to the printed version With the electronic version of the book it will be possible not only to incorporate corrections but also to make more substantial revisions and additions Readers are encouraged to send comments and suggestions as well as corrections to the email address posted on the web page xii Standard Notations Z , Q , R , C , H , O — the integers, rationals, reals, complexes, quaternions, and octonions Zn — the integers mod n Rn — n dimensional Euclidean space Cn — complex n space In particular, R0 = {0} = C0 , zero-dimensional vector spaces I = [0, 1] — the unit interval S n — the unit sphere in Rn+1 , all points of distance from the origin D n — the unit disk or ball in Rn , all points of distance ≤ from the origin ∂D n = S n−1 — the boundary of the n disk en — an n cell, homeomorphic to the open n disk D n − ∂D n In particular, D and e0 consist of a single point since R0 = {0} But S consists of two points since it is ∂D 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 A − B — set-theoretic difference, all points in A that are not in B iff — if and only if The aim of this short preliminary chapter is to introduce a few of the most common 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 between 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 equivalent if they have ‘the same shape’ in a sense that is much broader than homeomorphism 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 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 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 not move We can think of this shrinking process as taking place during a time interval ≤ t ≤ , and then it defines a family of functions ft : X→X parametrized by t ∈ I = [0, 1] , where ft (x) is the point to which a given point x ∈ X has moved at time t Appendix 530 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 531 (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 Appendix 532 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 Proposition A.17 If f : X →Y is a quotient map then so is f × 11 : X × Z →Y × Z whenever Z is locally compact This can be applied when Z = I to show that a homotopy defined on a quotient space is continuous Proof: Consider the diagram at the right, where W is Y × Z with the quotient topology from X × Z , with g the quotient map and h the identity Every open set in Y × Z is open in f × 11 −−−−− X ×Z − − − − − − − − − − − → Y ×Z g −−→ W −−− →−−−h W since f × 11 is continuous, so it will suffice to show that h is continuous Since g is continuous, so is the associated map g : X →W Z , by Proposition A.14 This implies that h : Y →W Z is continuous since f is a quotient map Applying Proposition A.14 again, we conclude that h is continuous 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.) 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fiber 69 Brown representability 448 action of a group 71, 457 BSO (n) 440 acyclic space 142 BSU (n) 440 Adams 427 bundle of groups 330 Adem relations 496, 501 Burnside problem 80 adjoint 395, 462 admissible monomial 499 cap product 239 Alexander 131, 177 Cartan formula 489, 490 Alexander duality 254 category 162 Alexander horned sphere 169, 170 Cayley graph, complex 77 amalgamation 456 ˇ Cech cohomology 257 aspherical space 343 ˇ Cech homology 257 attaching cells cell attaching spaces 13, 456 cell complex augmented chain complex 110 cellular approximation theorem 349 cellular chain complex 139 Barratt-Kahn-Priddy theorem 374 cellular cohomology 202 barycenter 119 cellular homology 139, 153 barycentric coordinates 103 cellular map 157, 270, 349 barycentric subdivision 119 chain 105, 108 base space 377 chain complex 106 basepoint 26, 28 chain homotopy 113 basepoint-preserving homotopy 36, 357, 421 chain map 111 basis 42 change of basepoint 28, 341 Betti number 130 characteristic map 7, 519 binomial coefficient 287, 491 circle 29 Bockstein homomorphism 303, 488 classifying space 165 Borel construction 323, 458, 503 closed manifold 231 Borel theorem 285 closure-finite 521 Borsuk–Ulam theorem 32, 38, 176 coboundary 198 Bott periodicity 384, 397 coboundary map 191, 197 boundary 106, 253 cochain 191, 197 540 Index cocycle 198 deck transformation 70 coefficients 153, 161, 198, 462 decomposable operation 497 cofiber 461 deformation retraction 2, 36, 346, 523 cofibration 460 deformation retraction, weak 18 cofibration sequence 398, 462 degree 134, 258 Cohen–Macaulay ring 228 ∆ complex (Delta-complex) 103 cohomology group 191, 198 diagonal 283 cohomology operation 488 diagram of spaces 456, 462 cohomology ring 211 dihedral group 75 cohomology theory 202, 314, 448, 454 dimension 6, 126, 231 cohomology with compact supports 242 cohomotopy groups 454 colimit 460, 462 collar 253 commutative diagram 111 commutative graded ring 215 commutativity of cup product 215 compact supports 242, 334 compact-open topology 529 direct limit 243, 311, 455, 460, 462 directed set 243 divided polynomial algebra 224, 286, 290 division algebra 173, 222, 428 dodecahedral group 142 Dold–Thom theorem 483 dominated 528 dual Hopf algebra 289 compactly generated topology 523, 531 complex of spaces 457, 462, 466 compression lemma 346 cone connected graded algebra 283 connected sum 257 contractible 4, 157 contravariant 163, 201 coproduct 283, 461 covariant 163 Eckmann–Hilton duality 460 edge 83 edgepath 86 EHP sequence 474 Eilenberg 131 Eilenberg–MacLane space 87, 365, 393, 410, 453, 475 ENR, Euclidean neighborhood retract 527 Euler characteristic 6, 86, 146 covering homotopy property 60 Euler class 438, 444 covering space 56, 321, 342, 377 exact sequence 113 covering space action 72 excess 499 covering transformation 70 excision 119, 201, 360 cross product 210, 218, 223, 268, 277, 278 excisive triad 476 cup product 249 Ext 195, 316, 317 CW approximation 352 extension lemma 348 CW complex 5, 519 extension problem 415 CW pair exterior algebra 217, 284 cycle 106 external cup product 210, 218 Index 541 face 103 holim 462 fiber 375 homologous cycles 106 fiber bundle 376, 431 homology 106 fiber homotopy equivalence 406 homology decomposition 465 fiber-preserving map 406 homology of groups 148, 423 fibration 375 homology theory 160, 314, 454 fibration sequence 409, 462 homotopy 3, 25 finitely generated homology 423, 527 homotopy equivalence 3, 10, 36, 346 finitely generated homotopy 364, 392, 423 homotopy extension property 14 five-lemma 129 homotopy fiber 407, 461, 479 fixed point 31, 73, 114, 179, 229, 493 homotopy group 340 flag 436, 447 homotopy group with coefficients 462 frame 301, 381 homotopy lifting property 60, 375, 379 free action 73 homotopy of attaching maps 13, 16 free algebra 227 homotopy type free group 42, 77, 85 Hopf 134, 173, 222, 281, 285 free product 41 Hopf algebra 283 free product with amalgamation 92 Hopf bundle 361, 375, 377, 378, 392 free resolution 193, 263 Hopf invariant 427, 447, 489, 490 Freudenthal suspension theorem 360 Hopf map 379, 380, 385, 427, 430, 474, function space 529 475, 498 functor 163 Hurewicz homomorphism 369, 486 fundamental class 236, 394 Hurewicz theorem 370, 372, 390 fundamental group 26 fundamental theorem of algebra 31 induced fibration 406 induced homomorphism 34, 110, 111, 118, Galois correspondence 63 201, 215 general linear group GLn 293 infinite loopspace 397 good pair 114 invariance of dimension 126 Gram-Schmidt orthogonalization 293, 382 invariance of domain 172 graph 6, 11, 83 inverse limit 312, 410, 462 graph of groups 92 inverse path 27 graph product of groups 92 isomorphism of actions 70 Grassmann manifold 226, 381, 435, 439, isomorphism of covering spaces 67 445 iterated mapping cylinder 457, 466 groups acting on spheres 75, 135, 391 Gysin sequence 438, 444 J (X ), James reduced product 224, 282, 288, 289, 467, 470 H–space 281, 419, 420, 422, 428 J –homomorphism 387 HNN extension 93 join 9, 20, 457 hocolim 460, 462 Jordan curve theorem 169 Index 542 K (G,1) space 87 mapping torus 53, 151, 457 k invariant 412, 475 maximal tree 84 Klein bottle 51, 74, 93, 102 Mayer–Vietoris axiom 449 Kă unneth formula 219, 268, 274, 275, 357, MayerVietoris sequence 149, 159, 161, 203 432 Milnor 408, 409 minimal chain complex 305 Lefschetz 131, 179, 229 Mittag–Leffler condition 320 Lefschetz duality 254 monoid 163 Lefschetz number 179 Moore space 143, 277, 312, 320, 391, 462, lens space 75, 88, 144, 251, 282, 304, 310, 391 Leray–Hirsch theorem 432 465, 475 Moore–Postnikov tower 414 morphism 162 Lie group 282 lift 29, 60 natural transformation 165 lifting criterion 61 naturality 127 lifting problem 415 n connected cover 415 limit 460, 462 n connected space, pair 346 lim-one 313, 411 nerve 257, 458 linking 46 nonsingular pairing 250 local coefficients: cohomology 328, 333 normal covering space 70 local coefficients: homology 328 nullhomotopic local degree 136 local homology 126, 256 object 162 local orientation 234 obstruction 417 local trivialization 377 obstruction theory 416 locally contractible 523, 525 octonion 173, 281, 294, 378, 498 locally finite homology 336 Ω spectrum 396 locally path-connected 62 open cover 459 long exact sequence: cohomology 200 orbit, orbit space 72, 457 long exact sequence: fibration 376 orientable manifold 234 long exact sequence: homology 114, 116, orientable sphere bundle 442 118 orientation 105, 234, 235 long exact sequence: homotopy 344 orientation class 236 loop 26 orthogonal group O (n) 292, 308, 435 loopspace 395, 408, 470 p adic integers 313 manifold 231, 527, 529 path 25 manifold with boundary 252 path lifting property 60 mapping cone 13, 182 pathspace 407 mapping cylinder 2, 182, 347, 457, 461 permutation 68 mapping telescope 138, 312, 457, 528 plus construction 374, 420 Index Poincar´ e 130 relative cycle 115 Poincar´ e conjecture 390 relative homology 115 Poincar´ e duality 241, 245, 253, 335 relative homotopy group 343 Poincar´ e series 230, 437 reparametrization 27 Pontryagin product 287, 298 retraction 3, 36, 114, 148, 525 543 Postnikov tower 354, 410 primary obstruction 419 Schoenflies theorem 169 primitive element 284, 298 semilocally simply-connected 63 principal fibration 412, 420 sheet 61 prism 112 short exact sequence 114, 116 product of CW complexes 8, 524 shrinking wedge 49, 54, 63, 79, 258 product of ∆ complexes 278 shuffle 278 product of paths 26 simplex 9, 102 product of simplices 278 simplicial approximation theorem 177 product space 34, 268, 343, 531 simplicial cohomology 202 projective plane 51, 102, 106, 212, 379 simplicial complex 107 projective space: complex 6, 140, 212, 226, simplicial homology 106, 128 229, 250, 282, 322, 380, 439, 491 projective space: quaternion 214, 226, 230, 250, 322, 378, 380, 439, 491, 492 projective space: real 6, 74, 88, 144, 154, 180, 212, 229, 250, 322, 439, 491 simplicial map 177 simply-connected 28 simply-connected manifold 430 singular complex 108 singular homology 108 properly discontinuous 72 singular simplex 108 pullback 406, 433, 461 skeleton 5, 519 Puppe sequence 398 slant product 280 pushout 461, 466 smash product 10, 223, 270 quasi-circle 79 quasifibration 479 quaternion 75, 173, 281, 294, 446 Quillen 374 quotient CW complex spectrum 454 sphere bundle 442, 444 Spin(n) 291 split exact sequence 147 stable homotopy group 384, 452 stable splitting 491 rank 42, 146 stable stem 384 realization 457 star 178 reduced cohomology 199 Steenrod algebra 496 reduced homology 110 Steenrod homology 257 reduced suspension 12, 395 Steenrod squares, powers 487 rel 3, 16 Stiefel manifold 301, 381, 436, 447, 493 relative boundary 115 subcomplex 7, 520 relative cohomology 199 subgraph 84 544 Index surface 5, 51, 88, 93, 102, 141, 207, 241, 390 transfer homomorphism 175, 321 tree 84 suspension 8, 137, 223, 466, 473 triple 118, 344 suspension spectrum 454 truncated polynomial algebra 284 symmetric polynomials 435 symmetric product 282, 365, 481 symplectic group Sp (n) 226, 382, 434 unique lifting property 62 unitary group U (n) 226, 382, 434 universal coefficient theorem 195, 264, 463 tensor algebra 288, 471 tensor product 218, 328 tensor product of chain complexes 273 Thom class 441, 510 Thom isomorphism 441 universal cover 59, 68 van Kampen 43 vector field 135, 493 vertex 83, 103 Thom space 441, 510 weak homotopy equivalence 352 Toda bracket 387 weak topology 5, 83, 521 topological group 281 wedge sum 10, 43, 126, 160, 202, 380, 466 Tor 263, 267 Whitehead product 381, 430 torsion coefficient 130 Whitehead tower 356 torus 34, 74, 102, 106, 227 Whitehead’s theorem 346, 367, 418 torus knot 47 Wirtinger presentation 55 total space 377 ... 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,... 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, ... 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