Algebraic Topology Math

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 49 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 212 A K¨ unneth Formula 214 Spaces with Polynomial Cohomology 220 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 K¨ 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 The Homotopy Extension Property 532 Simplicial CW Structures 533 Bibliography Index 539 545 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 There is also an Appendix dealing mainly with a number of matters of a pointset topological nature that arise in algebraic topology Since this is a textbook on algebraic topology, details involving point-set topology are often treated lightly or skipped entirely in the body of the text 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 [J¨ 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 ped- agogical viewpoint since it simplifies 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 Historically, ∆ complexes were first introduced by xi Preface Eilenberg and Zilber in 1950 under the name of semisimplicial complexes Soon after this, additional structure in the form of certain ‘degeneracy maps’ was introduced, leading to a very useful class of objects that came to be called simplicial sets The semisimplicial complexes of Eilenberg and Zilber then became ‘semisimplicial sets’, but in this book we have chosen to use the shorter term ‘ ∆ complex’ 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 includes 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 Note on the 2015 reprinting A large number of corrections are included in this reprinting In addition there are two places in the book where the material was rearranged to an extent requiring renumbering of theorems, etc In §3.2 starting on page 210 the renumbering is the following: old 3.11 3.12 3.13 3.14 3.15 3.16 3.17 3.18 3.19 3.20 3.21 new 3.16 3.19 3.14 3.11 3.13 3.15 3.20 3.16 3.17 3.21 3.18 And in §4.1 the following renumbering occurs in pages 352–355: old 4.13 4.14 4.15 4.16 4.17 new 4.17 4.13 4.14 4.15 4.16 xii Standard Notations Z , Q , R , C , H , O : the integers, rationals, reals, complexes, quaternions, and octonions Zn : the integers mod n Rn : C n n dimensional Euclidean space : 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 : the inclusion map A→B when A ⊂ B A − B : set-theoretic difference, all points in A that are not in B iff : if and only if There are also a few notations used in this book that are not completely standard The union of a set X with a family of sets Yi , with i ranging over some index set, is usually written simply as X ∪i Yi rather than something more elaborate such as X∪ i Yi Intersections and other similar operations are treated in the same way Definitions of mathematical terms are generally given within paragraphs of text, rather than displayed separately like theorems, and these definitions are indicated by the use of boldface type for the term being defined Some authors use italics for this purpose, but in this book italics usually denote simply emphasis, as in standard written prose Each term defined using the boldface convention is listed in the Index, with the page number where the definition occurs 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 regions in the plane bounded by one or more 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 Naturally we would like ft (x) to depend continuously on both t and x , and this will 536 Appendix Simplicial CW Structures the order-preserving injections, to the category of sets, namely the functor sending ∆n to Xn and the injection g to g ∗ Such a functor is exactly equivalent to a ∆ complex Explicitly, we can reconstruct the ∆ complex X from the functor by setting X= n ∗ n (Xn × ∆ )/(g (x), y) ∼ (x, g∗ (y)) for (x, y) ∈ Xn × ∆k , where g∗ is the linear inclusion ∆k →∆n sending the i th vertex of ∆k to the g(i) th vertex of ∆n , and we perform the indicated identifications letting g range over all order-preserving injections ∆k →∆n If we wish to generalize this to s∆ complexes, we will have to consider surjective linear maps ∆k →∆n as well as injections This corresponds to considering order- preserving surjections ∆k →∆n in addition to injections Every map of sets decomposes canonically as a surjection followed by an injection, so we may as well consider arbitrary order-preserving maps ∆k →∆n These form the morphisms in a category ∆∗ , with objects the ∆n ’s We are thus led to consider contravariant functors from ∆∗ to the category of sets Such a functor is called a simplicial set This terminology has the virtue that one can immediately define, for example, a simplicial group to be a contravariant functor from ∆∗ to the category of groups, and similarly for simplicial rings, simplicial modules, and so on One can even define simplicial spaces as contravariant functors from ∆∗ to the category of topological spaces and continuous maps For any space X there is an associated rather large simplicial set S(X) , the sin- gular complex of X , whose n simplices are all the continuous maps ∆n →X For a morphism g : ∆k →∆n the induced map g ∗ from n simplices of S(X) to k simplices of S(X) is obtained by composition with g∗ : ∆k →∆n We introduced S(X) in §2.1 in connection with the definition of singular homology and described it as a ∆ complex, but in fact it has the additional structure of a simplicial set In a similar but more restricted way, an s∆ complex X gives rise to a simplicial set ∆(X) whose k simplices are all the simplicial maps ∆k →X These are uniquely expressible as compositions σα q : ∆k →∆n →X of simplicial surjections q (preserv- ing orderings of vertices) with characteristic maps of simplices of X The maps g ∗ are obtained just as for S(X) , by composition with the maps g∗ : ∆k →∆n These examples ∆(X) in fact account for all simplicial sets: Proposition A.19 Every simplicial set is isomorphic to one of the form ∆(X) for some s∆ complex X which is unique up to isomorphism Here an isomorphism of simplicial sets means an isomorphism in the category of simplicial sets, where the morphisms are natural transformations between contravariant functors from ∆∗ to the category of sets This translates into just what one would expect, maps sending n simplices to n simplices that commute with the maps g ∗ Note that the proposition implies in particular that a nonempty simplicial Simplicial CW Structures Appendix 537 set contains simplices of all dimensions since this is evidently true for ∆(X) This is also easy to deduce directly from the definition of a simplicial set Thus simplicial sets are in a certain sense large infinite objects, but the proposition says that their essential geometrical core, an s∆ complex, can be much smaller Proof: Let Y be a simplicial set, with Yn its set of n simplices A simplex τ in Yn is called degenerate if it is in the image of g ∗ : Yk →Yn for some noninjective g : ∆n →∆k Since g can be factored as a surjection followed by an injection, there is no loss in requiring g to be surjective For example, in ∆(X) the degenerate simplices are those that are the simplicial maps ∆n →X that are not injective on the interior of ∆n Thus the main difference between X and ∆(X) is the degenerate simplices Every degenerate simplex of Y has the form g ∗ (τ) for some nondegenerate sim- plex τ and surjection g : ∆n →∆k We claim that such a g and τ are unique For suppose we have g1∗ (τ1 ) = g2∗ (τ2 ) with τ1 and τ2 nondegenerate and g1 : ∆n →∆k1 and g2 : ∆n →∆k2 surjective Choose order-preserving injections h1 : ∆k1 →∆n and h2 : ∆k2 →∆n with g1 h1 = 11 and g2 h2 = 11 Then g1∗ (τ1 ) = g2∗ (τ2 ) implies that ∗ ∗ ∗ ∗ ∗ ∗ ∗ h∗ g1 (τ1 ) = h2 g2 (τ2 ) = τ2 and h1 g2 (τ2 ) = h1 g1 (τ1 ) = τ1 , so the nondegeneracy of τ1 and τ2 implies that g1 h2 and g2 h1 are injective This in turn implies that k1 = k2 and g1 h2 = 11 = g2 h1 , hence τ1 = τ2 If g1 ≠ g2 then g1 (i) ≠ g2 (i) for some i , and if we choose h1 so that h1 g1 (i) = i , then g2 h1 g1 (i) = g2 (i) ≠ g1 (i) , contradicting g2 h1 = 11 and finishing the proof of the claim Just as we reconstructed a ∆ complex from its categorical description, we can associate to the simplicial set Y an s∆ complex |Y | , its geometric realization, by setting |Y | = n ∗ n (Yn × ∆ )/(g (y), z) ∼ (y, g∗ (z)) for (y, z) ∈ Yn × ∆k and g : ∆k →∆n Since every g factors canonically as a surjec- tion followed by an injection, it suffices to perform the indicated identifications just when g is a surjection or an injection Letting g range over surjections amounts to collapsing each simplex onto a unique nondegenerate simplex by a unique projection, by the claim in the preceding paragraph, so after performing the identifications just for surjections we obtain a collection of disjoint simplices, with one n simplex for each nondegenerate n simplex of Y Then doing the identifications as g varies over injections attaches these nondegenerate simplices together to form an s∆ complex, which is |Y | The quotient map from the collection of disjoint simplices to |Y | gives the collection of distinguished characteristic maps for the cells of |Y | If we start with an s∆ complex X and form |∆(X)| , then this is clearly the same as X In the other direction, if we start with a simplicial set Y and form ∆(|Y |) then there is an evident bijection between the n simplices of these two simplicial sets, and this commutes with the maps g ∗ so the two simplicial sets are equivalent ⊓ ⊔ 538 Appendix Simplicial CW Structures As we observed in the preceding proof, the geometric realization |Y | of a simplicial set Y can be built in two stages, by first collapsing all degenerate simplices by making the identifications (g ∗ (y), z) ∼ (y, g∗ (z)) as g ranges over surjections, and then glueing together these nondegenerate simplices by letting g range over injections We could equally well perform these two types of identifications in the opposite order If we first the identifications for injections, this amounts to regarding Y as a category-theoretic ∆ complex Y∆ by restricting Y , regarded as a functor from ∆∗ to sets, to the subcategory of ∆∗ consisting of injective maps, and then taking the geometric realization |Y∆ | to produce a geometric ∆ complex After doing this, if we perform the identifications for surjections g we obtain a natural quotient map |Y∆ |→|Y | This is a homotopy equivalence, but we will not prove this fact here The ∆ complex |Y∆ | is sometimes called the thick geometric realization of Y Since simplicial sets are very combinatorial objects, many standard constructions can be performed on them A good example is products For simplicial sets X and Y there is an easily-defined product simplicial set X × Y , having (X × Y )n = Xn × Yn and g ∗ (x, y) = g ∗ (x), g ∗ (y) The nice surprise about this definition is that it is compatible with geometric realization: the realization |X × Y | turns out to be homeomorphic to |X|× |Y | , the product of the CW complexes |X| and |Y | (with the compactly generated CW topology) The homeomorphism is just the product of the maps |X × Y |→|X| and |X × Y |→|Y | induced by the projections of X × Y onto its two factors As a very simple example, consider the case that X and Y are both ∆(∆1 ) Letting [v0 , v1 ] and [w0 , w1 ] be the two copies of ∆1 , the product X × Y has two nondegenerate simplices: ([v0 , v1 , v1 ], [w0 , w0 , w1 ]) = [(v0 , w0 ), (v1 , w0 ), (v1 , w1 )] ([v0 , v0 , v1 ], [w0 , w1 , w1 ]) = [(v0 , w0 ), (v0 , w1 ), (v1 , w1 )] These subdivide the square ∆1 × ∆1 into two simplices There are five nondegenerate simplices in X × Y , as shown in the figure One of these, the diagonal of the square, is the pair ([v0 , v1 ], [w0 , w1 ]) formed by the two nondegenerate simplices [v0 , v1 ] and [w0 , w1 ] , while the other four are pairs like ([v0 , v0 ], [w0 , w1 ]) where one factor is a degenerate simplex and the other is a nondegenerate simplex Obviously there are no nondegenerate n simplices in X × Y for n > It is not hard to see how this example generalizes to the product ∆p × ∆q Here one obtains the subdivision of the product into (p + q) simplices described in §3.B in terms of the shuffling operation Once one understands the case of a 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attaching spaces 13, 456 cell complex augmented chain complex 110 cellular approximation theorem 349 cellular chain complex 139 Barratt-Priddy-Quillen 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, 197 546 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 212 dihedral group 75 cohomology theory 202, 314, 448, 454 dimension 6, 126, 231 cohomology with compact supports 242 direct limit 243, 311, 455, 460, 462 cohomotopy groups 454 directed set 243 colimit 460, 462 divided polynomial algebra 224, 286, 290 collar 253 division algebra 173, 223, 428 commutative diagram 111 dodecahedral group 142 commutative graded ring 213 Dold–Thom theorem 483 commutativity of cup product 210 dominated 528 compact supports 242, 334 dual Hopf algebra 289 compact-open topology 529 compactly generated topology 523, 531 Eckmann–Hilton duality 460 complex of spaces 457, 462, 466 edge 83 compression lemma 346 edgepath 86 cone EHP sequence 474 connected graded algebra 283 Eilenberg 131 connected sum 257 Eilenberg–MacLane space 87, 365, 393, 410, contractible 4, 157 453, 475 contravariant 163, 201 ENR, Euclidean neighborhood retract 527 coproduct 283, 461 Euler characteristic 6, 86, 146 covariant 163 Euler class 438, 444 covering homotopy property 60 evenly covered 29, 56 covering space 29, 56, 321, 342, 377 exact sequence 113 covering space action 72 excess 499 covering transformation 70 excision 119, 201, 360 cross product 214, 219, 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 213, 284 cycle 106 external cup product 214 Index 547 face 103 H–space 281, 342, 419, 420, 422, 428 fiber 375 HNN extension 93 fiber bundle 376, 431 hocolim 460, 462 fiber homotopy equivalence 406 holim 462 fiber-preserving map 406 homologous cycles 106 fibration 375 homology 106 fibration sequence 409, 462 homology decomposition 465 finitely generated homology 423, 527 homology of groups 148, 423 finitely generated homotopy 364, 392, 423 homology theory 160, 314, 454 five-lemma 129 homotopy 3, 25 fixed point 31, 73, 114, 179, 229, 493 homotopy equivalence 3, 10, 36, 346 flag 436, 447 homotopy extension property 14 frame 301, 381 homotopy fiber 407, 461, 479 free action 73 homotopy group 340 free algebra 227 homotopy group with coefficients 462 free group 42, 77, 85 homotopy lifting property 60, 375, 379 free product 41 homotopy of attaching maps 13, 16 free product with amalgamation 92 homotopy type free resolution 193, 263 Hopf 134, 173, 222, 281, 285 Freudenthal suspension theorem 360 Hopf algebra 283 function space 529 Hopf bundle 361, 375, 377, 378, 392 functor 163 Hopf invariant 427, 447, 489, 490 fundamental class 236, 394 Hopf map 379, 380, 385, 427, 430, 474, fundamental group 26 fundamental theorem of algebra 31 475, 498 Hurewicz homomorphism 369, 486 Hurewicz theorem 366, 371, 390 Galois correspondence 63 general linear group GLn 293 induced fibration 406 good pair 114 induced homomorphism 34, 110, 111, 118, graded ring 212 201, 213 Gram-Schmidt orthogonalization 293, 382 infinite loopspace 397 graph 6, 11, 83 invariance of dimension 126 graph of groups 92 invariance of domain 172 graph product of groups 92 inverse limit 312, 410, 462 Grassmann manifold 227, 381, 435, 439, inverse path 27 445 isomorphism of actions 70 groups acting on spheres 75, 135, 391 isomorphism of covering spaces 67 Gysin sequence 438, 444 iterated mapping cylinder 457, 466 548 Index J (X ), James reduced product 224, 282, 288, 289, 467, 470 J –homomorphism 387 long exact sequence: homotopy 344 loop 26 loopspace 395, 408, 470 join 9, 20, 457, 467 Jordan curve theorem 169 manifold 231, 527, 529 manifold with boundary 252 K (G,1) space 87 mapping cone 13, 182 k invariant 412, 475 mapping cylinder 2, 182, 347, 457, 461 Klein bottle 51, 74, 93, 102 mapping telescope 138, 312, 457, 528 K¨ unneth formula 216, 268, 274, 275, 357, mapping torus 53, 151, 457 432 maximal tree 84 Mayer–Vietoris axiom 449 Lefschetz 131, 179, 229 Lefschetz duality 254 Lefschetz number 179 lens space 75, 88, 144, 251, 282, 304, 310, 391 Leray–Hirsch theorem 432 Lie group 282 lift 29, 60 lifting criterion 61 Mayer–Vietoris sequence 149, 159, 161, 203 Milnor 408, 409 minimal chain complex 305 Mittag–Leffler condition 320 monoid 163 Moore space 143, 277, 312, 320, 391, 462, 465, 475 Moore–Postnikov tower 414 morphism 162 lifting problem 415 limit 460, 462 natural transformation 165 lim-one 313, 411 naturality 127 linking 46 n connected cover 415 local coefficients: cohomology 328, 333 n connected space, pair 346 local coefficients: homology 328 nerve 257, 458 local degree 136 nonsingular pairing 250 local homology 126, 256 normal covering space 70 local orientation 234 nullhomotopic local trivialization 377 locally 62 object 162 locally compact 530 obstruction 417 locally contractible 63, 179, 254, 256, 522, obstruction theory 416 525 locally finite homology 336 octonion 173, 281, 294, 378, 498 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 Index orientation class 236 rank 42, 146 orthogonal group O (n) 292, 308, 435 realization 457 549 reduced cohomology 199 p adic integers 313 path 25 path lifting property 60 pathspace 407 permutation 68 plus construction 374, 420 Poincar´ e 130 Poincar´ e conjecture 390 Poincar´ e duality 241, 245, 253, 335 Poincar´ e series 230, 437 Pontryagin product 287, 298 Postnikov tower 354, 410 primary obstruction 419 primitive element 284, 298 principal fibration 412, 420 prism 112 product of CW complexes 8, 524 product of ∆ complexes 278 product of paths 26 product of simplices 278 product space 34, 268, 343, 531 projective plane 51, 102, 106, 208, 379 projective space: complex 6, 140, 212, 226, 230, 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, 230, 250, 322, 439, 491 properly discontinuous 72 pullback 406, 433, 461 Puppe sequence 398 pushout 461, 466 reduced homology 110 reduced suspension 12, 395 rel 3, 16 relative boundary 115 relative cohomology 199 relative cycle 115 relative homology 115 relative homotopy group 343 reparametrization 27 retraction 3, 36, 114, 148, 525 Schoenflies theorem 169 section 235, 438, 503 semilocally simply-connected 63 sheet 56, 61 short exact sequence 114, 116 shrinking wedge 49, 54, 63, 79, 258 shuffle 278 simplex 9, 102 simplicial approximation theorem 177 simplicial cohomology 202 simplicial complex 107 simplicial homology 106, 128 simplicial map 177 simply-connected 28 simply-connected manifold 430 singular complex 108 singular homology 108 singular simplex 108 skeleton 5, 519 slant product 280 smash product 10, 219, 270 quasi-circle 79 spectrum 454 quasifibration 479 sphere bundle 442, 444 quaternion 75, 173, 281, 294, 446 Spin(n) 291 Quillen 374 split exact sequence 147 quotient CW complex stable homotopy group 384, 452 550 Index stable splitting 491 Tor 263, 267 stable stem 384 torsion coefficient 130 star 178 torus 34, 74, 102, 106, 227 Steenrod algebra 496 torus knot 47 Steenrod homology 257 total space 377 Steenrod squares, powers 487 transfer homomorphism 175, 321 Stiefel manifold 301, 381, 436, 447, 493 tree 84 subcomplex 7, 520 triple 118, 344 subgraph 84 truncated polynomial algebra 284 surface 5, 51, 88, 93, 102, 141, 207, 241, 390 suspension 8, 137, 219, 466, 473 suspension spectrum 454 symmetric polynomials 435 unique lifting property 62 unitary group U (n) 227, 382, 434 universal coefficient theorem 195, 264, 463 universal cover 59, 68 symmetric product 282, 365, 481 van Kampen 43 symplectic group Sp (n) 227, 382, 434 vector field 135, 493 tensor algebra 288, 471 vertex 83, 103 tensor product 215, 328 weak homotopy equivalence 352 tensor product of chain complexes 273 weak topology 5, 83, 521 Thom class 441, 510 wedge sum 10, 43, 126, 160, 202, 380, 466 Thom isomorphism 441 Whitehead product 381, 430 Thom space 441, 510 Whitehead tower 356 Toda bracket 387 Whitehead’s theorem 346, 367, 418 topological group 281 Wirtinger presentation 55 ... matters of a pointset topological nature that arise in algebraic topology Since this is a textbook on algebraic topology, details involving point-set topology are often treated lightly or skipped entirely... 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... 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,