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allen hatcher copyright c 2002 by cambridge university press single paper or electronic copies for noncommercial personal use may be made without explicit permission from the author or publisher all other rights reserved.

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preface ix standard notations xii chapter 0 some underlying geometric notions homotopy and homotopy type 1 cell complexes 5 1 operations on spaces 8 two criteria for homotopy equivalence 10 the homotopy extension property 14 chapter 1 the fundamental group 1.1 basic constructions induced homomorphisms 34 21 25 paths and homotopy 25 the fundamental group of the circle 29 1.2 van kampen s theorem applications to cell complexes 50 40 free products of groups 41 the van kampen theorem 43 1.3 covering spaces 56 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 kg,1 spaces and graphs of groups 87.

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chapter 2 homology 97 2.1 simplicial and singular homology 102 complexes 102 simplicial homology 104 singular homology 108 homotopy invariance 110 exact sequences and excision 113 the equivalence of simplicial and singular homology 128 2.2 computations and applications homology with coefficients 153 134 degree 134 cellular homology 137 mayer-vietoris sequences 149 2.3 the formal viewpoint additional topics 160 axioms for homology 160 categories and functors 162 2.a homology and fundamental group 166 2.b classical applications 169 2.c simplicial approximation 177 chapter 3 cohomology 3.1 cohomology groups 3.2 cup product 185 190 the universal coefficient theorem 190 cohomology of spaces 197 206 the cohomology ring 211 a k¨nneth formula 218 u spaces with polynomial cohomology 224 3.3 poincar´ duality e 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¨nneth formula 268 u 3.c h­spaces and hopf algebras 281 3.d the cohomology of son 292 3.e bockstein homomorphisms 303 3.f limits and ext 311 3.g transfer homomorphisms 321 3.h local coefficients 327.

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chapter 4 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 fiber bundles 375 stable homotopy groups 384 360 excision for homotopy groups 360 the hurewicz theorem 366 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

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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 1 and 4 and homology and its mirror variant cohomology in chapters 2 and 3 these four chapters do not have to be read in this order 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 4 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 0 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

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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 k­theory 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 k­theory 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¨nich 1984 listed in the bibliography a 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 2 triangles 3 edges and 1 vertex by contrast a simplicial complex structure on the torus must have at least 14 triangles 21 edges and 7 vertices so complexes provide a significant improvement in efficiency which is nice from a pedagogical viewpoint since it cuts down on tedious calculations in examples a more fundamental reason for considering complexes is that they 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.

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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.

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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 1 from the origin d n the unit disk or ball in rn all points of distance 1 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 0 and e0 consist of a single point since r0 {0 but s 0 consists of two points since it is d 1 1 the identity function from a set to itself 1 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 ab 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.

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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 do this is to decompose a thick letter call it x into line segments connecting each point on the outer boundary of x to a unique point of the thin subletter x as indicated in the figure then we can shrink x to x by sliding each point of x x into x along the line segment that contains it points that are already in x do not move we can think of this shrinking process as taking place during a time interval 0 t 1 and then it defines a family of functions ft xx parametrized by t i [0 1 where ft x is the point to which a given point x x has moved at time t .

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2 chapter 0 some underlying geometric notions naturally we would like ft x to depend continuously on both t and x and this will be true if we have each x x x move along its line segment at constant speed so as to reach its image point in x at time t 1 while points x x are stationary as remarked earlier retraction of a space x onto a subspace a is a family of maps ft x x t i such 1 1 that f0 1 the identity map f1 x a and ft a 1 for all t the family ft should be continuous in the sense that the associated map x × i x x t ft x is continuous it is easy to produce many more examples similar to the letter examples with the deformation retraction ft obtained by sliding along line segments the figure on the left below shows such a deformation retraction of a m¨bius band onto its core circle o examples of this sort lead to the following general definition a deformation the three figures on the right show deformation retractions in which a disk with two smaller open subdisks removed shrinks to three different subspaces in all these examples the structure that gives rise to the deformation retraction can be described by means of the following definition for a map f x y the mapping cylinder mf is the quotient space of the disjoint union x × i tifying each x 1 x × i with f x y in the letter examples the space x is the outer boundary of the thick letter y is the thin letter and f x y sends y obtained by iden x×i y x f x y mf the outer endpoint of each line segment to its inner endpoint a similar description applies to the other examples then it is a general fact that a mapping cylinder mf deformation retracts to the subspace y by sliding each point x t along the segment {x}× i mf to the endpoint f x y not all deformation retractions arise in this way from mapping cylinders however for example the thick x deformation retracts to the thin x which in turn deformation retracts to the point of intersection of its two crossbars the net result is a deformation retraction of x onto a point during which certain pairs of points follow paths that merge before reaching their final destination later in this section we will describe a considerably more complicated example the so-called `house with two rooms where a deformation retraction to a point can be constructed abstractly but seeing the deformation with the naked eye is a real challenge.

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homotopy and homotopy type chapter 0 3 ciated map f x × i y given by f x t ft x is continuous one says that two maps f0 f1 x y are homotopic if there exists a homotopy ft connecting them f1 and one writes f0 homotopy which is simply any family of maps ft x y t i such that the asso a deformation retraction ft x x is a special case of the general notion of a from the identity map of x to a retraction of x onto a a map r x x such that 1 r x a and r a 1 one could equally well regard a retraction as a map x a retraction is a map r x x with r 2 r since this equation says exactly that r is the identity on its image retractions are the topological analogs of projection operators in other parts of mathematics not all retractions come from deformation retractions for example a space x always retracts onto any point x0 x via the constant map sending all of x to x0 but a space that deformation retracts onto a point must be path-connected since a deformation retraction of x to x0 gives a path joining each x x to x0 it is less trivial to show that there are path-connected spaces that do not deformation retract onto a point one would expect this to be the case for the letters `with holes a b d o p q r in chapter 1 we will develop techniques to prove this a homotopy ft x x that gives a deformation retraction of x onto a subspace 1 a has the property that ft a 1 for all t in general a homotopy ft x y whose restriction to a subspace a x is independent of t is called a homotopy relative to a or more concisely a homotopy rel a thus a deformation retraction of x onto a is a homotopy rel a from the identity map of x to a retraction of x onto a 1 r x a denotes the resulting retraction and i ax the inclusion we have r i 1 and ir fg if a space x deformation retracts onto a subspace a via ft x x then if 1 the latter homotopy being given by ft generalizing this situation a 1 1 the spaces x and y are said to be homotopy equivalent or to 1 in these terms a deformation retraction of x onto a subspace a is a homotopy restricting to the identity on the subspace a x from a more formal viewpoint a map f x y is called a homotopy equivalence if there is a map g y x such that 1 and gf 1 have the same homotopy type the notation is x formation retraction for example the three graphs y it is an easy exercise to check are all homotopy that this is an equivalence relation in contrast with the nonsymmetric notion of deequivalent since they are deformation retracts of the same space as we saw earlier but none of the three is a deformation retract of any other it is true in general that two spaces x and y are homotopy equivalent if and only if there exists a third space z containing both x and y as deformation retracts for the less trivial implication one can in fact take z to be the mapping cylinder mf of any homotopy equivalence f x y we observed previously that mf deformation retracts to y so what needs to be proved is that mf also deformation retracts to its other end x if f is a homotopy equivalence this is shown in corollary 0.21.

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4 chapter 0 some underlying geometric notions a space having the homotopy type of a point is called contractible this amounts to requiring that the identity map of the space be nullhomotopic that is homotopic to a constant map in general this is slightly weaker than saying the space deformation retracts to a point see the exercises at the end of the chapter for an example distinguishing these two notions let us describe now an example of a 2 dimensional subspace of r3 known as the house with two rooms which is contractible but not in any obvious way to build this space start with a box divided into two chambers by a horizontal rectangle where by a `rectangle we mean not just the four edges of a rectangle but also its interior access to the two chambers from outside the box is provided by two vertical tunnels the upper tunnel is made by punching out a square from the top of the box and another square directly below it from the middle horizontal rectangle then inserting four vertical rectangles the walls of the tunnel this tunnel allows entry to the lower chamber from outside the box the lower tunnel is formed in similar fashion providing entry to the upper chamber finally two vertical rectangles are inserted to form `support walls for the two tunnels the resulting space x thus consists of three horizontal pieces homeomorphic to annuli plus all the vertical rectangles that form the walls of the two chambers to see that x is contractible consider a closed neighborhood nx of x this clearly deformation retracts onto x if is sufficiently small in fact nx is the mapping cylinder of a map from the boundary surface of nx to x less obvious is the fact that nx is homeomorphic to d 3 the unit ball in r3 to see this imagine forming nx from a ball of clay by pushing a finger into the ball to create the upper tunnel then gradually hollowing out the lower chamber and similarly pushing a finger in to create the lower tunnel and hollowing out the upper chamber mathematically this process gives a family of embeddings ht d 3 r3 starting with the usual inclusion d 3 thus we have x r3 and ending with a homeomorphism onto nx nx d 3 point so x is contractible since homotopy if ft is a deformation retraction of the ball nx to a point x0 x and if r nxx equivalence is an equivalence relation in fact x deformation retracts to a point for is a retraction for example the end result of a deformation retraction of nx to x then the restriction of the composition r ft to x is a deformation retraction of x to x0 however it is quite a challenging exercise to see exactly what this deformation retraction looks like.

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cell complexes chapter 0 5 cell complexes a familiar way of constructing the torus s 1 × s 1 is by identifying opposite sides of a square more generally an orientable surface mg of genus g can be constructed from a polygon with 4g sides by identifying pairs of edges as shown in the figure in the first three cases g 1 2 3 the 4g edges of the polygon become a union of 2g circles in the surface all intersecting in a single point the interior of the polygon can be thought of as an open disk or a 2 cell attached to the union of the 2g circles one can also regard the union of the circles as being obtained from their common point of intersection by attaching 2g open arcs or 1 cells thus then attach a 2 cell a natural generalization of this is to construct a space by the following procedure 1 start with a discrete set x 0 whose points are regarded as 0 cells n 2 inductively form the n skeleton x n from x n-1 by attaching n cells e via maps abacdbccadddcbdefeaffabebacdcbaabbba the surface can be built up in stages start with a point attach 1 cells to this point s n-1 x n-1 this means that x n is the quotient space of the disjoint union x n-1 n d n of x n-1 with a collection of n disks d under the identifications n e n where each e is an n x x for x d thus as a set x n x n-1 open n disk 3 one can either stop this inductive process at a finite stage setting x x n for some n or one can continue indefinitely setting x open or closed in x n for each n a space x constructed in this way is called a cell complex or cw complex the explanation of the letters `cw is given in the appendix where a number of basic topological properties of cell complexes are proved the reader who wonders about various point-set topological questions lurking in the background of the following discussion should consult the appendix for details n x n in the latter case x is given the weak topology a set a x is open or closed iff a x n is

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6 chapter 0 some underlying geometric notions if x x n for some n then x is said to be finite-dimensional and the smallest such n is the dimension of x the maximum dimension of cells of x 0.1 a 1 dimensional cell complex x x 1 is what is called a graph in example algebraic topology it consists of vertices the 0 cells to which edges the 1 cells are attached the two ends of an edge can be attached to the same vertex example 0.2 the house with two rooms pictured earlier has a visually obvious 2 dimensional cell complex structure the 0 cells are the vertices where three or more of the depicted edges meet and the 1 cells are the interiors of the edges connecting these vertices this gives the 1 skeleton x 1 and the 2 cells are the components of the remainder of the space x x 1 if one counts up one finds there are 29 0 cells 51 1 cells and 23 2 cells with the alternating sum 29 51 23 equal to 1 this is the euler characteristic which for a cell complex with finitely many cells is defined to be the number of even-dimensional cells minus the number of odd-dimensional cells as we shall show in theorem 2.44 the euler characteristic of a cell complex depends only on its homotopy type so the fact that the house with two rooms has the homotopy type of a point implies that its euler characteristic must be 1 no matter how it is represented as a cell complex example 0.3 to regarding s n as the quotient space d n /d n and en the n cell being attached by the constant map s n-1 e0 this is equivalent the sphere s n has the structure of a cell complex with just two cells e0 example 0.4 real projective n space rpn is defined to be the space of all lines through the origin in rn+1 each such line is determined by a nonzero vector in rn+1 unique up to scalar multiplication and rpn is topologized as the quotient space of rn+1 {0 under the equivalence relation v v for scalars 0 we can restrict to vectors of length 1 so rpn is also the quotient space s n v -v the sphere with antipodal points identified this is equivalent to saying that rpn is the quotient space of a hemisphere d n with antipodal points of d n identified since d n with attaching an n cell with the quotient projection s n-1 rpn-1 as the attaching map it follows by induction on n that rpn has a cell complex structure e0 e1 ··· en with one cell ei in each dimension i n antipodal points identified is just rpn-1 we see that rpn is obtained from rpn-1 by example 0.5 union rp can view rp since rpn is obtained from rpn-1 by attaching an n cell the infinite as the space of lines through the origin in r n rpn becomes a cell complex with one cell in each dimension we n rn example 0.6 complex projective n space cpn is the space of complex lines through the origin in cn+1 that is 1 dimensional vector subspaces of cn+1 as in the case of rpn each line is determined by a nonzero vector in cn+1 unique up to scalar multiplication and cpn is topologized as the quotient space of cn+1 {0 under the

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