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.

Chapter 0. Some Underlying Geometric Notions

. . . 1 Homotopy and Homotopy Type 1. Cell Complexes 5.

Operations on Spaces 8. Two Criteria for Homotopy Equivalence 10.

The Homotopy Extension Property 14.

Chapter 1. The Fundamental Group

. . . 21

1.1. Basic Constructions

. . . 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

. . . 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. K(G,1) Spaces and Graphs of Groups 87.

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

. . . 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 3. 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 K¨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 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.

4.1. Homotopy Groups

. . . 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

. . . 533

Index

. . . 539

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 empha- sis 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 chap- ters, 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 ho- motopy. 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

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 familiar- ity with the content of the standard undergraduate courses in algebra and point-set topology. In particular, the reader should know about quotient spaces, or identifi- cation 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. 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 stan- dard picture of the torus as a square with opposite edges identified and divides the square into two triangles by cutting along a diagonal, then the result is a ∆ complex structure on the torus having 2 triangles, 3 edges, and 1 vertex. By contrast, it is known that a simplicial complex structure on the torus must have at least 14 trian- gles, 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. In spite of this naturality, ∆ complexes have appeared explicitly in the literature only rarely, and no standard name for the notion has emerged.

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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.

Z, Q, R, C, H, O: the integers, rationals, reals, complexes, quaternions, and Cayley octonions

Zn: the integers modn

Rⁿ: n dimensional Euclidean space Cⁿ: complexn space

I= [0, 1]: the unit interval

Sⁿ: the unit sphere in Rⁿ⁺¹, all vectors of length 1 Dⁿ: the unit disk or ball in Rⁿ, all vectors of length ≤ 1

∂Dⁿ= Sⁿ⁻¹: the boundary of then disk 11 : the identity function from a set to itself q: disjoint union of sets or spaces

× ,Q

: product of sets, groups, or spaces

≈: isomorphism

A⊂ B or B ⊃ A: set-theoretic containment, not necessarily proper iff : if and only if

somewhat informal, with no theorems or proofs until the last couple pages, and it should be read in this informal spirit, skipping bits here and there. In fact, this whole chapter could be skipped now, to be referred back to later for basic definitions.

To avoid overusing the word ‘continuous’ we adopt the convention that maps be- tween spaces are always assumed to be continuous unless otherwise stated.

Homotopy and Homotopy Type

One of the main ideas of algebraic topology is to consider two spaces to be equiv- alent if they have ‘the same shape’ in a sense that is much broader than homeo- morphism. To take an everyday example, the letters of the alphabet can be writ- ten either as unions of finitely many

straight and curved line segments, or in thickened forms that are compact subsurfaces of the plane bounded by simple closed curves. In each case the thin letter is a subspace of the thick

letter, and we can continuously shrink the thick letter to the thin one. A nice way to do this is to decompose a thick letter, call it X , into line segments connecting each point on the outer boundary of X to a unique point of the thin subletter X , as indi- cated 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:X

→

X parametrized by t∈ I = [0, 1] , where f_t(x) is the point to which a given point x ∈ X has moved at time t .

Naturally we would likef_t(x) to depend continuously on both t and x , and this will be true if we have each x ∈ X − X move along its line segment at constant speed so as to reach its image point in X at time t= 1, while points x ∈ X are stationary, as remarked earlier.

Examples of this sort lead to the following general definition. A deformation retraction of a space X onto a subspace A is a family of maps f_t:X

→

^{X , t∈ I , such}

that f₀ = ¹1 (the identity map), f₁(X) = A, and f_t||A = ¹1 for all t . The family f_t should be continuous in the sense that the associated mapX×I

→

^{X , (x, t)}

,

^{ft(x) ,}

is continuous.

It is easy to produce many more examples similar to the letter examples, with the deformation retractionf_t obtained by sliding along line segments. The figure on the left below shows such a deformation retraction of a M¨obius band onto its core circle.

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 mapf : X

→

Y , the mapping cylinder M_f is the quotient space of the disjoint union(X×I) q Y obtained by iden- tifying each (x, 1) ∈ X×I

with f (x) ∈ Y . In the let-

X× I X

Y Y

M_f f( )X

ter examples, the space X is the outer boundary of the thick letter, Y is the thin letter, and f : X

→

^{Y sends}

the outer endpoint of each line segment to its inner endpoint. A similar description applies to the other examples. Then it is a general fact that a mapping cylinder M_f deformation retracts to the subspaceY by sliding each point (x, t) along the segment {x}×I ⊂ Mf to the endpointf (x)∈ Y .

Not all deformation retractions arise in this way from mapping cylinders, how- ever. For example, the thick X deformation retracts to the thin X , which in turn 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.

A deformation retraction f_t:X

→

X is a special case of the general notion of a homotopy, which is simply any family of maps f_t:X

→

^{Y , t}∈ I , such that the asso- ciated map F : X×I

→

Y given by F (x, t) = f_t(x) is continuous. One says that two maps f₀, f₁:X

→

Y are homotopic if there exists a homotopy f_t connecting them, and one writesf₀' f1.

In these terms, a deformation retraction of X onto a subspace A is a homotopy from the identity map of X to a retraction of X onto A , a map r : X

→

X such that r (X)= A and r ||A =¹1 . One could equally well regard a retraction as a mapX

→

^A

restricting to the identity on the subspace A⊂ X . From a more formal viewpoint a retraction is a mapr : X

→

^{X with r2}= 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, every space X retracts onto any point x₀∈ X via the map sending all of X to x0. But a space that deformation retracts onto a point must certainly be path-connected, since a deforma- tion retraction of X to a point x₀ gives a path joining each x∈ X to x0. It is less trivial to show that there are path-connected 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 homotopyf_t:X

→

X that gives a deformation retraction of X onto a subspace A has the property that f_t||A =¹1 for all t . In general, a homotopy f_t: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 .

If a space X deformation retracts onto a subspace A via f_t:X

→

X , then if r : X

→

A denotes the resulting retraction and i : A

→

X the inclusion, we have r i=¹1 and ir ' ¹1 , the latter homotopy being given by f_t. Generalizing this situation, a mapf : X

→

Y is called a homotopy equivalence if there is a map g : Y

→

X such that f g'¹1 andgf '¹1 . The spaces X and Y are said to be homotopy equivalent or to have the samehomotopy type. The notation is X' Y . It is an easy exercise to check that this is an equivalence relation, in contrast with the nonsymmetric notion of de- formation retraction. For example, the three graphs are all homotopy equivalent 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 spacesX 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 M_f of any homotopy equivalence f : X

→

Y . We observed previously that M_f deformation retracts toY , so what needs to be proved is that M_f also deformation retracts to its other end X if f is a homotopy equivalence. This is shown in Corollary 0.21.

A space having the homotopy type of a point is calledcontractible. This amounts to requiring that the identity map of the space benullhomotopic, that is, homotopic to a constant map. In general, this is slightly weaker than saying the space deforma- tion 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 ofR³, 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 ε neighborhood N(X) of X . This clearly deformation retracts onto X if ε is sufficiently small. In fact, N(X) is the mapping cylinder of a map from the boundary surface of N(X) to X . Less obvious is the fact that N(X) is homeomorphic to D³, the unit ball in R³. To see this, imagine forming N(X) 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 embeddingsh_t:D³

→

^R3 starting with the usual inclusion D³

>

^R3 and ending with a homeomorphism onto N(X) .

Thus we have X ' N(X) = D³ ' point , so X is contractible since homotopy equivalence is an equivalence relation. In fact,X deformation retracts to a point. For iff_t is a deformation retraction of the ballN(X) to a point x₀∈ X and if r : N(X)

→

^X

is a retraction, for example the end result of a deformation retraction ofN(X) to X , then the restriction of the compositionr f_t 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.

Cell Complexes

A familiar way of constructing the torus S¹×S¹ is by identifying opposite sides of a square. More generally, an orientable surfaceM_g of genusg 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

b a

a a

b b

b

b b b c

a

a

a d

a c

c

c c

b

d

d d d e

e f

f a

e

f c d

b

a become a union of 2g circles

in the surface, all intersect- ing in a single point. The in- terior 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

the surface can be built up in stages: Start with a point, attach 1 cells to this point, 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 X⁰, whose points are regarded as 0 cells.

(2) Inductively, form then skeleton XⁿfromXⁿ⁻¹by attachingn cells e_αⁿvia maps ϕ_α:Sⁿ⁻¹

→

^Xn−1. This means thatXⁿ is the quotient space of the disjoint union Xⁿ⁻¹`

αDⁿ_α of Xⁿ⁻¹ with a collection of n disks Dⁿ_α under the identifications x ∼ ϕα(x) for x∈ ∂Dⁿα. Thus as a set, Xⁿ = Xⁿ⁻¹`

αe_αⁿ where each eⁿ_α is an open n disk.

(3) One can either stop this inductive process at a finite stage, setting X = Xⁿ for some n <∞, or one can continue indefinitely, setting X =S

nXⁿ. In the latter caseX is given the weak topology: A set A⊂ X is open (or closed) iff A ∩ Xⁿ is open (or closed) inXⁿ for eachn .

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.

If X = Xⁿ 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 .

E

xample 0.1. A 1 dimensional cell complex X = X¹ is what is called agraph in 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.

E

xample 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 X¹, and the 2 cells are the components of the remainder of the space, X− X¹. 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 theEuler 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.

E

xample 0.3. The sphere Sⁿhas the structure of a cell complex with just two cells,e⁰ and eⁿ, the n cell being attached by the constant map Sⁿ⁻¹

→

^e0. This is equivalent to regarding Sⁿ as the quotient space Dⁿ/∂Dⁿ.

E

xample 0.4. Real projective n space RPⁿ is defined to be the space of all lines through the origin inRⁿ⁺¹. Each such line is determined by a nonzero vector inRⁿ⁺¹, unique up to scalar multiplication, and RPⁿ is topologized as the quotient space of Rⁿ⁺¹− {0} under the equivalence relation v ∼ λv for scalars λ ≠ 0. We can restrict to vectors of length 1, so RPⁿ is also the quotient space Sⁿ/(v ∼ −v), the sphere with antipodal points identified. This is equivalent to saying thatRPⁿis the quotient space of a hemisphere Dⁿ with antipodal points of ∂Dⁿ identified. Since∂Dⁿ with antipodal points identified is just RPⁿ⁻¹, we see that RPⁿ is obtained fromRPⁿ⁻¹ by attaching an n cell, with the quotient projection Sⁿ⁻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.

E

xample 0.5. Since RPⁿ is obtained from RPⁿ⁻¹ by attaching an n cell, the infinite union RP^∞ =S

nRPⁿ becomes a cell complex with one cell in each dimension. We can view RP^∞ as the space of lines through the origin in R^∞=S

nRⁿ.

E

xample 0.6. Complex projective n spaceCPⁿ 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, andCPⁿ is topologized as the quotient space ofCⁿ⁺¹− {0} under the

equivalence relation v ∼ λv for λ ≠ 0. Equivalently, this is the quotient of the unit sphere S²ⁿ⁺¹⊂ Cⁿ⁺¹ with v ∼ λv for |λ| = 1. It is also possible to obtain CPⁿ as a quotient space of the diskD²ⁿunder the identifications v∼ λv for v ∈ ∂D²ⁿ, in the following way. The vectors inS²ⁿ⁺¹⊂ Cⁿ⁺¹ with last coordinate real and nonnegative are precisely the vectors of the form (w,p

1− |w|²) ∈ Cⁿ×C with |w| ≤ 1. Such vectors form the graph of the function w

,

^{p1− |w|2}. This is a diskD²ⁿ₊ bounded by the sphereS²ⁿ⁻¹⊂ S²ⁿ⁺¹ consisting of vectors(w, 0)∈ Cⁿ×C with |w| = 1. Each vector inS²ⁿ⁺¹ is equivalent under the identificationsv∼ λv to a vector in D²ⁿ₊ , and the latter vector is unique if its last coordinate is nonzero. If the last coordinate is zero, we have just the identifications v∼ λv for v ∈ S²ⁿ⁻¹.

From this description of CPⁿ as the quotient of D₊²ⁿ under the identifications v ∼ λv for v ∈ S²ⁿ⁻¹ it follows that CPⁿ is obtained from CPⁿ⁻¹ by attaching a cell e²ⁿ via the quotient map S²ⁿ⁻¹

→

^CPn−1. So by induction on n we obtain a cell structureCPⁿ= e⁰∪ e²∪ ··· ∪ e²ⁿ with cells only in even dimensions. Similarly,CP^∞ has a cell structure with one cell in each even dimension.

After these examples we return now to general theory. Each cell eⁿ_α in a cell complex X has a characteristic map Φα:D_αⁿ

→

X which extends the attaching map ϕ_α and is a homeomorphism from the interior of Dⁿ_α onto e_αⁿ. Namely, we can take Φ_α to be the composition D_αⁿ

>

^{Xn−1`αDnα}

→

^Xn

>

X where the middle map is the quotient map defining Xⁿ. For example, in the canonical cell structure on Sⁿ described in Example 0.3, a characteristic map for the n cell is the quotient map Dⁿ

→

^{Sn collapsing ∂Dn} to a point. For RPⁿ a characteristic map for the cell eⁱ is the quotient map Dⁱ

→

^{RPi⊂ RPn} identifying antipodal points of ∂Dⁱ, and similarly for CPⁿ.

Asubcomplex of a cell complex X is a closed subspace A ⊂ X that is a union of cells of X . Since A is closed, the characteristic map of each cell in A has image contained in A , and in particular the image of the attaching map of each cell in A is contained in A , so A is a cell complex in its own right. A pair (X, A) consisting of a cell complexX and a subcomplex A will be called a CW pair.

For example, each skeleton Xⁿ of a cell complex X is a subcomplex. Particular cases of this are the subcomplexesRP^k ⊂ RPⁿ and CP^k ⊂ CPⁿ for k≤ n. These are in fact the only subcomplexes ofRPⁿ and CPⁿ.

There are natural inclusions S⁰ ⊂ S¹ ⊂ ··· ⊂ Sⁿ, but these subspheres are not subcomplexes ofSⁿin its usual cell structure with just two cells. However, we can give Sⁿ a different cell structure in which each of the subspheres S^k is a subcomplex, by regarding eachS^kas being obtained inductively from the equatorialS^k−1by attaching twok cells, the components of S^k−S^k−1. The infinite-dimensional sphereS^∞=S

nSⁿ then becomes a cell complex as well. Note that the two-to-one quotient mapS^∞

→

^RP∞

that identifies antipodal points of S^∞ identifies the two n cells of S^∞ to the single n cell ofRP^∞.

In the examples of cell complexes given so far, the closure of each cell is a sub- complex, and more generally the closure of any collection of cells is a subcomplex.

Most naturally arising cell structures have this property, but it need not hold in gen- eral. For example, if we start withS¹ with its minimal cell structure and attach to this a 2 cell by a map S¹

→

^S1 whose image is a nontrivial subarc ofS¹, then the closure of the 2 cell is not a subcomplex since it contains only a part of the 1 cell.

Operations on Spaces

Cell complexes have a very nice mixture of rigidity and flexibility, with enough rigidity to allow many arguments to proceed in a combinatorial cell-by-cell fashion and enough flexibility to allow many natural constructions to be performed on them.

Here are some of those constructions.

Products. If X and Y are cell complexes, then X×Y has the structure of a cell complex with cells the products e^m_α×eβⁿ where e_α^m ranges over the cells of X and eⁿ_β ranges over the cells of Y . For example, the cell structure on the torus S¹×S¹ described at the beginning of this section is obtained in this way from the standard cell structure onS¹. In the general case there is one small complication, however: The topology on X×Y as a cell complex is sometimes slightly weaker than the product topology, with more open sets than the product topology has, though the two topologies coincide if either X or Y has only finitely many cells, or if both X and Y have countably many cells. This is explained in the Appendix. In practice this subtle point of point-set topology rarely causes problems.

Quotients. If (X, A) is a CW pair consisting of a cell complex X and a subcomplex A , then the quotient space X/A inherits a natural cell complex structure from X . The cells ofX/A are the cells of X− A plus one new 0 cell, the image of A in X/A. For a celle_αⁿofX−A attached by ϕ_α:Sⁿ⁻¹

→

^Xn−1, the attaching map for the correspond- ing cell in X/A is the composition Sⁿ⁻¹

→

^Xn−1

→

^{Xn−1/An−1.}

For example, if we giveSⁿ⁻¹any cell structure and buildDⁿfromSⁿ⁻¹by attach- ing ann cell, then the quotient Dⁿ/Sⁿ⁻¹isSⁿwith its usual cell structure. As another example, take X to be a closed orientable surface with the cell structure described at the beginning of this section, with a single 2 cell, and letA be the complement of this 2 cell, the 1 skeleton ofX . Then X/A has a cell structure consisting of a 0 cell with a 2 cell attached, and there is only one way to attach a cell to a 0 cell, by the constant map, soX/A is S².

Suspension. For a space X , the suspension SX is the quotient of X×I obtained by collapsing X×{0} to one point and X×{1} to an- other point. The motivating example is X = Sⁿ, when SX = Sⁿ⁺¹ with the two ‘suspension points’ at the north and south poles of Sⁿ⁺¹, the points(0,··· , 0, ±1). One can regard SX as a double cone

on X , the union of two copies of the cone CX= (X×I)/(X×{0}). If X is a CW com- plex, so are SX and CX as quotients of X×I with its product cell structure, I being given the standard cell structure of two 0 cells joined by a 1 cell.

Suspension becomes increasingly important the farther one goes into algebraic topology, though why this should be so is certainly not evident in advance. One especially useful property of suspension is that not only spaces but also maps can be suspended. Namely, a mapf : X

→

Y suspends to Sf : SX

→

SY , the quotient map of f×¹1 :X×I

→

^{Y×I .}

Join. The cone CX is the union of all line segments joining points of X to an external vertex, and similarly the suspensionSX is the union of all line segments joining points ofX to two external vertices. More generally, given X and a second space Y , one can define the space of all lines segments joining points in X to points in Y . This is thejoin X∗ Y , the quotient space of X×Y ×I under the identifications (x, y1, 0)∼ (x, y₂, 0) and (x₁, y, 1)∼ (x2, y, 1) . Thus we are collapsing the subspace X×Y ×{0}

to X and X×Y ×{1} to Y . For example, if X and Y are both closed intervals, then we are collapsing two opposite faces of a cube onto line segments so that the cube becomes a tetrahedron. In the general case, X ∗ Y X

I Y

contains copies ofX and Y at its two ‘ends,’

and every other point(x, y, t) in X∗ Y is on a unique line segment joining the point x∈ X ⊂ X ∗ Y to the point y ∈ Y ⊂ X ∗ Y , the segment obtained by fixing x and y and letting the coordinate t in (x, y, t) vary.

A nice way to write points of X∗ Y is as formal linear combinations t₁x+ t₂y with 0≤ t_i≤ 1 and t₁+t₂= 1, subject to the rules 0x+1y = y and 1x+0y = x that correspond exactly to the identifications defining X∗ Y . In much the same way, an iterated joinX₁∗···∗Xn can be regarded as the space of formal linear combinations t₁x₁+ ··· + tnx_n with 0≤ ti ≤ 1 and t1+ ··· + tn = 1, with the convention that terms 0t_i can be omitted. This viewpoint makes it easy to see that the join operation is associative. A very special case that plays a central role in algebraic topology is when eachX_i is just a point. For example, the join of two points is a line segment, the join of three points is a triangle, and the join of four points is a tetrahedron. The join ofn points is a convex polyhedron of dimension n− 1 called a simplex. Concretely, if the n points are the n standard basis vectors for Rⁿ, then their join is the space

∆ⁿ⁻¹= { (t1,··· , tn)∈ Rⁿ|| t1+ ··· + tn= 1 and ti≥ 0 }.

Another interesting example is when eachX_i isS⁰, two points. If we take the two points of X_i to be the two unit vectors along the i^thcoordinate axis in Rⁿ, then the joinX₁∗···∗X_n is the union of 2ⁿ copies of the simplex∆ⁿ⁻¹, and radial projection from the origin gives a homeomorphism between X₁∗ ··· ∗ X_n andSⁿ⁻¹.

If X and Y are CW complexes, then there is a natural CW structure on X∗ Y having the subspaces X and Y as subcomplexes, with the remaining cells being the product cells ofX×Y ×(0, 1). As usual with products, the CW topology on X ∗Y may be weaker than the quotient of the product topology on X×Y ×I .

Wedge Sum. This is a rather trivial but still quite useful operation. Given spaces X and Y with chosen points x₀∈ X and y₀∈ Y , then the wedge sum X ∨ Y is the quotient of the disjoint union Xq Y obtained by identifying x0 and y₀ to a single point. For example, S¹∨ S¹ is homeomorphic to the figure ‘8,’ two circles touching at a point.

More generally one could form the wedge sum W

αX_α of an arbitrary collection of spacesX_α by starting with the disjoint union `

αX_α and identifying points x_α∈ X_α to a single point. In case the spaces X_α are cell complexes and the points x_α are 0 cells, thenW

αX_α is a cell complex since it is obtained from the cell complex`

αX_α by collapsing a subcomplex to a point.

For any cell complexX , the quotient Xⁿ/Xⁿ⁻¹is a wedge sum ofn spheresW

αS_αⁿ, with one sphere for each n cell of X .

Smash Product. Like suspension, this is another construction whose importance be- comes evident only later. Inside a product space X×Y there are copies of X and Y , namelyX×{y0} and {x0}×Y for points x0∈ X and y0∈ Y . These two copies of X and Y in X×Y intersect only at the point (x0, y₀) , so their union can be identified with the wedge sumX∨ Y . The smash product X ∧ Y is then defined to be the quo- tient X×Y /X ∨ Y . One can think of X ∧ Y as a reduced version of X×Y obtained by collapsing away the parts that are not genuinely a product, the separate factorsX and Y .

The smash productX∧Y is a cell complex if X and Y are cell complexes with x₀ andy₀ 0 cells, assuming that we giveX×Y the cell-complex topology rather than the product topology in cases when these two topologies differ. For example,S^m∧Sⁿhas a cell structure with just two cells, of dimensions 0 andm+n, hence S^m∧Sⁿ= S^m+n. In particular, when m= n = 1 we see that collapsing longitude and meridian circles of a torus to a point produces a 2 sphere.

Two Criteria for Homotopy Equivalence

Earlier in this chapter the main tool we used for constructing homotopy equiva- lences was the fact that a mapping cylinder deformation retracts onto its ‘target’ end.

By repeated application of this fact one can often produce homotopy equivalences be- tween rather different-looking spaces. However, this process can be a bit cumbersome in practice, so it is useful to have other techniques available as well. We will describe two commonly used methods here. The first involves collapsing certain subspaces to points, and the second involves varying the way in which the parts of a space are put together.

Collapsing Subspaces

The operation of collapsing a subspace to a point usually has a drastic effect on homotopy type, but one might hope that if the subspace being collapsed already has the homotopy type of a point, then collapsing it to a point might not change the homotopy type of the whole space. Here is a positive result in this direction:

If (X, A) is a CW pair consisting of a CW complex X and a contractible subcomplex A , then the quotient map X

→

X/A is a homotopy equivalence.

A proof will be given later in Proposition 0.17, but for now let us look at some examples showing how this result can be applied.

E

xample 0.7: Graphs. The three graphs are homotopy equivalent since each is a deformation retract of a disk with two holes, but we can also deduce this from the collapsing criterion above since collapsing the middle edge of the first and third graphs produces the second graph.

More generally, suppose X is any graph with finitely many vertices and edges. If the two endpoints of any edge ofX are distinct, we can collapse this edge to a point, producing a homotopy equivalent graph with one fewer edge. This simplification can be repeated until all edges of X are loops, and then each component of X is either an isolated vertex or a wedge sum of circles.

This raises the question of whether two such graphs, having only one vertex in each component, can be homotopy equivalent if they are not in fact just isomorphic graphs. Exercise 12 at the end of the chapter reduces the question to the case of connected graphs. Then the task is to prove that a wedge sumW

mS¹ofm circles is not homotopy equivalent toW

nS¹ifm≠ n. This sort of thing is hard to do directly. What one would like is some sort of algebraic object associated to spaces, depending only on their homotopy type, and taking different values forW

mS¹and W

nS¹ifm≠ n. In fact the Euler characteristic does this sinceW

mS¹has Euler characteristic 1−m. But it is a rather nontrivial theorem that the Euler characteristic of a space depends only on its homotopy type. A different algebraic invariant that works equally well for graphs, and whose rigorous development requires less effort than the Euler characteristic, is the fundamental group of a space, the subject of Chapter 1.

E

xample 0.8. Consider the space X obtained from S² by attaching the two ends of an arc A to two distinct points on the sphere, say the north and south poles. Let B be an arc in S² joining the two points whereA attaches. Then X can be given a CW complex structure with the two endpoints of A and B as 0 cells, the interiors of A and B as 1 cells, and the rest of S² as a 2 cell. SinceA and B are contractible,

A

X

X/

X/

B A B

X/A and X/B are homotopy equivalent to X . The space X/A is the quotient S²/S⁰, the sphere with two points identified, and X/B is S¹∨ S². HenceS²/S⁰ and S¹∨ S² are homotopy equivalent, a fact which may not be entirely obvious at first glance.

E

xample 0.9. Let X be the union of a torus with n meridional disks. To obtain a CW structure on X , choose a longitudinal circle in the torus, intersecting each of the meridional disks in one point. These intersection points are then the 0 cells, the 1 cells are the rest of the longitudinal circle and the boundary circles of the meridional disks, and the 2 cells are the remaining regions of the torus and the interiors of the meridional disks. Collapsing each meridional disk to a point yields a homotopy

X Y Z

W equivalent space Y consisting of n 2 spheres, each tangent to its two neighbors, a

‘necklace with n beads.’ The third space Z in the figure, a strand of n beads with a string joining its two ends, collapses to Y by collapsing the string to a point, so this collapse is a homotopy equivalence. Finally, by collapsing the arc inZ formed by the front halves of the equators of the n beads, we obtain the fourth space W , a wedge sum of S¹ with n 2 spheres. (One can see why a wedge sum is sometimes called a

‘bouquet’ in the older literature.)

E

xample 0.10: Reduced Suspension. Let X be a CW complex and x₀∈ X a 0 cell.

Inside the suspension SX we have the line segment{x₀}×I , and collapsing this to a point yields a spaceΣX homotopy equivalent to SX , called the reduced suspension of X . For example, if we take X to be S¹∨ S¹ with x₀ the intersection point of the two circles, then the ordinary suspensionSX is the union of two spheres intersecting along the arc {x₀}×I , so the reduced suspension ΣX is S²∨ S², a slightly simpler space. More generally we have Σ(X ∨ Y ) = ΣX ∨ ΣY for arbitrary CW complexes X andY . Another way in which the reduced suspensionΣX is slightly simpler than SX is in its CW structure. In SX there are two 0 cells (the two suspension points) and an (n+ 1) cell eⁿ×(0, 1) for each n cell eⁿ ofX , whereas in ΣX there is a single 0 cell and an(n+ 1) cell for each n cell of X other than the 0 cell x₀.

The reduced suspension ΣX is actually the same as the smash product X ∧ S¹ since both spaces are the quotient ofX×I with X×∂I ∪ {x0}×I collapsed to a point.

Attaching Spaces

Another common way to change a space without changing its homotopy type in- volves the idea of continuously varying how its parts are attached together. A general definition of ‘attaching one space to another’ that includes the case of attaching cells

is the following. We start with a space X₀ and another space X₁ that we wish to attach to X₀ by identifying the points in a subspace A⊂ X₁ with points of X₀. The data needed to do this is a map f : A

→

^X0, for then we can form a quotient space of X₀q X1 by identifying each point a∈ A with its image f (a) ∈ X0. Let us de- note this quotient space by X₀tfX₁, the spaceX₀ with X₁ attached along A via f . When (X₁, A)= (Dⁿ, Sⁿ⁻¹) we have the case of attaching an n cell to X₀ via a map f : Sⁿ⁻¹

→

^X0.

Mapping cylinders are examples of this construction, since the mapping cylinder M_f of a mapf : X

→

Y is the space obtained from Y by attaching X×I along X×{1}

viaf . Closely related to the mapping cylinder M_f is themapping cone C_f = Y tfCX where CX is the cone (X×I)/(X×{0}) and we attach this to Y

along X×{1} via the identifications (x, 1) ∼ f (x). For exam- ple, whenX is a sphere Sⁿ⁻¹ the mapping coneC_f is the space obtained from Y by attaching an n cell via f : Sⁿ⁻¹

→

^{Y . A}

mapping cone C_f can also be viewed as the quotient M_f/X of

the mapping cylinder M_f with the subspace X= X×{0} collapsed to a point.

CX

Y

If one varies an attaching map f by a homotopy f_t, one gets a family of spaces whose shape is undergoing a continuous change, it would seem, and one might expect these spaces all to have the same homotopy type. This is often the case:

If (X₁, A) is a CW pair and the two attaching maps f , g : A

→

^X0 are homotopic, then X₀tfX₁' X0tgX₁.

Again let us defer the proof and look at some examples.

E

xample 0.11. Let us rederive the result in Example 0.8 that a sphere with two points

A S¹ S²

identified is homotopy equivalent to S¹∨ S². The sphere with two points identified can be obtained by attachingS² toS¹ by a map that wraps a closed arc A in S² aroundS¹, as shown in the figure. Since A is contractible, this attach- ing map is homotopic to a constant map, and attachingS² to S¹ via a constant map of A yields S¹∨ S². The result

then follows since (S², A) is a CW pair, S² being obtained from A by attaching a 2 cell.

E

xample 0.12. In similar fashion we can see that the necklace in Example 0.9 is homotopy equivalent to the wedge sum of a circle with n 2 spheres. The necklace can be obtained from a circle by attaching n 2 spheres along arcs, so the necklace is homotopy equivalent to the space obtained by attaching n 2 spheres to a circle at points. Then we can slide these attaching points around the circle until they all coincide, producing the wedge sum.

E

xample 0.13. Here is an application of the earlier fact that collapsing a contractible subcomplex is a homotopy equivalence: If (X, A) is a CW pair, consisting of a cell

complex X and a subcomplex A , then X/A ' X ∪ CA, the mapping cone of the inclusionA

>

X . For we have X/A= (X∪CA)/CA ' X∪CA since CA is a contractible subcomplex ofX∪ CA.

E

xample 0.14. If (X, A) is a CW pair and A is contractible in X , that is, the inclusion A

>

X is homotopic to a constant map, then X/A' X ∨ SA. Namely, by the previous example we haveX/A' X ∪ CA, and then since A is contractible in X , the mapping cone X∪ CA of the inclusion A

>

X is homotopy equivalent to the mapping cone of a constant map, which is X∨ SA. For example, Sⁿ/Sⁱ ' Sⁿ∨ Sⁱ⁺¹ for i < n , since Sⁱ is contractible in Sⁿ if i < n . In particular this gives S²/S⁰ ' S²∨ S¹, which is Example 0.8 again.

The Homotopy Extension Property

In this final section of the chapter we will actually prove a few things. In partic- ular we prove the two criteria for homotopy equivalence described above, along with the fact that any two homotopy equivalent spaces can be embedded as deformation retracts of the same space.

The proofs depend upon a technical property that arises in many other contexts as well. Consider the following problem. Suppose one is given a mapf₀:X

→

^{Y , and}

on a subspaceA⊂ X one is also given a homotopy f_t:A

→

^{Y of f0}||A that one would like to extend to a homotopyf_t:X

→

Y of the given f₀. If the pair(X, A) is such that this extension problem can always be solved, one says that(X, A) has the homotopy extension property. Thus (X, A) has the homotopy extension property if every map X×{0} ∪ A×I

→

Y can be extended to a map X×I

→

^{Y .}

In particular, the homotopy extension property for (X, A) implies that the iden- tity map X×{0} ∪ A×I

→

^X×{0} ∪ A×I extends to a map X×I

→

^X×{0} ∪ A×I , so X×{0} ∪ A×I is a retract of X×I . The converse is also true: If there is a retraction X×I

→

^X×{0} ∪ A×I , then by composing with this retraction we can extend every map X×{0} ∪ A×I

→

Y to a map X×I

→

Y . Thus the homotopy extension property for(X, A) is equivalent to X×{0} ∪ A×I being a retract of X×I . This implies for ex- ample that if(X, A) has the homotopy extension property, then so does (X×Z, A×Z) for any spaceZ , a fact that would not be so easy to prove directly from the definition.

If(X, A) has the homotopy extension property, then A must be a closed subspace of X , at least when X is Hausdorff. For if r : X×I

→

^X×I is a retraction onto the subspace X×{0} ∪ A×I , then the image of r is the set of points z ∈ X×I with r (z)= z , a closed set if X is Hausdorff, so X×{0}∪A×I is closed in X×I and hence A is closed in X .

A simple example of a pair (X, A) with A closed for which the homotopy exten- sion property fails is the pair (I, A) where A= {0, 1,¹/₂,¹/₃,¹/₄,···}. It is not hard to show that there is no continuous retraction I×I

→

I×{0} ∪ A×I . The breakdown of homotopy extension here can be attributed to the bad structure of (X, A) near 0 .

With nicer local structure the homotopy extension property does hold, as the next example shows.

E

xample 0.15. A pair (X, A) has the homotopy extension property if A has a map- ping cylinder neighborhood in X , by which we mean that there

is a map f : Z

→

A and a homeomorphism h from M_f onto a closed neighborhood N of A in X , with h ||A = ¹1 and with h(M_f−Z) an open neighborhood of A. Mapping cylinder neigh-

X

Ah( )Z

N

borhoods like this occur more frequently than one might think.

For example, the thick letters discussed at the beginning of the

chapter provide such neighborhoods of the thin letters, regarded as subspaces of the plane. To verify the homotopy extension property, notice first that I×I retracts onto I×{0} ∪ ∂I×I , hence Z ×I×I retracts onto Z ×I×{0} ∪ Z ×∂I×I , and this retraction induces a retraction of M_f×I onto Mf×{0} ∪ (Z q A)×I . Thus (Mf, Zq A) has the homotopy extension property. Hence so does the homeomorphic pair (N, h(Z)q A).

Now given a map X

→

Y and a homotopy of its restriction to A , we can take the con- stant homotopy onX−h(Mf−Z) and then extend over N by applying the homotopy extension property for (N, h(Z)q A) to the given homotopy on A and the constant homotopy onh(Z) .

Most applications of the homotopy extension property in this book will come from the following general result:

P

roposition 0.16. If (X, A) is a CW pair, then X×{0}∪A×I is a deformation retract of X×I , hence (X, A) has the homotopy extension property.

P

roof: There is a retraction r : Dⁿ×I

→

^{Dn×{0} ∪ ∂Dn}×I , for ex- ample the radial projection from the point (0, 2)∈ Dⁿ×R. Then settingr_t= tr + (1 − t)¹1 gives a deformation retraction ofDⁿ×I onto Dⁿ×{0} ∪ ∂Dⁿ×I . This deformation retraction gives rise to a deformation retraction ofXⁿ×I onto Xⁿ×{0} ∪ (Xⁿ⁻¹∪ Aⁿ)×I sinceXⁿ×I is obtained from Xⁿ×{0} ∪ (Xⁿ⁻¹∪ Aⁿ)×I by attach-

ing copies of Dⁿ×I along Dⁿ×{0} ∪ ∂Dⁿ×I . If we perform the deformation retrac- tion of Xⁿ×I onto Xⁿ×{0} ∪ (Xⁿ⁻¹∪ Aⁿ)×I during the t interval [1/2ⁿ⁺¹, 1/2ⁿ] , this infinite concatenation of homotopies is a deformation retraction of X×I onto X×{0} ∪ A×I . There is no problem with continuity of this deformation retraction at t= 0 since it is continuous on Xⁿ×I , being stationary there during the t interval [0, 1/2ⁿ⁺¹] , and CW complexes have the weak topology with respect to their skeleta so a map is continuous iff its restriction to each skeleton is continuous. tu

Now we can prove a generalization of the earlier assertion that collapsing a con- tractible subcomplex is a homotopy equivalence.