Allen Hatcher

Copyright c 2000 by Allen Hatcher

Paper or electronic copies for noncommercial use may be made freely without explicit permission from the author.

All other rights reserved.

Table of Contents

Chapter 0. Some Underlying Geometric Notions . . . . . 1

Homotopy and Homotopy Type 1. Cell Complexes 5.

Operations on Spaces 8. Two Criteria for Homotopy Equivalence 11.

The Homotopy Extension Property 14.

Chapter 1. The Fundamental Group . . . . . . . . . . . . . 21

1. Basic Constructions . . . . . . . . . . . . . . . . . . . . . . . 25Paths and Homotopy 25. The Fundamental Group of the Circle 28.

Induced Homomorphisms 34.

2. Van Kampen’s Theorem . . . . . . . . . . . . . . . . . . . . 39Free Products of Groups 39. The van Kampen Theorem 41.

Applications to Cell Complexes 49.

3. Covering Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 55Lifting Properties 59. The Classification of Covering Spaces 62.

Deck Transformations and Group Actions 69.

Additional TopicsA. Graphs and Free Groups 81.

B. K(G,1) Spaces and Graphs of Groups 86.

Chapter 2. Homology . . . . . . . . . . . . . . . . . . . . . . . 97

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. Computations and Applications . . . . . . . . . . . . . . . 134Degree 134. Cellular Homology 137. Mayer-Vietoris Sequences 149.

Homology with Coefficients 153.

3. The Formal Viewpoint . . . . . . . . . . . . . . . . . . . . . 160Axioms for Homology 160. Categories and Functors 162.

Additional TopicsA. Homology and Fundamental Group 166.

B. Classical Applications 168.

C. Simplicial Approximation 175.

Chapter 3. Cohomology . . . . . . . . . . . . . . . . . . . . . 183

1. Cohomology Groups . . . . . . . . . . . . . . . . . . . . . . 188The Universal Coefficient Theorem 188. Cohomology of Spaces 195.

2. Cup Product . . . . . . . . . . . . . . . . . . . . . . . . . . . 204The Cohomology Ring 209. A Künneth Formula 215.

Spaces with Polynomial Cohomology 221.

3. Poincaré Duality . . . . . . . . . . . . . . . . . . . . . . . . . 228Orientations and Homology 231. The Duality Theorem 237.

Connection with Cup Product 247. Other Forms of Duality 250.

Additional TopicsA. The Universal Coefficient Theorem for Homology 259.

B. The General Künneth Formula 266.

C. H–Spaces and Hopf Algebras 279.

D. The Cohomology of SO(n) 291.

E. Bockstein Homomorphisms 301.

F. Limits 309.

G. More About Ext 316.

H. Transfer Homomorphisms 320.

I. Local Coefficients 327.

Chapter 4. Homotopy Theory . . . . . . . . . . . . . . . . . 337

1. Homotopy Groups . . . . . . . . . . . . . . . . . . . . . . . 339Definitions and Basic Constructions 340. Whitehead’s Theorem 346.

Cellular Approximation 348. CW Approximation 351.

2. Elementary Methods of Calculation . . . . . . . . . . . . . 359Excision for Homotopy Groups 359. The Hurewicz Theorem 366.

Fiber Bundles 374. Stable Homotopy Groups 383.

3. Connections with Cohomology . . . . . . . . . . . . . . . . 392The Homotopy Construction of Cohomology 393. Fibrations 404.

Postnikov Towers 409. Obstruction Theory 415.

Additional TopicsA. Basepoints and Homotopy 421.

B. The Hopf Invariant 427.

C. Minimal Cell Structures 429.

D. Cohomology of Fiber Bundles 432.

E. The Brown Representability Theorem 448.

F. Spectra and Homology Theories 453.

G. Gluing Constructions 456.

H. Eckmann-Hilton Duality 461.

I. Stable Splittings of Spaces 468.

J. The Loopspace of a Suspension 471.

K. Symmetric Products and the Dold-Thom Theorem 477.

L. Steenrod Squares and Powers 489.

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521Topology of Cell Complexes 521. The Compact-Open Topology 531.

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 540

Preface

This book was written to be a readable introduction to Algebraic Topology with

rather broad coverage of the subject. Our viewpoint is quite classical in spirit, and

stays largely within the confines of pure Algebraic Topology. In a sense, the book

could have been written thirty years ago since virtually all its content is at least that

old. However, the passage of the intervening years has helped clarify what the most

important results and techniques are. For example, CW complexes have proved over

time to be the most natural class of spaces for Algebraic Topology, so they are em-

phasized 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, as-

pects 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 that it provides. At deeper levels, alge-

bra becomes increasingly important, so for the sake of balance it seems only fair to

emphasize geometry at the beginning.

Let us say something about the organization of the book. At the elementary level,

Algebraic Topology divides naturally into two channels, with the broad topic of Ho-

motopy on the one side and Homology on the other. We have divided this material

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, how-

ever. One could begin with Homology and perhaps continue on with Cohomology

before turning to Homotopy. In the other direction, one could postpone Homology

and Cohomology until after parts of Chapter 4. However, we have not pushed this

latter approach to its natural limit, in which Homology and Cohomology arise 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

our first priorities, we have delayed introducing this unifying viewpoint until later in

the book.

There is also a preliminary Chapter 0 introducing some of the basic geometric

concepts and constructions that play a central role in both the homological and ho-

motopical sides of the subject.

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 constraints of a first course. Altogether, these Additional Topics amount

to nearly half the book, and we have included them 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 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 famil-

iarity with the content of the standard undergraduate courses in algebra and point-set

topology. One topic that is not always a part of a first point-set topology course but

which is quite important for Algebraic Topology is quotient spaces, or identification

spaces as they are sometimes called. Good sources for this are the textbooks by Arm-

strong and Jänich listed in the Bibliography.

A book such as this one, whose aim is to present classical material from a fairly

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 simplicial complexes, which we call ∆ complexes. Thesehave made brief appearances in the literature previously, without a standard name

emerging. The idea is to weaken the condition that each simplex be embedded, to

require only that the interiors of simplices are embedded. (In addition, an ordering

of the vertices of each simplex is also part of the structure of a ∆ complex.) Forexample, 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, and1 vertex. By contrast, it is known that a simplicial complex structure on the torus

must have at least 14 triangles, 21 edges, and 7 vertices. So ∆ complexes providea 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 just seem to be very natural objects fromthe viewpoint of Algebraic Topology. They are the natural domain of definition for

simplicial homology, and a number of standard constructions produce ∆ complexesrather than simplicial complexes, for instance the singular complex of a space, or the

classifying space of a discrete group or category.

It is the author’s intention to keep this book available online permanently, as well

as publish it in the traditional manner for those who want the convenience of a bound

copy. With the electronic version it will be possible to continue making revisions and

additions, so comments and suggestions from readers will always be welcome. The

web address is:

http://www.math.cornell.edu/˜hatcher

One can also find here the parts of the other two books that are currently available.

Standard Notations

Rn : n dimensional Euclidean space, with real coordinatesCn : complex n spaceI = [0,1] : the unit intervalSn : the unit sphere in Rn+1 , all vectors of length 1Dn : the unit disk or ball in Rn , all vectors of length ≤ 1∂Dn = Sn−1 : the boundary of the n disk11: the identity function from a set to itself

q : disjoint union≈ : isomorphismZn : the integers modnA ⊂ B or B ⊃ A : set-theoretic containment, not necessarily proper

The aim of this short preliminary chapter is to introduce a few of the most com-

mon 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 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 written

either as unions of finitely many straight and curved line segments, or in thickened

forms that are compact subsurfaces of the plane bounded by simple closed curves.

In each case the thin letter is a subspace of the thick letter, and we can continuously

shrink the thick letter to the thin one. A nice way to 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

2 Chapter 0. Some Underlying Geometric Notions

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 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 willbe true if we have each x ∈ X − X move along its line segment at constant speed soas to reach its image point in X at time t = 1, while points x ∈ X are stationary, asremarked earlier.

These examples lead to the following general definition. A deformation retrac-

tion of a space X onto a subspace A is a family of maps ft :X→X , t ∈ I , suchthat f0 = 11 (the identity map), f1(X) = A , and ft ||A = 11 for all t . The family ftshould 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 first figurebelow shows such a deformation retraction of a Möbius band onto its core circle. The

other three figures 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 map-ping cylinder Mf is the quotient space of the disjoint union (X×I)q Y obtained byidentifying each (x,1) ∈ X×I with f(x) ∈ Y .

X × IX

Y Y

Mff X( )

In the letter examples, the space X is the outer boundary of the thick letter, Y is thethin letter, and the map f :X→Y sends the outer endpoint of each line segment toits 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 bysliding 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, how-

ever. For example, the thick X deformation retracts to the thin X , which in turn

Homotopy and Homotopy Type 3

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 ft :X→X is a special case of the general notion of ahomotopy, which is simply any family of maps ft :X→Y , t ∈ I , such that the asso-ciated map F :X×I→Y given by F(x, t) = ft(x) is continuous. One says that twomaps f0, f1 :X→Y are homotopic if there exists a homotopy ft connecting them,and one writes f0 ' f1 .

In these terms, a deformation retraction of X onto a subspace A is a homotopyfrom the identity map of X to a retraction of X onto A , a map r :X→X such thatr(X) = A and r ||A = 11. One could equally well regard a retraction as a map X→Arestricting to the identity on the subspace A ⊂ X . From a more formal viewpoint aretraction is a map r :X→X with r 2 = r , since this equation says exactly that r is theidentity 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 thatdeformation 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 lesstrivial 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 subspaceA has the property that ft ||A = 11 for all t . In general, a homotopy ft :X→Y whoserestriction to a subspace A ⊂ X is independent of t is called a homotopy relativeto A , or more concisely, a homotopy rel A . Thus, a deformation retraction of X ontoA 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 ft :X→X , then ifr :X→A denotes the resulting retraction and i :A→X the inclusion, we have ri = 11and ir ' 11, the latter homotopy being given by ft . Generalizing this situation, amap f :X→Y is called a homotopy equivalence if there is a map g :Y→X such thatfg ' 11 and gf ' 11. The spaces X and Y are said to be homotopy equivalent or tohave the same homotopy type. The notation is X ' Y . It is an easy exercise to checkthat 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.

4 Chapter 0. Some Underlying Geometric Notions

It is true in general that two spaces X and Y are homotopy equivalent if and onlyif there exists a third space Z containing both X and Y as deformation retracts. Forthe less trivial implication one can in fact take Z to be the mapping cylinder Mf ofany homotopy equivalence f :X→Y . We observed previously that Mf deformationretracts to Y , so what needs to be proved is that Mf also deformation retracts to itsother end X if f is a homotopy equivalence. This is shown in Corollary 0.21 near theend of this chapter.

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 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 of R3 , known as

the house with two rooms, which is contractible but not in any obvious way.

R

To build this space, start with a box divided into two chambers by a horizontal rect-

angle R , where by a ‘rectangle’ we mean not just the four edges of a rectangle butalso 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 R , then inserting four verticalrectangles, 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 horizontalpieces homeomorphic to annuli S1×I , plus all the vertical rectangles that form thewalls of the two chambers: the exterior walls, the walls of the tunnels, and the two

support walls.

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 . Lessobvious is the fact that N(X) is homeomorphic to D3 , the unit ball in R3 . To seethis, imagine forming N(X) from a ball of clay by pushing a finger into the ball to

Cell Complexes 5

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 :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 if ft is a deformation retractionof the ball N(X) to a point x0 ∈ X and if r :N(X)→X is a retraction, for examplethe end result of a deformation retraction of N(X) to X , then the restriction of thecomposition rft to X is a deformation retraction of X to x0 . However, it is noteasy to see exactly what this deformation retraction looks like! A slightly easier test

of geometric visualization is to find a nullhomotopy in X of the loop formed by ahorizontal cross section of one of the tunnels. We leave this as a puzzle for the

reader.

Cell Complexes

A familiar way of constructing the torus S1×S1 is by identifying opposite sidesof a square. More generally, an orientable surface Mg of genus g can be constructedfrom a 4g sided polygon by identifying pairs of edges, as shown in the figure for thecases g = 1,2,3.

aba

a

b b

b

b

b

bc

a

a

a

d

a

c

c

c

c

bd

d

dde

e

f

f

a

e

f

dcb

a

The 4g edges of the polygon become a union of 2g circles in the surface, all inter-secting in a single point. One can think of the interior of the polygon as an open

disk, or 2 cell, attached to the union of these circles. One can also regard the union

of the circles as being obtained from a point, their common point of intersection, by

6 Chapter 0. Some Underlying Geometric Notions

attaching 2g open arcs, or 1 cells. Thus the surface can be built up in stages: Startwith 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 the n skeleton Xn from Xn−1 by attaching n cells enα via

maps ϕα :Sn−1→Xn−1 . That is, Xn is the quotient space of the disjoint union

Xn−1∐αD

nα of X

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

∐αenα where each e

nα 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 = ⋃n Xn . In the lattercase X is given the weak topology: A set A ⊂ X is open (or closed) iff A∩Xn isopen (or closed) in Xn for each n .

A space X constructed in this way is called a cell complex, or more classically, aCW 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 that lurk in the background of

the following discussion should consult the Appendix for details.

Example 0.1. A 1 dimensional cell complex X = X1 is what is called a graph inalgebraic 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 obvious2 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 ofthe 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 isthe 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. The sphere Sn has 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 equivalentto regarding Sn as the quotient space Dn/∂Dn .

Example 0.4. Real projective n space RPn is defined to be the space of all linesthrough the origin in Rn+1 . Each such line is determined by a nonzero vector in Rn+1 ,

Cell Complexes 7

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 restrictto vectors of length 1, so RPn is also the quotient space Sn/(v ∼ −v) , the spherewith antipodal points identified. This is equivalent to saying that RPn is the quotient

space of a hemisphere Dn with antipodal points of ∂Dn identified. Since ∂Dn withantipodal points identified is just RPn−1 , we see that RPn is obtained from RPn−1 byattaching 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 ∪ ··· ∪ enwith one cell ei in each dimension i ≤ n .

Example 0.5. Since RPn is obtained from RPn−1 by attaching an n cell, the infiniteunion RP∞ = ⋃nRPn becomes a cell complex with one cell in each dimension. Wecan view RP∞ as the space of lines through the origin in R∞ = ⋃nRn .Example 0.6. Complex projective n space CPn is the space of complex lines throughthe origin in Cn+1 , that is, 1 dimensional vector subspaces of Cn+1 . As in the caseof RPn , each line is determined by a nonzero vector in Cn+1 , unique up to scalarmultiplication, and CPn is topologized as the quotient space of Cn+1−{0} under theequivalence relation v ∼ λv for λ ≠ 0. Equivalently, this is the quotient of the unitsphere S2n+1 ⊂ Cn+1 with v ∼ λv for |λ| = 1. It is also possible to obtain CPn as aquotient space of the disk D2n under the identifications v ∼ λv for v ∈ ∂D2n , in thefollowing way. The vectors in S2n+1 ⊂ Cn+1 with last coordinate real and nonnegativeare precisely the vectors of the form (w,

√1− |w|2 ) ∈ Cn×C with |w| ≤ 1. Such

vectors form the graph of the function w,√

1− |w|2 . This is a disk D2n+ boundedby the sphere S2n−1 ⊂ S2n+1 consisting of vectors (w,0) ∈ Cn×C with |w| = 1. Eachvector in S2n+1 is equivalent under the identifications v ∼ λv to a vector in D2n+ , andthe 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 D2n+ under the identifications

v ∼ λv for v ∈ S2n−1 it follows that CPn is obtained from CPn−1 by attaching acell e2n via the quotient map S2n−1→CPn−1 . So by induction on n we obtain a cellstructure CPn = e0∪ e2∪···∪e2n with cells only in even dimensions. Similarly, CP∞has a cell structure with one cell in each even dimension.

Each cell enα in a cell complex X has a characteristic map Φα :Dnα→X thatextends the attaching map ϕα and is a homeomorphism from the interior of D

nα

onto enα . Namely, we can take Φα to be the composition Dnα↩Xn−1∐αDnα→Xn↩Xwhere the middle map is the quotient map defining Xn . For example, in the canonicalcell structure on Sn described in Example 0.3, a characteristic map for the n cell isthe quotient map Dn→Sn collapsing ∂Dn to a point. For RPn a characteristic mapfor the cell ei is the quotient map Di→RPi ⊂ RPn identifying antipodal points of∂Di , and similarly for CPn .

8 Chapter 0. Some Underlying Geometric Notions

A subcomplex of a cell complex X is a closed subspace A ⊂ X that is a unionof cells of X . Since A is closed, the characteristic map of each cell in A has imagecontained in A , and in particular the image of the attaching map of each cell in A iscontained in A , so A is a cell complex in its own right. A pair (X,A) consisting of acell complex X and a subcomplex A we call a CW pair.

For example, each skeleton Xn of a cell complex X is a subcomplex. Particularcases of this are the subcomplexes RPk ⊂ RPn and CPk ⊂ CPn for k ≤ n . These arein fact the only subcomplexes of RPn and CPn .

There are natural inclusions S0 ⊂ S1 ⊂ ··· ⊂ Sn , but these subspheres are notsubcomplexes of Sn in its usual cell structure with just two cells. However, we can giveSn a different cell structure in which each of the subspheres Sk is a subcomplex, byregarding each Sk as being obtained inductively from the equatorial Sk−1 by attachingtwo k cells, the components of Sk−Sk−1 . The infinite-dimensional sphere S∞ = ⋃n Snthen becomes a cell complex as well. Note that the two-to-one quotient map S∞→RP∞that identifies antipodal points of S∞ identifies the two n cells of S∞ to the singlen cell of RP∞ .

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 with S1 with its minimal cell structure and attach to thisa 2 cell by a map S1→S1 whose image is a nontrivial subarc of S1 , then the closureof 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 complexwith cells the products emα ×enβ where emα ranges over the cells of X and enβ rangesover the cells of Y . For example, the cell structure on the torus S1×S1 described atthe beginning of this section is obtained in this way from the standard cell structure

on S1 . In the general case there is one small complication, however: The topology onX×Y as a cell complex is sometimes slightly weaker than the product topology, withmore 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 manycells. This is explained in the Appendix. In practice this subtle point of point-set

topology rarely causes problems.

Operations on Spaces 9

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 . Thecells of X/A are the cells of X−A plus one new 0 cell, the image of A in X/A . For a cellenα of X −A attached by ϕα :Sn−1→Xn−1 , the attaching map for the correspondingcell in X/A is the composition Sn−1→Xn−1→Xn−1/An−1 .

For example, if we give Sn−1 any cell structure and build Dn from Sn−1 by attach-ing an n cell, then the quotient Dn/Sn−1 is Sn with its usual cell structure. As anotherexample, take X to be a closed orientable surface with the cell structure described atthe beginning of this section, with a single 2 cell, and let A be the complement of this2 cell, the 1 skeleton of X . Then X/A has a cell structure consisting of a 0 cell witha 2 cell attached, and there is only one way to attach a cell to a 0 cell, by the constant

map, so X/A is S2 .

Suspension. For a space X , the suspension SX is the quotient of X×I obtainedby collapsing X×{0} to one point and X×{1} to another point. The motivatingexample is X = Sn , when SX = Sn+1 with the two ‘suspension points’ atthe north and south poles of Sn+1 , the points (0, ··· ,0,±1) . One canregard SX as a double cone on X , the union of two copies of the coneCX = (X×I)/(X×{0}) . If X is a CW complex, so are SX and CXas quotients of X×I with its product cell structure, I being given thestandard 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 map f :X→Y suspends to Sf :SX→SY , the quotient map off×11 :X×I→Y×I .Join. The cone CX is the union of all line segments joining points of X to an externalvertex, and similarly the suspension SX is the union of all line segments joining pointsof X to two external vertices. More generally, given X and a second space Y , one candefine the space of all lines segments joining points in X to points in Y . This isthe join 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, thenwe are collapsing two opposite faces of a

cube onto line segments so that it becomes

a tetrahedron. In the general case, X ∗ Y X I

Y

contains copies of X and Y at its two ‘ends,’ and every other point (x,y, t) in X∗Y ison 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.

10 Chapter 0. Some Underlying Geometric Notions

A nice way to write points of X∗Y is as formal linear combinations t1x+t2y with0 ≤ ti ≤ 1 and t1 + t2 = 1, subject to the rules 0x + 1y = y and 1x + 0y = x whichcorrespond exactly to the identifications that define X∗Y . In much the same way, aniterated join X1∗···∗Xn can be regarded as the space of formal linear combinationst1x1 + ··· + tnxn with 0 ≤ ti ≤ 1 and t1 + ··· + tn = 1, with the convention thatterms 0ti can be omitted. This viewpoint makes it easy to see that the join operationis associative. A very special case that plays a central role in algebraic topology is

when each Xi is just a point. For example, the join of two points is a line segment, thejoin of three points is a triangle, and the join of four points is a tetrahedron. The join

of n 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 each Xi is S0 , two points. If we take the two

points of Xi to be the two unit vectors along the ith coordinate axis in Rn , then the

join X1∗···∗Xn is the union of 2n copies of the simplex ∆n−1 , and radial projectionfrom the origin gives a homeomorphism between X1 ∗ ··· ∗Xn and Sn−1 .

If X and Y are CW complexes, then there is a natural CW structure on X ∗ Yhaving the subspaces X and Y as subcomplexes, with the remaining cells being theproduct cells of X×Y×(0,1) . As usual with products, the CW topology on X∗Y maybe 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 andY with chosen points x0 ∈ X and y0 ∈ Y , then the wedge sum X∨Y is the quotientof the disjoint union X q Y obtained by identifying x0 and y0 to a single point. Forexample, S1 ∨ S1 is homeomorphic to the figure ‘8,’ two circles touching at a point.More generally one could form the wedge sum

∨αXα of an arbitrary collection of

spaces Xα by starting with the disjoint union∐αXα and identifying points xα ∈ Xα

to a single point. In case the spaces Xα are cell complexes and the points xα are0 cells, then

∨αXα is a cell complex since it is obtained from the cell complex

∐αXα

by collapsing a subcomplex to a point.

For any cell complex X , the quotient Xn/Xn−1 is a wedge sum of n spheres∨αS

nα ,

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 ,namely X×{y0} and {x0}×Y for points x0 ∈ X and y0 ∈ Y . These two copies of Xand Y in X×Y intersect only at the point (x0, y0) , so their union can be identifiedwith the wedge sum X ∨ 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 obtainedby collapsing away the parts that are not genuinely a product, the separate factors Xand Y .

Two Criteria for Homotopy Equivalence 11

The smash product X∧Y is a cell complex if X and Y are cell complexes with x0and y0 0 cells, assuming that we give X×Y the cell-complex topology rather than theproduct topology in cases when these two topologies differ. For example, Sm∧Sn hasa cell structure with just two cells, of dimensions 0 and m+n , hence Sm∧Sn = Sm+n .In particular, when m = n = 1 we see that collapsing longitude and meridian circlesof 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

between rather different-looking spaces. However, this process can be a bit cumber-

some in practice, so it is useful to have other techniques available as well. Here is one

that can be quite helpful:

(1)If (X,A) is a CW pair consisting of a CW complex X and a contractible subcom-plex A , then the quotient map X→X/A is a homotopy equivalence.

A proof of this will be given later in Proposition 0.17, but let us look at some examples

now.

Example 0.7: Graphs. The three graphs are homotopy equivalentsince each is a deformation retract of a disk with two holes, but we can also deduce

this from statement (1) 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. Ifthe two endpoints of any edge of X 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, hence each component of X is either anisolated 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 sum∨mS

1 of m circles is nothomotopy equivalent to

∨nS

1 if m ≠ n . This sort of thing is hard to do directly. Whatone would like is some sort of algebraic object associated to spaces, depending only

on their homotopy type, and taking different values for∨mS

1 and∨nS

1 if m ≠ n . Infact the Euler characteristic does this since

∨mS

1 has Euler characteristic 1−m . But itis 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.

12 Chapter 0. Some Underlying Geometric Notions

Example 0.8. Consider the space X obtained from S2 by attaching the two ends ofan arc A to two distinct points on the sphere, say the north and south poles. Let Bbe an arc in S2 joining the two points where A attaches. Then X can be given a CWcomplex structure with the two endpoints of A and B as 0 cells, the interiors of Aand B as 1 cells, and the rest of S2 as a 2 cell. Since A and B are contractible, X/Aand X/B are homotopy equivalent to X . The space X/A is the quotient S2/S0 , thesphere with two points identified, and X/B is S1 ∨ S2 . Hence S2/S0 and S1 ∨ S2 arehomotopy equivalent, which might not have been entirely obvious a priori.

AB

X X/X/ BA

Example 0.9. Let X be the union of a torus with n meridional disks. To obtain a CWstructure on X , choose a longitudinal circle in X , intersecting each of the meridionaldisks 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.

X Y Z W Collapsing each meridional disk to a point yields a homotopy equivalent space Yconsisting of n 2 spheres, each tangent to its two neighbors, a ‘necklace with nbeads.’ The third space Z in the figure, a strand of n beads with a string joiningits two ends, collapses to Y by collapsing the string to a point, so this collapse is ahomotopy equivalence. Finally, by collapsing the arc in Z formed by the front halvesof 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’ inthe older literature.)

Example 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 apoint yields a space ΣX homotopy equivalent to SX , called the reduced suspensionof X . For example, if we take X to be S1 ∨ S1 with x0 the intersection point of thetwo circles, then the ordinary suspension SX is the union of two spheres intersectingalong the arc {x0}×I , so the reduced suspension ΣX is S2 ∨ S2 , a slightly simpler

Two Criteria for Homotopy Equivalence 13

space. More generally we have Σ(X ∨ Y) = ΣX ∨ ΣY for arbitrary CW complexes Xand Y . Another way in which the reduced suspension ΣX is slightly simpler than SXis 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 of X , whereas in ΣX there is a single 0 celland 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 ∧ S1since both spaces are the quotient of X×I with X×∂I∪{x0}×I collapsed to a point.

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 toattach to X0 by identifying the points in a subspace A ⊂ X1 with points of X0 . Thedata needed to do this is a map f :A→X0 , for then we can form a quotient spaceof X0 q X1 by identifying each point a ∈ A with its image f(a) ∈ X0 . Let us de-note this quotient space by X0 tf X1 , the space X0 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 mapf :Sn−1→X0 .

Mapping cylinders are examples of this construction, since the mapping cylinder

Mf of a map f :X→Y is the space obtained from Y by attaching X×I along X×{1}via f . Closely related to the mapping cylinder Mf is the mapping cone Cf = Y tf CXwhere CX is the cone (X×I)/(X×{0}) and we attach this to Y alongX×{1} via the identifications (x,1) ∼ f(x) . For example, when Xis a sphere Sn−1 the mapping cone Cf is the space obtained fromY by attaching an n cell via f :Sn−1→Y . A mapping cone Cf canalso be viewed as the quotient Mf/X of the mapping cylinder Mf with the subspaceX = X×{0} collapsed to a point.

CX

Y

Here is our second criterion for homotopy equivalence:

(2)If (X1, A) is a CW pair and we have two attaching maps f , g :A→X0 that arehomotopic, then X0 tf X1 ' X0 tg X1 .

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

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

A S1S2

identified is homotopy equivalent to S1∨S2 . The sphere withtwo points identified can be obtained by attaching S2 to S1

by a map that wraps a closed arc A in S2 around S1 , asshown in the figure. Since A is contractible, this attachingmap is homotopic to a constant map, and attaching S2 to S1

via a constant map of A yields S1∨S2 . The result then follows from (2) since (S2, A)is a CW pair, S2 being obtained from A by attaching a 2 cell.

14 Chapter 0. Some Underlying Geometric Notions

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

coincide, producing the wedge sum.

Example 0.13. Here is an application of the earlier fact that collapsing a contractiblesubcomplex is a homotopy equivalence: If (X,A) is a CW pair, consisting of a cellcomplex X and a subcomplex A , then X/A ' X ∪ CA , the mapping cone of theinclusion A↩X . For we have X/A = (X∪CA)/CA ' X∪CA since CA is a contractiblesubcomplex of X ∪ CA .

Example 0.14. If (X,A) is a CW pair and A is contractible in X , i.e., the inclusionA↩X is homotopic to a constant map, then X/A ' X∨SA . Namely, by the previousexample we have X/A ' X ∪CA , and then since A is contractible in X , the mappingcone X ∪CA of the inclusion A↩X is homotopy equivalent to the mapping cone ofa constant map, which is X ∨ SA . For example, Sn/Si ' Sn ∨ Si+1 for i < n , sinceSi is contractible in Sn if i < n . In particular this gives S2/S0 ' S2 ∨ S1 , which isExample 0.8 again.

The Homotopy Extension Property

In this final section of the chapter we shall actually prove a few things, in particular

the two criteria for homotopy equivalence described above and 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 map f0 :X→Y , andon a subspace A ⊂ X one is also given a homotopy ft :A→Y of f0 ||A that one wouldlike to extend to a homotopy ft :X→Y of the given f0 . If the pair (X,A) is such thatthis extension problem can always be solved, one says that (X,A) has the homotopyextension property. Thus (X,A) has the homotopy extension property if every mapX×{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 , soX×{0} ∪ A×I is a retract of X×I . The converse is also true: If there is a retractionX×I→X×{0} ∪ A×I , then by composing with this retraction we can extend everymap X×{0} ∪A×I→Y to a map X×I→Y . Thus the homotopy extension propertyfor (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 space Z , a fact that would not be so easy to prove directly from the definition.

The Homotopy Extension Property 15

If (X,A) has the homotopy extension property, then A must be a closed subspaceof X , at least when X is Hausdorff. For if r :X×I→X×I is a retraction onto thesubspace X×{0} ∪ A×I , then the image of r is the set of points z ∈ X×I withr(z) = z , a closed set if X is Hausdorff, so X×{0}∪A×I is closed in X×I and henceA 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 toshow that there is no continuous retraction I×I→I×{0} ∪A×I . The breakdown ofhomotopy 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.

Example 0.15. A pair (X,A) has the homotopy extension property if A has a map-ping cylinder neighborhood, in the following sense: There is a map f :Z→A and ahomeomorphism h from Mf onto a closed neighborhood N of A in X , with h||A = 11and with h(Mf − Z) an open neighborhood of A . Mapping cylinder neighborhoodslike this occur more frequently than one might think. For example, the thick let-

ters 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 retractsonto Z×I×{0} ∪ Z×∂I×I , and this retraction induces a retraction of Mf×I ontoMf×{0}∪(ZqA)×I . Thus (Mf ,ZqA) has the homotopy extension property, whichimplies that (X,A) does also since given a map X→Y and a homotopy of its restric-tion to A , we can take the constant homotopy on the closure of X−N and then applythe homotopy extension property for (Mf ,Z qA) to extend the homotopy over N .

Most applications of the homotopy extension property in this book will stem from

the following general result:

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

Proof: There is a retraction r :Dn×I→Dn×{0} ∪ ∂Dn×I , for examplethe radial projection from the point (0,2) ∈ Dn×R . Then settingrt = tr + (1 − t)11 gives a deformation retraction of Dn×I ontoDn×{0} ∪ ∂Dn×I . This deformation retraction gives rise to adeformation retraction of Xn×I onto Xn×{0} ∪ (Xn−1 ∪ An)×Isince Xn×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 ontoX×{0} ∪ A×I . (There is no problem with continuity of this deformation retractionat t = 0 since it is continuous on Xn×I , being stationary there during the t interval

16 Chapter 0. Some Underlying Geometric Notions

[0,1/2n+1] , and CW complexes have the weak topology with respect to their skeletaso a map is continuous iff its restriction to each skeleton is continuous.) tu

Now we can prove the following generalization of the earlier criterion (1) for ho-

motopy equivalence:

Proposition 0.17. If the pair (X,A) satisfies the homotopy extension property andA is contractible, then the quotient map q :X→X/A is a homotopy equivalence.

Proof: Let ft :X→X be a homotopy extending a contraction of A , with f0 = 11. Sinceft(A) ⊂ A for all t , the composition qft sends A to a point and hence factors as acomposition X q-----→X/A ft-----→X/A . Thus we have qft = f tq in the first of the followingtwo diagrams:

X

X/

X

A X/A

q q

ft

ft

X

X/

X

A X/A

q g q

f1

f1

−−−−−−−−−−−−→

−−−−−−−−→

−−−−−−→−−−−−−→

−−−−−−−−−−−−→

−−−−−−−−−−−−→

−−−−−−−−→

−−−−−−→−−−−−−→

−−

When t = 1 we have f1(A) equal to a point, the point to which A contracts, so f1induces a map g :X/A→X with gq = f1 . It follows that qg = f 1 since qg(x) =qgq(x) = qf1(x) = f 1q(x) = f 1(x) . The maps g and q are inverse homotopyequivalences since gq = f1 ' f0 = 11 via ft and qg = f 1 ' f 0 = 11 via f t . tu

Another application of the homotopy extension property, giving a slightly more

refined version of the criterion (2) for homotopy equivalence, is the following:

Proposition 0.18. If (X1, A) is a CW pair and we have attaching maps f , g :A→X0that are homotopic, then X0 tf X1 ' X0 tg X1 rel X0 .

Here the definition of W ' Z rel Y for pairs (W,Y) and (Z, Y) is that there aremaps ϕ :W→Z and ψ :Z→W restricting to the identity on Y , such that ψϕ ' 11and ϕψ ' 11 via homotopies that restrict to the identity on Y at all times.

Proof: If F :A×I→X0 is a homotopy from f to g , consider the space X0tF (X1×I) .This contains both X0 tf X1 and X0 tg X1 as subspaces. A deformation retractionof X1×I onto X1×{0}∪A×I as in Proposition 0.16 induces a deformation retractionof X0 tF (X1×I) onto X0 tf X1 . Similarly X0 tF (X1×I) deformation retracts ontoX0tgX1 . Both these deformation retractions restrict to the identity on X0 , so togetherthey give a homotopy equivalence X0 tf X1 ' X0 tg X1 rel X0 . tu

We finish this chapter with a technical result whose proof will involve several

applications of the homotopy extension property:

The Homotopy Extension Property 17

Proposition 0.19. Suppose (X,A) and (Y ,A) satisfy the homotopy extension prop-erty, and f :X→Y is a homotopy equivalence with f ||A = 11 . Then f is a homotopyequivalence rel A .

Corollary 0.20. If (X,A) satisfies the homotopy extension property and the inclusionA↩X is a homotopy equivalence, then A is a deformation retract of X .

Proof: Apply the proposition to the inclusion A↩X . tu

Corollary 0.21. A map f :X→Y is a homotopy equivalence iff X is a deformationretract of the mapping cylinder Mf . Hence, two spaces X and Y are homotopyequivalent iff there is a third space containing both X and Y as deformation retracts.

Proof: The inclusion i :X↩Mf is homotopic to the composition jf where j is theinclusion Y↩Mf , a homotopy equivalence. It then follows from Exercise 3 at theend of the chapter that i is a homotopy equivalence iff f is a homotopy equivalence.This gives the ‘if’ half of the first statement of the corollary. For the converse, the pair

(Mf ,X) satisfies the homotopy extension property by Example 0.15, so the ‘only if’implication follows from the preceding corollary. tu

Proof of 0.19: Let g :Y→X be a homotopy inverse for f , and let ht :X→X be ahomotopy from gf = h0 to 11 = h1 . We will use ht to deform g to a map g1 withg1 ||A = 11. Since f ||A = 11, we can view ht ||A as a homotopy from g ||A to 11. Thensince we assume (X,A) has the homotopy extension property, we can extend thishomotopy to a homotopy gt :Y→X from g = g0 to a map g1 with g1 ||A = 11.

Our next task is to construct a homotopy g1f ' 11 rel A . Since g ' g1 via gtwe have gf ' g1f via gtf . We also have gf ' 11 via ht , so since homotopy is anequivalence relation by Exercise 3 at the end of the Chapter, we have g1f ' 11. Anexplicit homotopy that shows this is

kt ={g1−2tf , 0 ≤ t ≤ 1/2h2t−1, 1/2 ≤ t ≤ 1

Note that the two definitions agree when t = 1/2 since f ||A = 11 and gt = ht on A .The homotopy kt ||A starts and ends with the identity, and its second half simplyretraces its first half, that is, kt = k1−t on A . In this situation we define a ‘homotopyof homotopies’ ktu :A→A by means of the figure to the right showing the parameterdomain I×I for the pairs (t,u) , with the t axis horizontal and theu axis vertical. On the bottom edge of the square we define kt0 = kt ||A .Below the ‘V’ we define ktu to be independent of u , and above the ‘V’ wedefine ktu to be independent of t . This is unambiguous since kt = k1−ton A . Since k0 = 11, we have ktu = 11 for (t,u) in the left, right, and top edges of thesquare. Since (X,A) has the homotopy extension property, so does (X×I,A×I) bythe initial remarks on the homotopy extension property. Viewing ktu as a homotopy

18 Chapter 0. Some Underlying Geometric Notions

of kt , we can therefore extend ktu :A→A to ktu :X→X with kt0 = kt :X→X . Nowif we restrict this ktu to the left, top, and right edges of the (t,u) square we get ahomotopy g1f ' 11 rel A .

Since g1 ' g , we have fg1 ' fg ' 11, so the preceding argument can be repeatedwith the pair f , g replaced by g1, f . The result is a map f1 :X→X with f1 ||A = 11and f1g1 ' 11 rel A . Hence f1 ' f1(g1f) = (f1g1)f ' f rel A and so fg1 ' f1g1 '11 rel A . Thus g1 is a homotopy inverse to f rel A . tu

Exercises

1. Construct an explicit deformation retraction of the torus with one point deleted

onto a graph consisting of two circles intersecting in a point, namely, longitude and

meridian circles of the torus.

2. Construct an explicit deformation retraction of Rn − {0} onto Sn−1 .3. (a) Show that the composition of homotopy equivalences X→Y and Y→Z is ahomotopy equivalence X→Z . Deduce that homotopy equivalence is an equivalencerelation.

(b) Show that the relation of homotopy among maps X→Y is an equivalence relation.(c) Show that a map homotopic to a homotopy equivalence is a homotopy equivalence.

4. A deformation retraction in the weak sense of a space X to a subspace A is ahomotopy ft :X→X such that f0 = 11, f1(X) ⊂ A , and ft(A) ⊂ A for all t . Showthat if X deformation retracts to A in this weak sense, then the inclusion A↩ X isa homotopy equivalence.

5. Show that if a space X deformation retracts to a point x ∈ X , then for eachneighborhood U of x in X there exists a neighborhood V ⊂ U of x such that theinclusion V↩U is nullhomotopic.

6. (a) Let X be the subspace of R2 consisting of the horizontal seg-ment [0,1]×{0} together with all the vertical segments {r}×[0,1−r]for r a rational number in [0,1] . Show that X deformation retractsto any point in the segment [0,1]×{0} , but not to any other point.[See the preceding problem.]

(b) Let Y be the subspace of R2 that is the union of an infinite number of copies of Xarranged as in the figure below. Show that Y is contractible but does not deformationretract onto any point.

Chapter 0. Some Underlying Geometric Notions 19

(c) Let Z be the zigzag subspace of Y homeomorphic to R indicated by the heavierline. Show there is a deformation retraction in the weak sense (see Exercise 4) of Yonto Z , but no true deformation retraction.

7. Fill in the details in the following construction from [Edwards] of a compact space

Y ⊂ R3 with the same properties as the space Y in Exercise 6, that is, Y is contractiblebut does not deformation retract to any point. To begin, let X be the union of an in-finite sequence of cones on the Cantor set arranged end-to-end, as in the figure. Next,

form the one-point compactification of X×R . Thisembeds i0.8 n R3 as a closed disk with curved ‘fins’

attached along circular arcs, and with the one-point

compactification of X as a cross-sectional slice. Fi-nally, Y is obtained from this by wrapping one morecone on the Cantor set around the boundary of the

disk. X Y

8. For n > 2, construct an n room analog of the house with two rooms.

9. Show that a retract of a contractible space is contractible.

10. Show that a space X is contractible iff every map f :X→Y , for arbitrary Y , isnullhomotopic. Similarly, show X is contractible iff every map f :Y→X is nullho-motopic.

11. Show that f :X→Y is a homotopy equivalence if there exist maps g,h :Y→Xsuch that fg ' 11 and hf ' 11. More generally, show that f is a homotopy equiva-lence if fg and hf are homotopy equivalences.

12. Show that a homotopy equivalence f :X→Y induces a bijection between the setof path-components of X and the set of path-components of Y , and that f restrictsto a homotopy equivalence from each path-component of X to the correspondingpath-component of Y .

13. Show that any two deformation retractions r 0t and r1t of a space X onto a

subspace A can be joined by a continuous family of deformation retractions r st ,0 ≤ s ≤ 1, of X onto A , where continuity means that the map X×I×I→X , (x, s, t),r st (x) , is continuous.

14. Given positive integers v , e , and f satisfying v − e + f = 2, construct a cellstructure on S2 having v 0 cells, e 1 cells, and f 2 cells.

15. Enumerate all the subcomplexes of S∞ , with the cell structure described in thissection, having two cells in each dimension.

16. Show that S∞ is contractible.

17. Construct a 2 dimensional cell complex that contains both an annulus S1×I anda Möbius band as deformation retracts.

18. Show that S1 ∗ S1 = S3 , and more generally Sm ∗ Sn = Sm+n+1 .

20 Chapter 0. Some Underlying Geometric Notions

19. Show that the space obtained from S2 by attaching n 2 cells along any collectionof n circles in S2 is homotopy equivalent to the wedge sum of n+ 1 2 spheres.20. Show that the subspace X ⊂ R3 formed by a Klein bottle in-tersecting itself in a circle, as shown in the figure, is homotopy

equivalent to S1 ∨ S1 ∨ S2 .21. If X is a connected space that is a union of a finite number of 2 spheres, anytwo of which intersect in at most one point, show that X is homotopy equivalent to awedge sum of S1 ’s and S2 ’s.

22. Let X be a finite graph lying in a half-plane P ⊂ R3 and intersecting the edgeof P in a subset of the vertices of X . Describe the homotopy type of the ‘surface ofrevolution’ obtained by rotating X about the edge line of P .

23. Show that a CW complex is contractible if it is the union of two contractible

subcomplexes whose intersection is also contractible.

24. Let X and Y be CW complexes with 0 cells x0 and y0 . Show that the quotientspaces X∗Y/(X∗{y0}∪{x0}∗Y) and S(X∧Y)/S({x0}∧{y0}) are homeomorphic,and deduce that X ∗ Y ' S(X ∧ Y) .25. Show that for a CW complex X with components Xα , the suspension SX ishomotopy equivalent to Y

∨αSXα where Y is a graph. In the case that X is a finite

graph, show that SX is homotopy equivalent to a wedge sum of circles and 2 spheres.

26. Use Corollary 0.20 to show that if (X,A) has the homotopy extension property,then X×I deformation retracts to X×{0} ∪ A×I . Deduce from this that Proposi-tion 0.18 holds more generally when (X,A) satisfies the homotopy extension prop-erty.

27. Given a pair (X,A) and a map f :A→B , define X/f to be the quotient spaceof X obtained by identifying points in A having the same image in B . Show that thequotient map X→X/f is a homotopy equivalence if f is a surjective homotopy equiv-alence and (X,A) has the homotopy extension property. [Hint: Consider X ∪Mf anduse the preceding problem.] When B is a point this gives another proof of Proposi-tion 0.17. Another interesting special case is when f is the projection A×I→A .28. Show that if (X1, A) satisfies the homotopy extension property, then so does everypair (X0 tf X1, X0) obtained by attaching X1 to a space X0 via a map f :A→X0 .29. In case the CW complex X is obtained from a subcomplex A by attaching a singlecell en , describe exactly what the extension of a homotopy ft :A→Y to X given bythe proof of Proposition 0.16 looks like. That is, for a point x ∈ en describe the pathft(x) for the extended ft .

One of the main techniques of algebraic topology is to study topological spaces

by forming algebraic images of them. Most often these algebraic images are groups,

but more elaborate structures such as rings, modules, and algebras also arise. The

mechanisms which create these images — the ‘lanterns’ of algebraic topology, one

might say — are known formally as functors and have the characteristic feature that

they form images not only of spaces but also of maps. Thus, continuous maps be-

tween spaces are projected onto homomorphisms between their algebraic images, so

topologically related spaces have algebraically related images.

With suitably constructed lanterns one might hope to be able to form images with

enough detail to reconstruct accurately the shapes of all spaces, or at least of large

and interesting classes of spaces. This is one of the main goals of algebraic topology,

and to a surprising extent this goal is achieved. Of course, the lanterns necessary to

do this are somewhat complicated pieces of machinery. But this machinery also has

a certain intrinsic beauty.

This first chapter introduces one of the simplest and most important of the func-

tors of algebraic topology, the fundamental group, which creates an algebraic image

of a space from the loops in the space, that is, paths starting and ending at the same

point.

The Idea of the Fundamental Group

To get a feeling for what the fundamental group is about, let us look at a few

preliminary examples before giving the formal definitions.

22 Chapter 1. The Fundamental Group

A B

Consider two linked circles A and B in R3 , as shown inthe figure. The fact that the circles are linked means, intu-

itively, that B cannot be separated from A by any continuousmotion of B , such as pushing, pulling, twisting, etc. We couldeven take B to be made of rubber or stretchable string and allow completely generalcontinuous deformations of B , staying in the complement of A at all times, and itwould still be impossible to pull B off A . At least that is what intuition suggests,and the fundamental group will give a way of making this intuition mathematically

rigorous.

A

A B−3

B2

Instead of having B link with A just once, we could makeit link A two or more times. As a further variation, by as-signing an orientation to B we can speak of B linking A apositive or a negative number of times, say positive when Bcomes forward through A and negative for the reverse direc-tion. Thus for each nonzero integer n we have an orientedcircle Bn linking A n times, where by ‘circle’ we mean a curvehomeomorphic to a circle. To complete the scheme, we could

let B0 be a circle which doesn’t link A at all.

Now, integers not only measure quantity, but they form a group under addition.

Can the group operation be mimicked geometrically with some sort of addition op-

eration on the oriented circles B linking A? An oriented circle B can be thought ofas a path traversed in time, starting and ending at the same point x0 , which we canchoose to be any point on the circle. Such a path starting and ending at the same point

is called a loop. Two different loops B and B′ both starting and ending at the samepoint x0 can be ‘added’ to form a new loop B + B′ which travels first around B , thenaround B′ . For example, if B1 and B

′1 are loops each linking A once in the positive

direction, then their sum B1 + B′1 is deformable to B2 , linking A twice:

A

B1

B1

x0

A

B2x0

´

In a similar way B1 + B−1 can be deformed to the loop B0 , unlinked from A :

A

B1x0

A B0

x0B−1

More generally, Bm + Bn is deformable to Bm+n for arbitrary integers m and n , asone can easily convince oneself of by drawing a few pictures like those above.

The Idea of the Fundamental Group 23

Note that in forming sums of loops we produce loops which pass through the

basepoint more than once. This is one reason why loops are defined merely as contin-

uous paths, which are allowed to pass through the same point many times. So if one is

thinking of a loop as something made of stretchable string, one has to give the string

the magical power of being able to pass through itself unharmed. However, we must

be sure not to allow our loops to intersect the fixed circle A at any time, otherwise wecould always unlink them from A .

Next we consider a slightly more complicated sort of linking, involving three cir-

cles forming a configuration known as the Borromean rings, shown in the figure on

the left below. The interesting feature here is that if any one of the three circles is

removed, the other two are not linked.

AA

BB

C C

In the same spirit as before, let us regard the third circle C as forming a loop in thecomplement of the other two, A and B , and we ask whether C can be continuouslydeformed to unlink it completely from A and B , always staying in the complementof A and B during the deformation. We can redraw the picture by pulling A and Bapart, dragging C along, and then we see C winding back and forth between A andB as shown in the second figure above. In this new position, if we start at the point ofC indicated by the dot and proceed in the direction given by the arrow, then we passin sequence: (1) forward through A , (2) forward through B , (3) backward through A ,and (4) backward through B . If we measure the linking of C with A and B by twointegers, then the ‘forwards’ and ‘backwards’ cancel and both integers are zero. This

reflects the fact that C is not linked with A or B individually.To get a more accurate measure of how C links with A and B together, we regard

the four parts (1)-(4) of C as an ordered sequence. Taking into account the directionsin which these segments of C pass through A and B , we may deform C to the suma+ b − a− b of four loops as in the next figure. We write the third and fourth loopsas the negatives of the first two since they can be deformed to the first two, but with

the opposite orientations, and as we saw in the preceding example, the sum of two

oppositely-oriented loops is deformable to a trivial loop, not linked with anything.

A Ba

b −b−a

A Bab −b−a

24 Chapter 1. The Fundamental Group

We would like to view the expression a+b−a−b as lying in a nonabelian group, sothat it is not automatically zero. Changing to the more usual multiplicative notation

for nonabelian groups, it would be written aba−1b−1 , the commutator of a and b .To shed further light on this example, suppose we modify it slightly so that the

circles A and B are now linked, as in the next figure.

A B

C

A B

C

The circle C can then be deformed into the position shown at the right, where it againrepresents the composite loop aba−1b−1 , where a and b are loops linking A andB . But it is apparent from the picture on the left that C can actually be unlinkedcompletely from A and B . So in this case the product aba−1b−1 should be trivial,i.e., a and b should commute.

The fundamental group of a space X will be defined so that its elements areloops in X starting and ending at a fixed basepoint x0 ∈ X , but two such loops areregarded as determining the same element of the fundamental group if one loop can

be continuously deformed to the other within the space X . (All loops which occurduring deformations must also start and end at x0 .) In the first example above, X isthe complement of the circle A , while in the other two examples X is the complementof the two circles A and B . In the second section in this chapter we will show:

— The fundamental group of the complement of the circle A in the first exampleis Z , with the loop B as a generator of this group. This amounts to saying thatevery loop in the complement of A can be deformed to one of the loops Bn , andthat Bn cannot be deformed to Bm if n ≠m .

— The fundamental group of the complement of the two unlinked circles A and B inthe second example is the nonabelian free group on two generators, represented

by the loops a and b linking A and B . In particular, the commutator aba−1b−1

is a nontrivial element of this group.

— The fundamental group of the complement of the two linked circles A and B inthe third example is the free abelian group Z×Z on two generators, representedby the loops a and b linking A and B .

As a result of these calculations, we have two ways to tell when a pair of circles Aand B is linked. The direct approach is given by the first example, where one circleis regarded as an element of the fundamental group of the complement of the other

circle. An alternative and somewhat more subtle method is given by the second and

third examples, where one distinguishes a pair of linked circles from a pair of unlinked

Section 1.1. Basic Constructions 25

circles by the fundamental group of their complement, which is abelian in one case and

nonabelian in the other. This method is much more general: One can often show that

two spaces are not homeomorphic by showing that their fundamental groups are not

isomorphic, since it will be an easy consequence of the definition of the fundamental

group that homeomorphic spaces have isomorphic fundamental groups.

This first section begins with the basic definitions and constructions, and then

proceeds quickly to an important calculation, the fundamental group of the circle,

using notions developed more fully in §1.3. More systematic methods of calculationare given in §1.2, sufficient to show for example that every group is realized as thefundamental group of some space. This idea is exploited in the Additional Topics

at the end of the chapter, which give some illustrations of how algebraic facts about

groups can be derived topologically.

Paths and Homotopy

There are two basic ingredients in the definition of the fundamental group: the

idea of continuous deformation of paths, and an operation for composing paths to

form new paths. We begin with the first of these.

Let X be a space and let I denote the unit interval [0,1] in R . By a path in Xwe mean a continuous map f : I→X . The idea of continuously deforming a path,keeping its endpoints fixed, is made precise by the following definition. A homotopy

of paths in X is a family ft : I→X , 0 ≤ t ≤ 1, such that(1) The endpoints ft(0) = x0 and ft(1) = x1 are independent of t .(2) The associated map F : I×I→X defined by F(s, t) = ft(s) is continuous.

When two paths f0 and f1 are connected in this way by a homotopy ft , we will saythey are homotopic and write f0 ' f1 .

x11

x0

0f

f

Example 1.1: Linear homotopies. Any two pathsf0 and f1 in R

n having the same two endpoints x0and x1 are homotopic via the homotopy ft(s) =(1− t)f0(s)+ tf1(s) . During this homotopy eachpoint f0(s) travels along the line segment to f1(s) at constant speed. This is becausethe line through f0(s) and f1(s) is linearly parametrized as f0(s)+t[f1(s)−f0(s)] =(1−t)f0(s)+tf1(s) , so as t goes from 0 to 1, ft(s) traces out the segment from f0(s)

26 Chapter 1. The Fundamental Group

to f1(s) . If f1(s) = f0(s) then this segment degenerates to a point, so ft(s) = f0(s)for all t . This happens in particular for s = 0 and s = 1, so ft is a path from x0 to x1for all t . Continuity of the homotopy ft as a map I×I→Rn follows from continuityof f0 and f1 since the algebraic operations in the definition of ft are continuous.

This construction shows more generally that for a convex subspace X ⊂ Rn , allpaths in X with given endpoints x0 and x1 are homotopic.

Before proceeding further we need to verify a technical property:

Proposition 1.2. The relation of homotopy on paths with fixed endpoints in any spaceis an equivalence relation.

Proof: The constant homotopy ft = f shows that f ' f . If f0 ' f1 via ft , thenf1 ' f0 via the homotopy f1−t . For transitivity, if f0 ' f1 via ft and f1 = g0 ' g1via gt , then f0 ' g1 via the homotopy ht which equals f2t for 0 ≤ t ≤ 1/2 andg2t−1 for 1/2 ≤ t ≤ 1. Continuity of the associated map H(s, t) = ht(s) comes fromthe elementary fact, which will be used frequently without explicit mention, that a

function defined on the union of two closed sets is continuous if its restriction to

each of the closed sets is continuous. In the case at hand, H is clearly continuous oneach of I×[0, 1/2] and I×[1/2,1], and on I×{1/2} the two definitions of H agree sincef1 = g0 , so H is continuous on I×I . tu

The equivalence class of a path f under the equivalence relation of homotopy isdenoted by [f ] and called the homotopy class of f .

Given two paths f , g : I→X such that f(1) = g(0) , we can define a ‘composition’or ‘product’ path f g which traverses first f then g , by the formula

f g(s) ={f(2s), 0 ≤ s ≤ 1/2g(2s − 1), 1/2 ≤ s ≤ 1

Thus the speed of traversal of f and g is doubled in order for f g to be traversedin unit time. This product operation respects homotopy classes since if f0 ' f1 andg0 ' g1 via homotopies ft and gt , and if f0(1) = g0(0) so that f0 g0 is defined,then ft gt is defined and provides a homotopy f0 g0 ' f1 g1 .

In particular, suppose we restrict attention to paths f : I→X with the same start-ing and ending point f(0) = f(1) = x0 ∈ X . Such paths are called loops, and thecommon starting and ending point x0 is referred to as the basepoint. The set ofhomotopy classes of loops in X at the basepoint x0 is denoted π1(X,x0) .

Proposition 1.3. π1(X,x0) is a group with respect to the product [f ][g] = [f g] .This group π1(X,x0) is called the fundamental group of X at the basepoint x0 .

Proof: By restricting attention to loops with a fixed basepoint x0 ∈ X we guaranteethat the product f g of any two such loops is defined. We have already observed

Section 1.1. Basic Constructions 27

that the homotopy class of f g depends only on the homotopy classes of f and g ,so the product [f ][g] = [f g] is well-defined. It remains to verify the three axiomsfor a group.

As a preliminary step, define a reparametrization of a path f to be a composi-tion fϕ where ϕ : I→I is any continuous map such that ϕ(0) = 0 and ϕ(1) = 1.Reparametrizing a path preserves its homotopy class since fϕ ' f via the homotopyfϕt where ϕt(s) = (1 − t)ϕ(s) + ts so that ϕ0 = ϕ and ϕ1(s) = s . Note that(1− t)ϕ(s)+ ts lies between ϕ(s) and s , hence is in I so fϕt is defined.

Given paths f , g,h with f(1) = g(0) and g(1) = h(0) , then the composed paths(f g) h and f (g h) are reparametrizations of each other, differing only in the speedsat which f and h are traversed. Hence (f g) h ' f (g h) . Restricting attention toloops at the basepoint x0 , this says the product in π1(X,x0) is associative.

Given a path f : I→X , let c be the constant path at f(1) , defined by c(s) = f(1)for all s ∈ I . Then f c is a reparametrization of f via the function ϕ(s) whichequals 2s on [0, 1/2] and 1 on [1/2,1], so f c ' f . Similarly, c f ' f where c is nowthe constant path at f(0) . Taking f to be a loop, we deduce that the homotopy classof the constant path at x0 is a two-sided identity in π1(X,x0) .

For a path f from x0 to x1 , the inverse path f from x1 back to x0 is definedby f(s) = f(1 − s) . Consider the homotopy ht = ft gt where ft is the path whichequals f on the interval [0, t] and which is stationary at f(t) on the interval [t,1] ,and gt is the inverse path of ft . Since f0 is the constant path c at x0 and f1 = f ,we see that ht is a homotopy from c c = c to f f . Now specialize to the case thatf is a loop at the basepoint x0 . We have shown that f f ' c , and replacing f by fgives f f ' c . So [f ] is a two-sided inverse for [f ] in π1(X,x0) . tu

In Chapter 4 we will see that π1(X,x0) is just the first in a sequence of groupsπn(X,x0) , called homotopy groups, which are defined in an entirely analogous fash-ion using the n dimensional cube In in place of I .

Example 1.4. A convex set X in Rn has π1(X,x0) = 0, the trivial group, for everybasepoint x0 ∈ X , since any two loops f0 and f1 based at x0 are homotopic via thelinear homotopy ft(s) = (1− t)f0(s)+ tf1(s) .

It is not so easy to show that a space has a nontrivial fundamental group since one

must somehow demonstrate the nonexistence of homotopies between certain loops.

We will tackle the simplest example shortly, showing that this is the case for the circle.

But first let us address a theoretical issue which the reader may be wondering about.

How does π1(X,x0) depend on the choice of the basepoint x0 ? Since π1(X,x0)involves only the path component of X containing x0 , it is clear that we can hope tofind a relation between π1(X,x0) and π1(X,x1) for two basepoints x0, x1 ∈ X onlyif x0 and x1 lie in the same path component of X . So let h : I→X be a path fromx0 to x1 , with the inverse path h(s) = h(1 − s) from x1 back to x0 . We can then

28 Chapter 1. The Fundamental Group

associate to each loop f based at x1 the loop h f h based at x0 . Strictly speaking,we should choose an order of forming the product h f h , either (h f) h or h (f h) ,but the two choices are homotopic and we are only interested in homotopy classes

here. Alternatively, to avoid any ambiguity we could define a general n fold productf1 ··· fn so that the path fi is traversed in the time interval

[ i−1n ,

in].

Proposition 1.5. The map βh :π1(X,x1)→π1(X,x0) defined by βh[f] = [h f h]is an isomorphism.

Proof: If ft is a homotopy of loops based at x1 then h ft h is a homotopy ofloops based at x0 , so βh is well-defined. Further, βh is a homomorphism sinceβh[f g] = [h f g h] = [h f h h g h] = βh[f]βh[g] . Finally, βh is an isomorphismwith inverse βh since βhβh[f] = βh[h f h] = [h h f h h] = [f ] , and similarlyβhβh[f] = [f ] . tu

Thus if X is path-connected, the group π1(X,x0) is, up to isomorphism, inde-pendent of the choice of basepoint x0 . In this case the notation π1(X,x0) is oftenabbreviated to π1(X) , or one could go further and write just π1X .

In general, a space is called simply-connected if it is path-connected and has

trivial fundamental group. The following result is probably the reason for this term.

Proposition 1.6. A space X is simply-connected iff there is a unique homotopy classof paths connecting any two points in X .

Proof: Path-connectedness is the existence of paths connecting every pair of points,so we need be concerned only with the uniqueness of connecting paths. Suppose

π1(X) = 0. If γ and η are two paths from x0 to x1 , then γ ' γ η η ' η , thelatter homotopy since the loop γ η is nullhomotopic if π1(X,x0) = 0. Conversely,if there is only one homotopy class of paths connecting a basepoint x0 to itself, thenπ1(X,x0) = 0. tu

The Fundamental Group of the Circle

Our first real theorem will be the calculation π1(S1) ≈ Z . Besides its intrinsic

interest, this basic result will have several immediate applications of some substance,

and it will be the starting point for many more calculations in the next section. In view

of the importance of this result, we can expect the proof to involve some genuine work,

which it does. To maximize the payoff for this work, the proof is written so that its

main technical steps apply in a more general setting. This setting, covering spaces, is

the topic of §1.3.Here is the theorem:

Section 1.1. Basic Constructions 29

Theorem 1.7. The map ψ :Z→π1(S1) sending an integer n to the homotopy classof the loop ωn(s) = (cos 2πns, sin 2πns) based at (1,0) is an isomorphism.

Proof: The idea is to compare paths in S1 with paths in R via the map p :R→S1given by p(s) = (cos 2πs, sin 2πs) . This map can be visualized geo-metrically by embedding R in R3 as the helix parametrized by s,(cos 2πs, sin 2πs, s) , and then p is the restriction to the helix of theprojection of R3 onto R2 , (x,y, z), (x,y) , as in the figure. Ob-serve that the loop ωn is the composition pω̃n where ω̃n : I→R isthe path ω̃n(s) = ns , starting at 0 and ending at n , winding aroundthe helix |n| times, upward if n > 0 and downward if n < 0. Therelation ωn = pω̃n is expressed by saying that ω̃n is a lift of ωn .

p

The definition of ψ can be restated by setting ψ(n) equal to the homotopy classof the loop pf̃ where f̃ is any path in R from 0 to n . Such an f̃ is homotopic toω̃n via the linear homotopy (1 − t)f̃ + tωn , hence pf̃ is homotopic to pω̃n = ωnand the new definition of ψ(n) agrees with the old one.

To verify that ψ is a homomorphism, let τm :R→R be the translation τm(x) =x +m . Then ω̃m (τmω̃n) is a path in R from 0 to m + n , so ψ(m + n) is thehomotopy class of the loop in S1 which is the image of this path under p . This imageis just ωm ωn , so ψ(m+n) = ψ(m) ψ(n) .

To show that ψ is an isomorphism we shall use two facts:

(a) For each path f : I→S1 starting at a point x0 ∈ S1 and each x̃0 ∈ p−1(x0) thereis a unique lift f̃ : I→R starting at x̃0 .

(b) For each homotopy ft : I→S1 of paths starting at x0 and each x̃0 ∈ p−1(x0)there is a unique lifted homotopy f̃t : I→R of paths starting at x̃0 .

Before proving these facts, let us see how they imply the theorem. To show that ψ issurjective, let f : I→S1 be a loop at the basepoint (1,0) , representing a given elementof π1(S

1) . By (a) there is a lift f̃ starting at 0. This path f̃ ends at some integer nsince pf̃ (1) = f(1) = (1,0) and p−1(1,0) = Z ⊂ R . By the extended definition of ψwe then have ψ(n) = [pf̃ ] = [f ] . Hence ψ is surjective.

To show that ψ is injective, suppose ψ(m) = ψ(n) , i.e., ωm ' ωn . Let ft be ahomotopy from ωm = f0 to ωn = f1 . By (b) this homotopy lifts to a homotopy f̃t ofpaths starting at 0. The uniqueness part of (a) implies that f̃0 = ω̃m and f̃1 = ω̃n .Since f̃t is a homotopy of paths, the endpoint f̃t(1) is independent of t . For t = 0this endpoint is m and for t = 1 it is n , so m = n .

It remains to prove (a) and (b). These can both be deduced from a third