zotero-db/storage/JC64FZNX/.zotero-ft-cache

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.

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
Standard Notations xii.
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 28. Induced Homomorphisms 33.
1.2. Van Kampen’s Theorem . . . . . . . . . . . . . . . . . . . 38
Free Products of Groups 39. The van Kampen Theorem 41. Applications to Cell Complexes 48.
1.3. Covering Spaces . . . . . . . . . . . . . . . . . . . . . . . . 54
Lifting Properties 58. The Classiﬁcation of Covering Spaces 61. Deck Transformations and Group Actions 68.
Additional Topics
1.A. Graphs and Free Groups 81. 1.B. K(G,1) Spaces and Graphs of Groups 85.

vi

Table of Contents

Chapter 2. Homology . . . . . . . . . . . . . . . . . . . . . . . 95
2.1. Simplicial and Singular Homology . . . . . . . . . . . . . 100
∆ Complexes 100. Simplicial Homology 102. Singular Homology 106. Homotopy Invariance 108. Exact Sequences and Excision 111. The Equivalence of Simplicial and Singular Homology 126.
2.2. Computations and Applications . . . . . . . . . . . . . . 132
Degree 132. Cellular Homology 135. Mayer-Vietoris Sequences 147. Homology with Coefﬁcients 151.
2.3. The Formal Viewpoint . . . . . . . . . . . . . . . . . . . . 158
Axioms for Homology 158. Categories and Functors 160.
Additional Topics
2.A. Homology and Fundamental Group 164. 2.B. Classical Applications 167. 2.C. Simplicial Approximation 175.
Chapter 3. Cohomology . . . . . . . . . . . . . . . . . . . . . 183
3.1. Cohomology Groups . . . . . . . . . . . . . . . . . . . . . 188
The Universal Coefﬁcient Theorem 188. Cohomology of Spaces 195.
3.2. Cup Product . . . . . . . . . . . . . . . . . . . . . . . . . . 204
The Cohomology Ring 209. A Ku¨nneth Formula 216. Spaces with Polynomial Cohomology 222.
3.3. Poincar´e Duality . . . . . . . . . . . . . . . . . . . . . . . . 228
Orientations and Homology 231. The Duality Theorem 237. Connection with Cup Product 247. Other Forms of Duality 250.
Additional Topics
3.A. Universal Coefﬁcients for Homology 259. 3.B. The General Ku¨nneth Formula 266. 3.C. H–Spaces and Hopf Algebras 279. 3.D. The Cohomology of SO(n) 290. 3.E. Bockstein Homomorphisms 301. 3.F. Limits and Ext 309. 3.G. Transfer Homomorphisms 319. 3.H. Local Coefﬁcients 325.

Table of Contents

vii

Chapter 4. Homotopy Theory . . . . . . . . . . . . . . . . . 335
4.1. Homotopy Groups . . . . . . . . . . . . . . . . . . . . . . 337
Deﬁnitions and Basic Constructions 338. Whitehead’s Theorem 344. Cellular Approximation 346. CW Approximation 350.
4.2. Elementary Methods of Calculation . . . . . . . . . . . . 358
Excision for Homotopy Groups 358. The Hurewicz Theorem 364. Fiber Bundles 373. Stable Homotopy Groups 382.
4.3. Connections with Cohomology . . . . . . . . . . . . . . 391
The Homotopy Construction of Cohomology 391. Fibrations 403. Postnikov Towers 408. Obstruction Theory 413.
Additional Topics
4.A. Basepoints and Homotopy 419. 4.B. The Hopf Invariant 425. 4.C. Minimal Cell Structures 427. 4.D. Cohomology of Fiber Bundles 429. 4.E. The Brown Representability Theorem 446. 4.F. Spectra and Homology Theories 450. 4.G. Gluing Constructions 454. 4.H. Eckmann-Hilton Duality 458. 4.I. Stable Splittings of Spaces 464. 4.J. The Loopspace of a Suspension 468. 4.K. The Dold-Thom Theorem 473. 4.L. Steenrod Squares and Powers 485.
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517
Topology of Cell Complexes 517. The Compact-Open Topology 527.
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537

This book was written to be a readable introduction to algebraic topology with rather broad coverage of the subject. The viewpoint is quite classical in spirit, and stays well within the conﬁnes of pure algebraic topology. In a sense, the book could have been written thirty or forty years ago since virtually everything in it is at least that old. However, the passage of the intervening years has helped clarify what are the most important results and techniques. For example, CW complexes have proved over time to be the most natural class of spaces for algebraic topology, so they are emphasized here much more than in the books of an earlier generation. This emphasis also illustrates the book’s general slant towards geometric, rather than algebraic, aspects of the subject. The geometry of algebraic topology is so pretty, it would seem a pity to slight it and to miss all the intuition it provides.
At the elementary level, algebraic topology separates naturally into the two broad channels of homology and homotopy. This material is here divided into four chapters, roughly according to increasing sophistication, with homotopy split between Chapters 1 and 4, and homology and its mirror variant cohomology in Chapters 2 and 3. These four chapters do not have to be read in this order, however. One could begin with homology and perhaps continue with cohomology before turning to homotopy. In the other direction, one could postpone homology and cohomology until after parts of Chapter 4. If this latter strategy is pushed to its natural limit, homology and cohomology can be developed just as branches of homotopy theory. Appealing as this approach is from a strictly logical point of view, it places more demands on the reader, and since readability is one of the ﬁrst priorities of the book, this homotopic interpretation of homology and cohomology is described only after the latter theories have been developed independently of homotopy theory.
Preceding the four main chapters there is a preliminary Chapter 0 introducing some of the basic geometric concepts and constructions that play a central role in both the homological and homotopical sides of the subject. This can either be read before the other chapters or skipped and referred back to later for speciﬁc topics as they become needed in the subsequent chapters.
Each of the four main chapters concludes with a selection of additional topics that the reader can sample at will, independent of the basic core of the book contained in the earlier parts of the chapters. Many of these extra topics are in fact rather important in the overall scheme of algebraic topology, though they might not ﬁt into the time

x

Preface

constraints of a ﬁrst course. Altogether, these additional topics amount to nearly half the book, and they are included here both to make the book more comprehensive and to give the reader who takes the time to delve into them a more substantial sample of the true richness and beauty of the subject.
Not included in this book is the important but somewhat more sophisticated topic of spectral sequences. It was very tempting to include something about this marvelous tool here, but spectral sequences are such a big topic that it seemed best to start with them afresh in a new volume. This is tentatively titled ‘Spectral Sequences in Algebraic Topology’ and is referred to herein as [SSAT]. There is also a third book in progress, on vector bundles, characteristic classes, and K–theory, which will be largely independent of [SSAT] and also of much of the present book. This is referred to as [VBKT], its provisional title being ‘Vector Bundles and K–Theory.’
In terms of prerequisites, the present book assumes the reader has some familiarity with the content of the standard undergraduate courses in algebra and point-set topology. In particular, the reader should know about quotient spaces, or identiﬁcation spaces as they are sometimes called, which are quite important for algebraic topology. Good sources for this concept are the textbooks [Armstrong 1983] and [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 standard picture of the torus as a square with opposite edges identiﬁed and divides the square into two triangles by cutting along a diagonal, then the result is a ∆ complex structure on the torus having 2 triangles, 3 edges, and 1 vertex. By contrast, it is known that a simplicial complex structure on the torus must have at least 14 triangles, 21 edges, and 7 vertices. So ∆ complexes provide a signiﬁcant improvement in efﬁciency, which is nice from a pedagogical viewpoint since it cuts down on tedious calculations in examples. A more fundamental reason for considering ∆ complexes is that they seem to be very natural objects from the viewpoint of algebraic topology. They are the natural domain of deﬁnition for simplicial homology, and a number of standard constructions produce ∆ complexes rather than simplicial complexes, for instance the singular complex of a space, or the classifying space of a discrete group or category. In spite of this naturality, ∆ complexes have appeared explicitly in the literature only rarely, and no standard name for the notion has emerged.

Preface

xi

This book will remain available online in electronic form after it has been printed in the traditional fashion. The web address is
http://www.math.cornell.edu/˜hatcher
One can also ﬁnd here the parts of the other two books in the sequence that are currently available. Although the present book has gone through countless revisions already, including corrections of many small errors both typographical and mathematical that were found by careful readers of earlier versions, it is a virtual certainty that some errors remain, so the web page will contain also a list of corrections. Readers are encouraged to submit their candidates for entries on this list to the email address posted on the web page. With the electronic version of the book it will be possible to continue making revisions and additions as well as corrections, so comments and suggestions from readers will always be welcome.

xii
Standard Notations
Z , Q , R , C , H , O : the integers, rationals, reals, complexes, quaternions, and Cayley octonions
Zn : the integers mod n Rn : n dimensional Euclidean space Cn : complex n 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 the n disk 11 : the identity function from a set to itself
: disjoint union of sets or spaces × , : product of sets, groups, or spaces ≈ : isomorphism A ⊂ B or B ⊃ A : set-theoretic containment, not necessarily proper iff : if and only if

The aim of this short preliminary chapter is to introduce a few of the most common geometric concepts and constructions in algebraic topology. The exposition is somewhat informal, with no theorems or proofs until the last couple pages, and it should be read in this informal spirit, skipping bits here and there. In fact, this whole chapter could be skipped now, to be referred back to later for basic deﬁnitions.
To avoid overusing the word ‘continuous’ we adopt the convention that maps between spaces are always assumed to be continuous unless otherwise stated.
Homotopy and Homotopy Type
One of the main ideas of algebraic topology is to consider two spaces to be equivalent if they have ‘the same shape’ in a sense that is much broader than homeomorphism. To take an everyday example, the letters of the alphabet can be written either as unions of ﬁnitely many straight and curved line segments, or in thickened forms that are compact subsurfaces of the plane bounded by simple closed curves. In each case the thin letter is a subspace of the thick letter, and we can continuously shrink the thick letter to the thin one. A nice way to do this is to decompose a thick letter, call it X , into line segments connecting each point on the outer boundary of X to a unique point of the thin subletter X , as indicated in the ﬁgure. Then we can shrink X to X by sliding each point of X − X into X along the line segment that contains it. Points that are already in X do not move.
We can think of this shrinking process as taking place during a time interval
0 ≤ t ≤ 1 , and then it deﬁnes a family of functions ft : 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 .

2 Chapter 0

Some Underlying Geometric Notions

Naturally we would like ft(x) to depend continuously on both t and x , and this will be true if we have each x ∈ X − X move along its line segment at constant speed so as to reach its image point in X at time t = 1 , while points x ∈ X are stationary, as
remarked earlier.
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 ft : X→X , t ∈ I , such
that f0 = 11 (the identity map), f1(X) = A , and ft || A = 11 for all t . The family ft
should be continuous in the sense that the associated map X × I→X , (x, t) ft(x) ,
is continuous.
It is easy to produce many more examples similar to the letter examples, with the
deformation retraction ft obtained by sliding along line segments. The ﬁgure on the left below shows such a deformation retraction of a M¨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 map f : X→Y , the mapping

cylinder Mf is the quotient space of the disjoint union (X × I) Y obtained by identifying each (x, 1) ∈ X × I

with f (x) ∈ Y . In the let-

X

ter examples, the space X is the outer boundary of the

X×I

f (X )

Mf

thick letter, Y is the thin

Y

Y

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 Mf deformation retracts to the subspace Y by sliding each point (x, t) along the segment

{x}× I ⊂ Mf to the endpoint f (x) ∈ Y . Not all deformation retractions arise in this way from mapping cylinders, 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.

Homotopy and Homotopy Type

Chapter 0 3

A deformation retraction ft : X→X is a special case of the general notion of a homotopy, which is simply any family of maps ft : X→Y , t ∈ I , such that the associated map F : X × I→Y given by F (x, t) = ft(x) is continuous. One says that two maps f0, f1 : X→Y are homotopic if there exists a homotopy ft connecting them,
and one writes f0 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 map X→A
restricting to the identity on the subspace A ⊂ X . From a more formal viewpoint a
retraction is a map r : X→X with r 2 = r , since this equation says exactly that r is the

identity on its image. Retractions are the topological analogs of projection operators

in other parts of mathematics.

Not all retractions come from deformation retractions. For example, 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 homotopy ft : X→X that gives a deformation retraction of X onto a subspace A has the property that ft || A = 11 for all t . In general, a homotopy ft : X→Y whose
restriction to a subspace A ⊂ X is independent of t is called a homotopy relative

to A , or more concisely, a homotopy rel A . Thus, a deformation retraction of X onto

A is a homotopy rel A from the identity map of X to a retraction of X onto A .
If a space X deformation retracts onto a subspace A via ft : 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 ft . Generalizing this situation, a
map f : X→Y is called a homotopy equivalence if there is a map g : Y →X such that

f g 11 and gf 11 . The spaces X and Y are said to be homotopy equivalent or to

have the same homotopy 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 spaces X and Y are homotopy equivalent if and only

if there exists a third space Z containing both X and Y as deformation retracts. For

the less trivial implication one can in fact take Z to be the mapping cylinder Mf of
any homotopy equivalence f : X→Y . We observed previously that Mf deformation
retracts to Y , so what needs to be proved is that Mf also deformation retracts to its
other end X if f is a homotopy equivalence. This is shown in Corollary 0.21.

4 Chapter 0

Some Underlying Geometric Notions

A space having the homotopy type of a point is called contractible. This amounts to requiring that the identity map of the space be nullhomotopic, that is, homotopic to a constant map. In general, this is slightly weaker than saying the space deformation retracts to a point; see the exercises at the end of the chapter for an example distinguishing these two notions.
Let us describe now an example of a 2 dimensional subspace of R3 , known as the house with two rooms, which is contractible but not in any obvious way. To build this

=

∪

∪

space, start with a box divided into two chambers by a horizontal rectangle, where by a ‘rectangle’ we mean not just the four edges of a rectangle but also its interior. Access to the two chambers from outside the box is provided by two vertical tunnels. The upper tunnel is made by punching out a square from the top of the box and another square directly below it from the middle horizontal rectangle, then inserting four vertical rectangles, the walls of the tunnel. This tunnel allows entry to the lower chamber from outside the box. The lower tunnel is formed in similar fashion, providing entry to the upper chamber. Finally, two vertical rectangles are inserted to form ‘support walls’ for the two tunnels. The resulting space X thus consists of three horizontal pieces homeomorphic to annuli plus all the vertical rectangles that form the walls of the two chambers.
To see that X is contractible, consider a closed 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 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 retraction of the ball N(X) to a point x0 ∈ X and if r : N(X)→X
is a retraction, for example the end result of a deformation retraction of N(X) to X , then the restriction of the composition r ft to X is a deformation retraction of X to x0 . However, it is quite a challenging exercise to see exactly what this deformation retraction looks like.

Cell Complexes

Chapter 0 5

Cell Complexes

A familiar way of constructing the torus S1 × S1 is by identifying opposite sides

of a square. More generally, an orientable surface Mg of genus g can be constructed

from a polygon with 4g sides

by identifying pairs of edges,

a

as shown in the ﬁgure in the ﬁrst three cases g = 1, 2, 3 . The 4g edges of the polygon become a union of 2g circles in the surface, all intersecting in a single point. The in-

ba

b

c

b

b

a

c

d

b

c

a

terior of the polygon can be thought of as an open disk, or a 2 cell, attached to the

a

d

d

b

a

dc d

union of the 2g circles. One

c

d

e

c

can also regard the union of

b

the circles as being obtained

e

f

b

from their common point of

e

a

intersection, by attaching 2g open arcs, or 1 cells. Thus

af

f ab

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 the n skeleton Xn from Xn−1 by attaching n cells eαn via maps
→ ϕα : Sn−1 Xn−1 . This means that Xn is the quotient space of the disjoint union
Xn−1 α Dαn of Xn−1 with a collection of n disks Dαn under the identiﬁcations x ∼ ϕα(x) for x ∈ ∂Dαn . Thus as a set, Xn = Xn−1 α eαn where each eαn 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 latter case X is given the weak topology: A set A ⊂ X is open (or closed) iff A ∩ Xn is open (or closed) in Xn for each n .

A space X constructed in this way is called a cell complex or CW complex. The explanation of the letters ‘CW’ is given in the Appendix, where a number of basic topological properties of cell complexes are proved. The reader who wonders about various point-set topological questions lurking in the background of the following discussion should consult the Appendix for details.

6 Chapter 0

Some Underlying Geometric Notions

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 .
Example 0.1. A 1 dimensional cell complex X = X1 is what is called a graph 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.
Example 0.2. The house with two rooms, pictured earlier, has a visually obvious
2 dimensional cell complex structure. The 0 cells are the vertices where three or more
of the depicted edges meet, and the 1 cells are the interiors of the edges connecting these vertices. This gives the 1 skeleton 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
the Euler characteristic, which for a cell complex with ﬁnitely many cells is deﬁned
to be the number of even-dimensional cells minus the number of odd-dimensional
cells. As we shall show in Theorem 2.44, the Euler characteristic of a cell complex
depends only on its homotopy type, so the fact that the house with two rooms has the
homotopy type of a point implies that its Euler characteristic must be 1, no matter
how it is represented as a cell complex.
Example 0.3. 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 equivalent
to regarding Sn as the quotient space Dn/∂Dn .
Example 0.4. Real projective n space RPn is deﬁned to be the space of all lines
through the origin in Rn+1 . Each such line is determined by a nonzero vector in Rn+1 , unique up to scalar multiplication, and RPn is topologized as the quotient space of Rn+1 − {0} under the equivalence relation v ∼ 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 that RPn is 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 from RPn−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 .
Example 0.5. Since RPn is obtained from RPn−1 by attaching an n cell, the inﬁnite
union RP∞ = n RPn becomes a cell complex with one cell in each dimension. We can view RP∞ as the space of lines through the origin in R∞ = n Rn .
Example 0.6. Complex projective n space CPn is the space of complex lines through
the origin in Cn+1 , that is, 1 dimensional vector subspaces of Cn+1 . As in the case of RPn , each line is determined by a nonzero vector in Cn+1 , unique up to scalar multiplication, and CPn is topologized as the quotient space of Cn+1 − {0} under the

Cell Complexes

Chapter 0 7

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 disk D2n under the identiﬁcations v ∼ λv for v ∈ ∂D2n , in the

following way. The vectors in S2n+1 ⊂ Cn+1 with last coordinate real and nonnegative

are 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 D+2n bounded

by the sphere S2n−1 ⊂ S2n+1 consisting of vectors (w, 0) ∈ Cn × C with |w| = 1 . Each

vector in S2n+1 is equivalent under the identiﬁcations v ∼ λv to a vector in D+2n , 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

structure CPn = 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 eαn in a cell
complex X has a characteristic map → Φα : Dαn X which extends the attaching map
ϕα and is a homeomorphism from the interior of Dαn onto eαn . Namely, we can take
→ Φα to be the composition Dαn Xn−1 α Dαn 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 .
A subcomplex 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 complex X 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 subcomplexes RPk ⊂ RPn and CPk ⊂ CPn for k ≤ n . These are in fact the only subcomplexes of RPn and CPn .
There are natural inclusions S0 ⊂ S1 ⊂ ··· ⊂ Sn , but these subspheres are not subcomplexes of Sn in its usual cell structure with just two cells. However, we can give Sn a different cell structure in which each of the subspheres Sk is a subcomplex, by regarding each Sk as being obtained inductively from the equatorial Sk−1 by attaching two k cells, the components of Sk−Sk−1 . The inﬁnite-dimensional sphere S∞ = n Sn
then 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 single n cell of RP∞ .

8 Chapter 0

Some Underlying Geometric Notions

In the examples of cell complexes given so far, the closure of each cell is a subcomplex, 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 general. For example, if we start with S1 with its minimal cell structure and attach to this
a 2 cell by a map S1→S1 whose image is a nontrivial subarc of S1 , 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 eαm × eβn where eαm ranges over the cells of X and eβn 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 on S1 . 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 of X/A are the cells of X − A plus one new 0 cell, the image of A in X/A . For a
→ cell eαn of X − A attached by ϕα : 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 give Sn−1 any cell structure and build Dn from Sn−1 by attaching an n cell, then the quotient Dn/Sn−1 is Sn with 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 let A be the complement of this 2 cell, the 1 skeleton of X . 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, so X/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 another 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

Operations on Spaces

Chapter 0 9

on X , the union of two copies of the cone CX = (X × I)/(X × {0}) . If X is a CW complex, 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 map f : 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 suspension SX is the union of all line segments joining points

of X 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

the 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, 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 contains copies of X and Y at its two ‘ends,’

Y I

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 ≤ ti ≤ 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 join X1 ∗ ··· ∗ Xn can be regarded as the space of formal linear combinations t1x1 + ··· + tnxn with 0 ≤ ti ≤ 1 and t1 + ··· + tn = 1 , with the convention that terms 0ti 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 each Xi 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

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 projection from the origin gives a homeomorphism between X1 ∗ ··· ∗ Xn and Sn−1 .

10 Chapter 0

Some Underlying Geometric Notions

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 of X × 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 X 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 α Xα of an arbitrary collection of spaces Xα by starting with the disjoint union α Xα and identifying points xα ∈ Xα to a single point. In case the spaces Xα are cell complexes and the points xα are 0 cells, then α Xα is a cell complex since it is obtained from the cell complex α Xα by collapsing a subcomplex to a point.
For any cell complex X , the quotient Xn/Xn−1 is a wedge sum of n spheres α Sαn , with one sphere for each n cell of X .
Smash Product. Like suspension, this is another construction whose importance becomes 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 X and Y in X × Y intersect only at the point (x0, y0) , so their union can be identiﬁed with the wedge sum X ∨ Y . The smash product X ∧ Y is then deﬁned to be the quotient 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 factors X and Y .
The smash product X ∧ Y is a cell complex if X and Y are cell complexes with x0 and y0 0 cells, assuming that we give X × Y the cell-complex topology rather than the product topology in cases when these two topologies differ. For example, Sm ∧Sn has a 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 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 equivalences 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 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.

Two Criteria for Homotopy Equivalence Chapter 0 11

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.

Example 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 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, 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 sum m S1 of m circles is not homotopy equivalent to n S1 if m ≠ 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 for m S1 and n S1 if m ≠ n . In fact the Euler characteristic does this since m S1 has 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.

Example 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

B

A

X

joining the two points where A attaches. Then X can be given a CW complex structure with the two endpoints of A and B as 0 cells, the

X/A

X/B

interiors of A and B as 1 cells, and the rest of S2 as a 2 cell. Since A and B are contractible,

12 Chapter 0

Some Underlying Geometric Notions

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 . Hence S2/S0 and S1 ∨ S2 are homotopy equivalent, a fact which may not be entirely obvious at ﬁrst glance.
Example 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 in Z 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.)

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 a point yields a space ΣX 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 suspension SX 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
and Y . Another way in which the reduced suspension ΣX 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 of X , 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 of X × I with X × ∂I ∪ {x0}× I collapsed to a point.

Attaching Spaces
Another common way to change a space without changing its homotopy type involves 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

Two Criteria for Homotopy Equivalence Chapter 0 13

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 X0 X1 by identifying each point a ∈ A with its image f (a) ∈ X0 . Let us de-
note this quotient space by X0 f 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 map
→ f : 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 f CX
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, when X is a sphere Sn−1 the mapping cone Cf is the space
obtained from Y by attaching an n cell via → f : Sn−1 Y . A

CX Y

mapping cone Cf can also be viewed as the quotient Mf /X of the mapping cylinder Mf with the subspace X = X × {0} collapsed to a point.
If one varies an attaching map f by a homotopy ft , 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
X0 f X1 X0 g 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

identiﬁed is homotopy equivalent to S1 ∨ S2 . The sphere

with two points identiﬁed can be obtained by attaching S2 S 2

to S1 by a map that wraps a closed arc A in S2 around S1 , as shown in the ﬁgure. Since A is contractible, this attach-

A

S1

ing map is homotopic to a constant map, and attaching S2

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.

Example 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.

Example 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

14 Chapter 0

Some Underlying Geometric Notions

complex X and a subcomplex A , then X/A X ∪ CA , the mapping cone of the inclusion A X . For we have X/A = (X∪CA)/CA X∪CA since CA is a contractible subcomplex of X ∪ CA .
Example 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 have X/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 particular 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 map f0 : X→Y , and on a subspace A ⊂ X one is also given a homotopy ft : A→Y of f0 || A that one would like to extend to a homotopy ft : 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 example 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.
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 extension 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 .

The Homotopy Extension Property

Chapter 0 15

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 a
homeomorphism h from Mf onto a closed neighborhood N of A in X , with h || A = 11 and with h(Mf − Z) an open neighborhood of A . Mapping cylinder neighborhoods 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 Mf × I onto Mf × {0} ∪ (Z A)× I . Thus (Mf , Z A) has the homotopy extension property, which
implies 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 apply the homotopy extension property for (Mf , Z A) 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 retract
of X × I , hence (X, A) has the homotopy extension property.
Proof: 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 setting rt = tr + (1 − t)11 gives a deformation retraction of Dn × I onto Dn × {0} ∪ ∂Dn × I . This deformation retraction gives rise to a deformation retraction of Xn × I onto Xn × {0} ∪ (Xn−1 ∪ An)× I since Xn × I is obtained from Xn × {0} ∪ (Xn−1 ∪ An)× I by attaching copies of Dn × I along Dn × {0} ∪ ∂Dn × I . If we perform the deformation retraction 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.
Now we can prove the following generalization of the earlier assertion that collapsing a contractible subcomplex is a homotopy equivalence:
Proposition 0.17. If the pair (X, A) satisﬁes the homotopy extension property and A is contractible, then the quotient map q : X→X/A is a homotopy equivalence.

16 Chapter 0

Some Underlying Geometric Notions

Proof: Let ft : X→X be a homotopy extending a contraction of A , with f0 = 11 . Since ft(A) ⊂ A for all t , the composition qft : X→X/A sends A to a point and hence factors as a composition X →q X/A→X/A . Denoting the latter map by f t : X/A→X/A ,

−−−−−−→ −−−−−−→ −−−−−−→ −−−−−−→

we have qft = f tq in the ﬁrst of the two diagrams at the right. When t = 1 we have
f1(A) equal to a point, the point to which A
contracts, so f1 induces a map g : X/A→X
with gq = f1 , as in the second diagram. It

X −−−−−−−−f−t−−−→X

q

q

X/A−−−−−−f−−t−→X/A

X −−−−−−−−f−1−−−→X

q
X/A

−−−−−−−−−−−−g−f−−1−−−→−−−→X/Aq

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 homotopy equivalences since gq = f1 f0 = 11 via ft and

qg = f 1 f 0 = 11 via f t .

Another application of the homotopy extension property, giving a slightly more reﬁned version of one of our earlier criteria for homotopy equivalence, is the following:
Proposition 0.18. If (X1, A) is a CW pair and we have attaching maps f , g : A→X0
that are homotopic, then X0 f X1 X0 g X1 rel X0 .
Here the deﬁnition of W Z rel Y for pairs (W , Y ) and (Z, Y ) is that there are
maps ϕ : W →Z and ψ : Z→W restricting to the identity on Y , such that ψϕ 11
and 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 X0 F (X1 × I) .
This contains both X0 f X1 and X0 g X1 as subspaces. A deformation retraction of X1 × I onto X1 × {0} ∪ A× I as in Proposition 0.16 induces a deformation retraction of X0 F (X1 × I) onto X0 f X1 . Similarly X0 F (X1 × I) deformation retracts onto X0 g X1 . Both these deformation retractions restrict to the identity on X0 , so together they give a homotopy equivalence X0 f X1 X0 g X1 rel X0 .

We ﬁnish this chapter with a technical result whose proof will involve several applications of the homotopy extension property:
Proposition 0.19. Suppose (X, A) and (Y , A) satisfy the homotopy extension property, and f : X→Y is a homotopy equivalence with f || A = 11 . Then f is a homotopy
equivalence rel A .
Corollary 0.20. If (X, A) satisﬁes the homotopy extension property and the inclusion
A X is a homotopy equivalence, then A is a deformation retract of X .
Proof: Apply the proposition to the inclusion A X .
Corollary 0.21. A map f : X→Y is a homotopy equivalence iff X is a deformation
retract of the mapping cylinder Mf . Hence, two spaces X and Y are homotopy equivalent iff there is a third space containing both X and Y as deformation retracts.

The Homotopy Extension Property

Chapter 0 17

Proof: The inclusion i : X Mf is homotopic to the composition jf where j is the
inclusion Y Mf , a homotopy equivalence. It then follows from Exercise 3 at the end 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.

Proof of 0.19: Let g : Y →X be a homotopy inverse for f , and let ht : X→X be a
homotopy from gf = h0 to 11 = h1 . We will use ht to deform g to a map g1 with g1 || A = 11 . Since f || A = 11 , we can view ht || A as a homotopy from g || A to 11 . Then since we assume (X, A) has the homotopy extension property, we can extend this
homotopy 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 gt we have gf g1f via gtf . We also have gf 11 via ht , so since homotopy is an equivalence relation, we have g1f 11 . An explicit homotopy from g1f to 11 is

kt =

g1−2t f , h2t−1,

0 ≤ t ≤ 1/2 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 simply re-
traces its ﬁrst half, that is, kt = k1−t on A . In this situation we deﬁne a ‘homotopy of
homotopies’ ktu : A→A by means of the ﬁgure to the right showing
the parameter domain I × I for the pairs (t, u) , with the t axis hori-

zontal and the u 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’ we deﬁne ktu to be independent of t . This is unambiguous since kt = k1−t on A . Since k0 = 11 , we have ktu = 11 for (t, u) in the left, right, and top edges of the square. Since (X, A) has the homotopy exten-

sion property, so does (X × I, A× I) by the initial remarks on the homotopy extension
property. Viewing ktu as a homotopy of kt , we can therefore extend ktu : A→A to ktu : X→X with kt0 = kt : X→X . Now if we restrict this ktu to the left, top, and right
edges of the (t, u) square, we get a homotopy g1f 11 rel A .
Since g1 g , we have f g1 f g 11 , so the preceding argument can be repeated
with the pair f , g replaced by g1, f . The result is a map f1 : X→X with f1 || A = 11
and f1g1 11 rel A . Hence f1 f1(g1f ) = (f1g1)f f rel A . From this we deduce
that f g1 f1g1 11 rel A . Thus g1 is a homotopy inverse to f rel A .

18 Chapter 0

Some Underlying Geometric Notions

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 a homotopy equivalence X→Z . Deduce that homotopy equivalence is an equivalence
relation.
(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 a
homotopy ft : X→X such that f0 = 11 , f1(X) ⊂ A , and ft(A) ⊂ A for all t . Show
that if X deformation retracts to A in this weak sense, then the inclusion A X is a homotopy equivalence.
5. Show that if a space X deformation retracts to a point x ∈ X , then for each neighborhood U of x in X there exists a neighborhood V ⊂ U of x such that the inclusion V U is nullhomotopic. 6. (a) Let X be the subspace of R2 consisting of the horizontal segment [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 retracts to 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 X arranged as in the ﬁgure below. Show that Y is contractible but does not deformation retract onto any point.

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

7. Fill in the details in the following construction from [Edwards 1999] of a compact space Y ⊂ R3 with the same properties as the space Y in Exercise 6, that is, Y is contractible but does not deformation retract to any point. To begin, let X be the union of an inﬁnite se-

quence of cones on the Cantor set arranged end-to-end,

as in the ﬁgure. Next, form the one-point compactiﬁca- X

Y

tion of X × R . This embeds in R3 as a closed disk with curved ‘ﬁns’ attached along

Exercises

Chapter 0 19

circular arcs, and with the one-point compactiﬁcation of X as a cross-sectional slice. The desired space Y is then obtained from this subspace of R3 by wrapping one more cone on the Cantor set around the boundary of the disk.
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 , is nullhomotopic. 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 →X
such that f g 11 and hf 11 . More generally, show that f is a homotopy equivalence if f g and hf are homotopy equivalences.
12. Show that a homotopy equivalence f : X→Y induces a bijection between the set
of path-components of X and the set of path-components of Y , and that f restricts to a homotopy equivalence from each path-component of X to the corresponding pathcomponent of Y . Prove also the corresponding statement with components instead of path-components. Deduce from this that if the components and path-components of a space coincide, then the same is true for any homotopy equivalent space. 13. Show that any two deformation retractions rt0 and rt1 of a space X onto a subspace A can be joined by a continuous family of deformation retractions rts ,
0 ≤ s ≤ 1 , of X onto A , where continuity means that the map X × I × I→X sending
(x, s, t) to rts(x) is continuous.
14. Given positive integers v , e , and f satisfying v − e + f = 2 , construct a cell structure 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 this section, 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 and a M¨obius band as deformation retracts. 18. Show that S1 ∗ S1 = S3 , and more generally Sm ∗ Sn = Sm+n+1 . 19. Show that the space obtained from S2 by attaching n 2 cells along any collection of 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 intersecting 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, any two of which intersect in at most one point, show that X is homotopy equivalent to a wedge sum of S1 ’s and S2 ’s.

20 Chapter 0

Some Underlying Geometric Notions

22. Let X be a ﬁnite graph lying in a half-plane P ⊂ R3 and intersecting the edge of P in a subset of the vertices of X . Describe the homotopy type of the ‘surface of revolution’ 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 quotient spaces X ∗ Y /(X ∗ {y0} ∪ {x0} ∗ Y ) and S(X ∧ Y )/S({x0} ∧ {y0}) are homeomorphic, and deduce that X ∗ Y S(X ∧ Y ) .
25. If X is a CW complex with components Xα , show that the suspension SX is homotopy equivalent to Y α SXα for some graph Y . 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 Proposition 0.18 holds more generally when (X, A) satisﬁes the homotopy extension property.
27. Given a pair (X, A) and a map f : A→B , deﬁne X/f to be the quotient space
of X obtained by identifying points in A having the same image in B . Show that the
quotient 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 and use 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 every
→ pair (X0 f 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 single
cell en , describe exactly what the extension of a homotopy ft : A→Y to X given by
the proof of Proposition 0.16 looks like. That is, for a point x ∈ en , describe the path ft(x) for the extended ft .

Algebraic topology can be roughly deﬁned as the study of techniques for forming algebraic images of topological spaces. Most often these algebraic images are groups, but more elaborate structures such as rings, modules, and algebras also arise. The mechanisms that 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 between 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 functors of algebraic topology, the fundamental group, which creates an algebraic image of a space from the loops in the space, the paths in the space 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

Consider two linked circles A and B in R3 , as shown

in the ﬁgure. Our experience with actual links and chains

tells us that since the two circles are linked, it is impossi-

ble to separate B from A by any continuous motion of B , A such as pushing, pulling, or twisting. We could even take

B

B to be made of rubber or stretchable string and allow completely general continu-

ous deformations of B , staying in the complement of A at all times, and it would

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.

Instead of having B link with A just once, we could

make it link with A two or more times, as in the ﬁgures to the

right. As a further variation, by assigning an orientation to B

we can speak of B linking A a positive or a negative number A

B2

of times, say positive when B comes forward through A and

negative for the reverse direction. Thus for each nonzero

integer n we have an oriented circle Bn linking A n times,

where by ‘circle’ we mean a curve homeomorphic to a circle. A To complete the scheme, we could let B0 be a circle not

B−3

linked to 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

of as a path traversed in time, starting and ending at the same point x0 , which we can choose 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 end-

ing at the same point x0 can be ‘added’ to form a new loop B + B that travels ﬁrst

around B , then around B . For example, if B1 and B1 are loops each linking A once in

the positive direction,

then their sum B1 + B1 is deformable to B2 , linking A twice. Similarly, B1 + B−1 can be A

B1 x0

B1

A

B2 x0

deformed to the loop
B0 , unlinked from A . More generally, we see that Bm + Bn can be A

B1

x0

B−1

A

x0 B0

deformed to Bm+n for

arbitrary integers m and n .

Note that in forming sums of loops we produce loops that pass through the basepoint more than once. This is one reason why loops are deﬁned merely as continuous

The Idea of the Fundamental Group

23

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 we could 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 at the left in the ﬁg-

ure below. The interesting feature here is that if any one of the three circles is removed,

the other two are not

linked. In the same A

B

spirit as before, let us

A

B

regard one of the cir-

cles, say C , as a loop

in the complement of

C

C

the other two, A and

B , and we ask whether C can be continuously deformed to unlink it completely from

A and B , always staying in the complement of A and B during the deformation. We

can redraw the picture by pulling A and B apart, dragging C along, and then we see

C winding back and forth between A and B as shown in the second ﬁgure above.

In this new position, if we start at the point of C indicated by the dot and proceed

in the direction given by the arrow, then we pass in 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 two integers, 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 re-

gard the four parts (1)–(4) of C as an ordered sequence. Taking into account the

directions in which these segments of C pass

through A and B , we may deform C to the sum A

a + b − a − b of four loops as in the ﬁgure. We

a

write the third and fourth loops as the nega-

−a

tives of the ﬁrst two since they can be deformed

B b −b

to the ﬁrst two, but with the opposite orienta-

A

tions, and as we saw in the preceding exam-

a

ple, the sum of two oppositely oriented loops is deformable to a trivial loop, not linked with

−a

B b −b

anything. We would like to view the expression a + b − a − b as lying in a nonabelian group, so that 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 .

24 Chapter 1

The Fundamental Group

To shed further light on this example, suppose we modify it slightly so that the cir-

cles A and B are now linked, as in the next ﬁgure. The circle C can then be deformed

into the position shown at

the right, where it again rep-

A

B

A

B

resents the composite loop

aba−1b−1 , where a and b

are loops linking A and B .

But from the picture on the

C

C

left it is apparent that C can

actually be unlinked completely from A and B . So in this case the product aba−1b−1

should be trivial.

The fundamental group of a space X will be deﬁned so that its elements are loops in X starting and ending at a ﬁxed basepoint x0 ∈ X , but two such loops are regarded 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 that occur during deformations must also start and end at x0 .) In the ﬁrst example above, X is the complement of the circle A , while in the other two examples X is the complement of 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 example is inﬁnite cyclic with the loop B as a generator. This amounts to saying that every loop in the complement of A can be deformed to one of the loops Bn , and that Bn cannot be deformed to Bm if n ≠ m . The fundamental group of the complement of the two unlinked circles A and B in the 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 in the third example is the free abelian group on two generators, represented by 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 A and B is linked. The direct approach is given by the ﬁrst example, where one circle is 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 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.

Basic Constructions

Section 1.1 25

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 calculation are given in §1.2, sufﬁcient to show for example that every group is realized as the fundamental 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

The fundamental group of a space X will be deﬁned in terms of loops in X and

continuous deformations of these loops, but it is useful to consider also the more

general notion of paths and their deformations. By a path in X we mean a continuous
map f : I→X where I is the unit interval [0, 1] . 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 .

f0

(2) The associated map F : I × I→X deﬁned by x0

x1

F (s, t) = ft(s) is continuous.

f1

When two paths f0 and f1 are connected in this way by a homotopy ft , they are said to be homotopic. The notation for this is f0 f1 .

Example 1.1: Linear Homotopies. Any two paths f0 and f1 in Rn having the same
endpoints x0 and x1 are homotopic via the homotopy ft(s) = (1 − t)f0(s) + tf1(s) .

During this homotopy each point f0(s) travels along the line segment to f1(s) at con-

stant speed. This is because the 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) to f1(s) . If f1(s) = f0(s) then this segment de-

generates to a point, so ft(s) = f0(s) for all t . This happens in particular for s = 0

and map

s I

= ×

I1→, sRoneafcohllofwt sisfraompatchonfrtionmuitxy0otfofx0 1a.nCdofn1tinsuinitcye

of the homotopy ft as a the algebraic operations

in the deﬁnition of ft are continuous. This construction shows more generally that for a convex subspace X ⊂ Rn , all

paths 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 space
is an equivalence relation.

26 Chapter 1

The Fundamental Group

Proof: The constant homotopy ft = f shows that f f . If f0 f1 via ft , then
f1 f0 via the homotopy f1−t . For transitivity, if f0 f1 via ft and f1 = g0 g1 via gt , then f0 g1 via the homotopy ht that equals f2t for 0 ≤ t ≤ 1/2 and g2t−1 for 1/2 ≤ t ≤ 1. These two deﬁnitions agree for t = 1/2 since we assume f1 = g0 . Continuity of the associated map H(s, t) = ht(s) comes from the elementary fact,
which will be used frequently without explicit mention, that a function deﬁned on the
union of two closed sets is continuous if it is continuous when restricted to each of
the closed sets separately. In the case at hand, H is clearly continuous on I × [0, 1/2] and on I × [1/2, 1], so H is continuous on I × I .

The equivalence class of a path f under the equivalence relation of homotopy is

denoted [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 that traverses ﬁrst f then g by the formula

f g(s) =

f (2s), g(2s − 1),

0 ≤ s ≤ 1/2 1/2 ≤ s ≤ 1

Thus the speed of traversal of f and g is doubled in order for f g to be traversed

in unit time. This product operation respects homotopy classes since if f0 f1 and g0 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 In particular, suppose we restrict attention to paths

f1 f :I

→g1X.

with

the

same

start-

ing and ending point f (0) = f (1) = x0 ∈ X . Such paths are called loops, and the

common starting and ending point x0 is referred to as the basepoint. The set of

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

Proof: By restricting attention to loops with a ﬁxed basepoint x0 ∈ X we guarantee
that the product f g of any two such loops is deﬁned. We have already observed
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 axioms
for a group.
As a preliminary step, deﬁne a reparametrization of a path f to be a composi-
tion f ϕ 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 f ϕ f via the homotopy f ϕt 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 f ϕt is deﬁned.

Basic Constructions

Section 1.1 27

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 speeds
at which f and h are traversed. Hence (f g) h f (g h) . Restricting attention to
loops 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) that equals 2s on [0, 1/2] and 1 on [1/2, 1], so f c f . Similarly, c f f where c is now the constant path at f (0) . Taking f to be a loop, we deduce that the homotopy class of
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 deﬁned by f (s) = f (1 − s) . Consider the homotopy ht = ft gt where ft is the path that equals f on the interval [0, t] and that 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 . Thus f f c , and replacing f by f gives f f c for c the constant path at x1 . Specializing to the case that f is a loop at the basepoint x0 , we deduce that [ f ] is a two-sided inverse for [f ] in π1(X, x0) .

Example 1.4. A convex set X in Rn has π1(X, x0) = 0 , the trivial group, for every
basepoint x0 ∈ X , since any two loops f0 and f1 based at x0 are homotopic via the linear 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, computing the fundamental group of the circle.

It is natural to ask about the dependence of π1(X, x0) on the choice of the base-

point x0 . Since π1(X, x0) involves only the path-component of X containing x0 , it

is clear that we can hope to ﬁnd a relation between π1(X, x0) and π1(X, x1) for two

basepoints
let h : I→X

x0 be

and x1 only if a path from x0

x0 and x1 to x1 , with

lie in the same path-component the inverse path

of

X.

So

h(s) = h(1 − s) from x1 back to x0 . We can then associate

h

to each loop f based at x1 the loop h f h based at x0 . x0

x1

f

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 gen-

eral n fold product f1 ··· fn in which the path fi is traversed in the time interval

i−1 n

,

i n

.

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

28 Chapter 1

The Fundamental Group

Proof: If ft is a homotopy of loops based at x1 then h ft h is a homotopy of
loops based at x0 , so βh is well-deﬁned. Further, βh is a homomorphism since βh[f g] = [h f g h] = [h f h h g h] = βh[f ]βh[g] . Finally, βh is an isomorphism with inverse βh since βhβh[f ] = βh[h f h] = [h h f h h] = [f ] , and similarly βhβh[f ] = [f ] .
Thus if X is path-connected, the group π1(X, x0) is, up to isomorphism, independent of the choice of basepoint x0 . In this case the notation π1(X, x0) is often abbreviated 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 class
of 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 via nullhomotopies of the loops η η and γ η , using the assumption π1(X, x0) = 0 in the latter case. Conversely, if there is only one homotopy class of paths connecting a basepoint x0 to itself, then π1(X, x0) = 0 .

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. It should be no surprise then that the proof will involve some genuine work. To maximize the payoff for this work, the proof is written so that its main technical steps apply in the more general setting of covering spaces, the main topic of §1.3.

Theorem 1.7. The map ψ : Z→π1(S1) sending an integer n to the homotopy class
of the loop ωn(s) = (cos 2π ns, sin 2π ns) 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→S1 given by p(s) = (cos 2π s, sin 2π s) . This map can be visu-
alized geometrically by embedding R in R3 as the helix parametrized

by s (cos 2π s, sin 2π s, s) , and then p is the restriction to the he-

lix of the projection of R3 onto R2 , (x, y, z) (x, y) , as in the

ﬁgure. Observe
ωn : I→R is the

that the loop ωn path ωn(s) = ns

is the composition , starting at 0 and

pωn where ending at n ,

p

winding around the helix |n| times, upward if n > 0 and downward

if n < 0 . The relation ωn = pωn is expressed by saying that ωn is a lift of ωn .

Basic Constructions

Section 1.1 29

The deﬁnition of ψ can be reformulated by setting ψ(n) equal to the homotopy

class of the loop pf for f any path in R from 0 to n . Such an f is homotopic to
ωn via the linear homotopy (1 − t)f + tωn , hence pf is homotopic to pωn = ωn
and 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 ωm (τmωn) is a path in R from 0 to m + n , so ψ(m + n) is the homotopy class of the loop in S1 that is the image of this path under p . This image
is 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 is a unique lift f : I→R

starting at a point starting at x0 .

x0

∈ S1

and each

x0

∈ p−1(x0)

there

(b)

For each homotopy ft : I→S1 of
there is a unique lifted homotopy

paths starting at
ft : I→R of paths

x0 and each x0 starting at x0 .

∈

p−1(x0)

Before proving these facts, let us see how they imply the theorem. To show that ψ is
surjective, let f : I→S1 be a loop at the basepoint (1, 0) , representing a given element
of π1(S1) . By (a) there is a lift f starting at 0 . This path f ends at some integer n since pf (1) = f (1) = (1, 0) and p−1(1, 0) = Z ⊂ R . By the extended deﬁnition of ψ
we then have ψ(n) = [pf ] = [f ] . Hence ψ is surjective.
To show that ψ is injective, suppose ψ(m) = ψ(n) , which means ωm ωn . Let ft be a homotopy from ωm = f0 to ωn = f1 . By (b) this homotopy lifts to a homotopy ft of paths starting at 0 . The uniqueness part of (a) implies that f0 = ωm and f1 = ωn . Since ft is a homotopy of paths, the endpoint ft(1) is independent of t . For t = 0 this endpoint is m and for t = 1 it is n , so m = n .

It remains to prove (a) and (b). Both statements can be deduced from a more general assertion:
(c) Given a map F : Y × I→S1 and a map F : Y × {0}→R lifting F |Y × {0} , then there is a unique map F : Y × I→R lifting F and restricting to the given F on Y × {0} .

Statement (a) is the special case that Y is a point, and (b) is obtained by applying (c)

with Y F (s, t) of (a).

T==hIfenti(ns(ct)h) eagsifvouelsslouawaliu.nngAiqwuuaenyi.FqTu:hIee×lhiIfo→tmFRo:t.Io×pT{yh0ef}t→reisnRtr(bici)stgiooivnbestsaiFFn|e:{dI0×}b×Iy→IanaSn1adpbpyFl|si{ce1at}ttii×onngI

are paths lifting the constant path at x0 , hence they must also be constant by the uniqueness part of (a). So ft(s) = F (s, t) is a homotopy of paths lifting ft .
We shall prove (c) using just one special property of the projection p : R→S1 ,

namely:

There is an open cover {Uα} of S1 such that for each α , p−1(Uα) can be (∗) decomposed as a disjoint union of open sets each of which is mapped homeo-

morphically onto Uα by p .

30 Chapter 1

The Fundamental Group

For example, we could take the cover {Uα} to consist of any two open arcs in S1 whose union is S1 .
To prove (c) we will ﬁrst construct F : N × I→R for N some neighborhood in Y
of a given point y0 ∈ Y . Since F is continuous, every point (y0, t) has a product neighborhood Nt × (at, bt) such that F Nt × (at, bt) ⊂ Uα for some α . By compactness of {y0}× I , ﬁnitely many such products Nt × (at, bt) cover {y0}× I , so we can choose a single neighborhood N of y0 and a partition 0 = t0 < t1 < ··· < tm = 1 of I so that for each i , F (N × [ti, ti+1]) is contained in some Uαi . Assume inductively that F has been constructed on N × [0, ti] . We have F (N × [ti, ti+1]) ⊂ Uαi , so by (∗) there is an open set Uαi ⊂ R projecting homeomorphically onto Uαi by p and
containing the point F (y0, ti) . After replacing N by a smaller neighborhood of y0 we may assume that F (N × {ti}) ⊂ Uαi , namely, replace N × {ti} by its intersection with (F || N × {ti})−1(Uαi ) . Now we can deﬁne F on N × [ti, ti+1] to be the composition of
→ F with the homeomorphism p−1 : Uαi Uαi . After ﬁnitely many repetitions of this → induction step we eventually get a lift F : N × I R for some neighborhood N of y0 .
Next we show the uniqueness part of (c) in the special case that Y is a point. In this
case we can omit Y from the notation. So suppose F and F are two lifts of F : I→S1
such that F (0) = F (0) . As before, choose a partition 0 = t0 < t1 < ··· < tm = 1 of
I so that for each i , F ([ti, ti+1]) is contained in some Uαi . Assume inductively that F = F on [0, ti] . Since [ti, ti+1] is connected, so is F ([ti, ti+1]) , which must therefore
lie in a single one of the disjoint open sets Uαi projecting homeomorphically to Uαi as in (∗) . By the same token, F ([ti, ti+1]) lies in a single Uαi , in fact in the same one that contains F ([ti, ti+1]) since F (ti) = F (ti) . Because p is injective on Uαi and pF = F , it follows that F = F on [ti, ti+1] , and the induction step is ﬁnished.
The last step in the proof of (c) is to observe that since the F ’s constructed above
on sets of the form N × I are unique when restricted to each segment {y}× I , they
must agree whenever two such sets N × I overlap. So we obtain a well-deﬁned lift F
on all of Y × I . This F is continuous since it is continuous on each N × I , and it is
unique since it is unique on each segment {y}× I .

Now we turn to some applications of this theorem. Although algebraic topology is usually ‘algebra serving topology,’ the roles are reversed in the following proof of the Fundamental Theorem of Algebra.

Theorem 1.8. Every nonconstant polynomial with coefﬁcients in C has a root in C .

Proof: We may assume the polynomial is of the form p(z) = zn + a1zn−1 + ··· + an .
If p(z) has no roots in C , then for each real number r ≥ 0 the formula

fr (s)

=

p(r e2πis )/p(r ) |p(r e2πis )/p(r )|

deﬁnes a loop in the unit circle S1 ⊂ C based at 1 . As r varies, fr is a homotopy of loops based at 1 . Since f0 is the trivial loop, we deduce that the class [fr ] ∈ π1(S1)

Basic Constructions

Section 1.1 31

is zero for all r . Now ﬁx a large value of r , bigger than 1 + |a1| + ··· + |an| . Then for |z| = r we have
|zn| = r n = r · r n−1 > (|a1| + ··· + |an|)|zn−1| ≥ |a1zn−1 + ··· + an|
from which it follows that the polynomial pt(z) = zn + t(a1zn−1 + ··· + an) has no roots on the circle |z| = r when 0 ≤ t ≤ 1 . Replacing p by pt in the formula for fr above and letting t go from 1 to 0 , we obtain a homotopy from the loop fr to the loop ωn(s) = e2πins . By Theorem 1.7, ωn represents n times a generator of the inﬁnite cyclic group π1(S1) . Since we have shown that [ωn] = [fr ] = 0 , we conclude that n = 0 . Thus the only polynomials without roots in C are constants.

For the next result we use the standard notation Dn for the closed unit disk in Rn , all vectors x of length |x| ≤ 1 . Thus the boundary of Dn is the unit sphere Sn−1 .

Theorem 1.9. Every continuous map h : D2→D2 has a ﬁxed point, that is, a point
x with h(x) = x .

Proof: Suppose on the contrary that h(x) ≠ x for all x ∈ D2 .
Then we can deﬁne a map r : D2→S1 by letting r (x) be the
point of S1 where the ray in R2 starting at h(x) and passing
through x leaves D2 . Continuity of r is clear since small per-

h(x) x

turbations of x produce small perturbations of h(x) , hence r(x)

also small perturbations of the ray through these two points.
The crucial property of r , besides continuity, is that r (x) = x if x ∈ S1 . Thus r is
a retraction of D2 onto S1 . We will show that no such retraction can exist.
Let f0 be any loop in S1 . In D2 there is a homotopy of f0 to a constant loop, for example the linear homotopy ft(s) = (1 − t)f0(s) + tx0 where x0 is the basepoint of f0 . Since the retraction r is the identity on S1 , the composition r ft is then a homotopy in S1 from r f0 = f0 to the constant loop at x0 . But this contradicts the fact that π1(S1) is nonzero.

This theorem was ﬁrst proved by Brouwer around 1910, one of the early triumphs of algebraic topology. Brouwer in fact proved the corresponding result for Dn , and we shall obtain this generalization in Corollary 2.11 using homology groups in place of π1 . One could also use the higher homotopy group πn . Brouwer’s original proof used neither homology nor homotopy groups, which had not been invented at the
time. Instead it used the notion of degree for maps Sn→Sn , which we shall deﬁne in
§2.2 using homology but which Brouwer deﬁned directly in more geometric terms.
These proofs are all arguments by contradiction, and so they show just the existence of ﬁxed points without giving any clue as to how to ﬁnd one in explicit cases. Our proof of the Fundamental Theorem of Algebra was similar in this regard. There

32 Chapter 1

The Fundamental Group

exist other proofs of the Brouwer ﬁxed point theorem that are somewhat more constructive, for example the elegant and quite elementary proof by Sperner in 1928, which is explained very nicely in [Aigner-Ziegler 1999].
The techniques used to calculate π1(S1) can be applied to prove the Borsuk–Ulam theorem in dimension two:
Theorem 1.10. For every continuous map f : S → 2 R2 there exists a pair of antipodal
points x and −x in S2 with f (x) = f (−x) .

It may be that there is only one such pair of antipodal points x , −x , for example
if f is simply orthogonal projection of the standard sphere S2 ⊂ R3 onto a plane.
The Borsuk–Ulam theorem holds also for maps S → n Rn , as we show in Proposi-
tion 2B.6. The proof for n = 1 is easy since the difference f (x) − f (−x) changes sign

as x goes halfway around the circle, hence this difference must be zero for some x . For n ≥ 2 the theorem is certainly less obvious. Is it apparent, for example, that

at every instant there must be a pair of antipodal points on the surface of the earth

having the same temperature and the same barometric pressure?

The theorem says in particular that there is no one-to-one continuous map from S2 to R2 , so S2 is not homeomorphic to a subspace of R2 , an intuitively obvious fact

that is not easy to prove directly.

Proof: If the conclusion is false for f : S2→R2 , we can deﬁne a map g : S2→S1 by

→ g(x)

=

f (x)−f (−x) |f (x)−f (−x)|

.

Deﬁne

a

loop

η:I

S2 ⊂ R3 by η(s) = (cos 2π s, sin 2π s, 0) , and

→ let h : I S1 be the composed loop gη . Since g(−x) = −g(x) , we have the relation

h(s + 1/2) = π1(S1) , the

−h(s) for all s loop h can be

in the interval lifted to a path

[0, h:

1I/→2].RA.sTwhee

showed in the calculation of equation h(s + 1/2) = −h(s)

implies that h(s + 1/2) = h(s) + q/2 for some odd integer q . By solving the equation

h(s + 1/2) = h(s) + q/2 for q we see that q depends continuously on s ∈ [0, 1/2], so

q must be a constant independent of s since it is constrained to integer values. In

particular, we have h(1) = h(1/2) + q/2 = h(0) + q. This means that h represents q

times But h

a generator of π1(S1) was the composition

. Since q is odd, we conclude that h
gη : I→S2→S1 , and η is obviously

is not nullhomotopic. nullhomotopic in S2 ,

so gη is nullhomotopic in S1 by composing a nullhomotopy of η with g . Thus we

have arrived at a contradiction.

Corollary 1.11. Whenever S2 is expressed as the union of three closed sets A1 , A2 ,
and A3 , then one of these sets must contain a pair of antipodal points x and −x .

→ Proof: Let di : S2 R measure distance to Ai , that is, di(x) = infy∈Ai |x − y| . This

is a continuous function, so we may apply the Borsuk–Ulam theorem to the map

S → 2 R2 , x

d1(x), d2(x) , obtaining a pair of antipodal points x and −x with

d1(x) = d1(−x) and d2(x) = d2(−x) . If either of these two distances is zero, then

x and −x both lie in the same set A1 or A2 since these are closed sets. On the other

Basic Constructions

Section 1.1 33

hand, if the distances from x and −x to A1 and A2 are both strictly positive, then x and −x lie in neither A1 nor A2 so they must lie in A3 .

To see that the number ‘three’ in this result is best possible, consider a sphere inscribed in a tetrahedron. Projecting the four faces of the tetrahedron radially onto the sphere, we obtain a cover of S2 by four closed sets, none of which contains a pair of antipodal points.
Assuming the higher-dimensional version of the Borsuk–Ulam theorem, the same arguments show that Sn cannot be covered by n + 1 closed sets without antipodal pairs of points, though it can be covered by n + 2 such sets. Even the case n = 1 is somewhat interesting: If the circle is covered by two closed sets, one of them must contain a pair of antipodal points.
The following simple fact will allow us to compute the fundamental groups of a few more spaces.
Proposition 1.12. π1(X × Y ) is isomorphic to π1(X)× π1(Y ) if X and Y are path-
connected.
Proof: A basic property of the product topology is that a map f : Z→X × Y is continuous iff the maps g : Z→X and h : Z→Y deﬁned by f (z) = (g(z), h(z)) are both
continuous. Hence a loop in X × Y based at (x0, y0) is equivalent to a pair of loops in X and Y based at x0 and y0 respectively. Similarly, a homotopy of a loop in X × Y is equivalent to a pair of homotopies of the corresponding loops in X and Y . Thus we obtain a bijection π1 X × Y , (x0, y0) ≈ π1(X, x0)× π1(Y , y0) , and this is obviously a group isomorphism.
Example 1.13: The Torus. By the proposition we have an isomorphism π1(S1 × S1) ≈
Z× Z . Under this isomorphism a pair (p, q) ∈ Z× Z corresponds to a loop that winds p times around one S1 factor of the torus and q times around the other S1 factor, for example the loop ωpq(s) = (ωp(s), ωq(s)) . More generally, the n dimensional torus, which is the product of n circles, has fundamental group isomorphic to the product of n copies of Z . This follows by induction on n .

Induced Homomorphisms

FmoorrbpSrhueipvspimtoysϕwe ∗eϕ:wπ: rX1i(t→eX,ϕYx0:is()X→a,mxπa01)p(→Yta,k(yYi0n,)gy, td0h)eeﬁibnnaetsdheipbsoysicintotumaxtp0ioo∈nsi.XnTgtholoetnohpeϕsbfains:edIp→uocieXnst

y0 ∈ Y . a homobased at

x0 with ϕ , that is, ϕ∗[f ] = [ϕf ] . This induced map ϕ∗ is well-deﬁned since a

homotopy ft of loops based at x0 yields a composed homotopy ϕft of loops based at y0 , so ϕ∗[f0] = [ϕf0] = [ϕf1] = ϕ∗[f1] . Furthermore, ϕ∗ is a homomorphism since ϕ(f g) = (ϕf ) (ϕg) .

34 Chapter 1

The Fundamental Group

Two basic properties of induced homomorphisms are:
→ → (ϕψ)∗ = ϕ∗ψ∗ for a composition (X, x0) ψ (Y , y0) ϕ (Z, z0) . 11∗ = 11 , which is a concise way of saying that the identity map 11 : X→X induces → the identity map 11 : π1(X, x0) π1(X, x0) .
The ﬁrst of these follows since composition of maps is associative, (ϕψ)f = ϕ(ψf ) , and the second is obvious. These two properties of induced homomorphisms are what makes the fundamental group a functor. The formal deﬁnition of a functor requires the introduction of certain other preliminary concepts, however, so we postpone this until it is needed in §2.3.
If ϕ is a homeomorphism with inverse ψ then ϕ∗ is an isomorphism with inverse ψ∗ since ϕ∗ψ∗ = (ϕψ)∗ = 11∗ = 11 and similarly ψ∗ϕ∗ = 11 . We will use this fact in the following calculation of the fundamental groups of higher-dimensional spheres:
Proposition 1.14. π1(Sn) = 0 if n ≥ 2 .
Proof: Let f be a loop in Sn at a chosen basepoint x0 . If the image of f is disjoint
from some other point x ∈ Sn then f is nullhomotopic since Sn − {x} is homeomorphic to Rn , which is simply-connected. So it will sufﬁce to homotope f to be nonsurjective. To do this we will look at a small open ball B about any point x ≠ x0 in Sn and see that the number of times that f enters B , passes through x , and leaves B is ﬁnite, and each of these portions of f can be pushed off x without changing the rest of f .
The set f −1(B) is open in (0, 1) , hence is the union of at most countably many disjoint open intervals (ai, bi) . The compact set f −1(x) is contained in the union of these intervals, so it must be contained in the union of ﬁnitely many of them. Consider one of the intervals (ai, bi) meeting f −1(x) . The path fi obtained by restricting f to [ai, bi] lies in the closure of B , and its endpoints f (ai) and f (bi) lie in the boundary of B . If n ≥ 2 , we can choose a path gi from f (ai) to f (bi) in the closure of B but disjoint from x . For example, we could choose gi to lie in the boundary of B , which is a sphere of dimension n − 1 , hence path-connected if n ≥ 2 . Since the closure of B is homeomorphic to a convex set in Rn and hence simply-connected, the path fi is homotopic to gi by Proposition 1.6, so we may homotope f by deforming fi to gi . After repeating this process for each of the intervals (ai, bi) that meet f −1(x) , we obtain a loop g homotopic to the original f and with g(I) disjoint from x .
Example 1.15. For x ∈ Rn we have Rn − {x} homeomorphic to Sn−1 × R , so by
Proposition 1.12, π1(Rn − {x}) is isomorphic to π1(Sn−1)× π1(R) , hence is Z for n = 2 and trivial for n > 2 .
Corollary 1.16. R2 is not homeomorphic to Rn for n ≠ 2 .

Basic Constructions

Section 1.1 35

Proof: Suppose f : R2→Rn is a homeomorphism. The case n = 1 is easily disposed
of since R2 − {0} is path-connected but the homeomorphic space Rn − {f (0)} is not path-connected when n = 1 . When n > 2 we cannot distinguish R2 − {0} from Rn − {f (0)} by the number of path-components, but by Example 1.15 we can distin-
guish them by their fundamental groups.

The more general statement that Rm is not homeomorphic to Rn if m ≠ n can
be proved in the same way using either the higher homotopy groups or homology groups. In fact, nonempty open sets in Rm and Rn can be homeomorphic only if m = n , as we will show in Theorem 2.19 using homology.

Induced homomorphisms allow certain relations between spaces to be transformed into relations between their fundamental groups. For example:

Proposition 1.17. If a space X retracts onto a subspace A , then the homomorphism → i∗ : π1(A, x0) π1(X, x0) induced by the inclusion i : A X is injective. If A is a

deformation retract of X , then i∗ is an isomorphism.

Proof: If r : X→A is a retraction, then r i = 11 , hence r∗i∗ = 11 , which implies that i∗

is injective. If rt : X→X
and r1(X) ⊂ A , then for

is a any

deformation retraction
loop f : I→X based at

of x0

X ∈

onto A , so r0 = 11 , A the composition

rt|A = 11 , rtf gives

a homotopy of f to a loop in A , so i∗ is also surjective.

This gives another way of seeing that S1 is not a retract of D2 , a fact we showed

earlier in the proof of the Brouwer ﬁxed point theorem, since the inclusion-induced
map → π1(S1) π1(D2) is a homomorphism Z→0 that cannot be injective.
The exact group-theoretic analog of a retraction is a homomorphism ρ of a group

G onto a subgroup H such that ρ restricts to the identity on H . In the notation

above, if we identify π1(A) with its image under i∗ , then r∗ is such a homomorphism

ρfro: Gm→πH1(Xis)

onto quite

the subgroup π1(A) . a strong condition on

The existence of a retracting homomorphism H . If H is a normal subgroup, it implies that

G is the direct product of H and the kernel of ρ . If H is not normal, then G is what

is called in group theory the semi-direct product of H and the kernel of ρ .

t

∈

Recall from Chapter 0 the general deﬁnition of a homotopy as
I , such that the associated map Φ : X × I→Y , Φ(x, t) = ϕt(x) ,

a family ϕt : X→Y ,
is continuous. If ϕt

takes a subspace A ⊂ X to a subspace B ⊂ Y for all t , then we speak of a homotopy of

ϕmtap: (sXo,fxp0)a→irs(, Yϕ, ty:0()Xi,sAt)h→e c(aYs,eBt)h.aItnϕpta(rxti0c)ul=ary, 0a

basepoint-preserving homotopy for all t . Another basic property

of induced homomorphisms is their invariance under such homotopies:

→ If ϕt : (X, x0) (Y , y0) is a basepoint-preserving homotopy, then ϕ0∗ = ϕ1∗ .

This holds since ϕ0∗[f ] = [ϕ0f ] = [ϕ1f ] = ϕ1∗[f ] , the middle equality coming from the homotopy ϕtf .

36 Chapter 1

The Fundamental Group

(Y

,

There is a notion of homotopy equivalence
y0) if there are maps ϕ : (X, x0)→(Y , y0)

for spaces and ψ : (Y

with basepoints: (X
, y0)→(X, x0) with

, x0) homo-

topies ϕψ 11 and ψϕ 11 through maps ﬁxing the basepoints. In this case the

induced maps on π1 satisfy ϕ∗ψ∗ = (ϕψ)∗ = 11∗ = 11 and likewise ψ∗ϕ∗ = 11 , so ϕ∗ and ψ∗ are inverse isomorphisms π1(X, x0) ≈ π1(Y , y0) . This somewhat formal

argument gives another proof that a deformation retraction induces an isomorphism

on fundamental groups, since if X deformation retracts onto A then (X, x0) (A, x0) for any choice of basepoint x0 ∈ A .

Having to pay so much attention to basepoints when dealing with the fundamental

group is something of a nuisance. For homotopy equivalences one does not have to

be quite so careful, as the conditions on basepoints can actually be dropped:

Proposition 1.18. If ϕ : X→Y is a homotopy equivalence, then the induced homo→ morphism ϕ∗ : π1(X, x0) π1 Y , ϕ(x0) is an isomorphism for all x0 ∈ X .
The proof will use a simple fact about homotopies that do not ﬁx the basepoint:

Lemma 1.19. If ϕt : X→Y is a homotopy and
h is the path ϕt(x0) formed by the images of a basepoint x0 ∈ X , then the three maps in the diagram at the right satisfy ϕ0∗ = βhϕ1∗ .

π1(

X,

x0

)

−−−−−−−ϕϕ−−−−10−−∗−∗−−→→

π1( π1(

−−−−−→

Y, Y,

ϕ1(
βh
ϕ0 (

x0 x0

) )

) )

Proof: Let ht be the restriction of h to the interval [0, t] , with a reparametrization
so that the domain of ht is still [0, 1] . Explicitly, we can take ht(s) = h(ts) . Then if
f is a loop in X at the basepoint x0 , the formula ht (ϕtf ) ht deﬁnes a homotopy of loops at ϕ0(x0) . Restricting this homotopy to t = 0 and t = 1 , we see that ϕ0∗([f ]) = βh ϕ1∗([f ]) .

Proof of 1.18: Let ψ : Y →X be a homotopy-inverse for ϕ , so that ϕψ
ψϕ 11 . Consider the maps

11 and

→ → → π1(X, x0) ϕ∗ π1 Y , ϕ(x0) ψ∗ π1 X, ψϕ(x0) ϕ∗ π1 Y , ϕψϕ(x0)

The composition of the ﬁrst two maps is an isomorphism since ψϕ 11 implies that ψ∗ϕ∗ = βh for some h , by the lemma. In particular, since ψ∗ϕ∗ is an isomorphism, ϕ∗ is injective. The same reasoning with the second and third maps shows that ψ∗ is injective. Thus the ﬁrst two of the three maps are injections and their composition
is an isomorphism, so the ﬁrst map ϕ∗ must be surjective as well as injective.

Basic Constructions

Section 1.1 37

Exercises

1. Show that composition of paths satisﬁes the following cancellation property: If f0 g0 f1 g1 and g0 g1 then f0 f1 .
2. Show that the change-of-basepoint homomorphism βh depends only on the homotopy class of h .

3. For a path-connected space X , show that π1(X) is abelian iff all basepoint-change homomorphisms βh depend only on the endpoints of the path h .

4. A subspace X ⊂ Rn is said to be star-shaped if there is a point x0 ∈ X such that, for each x ∈ X , the line segment from x0 to x lies in X . Show that if a subspace X ⊂ Rn is locally star-shaped, in the sense that every point of X has a star-shaped

neighborhood in X , then every path in X is homotopic in X to a piecewise linear

path, that is, a path consisting of a ﬁnite number of straight line segments traversed

at constant speed. Show this applies in particular when X is open or when X is a

union of ﬁnitely many closed convex sets.

5. Show that every homomorphism homomorphism ϕ∗ of a map ϕ : S1

→π1S(S1

1
.

)→π1

(S

1

)

can

be

realized

as

the

induced

6. Given a space X and show that the map π1(A,

a path-connected subspace
x0)→π1(X, x0) induced by

A containing the the inclusion A

basepoint x0 , X is surjective

iff every path in X with endpoints in A is homotopic to a path in A .

7. Show that for a space X , the following three conditions are equivalent:
(a) Every map S1→X is homotopic to a constant map, with image a point. (b) Every map S1→X extends to a map D2→X .
(c) π1(X, x0) = 0 for all x0 ∈ X .
Deduce that a space X is simply-connected iff all maps S1→X are homotopic. [In
this problem, ‘homotopic’ means ‘homotopic without regard to basepoints.’]

w8m.iatWphseno(cSac1no,nsr0de)gi→taiordn(Xsπ,ox1n(0Xb) .a, xsLe0ep)to[aiSns1tts,hX. eT]shbeuetsotthfheebrsaeesteispoaof ihnnoat-tmpuroreatsolepmryvaicpnlgaΦshs: oeπsm1(ooXfto,mxpa0yp)c→slaSs[1sS→e1s, XXo]f,
obtained by ignoring basepoints. Show that Φ is onto if X is path-connected, and that Φ([f ]) = Φ([g]) iff [f ] and [g] are conjugate in π1(X, x0) . Hence Φ induces a oneto-one correspondence between [S1, X] and the set of conjugacy classes in π1(X) , when X is path-connected.
9. Deﬁne f : S1 × I→S1 × I by f (θ, s) = (θ + 2π s, s) , so f restricts to the identity
on the two boundary circles of S1 × I . Show that f is homotopic to the identity by
a homotopy ft that is stationary on one of the boundary circles, but not by any homotopy ft that is stationary on both boundary circles. [Consider what f does to the path s (θ0, s) for ﬁxed θ0 ∈ S1 .]

38 Chapter 1

The Fundamental Group

10. Does the Borsuk–Ulam theorem hold for the torus? In other words, for every map
f → : S1 × S1 R2 must there exist (x, y) ∈ S1 × S1 such that f (x, y) = f (−x, −y) ?

11. Let A1 , A2 , A3 be compact sets in R3 . Use the Borsuk–Ulam theorem to show that there is one plane P ⊂ R3 that simultaneously divides each Ai into two pieces of

equal measure.
12. Given a map f : X→Y and a path h : I→X

π1( X, x1) −−−−−−−β−−h−−→ π1( X, x0)

−−−−−→ −−−−−→

from x0 to x1 , show that f∗βh = βf hf∗ in the diagram to the right.

f∗

f∗

π1( Y, f(x1)) −−−−−−β−−−f−−h→ π1( Y, f(x0))

13. Show, using fundamental groups and induced homomorphisms, that there is no
retraction of the M¨obius band onto its boundary circle.
14. Construct inﬁnitely many nonhomotopic retractions S1 ∨ S1→S1 .

15. the

iInf cXlu0siisonthXe 0path-XcoimndpuocneesnatnofisaosmpoarcpehXismconπt1a(iXni0n, gx0th)→e bπas1e(pXo, ixn0t)x. 0

,

show

that

16. Using the technique in the proof of Proposition 1.14, show that if a space X is obtained from a path-connected subspace A by attaching a cell en with n ≥ 2 , then

the inclusion A X induces a surjection on π1 .

17. Modify the proof of Proposition 1.14 to give an elementary proof that π1(S1)

is cyclic, generated by the standard loop winding once around the circle. [The more

difﬁcult part of the calculation of π1(S1) is therefore the fact that no iterate of this

loop is nullhomotopic.]

18. Suppose ft : X→X is a homotopy such that f0 and f1 are each the identity map.
Use Lemma 1.19 to show that for any x0 ∈ X , the loop ft(x0) represents an element of

the center of π1(X, x0) . an element of the center

[One can interpret the result as saying that a loop
of π1(X) if it extends to a loop of maps X→X .]

represents

The van Kampen theorem gives a method for computing the fundamental groups of spaces that can be decomposed into simpler spaces whose fundamental groups are already known. By systematic use of this theorem one can compute the fundamental groups of a very large number of spaces. We shall see for example that for every group G there is a space XG whose fundamental group is isomorphic to G .
To give some idea of how one might hope to compute fundamental groups by decomposing spaces into simpler pieces, let us look at an example. Consider the space X formed by two circles A and B intersecting in a single point, which we choose as the basepoint x0 . By our preceding calculations we know that π1(A) is inﬁnite cyclic,

Van Kampen’s Theorem

Section 1.2 39

generated by a loop a that goes once around A . Similarly, π1(B) is a copy of Z generated by a loop b going once around B . Each product of powers of a and b then gives an element of π1(X) . For example, the product a5b2a−3ba2 is the loop that goes ﬁve times around A , then twice around B , then three times around A in the
opposite direction, then once around B , then twice around A . The set of all words like a5b2a−3ba2 , consisting of powers of a alternating with powers of b , forms a group usually denoted Z∗Z . Multiplication in this group is deﬁned just as one would expect, e.g., (b4a5b2a−3)(a4b−1ab3) = b4a5b2ab−1ab3 . The identity element is the empty word, and inverses are what they have to be, e.g., (a2b−3aba−2)−1 = a2b−1a−1b3a−2 .
It would be very nice if such words in a and b corresponded exactly to elements of π1(X) , so that π1(X) was isomorphic to the group Z ∗ Z . The van Kampen theorem will imply that this is indeed the case. Similarly, if X is the union of three circles touching at a single point, the van Kampen theorem will imply that π1(X) is Z ∗ Z ∗ Z , the group consisting of words in powers of three letters a , b , c . The generalization
to a union of any number of circles touching at one point will also follow as a special
case of the van Kampen theorem. The group Z ∗ Z is an example of a general construction called the free product
of groups. The statement of van Kampen’s theorem will be in terms of free products,
so before stating the theorem we should describe exactly what free products are, in
case the reader has not seen this algebraic construction previously.
Free Products of Groups
Suppose one is given a collection of groups Gα and one wishes to construct a single group containing all these groups as subgroups. One way to do this would be
to take the product group α Gα , whose elements can be regarded as the functions α gα ∈ Gα . Or one could restrict to functions taking on nonidentity values at most ﬁnitely often, forming the direct sum α Gα . Both these constructions produce groups containing all the Gα ’s as subgroups, but with the property that elements of different subgroups Gα commute with each other. In the realm of nonabelian groups this commutativity is unnatural, and so one would like a ‘nonabelian’ version of α Gα or α Gα . Since the sum α Gα is smaller and presumably simpler than α Gα , it should be easier to construct a nonabelian version of α Gα , and this is what the free product ∗α Gα achieves.
Here is the precise deﬁnition. As a set, the free product ∗α Gα consists of all words g1g2 ··· gm of arbitrary ﬁnite length m ≥ 0 , where each letter gi belongs to a group Gαi and is not the identity element of Gαi , and adjacent letters gi and gi+1 belong to different groups Gα , that is, αi ≠ αi+1 . Words satisfying these conditions are called reduced, the idea being that nonreduced words can always be sim-
pliﬁed until they are reduced by writing adjacent letters that lie in the same Gαi as a single letter and by canceling trivial letters. The empty word is allowed, and will

40 Chapter 1

The Fundamental Group

be the identity element of ∗α Gα . The group operation in ∗α Gα is juxtaposition, (g1 ··· gm)(h1 ··· hn) = g1 ··· gmh1 ··· hn . This product may not be reduced, how-
ever: If gm and h1 belong to the same Gα , they should be combined into a single letter
(gmh1) according to the multiplication in Gα , and if this new letter gmh1 happens to
be the identity of Gα , it should be canceled from the product. This may allow gm−1
and h2 to be combined, and possibly canceled too. Repetition of this process eventually produces a reduced word. For example, in the product (g1 ··· gm)(gm −1 ··· g1−1) everything cancels and we get the identity element of ∗α Gα , the empty word.

Verifying directly that this multiplication is associative would be rather tedious,

but there is an indirect approach that avoids most of the work. Let W be the set of

reduced words g1 ··· associate the function

gm Lg

:

as above, including the empty
W →W given by multiplication

word. on the

To each g left, Lg(g1

∈ Gα we ··· gm) =

gg1 ··· gm where we combine g with g1 if g1 ∈ Gα to make gg1 ··· gm a reduced

word. A key property of the association g Lg is the formula Lgg = LgLg for

g, g ∈ Gα , that is, g(g (g1 ··· gm)) = (gg )(g1 ··· gm) . This special case of asso-

ciativity follows rather trivially from associativity in Gα . The formula Lgg = LgLg

implies that Lg is invertible with inverse Lg−1 . Therefore the association g Lg de-

ﬁnes a homomorphism generally, we can deﬁne

from L:W

→GαP

to the group (W ) by L(g1

P (W ) ··· gm

of all permutations of W . More ) = Lg1 ··· Lgm for each reduced

word g1 ··· gm . This function L is injective since the permutation L(g1 ··· gm) sends

the empty word to g1 ··· gm . The product operation in W corresponds under L to

composition in P (W ) , because of the relation Lgg = LgLg . Since composition in

P (W ) is associative, we conclude that the product in W is associative.

In particular, we have the free product Z ∗ Z as described earlier. This is an

example of a free group, the free product of any number of copies of Z , ﬁnite or

inﬁnite. The elements of a free group are uniquely representable as reduced words in

powers of generators for the various copies of Z , with one generator for each Z , just as in the case of Z ∗ Z . These generators are called a basis for the free group, and the

number of basis elements is the rank of the free group. The abelianization of a free

group is a free abelian group with basis the same set of generators, so since the rank

of a free abelian group is well-deﬁned, independent of the choice of basis, the same

is true for the rank of a free group.

An interesting example of a free product that is not a free group is Z2 ∗ Z2 . This is like Z ∗ Z but simpler since a2 = e = b2 , so powers of a and b are not needed, and

Z2 ∗ Z2 consists of just the alternating words in a and b : a , b , ab , ba , aba , bab ,

abab , can be

baba , ababa, ··· , together with the empty word. elucidated by looking at the homomorphism ϕ : Z2

∗ThZe2s→truZc2tuarsesoocfiaZt2in∗g

Z2 to

each word its length mod 2 . Obviously ϕ is surjective, and its kernel consists of the

words of even length. These form an inﬁnite cyclic subgroup generated by ab since ba = (ab)−1 in Z2 ∗ Z2 . In fact, Z2 ∗ Z2 is the semi-direct product of the subgroups

Van Kampen’s Theorem

Section 1.2 41

Z and Z2 generated by ab and a , with the conjugation relation a(ab)a−1 = (ab)−1 . This group is sometimes called the inﬁnite dihedral group.

For a general free product ∗α Gα , each group Gα is naturally identiﬁed with a subgroup of ∗α Gα , the subgroup consisting of the empty word and the nonidentity one-letter words g ∈ Gα . From this viewpoint, the empty word is the common iden-
tity element of all the subgroups Gα , which are otherwise disjoint. A consequence of associativity is that any product g1 ··· gm of elements gi in the groups Gα has a unique reduced form, the element of ∗α Gα obtained by performing the multipli-
cations in any order. In fact, any sequence of reduction operations on an unreduced
product g1 ··· gm , combining adjacent letters gi and gi+1 that lie in the same Gα or
canceling a gi that is the identity, can be viewed as a way of inserting parentheses into g1 ··· gm and performing the resulting sequence of multiplications. Thus asso-
ciativity implies the nonobvious fact that any two sequences of reduction operations

performed on the same unreduced word always yield the same reduced word in the

end.

A basic property of the free product
phisms ϕα : Gα→H extends uniquely to

∗α Gα is that any a homomorphism

collection of homomor-
ϕ : ∗α Gα→H . Namely,

the value of ϕ on a word g1 ··· gn with gi ∈ Gαi must be ϕα1 (g1) ··· ϕαn (gn) , and

using this formula to deﬁne ϕ gives a well-deﬁned homomorphism since the process

of reducing an unreduced product in ∗α Gα does not affect its image under ϕ . For example, for a free product G ∗ H the inclusions G G× H and H G× H induce
a surjective homomorphism G ∗ H→G× H .

The van Kampen Theorem

Suppose a space X is decomposed as the union of a collection of path-connected

open subsets Aα , each of preceding paragraph, the sions Aα X extend to

ahwohhmoicmohmocmoonroptrahpiinhssmistmshejΦαb::aπ∗seα1p(πAo1iαn()At→αx)0π→∈1(πXX1).(XBiny)d.tuhTceheredevmbayanrthkKesamiinnpctlehune-

theorem will say that Φ is very often surjective, but we can expect Φ to have a nontriv-
→ ial kernel in general. For if iαβ : π1(Aα ∩ Aβ) π1(Aα) is the homomorphism induced
by the inclusion Aα ∩ Aβ Aα then jαiαβ = jβiβα , both these compositions being induced by the inclusion Aα ∩ Aβ X , so the kernel of Φ contains all the elements of the form iαβ(ω)iβα(ω)−1 for ω ∈ π1(Aα ∩ Aβ) . Van Kampen’s theorem asserts

that under fairly broad hypotheses this gives a full description of Φ :

Theorem 1.20. If X is the union of path-connected open sets Aα each containing → the basepoint x0 ∈ X and if each intersection Aα ∩ Aβ is path-connected, then
Φ : ∗α π1(Aα) π1(X) is surjective. If in addition each intersection Aα ∩ Aβ ∩ Aγ
is path-connected, then the kernel of Φ is the normal subgroup N generated by all elements of the form iαβ(ω)iβα(ω)−1 , and so Φ induces an isomorphism π1(X) ≈ ∗α π1(Aα)/N .

42 Chapter 1

The Fundamental Group

Example 1.21: Wedge Sums. In Chapter 0 we deﬁned the wedge sum α Xα of a
collection of spaces Xα with basepoints xα ∈ Xα to be the quotient space of the

disjoint union α Xα in which all the basepoints xα are identiﬁed to a single point.

If each xα is a deformation retract of an open neighborhood Uα in Xα , then Xα is a deformation retract of its open neighborhood Aα = Xα β≠α Uβ . The intersection

of two or Kampen’s

more distinct theorem then

Aα ’s is implies

α Uα , that Φ :

which deformation retracts to a point. Van
→ ∗α π1(Xα) π1( α Xα) is an isomorphism.

Thus for a wedge sum α Sα1 of circles, π1( α Sα1) is a free group, the free product of copies of Z , one for each circle Sα1 . In particular, π1(S1 ∨S1) is the free group Z∗Z ,

as in the example at the beginning of this section.

It is true more generally that the fundamental group of any connected graph is free, as we show in §1.A. Here is an example illustrating the general technique.

Example 1.22. Let X be the graph shown in the ﬁgure, consist-
ing of the twelve edges of a cube. The seven heavily shaded edges form a maximal tree T ⊂ X , a contractible subgraph containing all
the vertices of X . We claim that π1(X) is the free product of ﬁve copies of Z , one for each edge not in T . To deduce this from van Kampen’s theorem, choose for each edge eα of X − T an open neighborhood Aα of T ∪ eα in X that deformation retracts onto T ∪ eα . The intersection of two or more Aα ’s deformation retracts onto T , hence is contractible. The Aα ’s form a cover of X satisfying the hypotheses of van Kampen’s theorem, and since the intersection of any two of them is simply-connected we obtain an isomorphism π1(X) ≈ ∗α π1(Aα) . Each Aα deformation retracts onto a circle, so π1(X) is free on ﬁve generators, as claimed. As explicit generators we can choose for each edge eα of X − T a loop fα that starts at a basepoint in T , travels in T to one end of eα , then across eα , then back to the basepoint along a path in T .
Notice in this example that the graph is embedded in the plane with ﬁve bounded
complementary regions. The ﬁve boundary loops of these regions, when connected
to a common basepoint by paths, also form free generators for π1(X) . This can be shown by an inductive argument using van Kampen’s theorem.

Van Kampen’s theorem is often applied when there are just two sets Aα and Aβ in the cover of X , so the condition on triple intersections Aα ∩ Aβ ∩ Aγ is vacuous and one obtains an isomorphism π1(X) ≈ π1(Aα) ∗ π1(Aβ) /N , under the assumption that Aα ∩ Aβ is path-connected. The proof in this special case is virtually identical
with the proof in the general case, however.
One can see that the intersections Aα ∩ Aβ need to be path-connected by considering the example of S1 decomposed as the union of two open arcs. In this case
Φ is not surjective. For an example showing that triple intersections Aα ∩ Aβ ∩ Aγ
need to be path-connected, let X be the suspension of three points a , b , c , and let

Van Kampen’s Theorem

Section 1.2 43

Aα, Aβ , and Aγ be the complements of these three points. The theo-

rem does apply to the covering {Aα, Aβ} , so there are isomorphisms π1(X) ≈ π1(Aα) ∗ π1(Aβ) ≈ Z ∗ Z since Aα ∩ Aβ is contractible.

a

If we tried to use the covering {Aα, Aβ, Aγ } , which has each of the

bc

twofold intersections path-connected but not the triple intersection, then we would

get π1(X) ≈ Z ∗ Z ∗ Z , but this is not isomorphic to Z ∗ Z since it has a different

abelianization.

Proof of van Kampen’s theorem: First we show Φ is surjective. Given a loop f : I→X
at the basepoint x0 , we claim there is a partition 0 = s0 < s1 < ··· < sm = 1 of I
such that each subinterval [si−1, si] is mapped by f to a single Aα . Namely, since f is continuous, each s ∈ I has an open neighborhood Vs in I mapped by f to some
Aα . We may in fact take Vs to be an interval whose closure is mapped to a single
Aα . Compactness of I implies that a ﬁnite number of these intervals cover I . The
endpoints of this ﬁnite set of intervals then deﬁne the desired partition of I .

Denote the Aα containing f ([si−1, si]) by Ai , and let fi be the path obtained

by restricting f to [si−1, si] . Then f is the composition f1 ··· fm with fi a path

in Ai . Since Ai ∩ Ai+1 is path-connected, we may

choose a path gi in Ai ∩ Ai+1 from x0 to the point f (si) ∈ Ai ∩ Ai+1 . Consider the loop

g1

f1

(f1 g1) (g1 f2 g2) (g2 f3 g3) ··· (gm−1 fm)
which is homotopic to f . This loop is a composition of loops each lying in a single Ai , the loops indicated

f2 Aα

x0 g2

f3 Aβ

by the parentheses. Hence [f ] is in the image of Φ , and Φ is surjective.

The harder part of the proof is to show that the kernel of Φ is N . It may clarify matters to introduce some terminology. By a factorization of an element [f ] ∈ π1(X) we shall mean a formal product [f1] ··· [fk] where:
— Each fi is a loop in some Aα at the basepoint x0 , and [fi] ∈ π1(Aα) is the
homotopy class of fi . — The loop f is homotopic to f1 ··· fk in X .
A factorization of [f ] is thus a word in ∗α π1(Aα) , possibly unreduced, that is mapped to [f ] by Φ . The proof of surjectivity of Φ showed that every [f ] ∈ π1(X)
has a factorization.

We will be concerned now with the uniqueness of factorizations. Call two factor-

izations of [f ] equivalent if they are related by a sequence of the following two sorts

of moves or their inverses:

— Combine adjacent terms [fi][fi+1] into a single term [fi fi+1] if [fi] and [fi+1]
lie in the same group π1(Aα) . — Regard the term [fi] ∈ π1(Aα) as lying in the group π1(Aα ) rather than π1(Aα)
if fi is a loop in Aα ∩ Aα .

44 Chapter 1

The Fundamental Group

The ﬁrst move does not change the element of ∗α π1(Aα) deﬁned by the factorization. The second move does not change the image of this element in the quotient group Q = ∗α π1(Aα)/N , by the deﬁnition of N . So equivalent factorizations give the same element of Q .

If we can show that any two factorizations of [f ] are equivalent, this will say that
the map Q→π1(X) induced by Φ is injective, hence the kernel of Φ is exactly N , and
the proof will be complete.

Let [f1] ··· [fk] and paths f1 ··· fk and f1

[f1] ··· [f ] be two factorizations ··· f are then homotopic, so let

of [f ] . The composed
F : I × I→X be a homo-

topy from f1 ··· fk to f1 ··· f . There exist partitions 0 = s0 < s1 < ··· < sm = 1

and 0 = t0 < t1 < ··· < tn = 1 such that each rectangle Rij = [si−1, si]× [tj−1, tj]

is mapped by F into a single Aα , which we label Aij . These partitions may be obtained by covering I × I by ﬁnitely many rectangles [a, b]× [c, d] each mapping to a

single Aα , using a compactness argument, then partitioning I × I by the union of all

the horizontal and vertical lines containing edges of these rectangles. We may assume

the s partition subdivides the partitions giving the products f1 ··· fk and f1 ··· f . Since F maps a neighborhood of Rij to Aij , we may perturb the vertical sides of the rectangles Rij so that each point of I × I lies in at most three Rij ’s. We may assume there are at least three rows of rectangles, so we can do this perturbation just on the rectangles

9 10 11 12 5 6 78 1 23 4

in the intermediate rows, leaving the top and bottom rows unchanged. Let us relabel the new rectangles R1, R2, ··· , Rmn as in the ﬁgure.
If γ is a path in I × I from the left edge to the right edge, then the restriction F || γ is a loop at the basepoint x0 since F maps both the left and right edges of I × I to x0 . Let γr be the path separating the ﬁrst r rectangles R1, ··· , Rr from the remaining rectangles. Thus γ0 is the bottom edge of I × I and γmn is the top edge. We pass
from γr to γr +1 by pushing across the rectangle Rr +1 .
Let us call the corners of the Rr ’s vertices. For each vertex v with F (v) ≠ x0 , let
gv be a path from x0 to F (v) . We can choose gv to lie in the intersection of the two or
three Aij ’s corresponding to the Rr ’s containing v since we assume the intersection of any two or three Aij ’s is path-connected. If we insert into F || γr the appropriate
paths gv gv at successive vertices, as in the proof of surjectivity of Φ , then we obtain a factorization of [F || γr ] by regarding the loop corresponding to a horizontal or vertical
segment between adjacent vertices as lying in the Aij for either of the Rs ’s containing
the segment. Different choices of these containing Rs ’s change the factorization of [F || γr ] to an equivalent factorization. Furthermore, the factorizations associated to
successive paths γr and γr +1 are equivalent since pushing γr across Rr +1 to γr +1 changes F || γr to F || γr +1 by a homotopy within the Aij corresponding to Rr +1 , and
we can choose this Aij for all the segments of γr and γr +1 in Rr +1 .

Van Kampen’s Theorem

Section 1.2 45

We can arrange that the factorization associated to γ0 is equivalent to the factorization [f1] ··· [fk] by choosing the path gv for each vertex v along the lower edge of I × I to lie not just in the two Aij ’s corresponding to the Rs ’s containing v , but also
to lie in the Aα for the fi containing v in its domain. In case v is the common endpoint of the domains of two consecutive fi ’s we have F (v) = x0 , so there is no need
to choose a gv . In similar fashion we may assume that the factorization associated to the ﬁnal γmn is equivalent to [f1] ··· [f ] . Since the factorizations associated to all the γr ’s are equivalent, we conclude that the factorizations [f1] ··· [fk] and [f1] ··· [f ] are equivalent.

Example 1.23: Linking of Circles. We can apply van Kampen’s theorem to calculate

the fundamental groups of three spaces discussed in the introduction to this chapter, the complements in R3 of a single circle, two unlinked circles, and two linked circles.

The complement R3 −A of a single circle A

deformation retracts onto a wedge sum S1 ∨ S2

embedded in R3 − A as shown in the ﬁrst of A

A

the two ﬁgures at the right. It may be easier

to see that R3 − A deformation retracts onto

the the union of S2 with a diameter, as in the

second ﬁgure, where points outside S2 deformation retract onto S2 , and points inside

S2 and not in A can be pushed away from A toward S2 or the diameter. Having

this deformation retraction in mind, one can then see how it must be modiﬁed if

the two endpoints of the diameter are gradually moved toward each other along the equator until they coincide, forming the S1 summand of S1 ∨ S2 . Another way of seeing the deformation retraction of R3 − A onto S1 ∨ S2 is to note ﬁrst that an open ε neighborhood of S1 ∨ S2 obviously deformation retracts onto S1 ∨ S2 if ε is sufﬁciently small. Then observe that this neighborhood is homeomorphic to R3 − A by a homeomorphism that is the identity on S1 ∨ S2 . In fact, the neighborhood can be gradually enlarged by homeomorphisms until it becomes all of R3 − A .

In any event, once we see that R3 − A deformation retracts to S1 ∨ S2 , then we immediately obtain isomorphisms π1(R3 − A) ≈ π1(S1 ∨ S2) ≈ Z since π1(S2) = 0 .

In similar fashion, the complement R3 − (A ∪ B)

of two unlinked circles A and B deformation retracts

onto S1∨S1∨S2∨S2 , as in the ﬁgure to the right. From

A

B

this we get π1 R3 − (A ∪ B) ≈ Z ∗ Z . On the other hand, if A

and B are linked, then R3 − (A ∪ B) deformation retracts onto

the wedge sum of S2 and a torus S1 × S1 separating A and B ,

as shown in the ﬁgure to the left, hence π1 R3 − (A ∪ B) ≈ π1(S1 × S1) ≈ Z× Z .

46 Chapter 1

The Fundamental Group

Example 1.24: Torus Knots. For relatively prime positive integers m and n , the
→ torus knot K = Km,n ⊂ R3 is the image of the embedding f : S1 S1 × S1 ⊂ R3 ,
f (z) = (zm, zn) , where the torus S1 × S1 is embedded in R3 in the standard way.

The knot K winds around the torus a total of m

times in the longitudinal direction and n times in

the meridional direction, as shown in the ﬁgure for the cases (m, n) = (2, 3) and (3, 4) . One needs to

assume that m and n are relatively prime in order

for the map f to be injective. Without this assumption f would be d –to–1 where

d is the greatest common divisor of m and n , and the image of f would be the

knot Km/d,n/d . One could also allow negative values for m or n , but this would only
change K to a mirror-image knot.
Let us compute π1(R3 − K) . It is slightly easier to do the calculation with R3 replaced by its one-point compactiﬁcation S3 . An application of van Kampen’s theorem shows that this does not affect π1 . Namely, write S3 − K as the union of R3 − K and
an open ball B formed by the compactiﬁcation point together with the complement of a large closed ball in R3 containing K . Both B and B ∩ (R3 − K) are simply-connected, the latter space being homeomorphic to S2 × R . Hence van Kampen’s theorem implies that the inclusion R3 − K S3 − K induces an isomorphism on π1 .
We compute π1(S3 − K) by showing that it deformation retracts onto a 2 dimensional complex X = Xm,n homeomorphic to the quotient space of a cylinder S1 × I under the identiﬁcations (z, 0) ∼ (e2πi/mz, 0) and (z, 1) ∼ (e2πi/nz, 1) . If we let Xm and Xn be the two halves of X formed by the quotients of S1 × [0, 1/2] and S1 × [1/2, 1], then Xm and Xn are the mapping cylinders of z zm and z zn . The intersection Xm ∩ Xn is the circle S1 × {1/2}, the domain end of each mapping cylinder.
To obtain an embedding of X in S3 − K as a deformation retract we will use the standard decomposition of S3 into two solid tori S1 × D2 and D2 × S1 , the result of writing S3 = ∂D4 = ∂(D2 × D2) = ∂D2 × D2 ∪ D2 × ∂D2 . Geometrically, the ﬁrst solid torus S1 × D2 can be identiﬁed with the compact region in R3 bounded by the standard torus S1 × S1 containing K , and the second solid torus D2 × S1 is then the

closure of the complement of the ﬁrst solid torus, together with the compactiﬁcation point at inﬁnity. Notice that meridional circles in S1 × S1 bound disks in the ﬁrst solid

torus, while it is longitudinal circles that bound disks in the second solid torus.

In the ﬁrst solid torus, K intersects each of the meridian

K

circles {x}× ∂D2 in m equally spaced points, as indicated in

the ﬁgure, which shows a meridian disk {x}× D2 . These m

points can be separated by a union of m radial line segments.

Letting x vary, these radial segments then trace out a copy of K

K

the mapping cylinder Xm in the ﬁrst solid torus. Symmetrically,

Van Kampen’s Theorem

Section 1.2 47

there is a copy of the other mapping cylinder Xn in the second solid torus. The complement of K in the ﬁrst solid torus deformation retracts onto Xm by ﬂowing within each meridian disk as shown. In similar fashion the complement of K in the
second solid torus deformation retracts onto Xn . These two deformation retractions do not agree on their common domain of deﬁnition S1 × S1 − K , but this is easy to correct by distorting the ﬂows in the two solid tori so that in S1 × S1 − K both ﬂows
are orthogonal to K . After this modiﬁcation we now have a well-deﬁned deformation retraction of S3 − K onto X . Another way of describing the situation would be to
say that for an open ε neighborhood N of K bounded by a torus T , the complement
S3 − N is the mapping cylinder of a map T →X .
To compute π1(X) we apply van Kampen’s theorem to the decomposition of X as the union of Xm and Xn , or more properly, open neighborhoods of these two sets that deformation retract onto them. Both Xm and Xn are mapping cylinders that deformation retract onto circles, and Xm ∩ Xn is a circle, so all three of these spaces have fundamental group Z . A loop in Xm ∩ Xn representing a generator of π1(Xm ∩ Xn) is homotopic in Xm to a loop representing m times a generator, and in Xn to a loop representing n times a generator. Van Kampen’s theorem then says that π1(X) is the quotient of the free group on generators a and b obtained by factoring out the normal subgroup generated by the element amb−n .
Let us denote by Gm,n this group π1(Xm,n) deﬁned by two generators a and b and one relation am = bn . If m or n is 1 , then Gm,n is inﬁnite cyclic since in these cases the relation just expresses one generator as a power of the other. To
describe the structure of Gm,n when m, n > 1 let us ﬁrst compute the center of Gm,n , the subgroup consisting of elements that commute with all elements of Gm,n . The element am = bn commutes with a and b , so the cyclic subgroup C generated
by this element lies in the center. In particular, C is a normal subgroup, so we can pass to the quotient group Gm,n/C , which is the free product Zm ∗ Zn . According to Exercise 1 at the end of this section, a free product of nontrivial groups has trivial
center. From this it follows that C is exactly the center of Gm,n . As we will see in Example 1.44, the elements a and b have inﬁnite order in Gm,n , so C is inﬁnite cyclic.
We will show now that the integers m and n are uniquely determined by the group Zm ∗ Zn , hence also by Gm,n . The abelianization of Zm ∗ Zn is Zm × Zn , of order mn , so the product mn is uniquely determined by Zm ∗ Zn . To determine m and n individually, we use another assertion from Exercise 1 at the end of the section, that all torsion elements of Zm ∗ Zn are conjugate to elements of the subgroups Zm and Zn , hence have order dividing m or n . Thus the maximum order of torsion elements of Zm ∗ Zn is the larger of m and n . The larger of these two numbers is therefore uniquely determined by the group Zm ∗ Zn , hence also the smaller since the product is uniquely determined.
The preceding analysis of π1(Xm,n) did not need the assumption that m and n

48 Chapter 1

The Fundamental Group

are relatively prime, which was used only to relate Xm,n to torus knots. An interesting fact is that Xm,n can be embedded in R3 only when m and n are relatively prime. This is shown in the remarks following Corollary 3.45. For example, X2,2 is the Klein bottle since it is the union of two copies of the M¨obius band X2 with their boundary circles identiﬁed, so this nonembeddability statement generalizes the fact that the Klein bottle cannot be embedded in R3 .
There exist algorithms to compute π1(R3 − K) for an arbitrary smooth or piecewise linear knot K , but the problem of determining when two of these fundamental
groups are isomorphic is generally much more difﬁcult than in the special case of
torus knots. A large part of knot theory is devoted to this problem; see [Rolfsen 1976]
for an introduction to this subject.

Example 1.25: The Shrinking Wedge of Circles. Consider the sub-
space X ⊂ R2 that is the union of the circles Cn of radius 1/n and center (1/n, 0) for n = 1, 2, ··· . At ﬁrst glance one might confuse
X with the wedge sum of an inﬁnite sequence of circles, but we will

show that X has a much larger fundamental group than the wedge
Etshuaecmhb. arCsnoenpinsoididnuetc.reTtshheaerspeurtrrojadeccutticiotonnosfρrtnnhe:: πXρ1→n(X’sC)in→s caπohl1lo(aCmpnsoi)mn≈goarZpll,hCwisihm’sereρex:cwπeep1(ttXaCk)ne→ttohet∞hoZeriogtroiingtihanes.
direct product (not the direct sum) of inﬁnitely many copies of Z , and ρ is surjective
since for every sequence of integers kn we can construct a loop f : I→X that wraps
kn times around Cn in the time interval [1 − 1/n, 1 − 1/n+1]. This inﬁnite composition
of loops is certainly continuous at each time less than 1 , and it is continuous at time

1 since every neighborhood of the basepoint in X contains all but ﬁnitely many of the

circles Cn . Since π1(X) maps onto the uncountable group ∞ Z , it is uncountable. On the other hand, the fundamental group of a wedge sum of countably many circles

is countably generated, hence countable.

it

is

The group π1(X) nonabelian, since

is actually far more complicated than ∞ Z . For
the retraction X→C1 ∪ ··· ∪ Cn that collapses all

one thing, the circles

smaller than Cn to the basepoint induces a surjection from π1(X) to a free group on

n generators. For a complete description of π1(X) see [Cannon & Conner 2000].

A theorem in [Shelah 1988] asserts that for a compact metric space that is path-

connected and locally path-connected, the fundamental group is either ﬁnitely gener-

ated or uncountable.

Applications to Cell Complexes
For the remainder of this section we shall be interested in 2 dimensional cell complexes, analyzing how the fundamental group is affected by attaching 2 cells. According to an exercise at the end of this section, attaching cells of higher dimension has no effect on π1 , so all the interest lies in how the 2 cells are attached.

Van Kampen’s Theorem

Section 1.2 49

taϕhtαaϕn: SSαSu1(1p→s→0p)oXtXsh,e.pawtrFoeowdraeutdtscaihifcnfahegllraaecncasoptlllaαlϕecc’esαtYi,tohe.nveIeofbnfsa02tsheiscopeuaolglibnshatesstα2eeϕpcthooαni(anisctp0aoa)lftloyhSf-l1cotohtonhepnessneeaclϕrtoeeoαdmpdssapeptϕaescrαeIm→mXinavXeyisanramaolttoahoapeplsrl
coincide. To remedy this, choose a basepoint x0 ∈ X and a path γα in X from x0 to

ϕα(s0) for each α . Then γαϕαγα is a loop at x0 . This loop may not be nullhomotopic

in X , but it will certainly be nullhomotopic after the cell eα2 is attached. Thus the

normal subgroup N ⊂ lies in the kernel of the

π1(X, x0) map π1(X

generated by
, x0)→π1(Y ,

all x0)

the loops γαϕαγα for varying induced by the inclusion X

α Y.

Proposition 1.26. The inclusion X Y induces a surjection → π1(X, x0) π1(Y , x0)
whose kernel is N . Thus π1(Y ) ≈ π1(X)/N .

It follows that N is independent of the choice of the paths γα , but this can also be
seen directly: If we replace γα by another path ηα having the same endpoints, then γαϕαγα changes to ηαϕαηα = (ηαγα)γαϕαγα(γαηα) , so γαϕαγα and ηαϕαηα
deﬁne conjugate elements of π1(X, x0) .

Proof: Let us expand Y to a slightly larger space Z that deformation retracts onto Y

and is more convenient for applying van Kampen’s theorem. The space Z is obtained

from Y by attaching rectangular strips Sα = I × I , with the lower edge I × {0} attached

along γα , the right edge {1}× I attached

along an arc in eα2 , and all the left edges

yα

{0}× I of the different strips identiﬁed

together. The top edges of the strips are

X

eα2 Sα

not attached to anything, and this allows

x0

γα

us to deformation retract Z onto Y .

In each cell eα2 choose a point yα not in the arc along which Sα is attached. Let A = Z − α{yα} and let B = Z − X . Then A deformation retracts onto X , and B is contractible. Since π1(B) = 0 , van Kampen’s theorem applied to the cover {A, B} says

bthyatthπe 1im(Za)giesoisfotmheomrpahpicπto1(tAhe∩qBu)o→tieπn1t(oAf)π. 1S(oAit)

by the normal subgroup remains only to see that

generated π1(A ∩ B)

is generated by the loops γαϕαγα , or rather by loops in A ∩ B homotopic to these

loops. This can be shown by another application of van Kampen’s theorem, this time

to the cover of A ∩ B by the open sets Aα = A ∩ B − β≠α eβ2 . Since Aα deformation retracts onto a circle in eα2 −{yα} , we have π1(Aα) ≈ Z generated by a loop homotopic

to γαϕαγα , and the result follows.

As a ﬁrst application we compute the fundamental group of the orientable surface Mg of genus g . This has a cell structure with one 0 cell, 2g 1 cells, and one 2 cell, as we saw in Chapter 0. The 1 skeleton is a wedge sum of 2g circles, with fundamental group free on 2g generators. The 2 cell is attached along the loop given by the

50 Chapter 1

The Fundamental Group

product of the commutators of these generators, say [a1, b1] ··· [ag, bg] . Therefore
π1(Mg) ≈ a1, b1, ··· , ag, bg || [a1, b1] ··· [ag, bg]
where gα || rβ denotes the group with generators gα and relators rβ , in other words, the free group on the generators gα modulo the normal subgroup generated by the words rβ in these generators.
Corollary 1.27. The surface Mg is not homeomorphic, or even homotopy equivalent,
to Mh if g ≠ h .
Proof: The abelianization of π1(Mg) is the direct sum of 2g copies of Z . So if
Mg Mh then π1(Mg) ≈ π1(Mh) , hence the abelianizations of these groups are isomorphic, which implies g = h .

Nonorientable surfaces can be treated in the same way. If we attach a 2 cell to the
wedge sum of g circles by the word a21 ··· a2g we obtain a nonorientable surface Ng . For example, N1 is the projective plane RP2 , the quotient of D2 with antipodal points of ∂D2 identiﬁed. And N2 is the Klein bottle, though the more usual representation

−N−1−−−R−P−2 :

a

a
−−N−−2 :

b

c

b

c

c

b

a

a

a

a

of the Klein bottle is as a square with opposite sides identiﬁed via the word aba−1b .

If one cuts the square along a diagonal and reassembles the resulting two triangles

as shown in the ﬁgure, one obtains the other representation as a square with sides identiﬁed via the word a2c2 . By the proposition, π1(Ng) ≈ a1, ··· , ag || a21 ··· a2g . This abelianizes to the direct sum of Z2 with g − 1 copies of Z since in the abelianization we can rechoose the generators to be a1, ··· , ag−1 and a1 + ··· + ag , with 2(a1 + ··· + ag) = 0 . Hence Ng is not homotopy equivalent to Nh if g ≠ h , nor is
Ng homotopy equivalent to any orientable surface Mh .

Here is another application of the preceding proposition:

Corollary 1.28. For every group G there is a 2 dimensional cell complex XG with
π1(XG) ≈ G .
Proof: Choose a presentation G = gα || rβ . This exists since every group is a
quotient of a free group, so the gα ’s can be taken to be the generators of this free group with the rβ ’s generators of the kernel of the map from the free group to G . Now construct XG from α Sα1 by attaching 2 cells eβ2 by the loops speciﬁed by the words rβ .

Van Kampen’s Theorem

Section 1.2 51

Example 1.29. If G = a || an = Zn then XG is S1 with a cell e2 attached by the
map z zn , thinking of S1 as the unit circle in C . When n = 2 we get XG = RP2 , but for n > 2 the space XG is not a surface since there are n ‘sheets’ of e2 attached at each point of the circle S1 ⊂ XG . For example, when n = 3 one can construct a neighborhood N of S1 in XG by taking the product of the letter ‘ Y ’ with the interval I ,
and then identifying the two ends of this product
via a one-third twist as shown in the ﬁgure. The
boundary of N consists of a single circle, formed
by the three endpoints of each ‘ Y ’ cross section of N . To complete the construction
of XG from N one attaches a disk along the boundary circle of N . This cannot be done in R3 , though it can in R4 . For n = 4 one would use the letter ‘ X ’ instead of
‘ Y ,’ with a one-quarter twist instead of a one-third twist. For larger n one would use an n pointed ‘asterisk’ and a 1/n twist.

Exercises
1. Show that the free product G ∗ H of nontrivial groups G and H has trivial center, and that the only elements of G ∗ H of ﬁnite order are the conjugates of ﬁnite-order elements of G and H .
2. Let X ⊂ Rm be the union of convex open sets X1, ··· , Xn such that Xi ∩Xj ∩Xk ≠ ∅ for all i, j, k . Show that X is simply-connected.
3. Show that the complement of a ﬁnite set of points in Rn is simply-connected if n ≥ 3.
4. Let X ⊂ R3 be the union of n lines through the origin. Compute π1(R3 − X) . 5. Let X ⊂ R2 be a ﬁnite graph that is the union of the edges of a convex polygon and a ﬁnite number of line segments having endpoints on these edges. (a) Show that π1(X) is free with a basis consisting of loops formed by the boundaries
of the bounded complementary regions of X , joined to a basepoint by paths in X . (b) Show this is true for all choices of paths to the basepoint.
6. Suppose a space Y is obtained from a path-connected subspace X by attaching n cells for a ﬁxed n ≥ 3 . Show that the inclusion X Y induces an isomorphism on π1 . [See the proof of Proposition 1.26.] Apply this to show that the complement of a discrete subspace of Rn is simply-connected if n ≥ 3 .
7. Let X be the quotient space of S2 obtained by identifying the north and south poles to a single point. Put a cell complex structure on X and use this to compute π1(X) .

52 Chapter 1

The Fundamental Group

8. Compute the fundamental group of the space obtained from two tori S1 × S1 by identifying a circle S1 × {x0} in one torus with the corresponding circle S1 × {x0} in the other torus.

9. In the surface Mg of genus g , let

C be a circle that separates Mg into

two compact subsurfaces Mh and Mk

C

obtained from the closed surfaces Mh and Mk by deleting an open disk from

Mh

C

Mk

each. Show that Mh does not retract onto its boundary circle C , and hence Mg does

not retract onto C . [Hint: abelianize π1 .] But show that Mg does retract onto the

nonseparating circle C in the ﬁgure.

10. Consider two arcs α and β embedded in D2 × I as shown in the ﬁgure. The loop γ is obviously nullhomotopic

α

β

in D2 × I , but show that there is no nullhomotopy of γ in

the complement of α ∪ β .

γ

11. The mapping torus Tf of a map f : X→X is the quotient of X × I obtained
by identifying each point (x, 0) with (f (x), 1) . In the case X = S1 ∨ S1 with f

basepoint-preserving, compute a
map → f∗ : π1(X) π1(X) . Do the

presentation for π1(Tf ) same when X = S1 × S1 .

in terms [One way

of to

the induced do this is to

regard Tf as built from X ∨ S1 by attaching cells.]

12. The Klein bottle is usually pictured as a subspace of R3 like the subspace X ⊂ R3 shown in

the ﬁrst ﬁgure at the right. If one wanted a model

that could actually function as a bottle, one would

delete the open disk bounded by the circle of selfintersection of X , producing a subspace Y ⊂ X . Show that π1(X) ≈ Z ∗ Z and that π1(Y ) has the presentation a, b, c || aba−1b−1cbεc−1 for ε = ±1 . (Changing the
sign of ε gives an isomorphic group, as it happens.) Show also that π1(Y ) is isomorphic to π1(R3 −Z) for Z the graph shown in the ﬁgure. The groups π1(X) and π1(Y )
are not isomorphic, but this is not easy to prove; see the discussion in Example 1B.13.

13. The space Y in the preceding exercise can be obtained from a disk with two holes by identifying its three boundary circles. There are only two essentially different ways of identifying the three boundary circles. Show that the other way yields a space Z with π1(Z) not isomorphic to π1(Y ) . [Abelianize the fundamental groups to show they are not isomorphic.] 14. Consider the quotient space of a cube I3 obtained by identifying each square face with the opposite square face via the right-handed screw motion consisting of a translation by one unit in the direction perpendicular to the face combined with a one-quarter twist of the face about its center point. Show this quotient space X is a

Van Kampen’s Theorem

Section 1.2 53

cell complex with two 0 cells, four 1 cells, three 2 cells, and one 3 cell. Using this structure, show that π1(X) is the quaternion group {±1, ±i, ±j, ±k} , of order eight. 15. Given a space X with basepoint x0 ∈ X , we may construct a CW complex L(X) having a single 0 cell, a 1 cell for each loop in X based at x0 , and a 2 cell for each map of a standard triangle T into X taking the three vertices to the basepoint. Such

a 2 cell is attached to the three 1 cells that are the loops obtained by restricting the

map to the three edges of T . Show that isomorphism induced by a natural map

π1 L(X

L(X )
)→X

.

is

isomorphic

to

π1(X, x0)

via

an

16. Show that the fundamental group of the surface of inﬁnite genus shown below is free on an inﬁnite number of generators.

17. Show that π1(R2 − Q2) is uncountable.
18. In this problem we use the notions of suspension, reduced suspension, cone, and
mapping cone deﬁned in Chapter 0. Let X be the subspace of R consisting of the sequence 1, 1/2, 1/3, 1/4, ··· together with its limit point 0 . (a) For the suspension SX , show that π1(SX) is free on a countably inﬁnite set of
generators, and deduce that π1(SX) is countable. In contrast to this, the reduced suspension ΣX , obtained from SX by collapsing the segment {0}× I to a point, is
the shrinking wedge of circles in Example 1.25, with an uncountable fundamental
group.
(b) Let C be the mapping cone of the quotient map SX→ΣX . Show that π1(C) is un-
countable by constructing a homomorphism from π1(C) onto ∞ Z/ ∞ Z . Note that C is the reduced suspension of the cone CX . Thus the reduced suspension
of a contractible space need not be contractible, unlike the unreduced suspension.
19. Show that the subspace of R3 that is the union of the spheres of radius 1/n and center (1/n, 0, 0) for n = 1, 2, ··· is simply-connected. 20. Let X be the subspace of R2 that is the union of the circles Cn of radius n and center (n, 0) for n = 1, 2, ··· . Show that π1(X) is the free group ∗n π1(Cn) , the same as for the inﬁnite wedge sum ∞ S1 . Show that X and ∞ S1 are in fact homotopy equivalent, but not homeomorphic.
21. Show that the join X ∗ Y of two nonempty spaces X and Y is simply-connected
if X is path-connected.

54 Chapter 1

The Fundamental Group

We come now to the second main topic of this chapter, covering spaces. We have in fact already encountered one example of a covering space in our calculation
of π1(S1) . This was the map R→S1 that we pictured as the projection of a helix
onto a circle, with the helix lying above the circle, ‘covering’ it. A number of things we proved for this covering space are valid for all covering spaces, and this allows covering spaces to serve as a useful general tool for calculating fundamental groups. But the connection between the fundamental group and covering spaces runs much deeper than this, and in many ways they can be regarded as two viewpoints toward the same thing. This means that algebraic features of the fundamental group can often be translated into the geometric language of covering spaces. This is exempliﬁed in one of the main results in this section, giving an exact correspondence between the various connected covering spaces of a given space X and subgroups of π1(X) . This is strikingly reminiscent of Galois theory, with its correspondence between ﬁeld extensions and subgroups of the Galois group.

Let us begin with the deﬁnition. A covering space of a space X is a space X
together with a map p : X→X satisfying the following condition: There exists an
open cover {Uα} of X such that for each α , p−1(Uα) is a disjoint union of open sets
in X , each of which is mapped by p homeomorphically onto Uα . We do not require p−1(Uα) to be nonempty, so p need not be surjective.
In the helix example one has p : R→S1 given by p(t) = (cos 2π t, sin 2π t) , and
the cover {Uα} can be taken to consist of any two open arcs whose union is S1 . A related example is the helicoid surface S ⊂ R3 consisting of points of the form
(s cos 2π t, s sin 2π t, t) for (s, t) ∈ (0, ∞)× R . This projects onto R2 − {0} via the
map (x, y, z) (x, y) , and this projection deﬁnes a covering space p : S→R2 − {0}
since for each open disk U in R2 − {0} , p−1(U ) consists of countably many disjoint

open disks in S , each mapped homeomorphically onto U by p .
Another example is the map p : S1→S1 , p(z) = zn where we
view z as a complex number with |z| = 1 and n is any positive

integer. The closest one can come to realizing this covering space

as a linear projection in 3 space analogous to the projection of the

helix is to draw a circle wrapping around a cylinder n times and

p

intersecting itself in n − 1 points that one has to imagine are not

really intersections. For an alternative picture without this defect, embed S1 in the boundary torus of a solid torus S1 × D2 so that it winds n times monotonically around the S1 factor without self-intersections, then restrict the pro-
jection → S1 × D2 S1 × {0} to this embedded circle. The ﬁgure for Example 1.29 in the
preceding section illustrates the case n = 3 .

Covering Spaces

Section 1.3 55

As our general theory will show, these examples for n ≥ 1 together with the helix example exhaust all the connected coverings spaces of S1 . There are many other disconnected covering spaces of S1 , such as n disjoint circles each mapped homeomorphically onto S1 , but these disconnected covering spaces are just disjoint
unions of connected ones. We will usually restrict our attention to connected covering
spaces as these contain most of the interesting features of covering spaces.

The covering spaces of S1 ∨ S1 form a remarkably rich family illustrating most of

the general theory very concretely, so let us look at a few of these covering spaces to get an idea of what is going on. To abbreviate notation, set X = S1 ∨ S1 . We view this

as a graph with one vertex and two edges. We label the edges

a and b and we choose orientations for a and b . Now let b

a

X be any other graph with four edges meeting at each vertex,

and suppose the edges of X have been assigned labels a and b and orientations in

such a way that the local picture near each vertex is the same as in X , so there is an

a edge oriented toward the vertex, an a edge oriented away from the vertex, a b edge

oriented toward the vertex, and a b edge oriented away from the vertex. To give a

name to this structure, let us call X a 2 oriented graph.

The table on the next page shows just a small sample of the inﬁnite variety of

possible examples.
Given a 2 oriented graph X we can construct a map p : X→X sending all vertices

of X to the vertex of X and sending each edge of X to the edge of X with the same

label by a map that is a homeomorphism on the interior of the edge and preserves

orientation. It is clear that the covering space condition is satisﬁed for p . The con-

verse is also true: Every covering space of X is a graph that inherits a 2 orientation

from X .

As the reader will discover by experimentation, it seems that every graph having

four edges incident at each vertex can be 2 oriented. This can be proved for ﬁnite

graphs as follows. A very classical and easily shown fact is that every ﬁnite connected

graph with an even number of edges incident at each vertex has an Eulerian circuit,

a loop traversing each edge exactly once. If there are four edges at each vertex, then

labeling the edges of an Eulerian circuit alternately a and b produces a labeling with

two a and two b edges at each vertex. The union of the a edges is then a collection

of disjoint circles, as is the union of the b edges. Choosing orientations for all these

circles gives a 2 orientation.

It is a theorem in graph theory that inﬁnite graphs with four edges incident at each

vertex can also be 2 oriented; see Chapter 13 of [Koenig 1990] for a proof. There is

also a generalization to n oriented graphs, which are covering spaces of the wedge

sum of n circles.

56 Chapter 1

The Fundamental Group

(1) a

Some Covering Spaces of S1 ∨ S1

b a
b a, b 2, bab 1

(2) aa

b b

a2, b 2, ab

( 3)

(4)

b

a

b

a

a

b

a

b

b

a

b

a

a2, b 2, aba 1, bab 1

a, b2, ba2b 1, baba 1b 1

(5)

a

(6) a

b bb

a3, b 3, ab 1, b 1a

a

a

b bb

a3, b 3, ab, ba

a

a

(7) a
a

a

b b

b b

a

a4, b4, ab , ba , a2b2

(8) a
b

ab ba

b a2, b 2, (ab) 2, (ba) 2, ab2a
a

(9) ab

b

a

ab a

b a2, b4, ab, ba2b 1, bab 2

( 10 )

a

a

a

bb bb b

a

a

a

b 2nab 2n 1, b 2n + 1ab 2n | n ∈ Z

( 11 )

a

a

a

( 12 )

a

b

b

b

b

bb

a

bnab n | n ∈ Z

( 13 ) b a

( 14 )

a

a

b

b

b

ab

a , bab 1

Covering Spaces

Section 1.3 57

A simply-connected covering space of X can be constructed in the following way. Start with the open intervals (−1, 1) in the coordinate axes of R2 . Next, for a ﬁxed number λ , 0 < λ < 1/2, for example λ = 1/3, adjoin four open segments of length 2λ , at distance λ from the ends of the previous seg-
ments and perpendicular to them, the new shorter seg-
ments being bisected by the older ones. For the third stage, add perpendicular open segments of length 2λ2 at distance λ2 from the endpoints of all the previous
segments and bisected by them. The process is now repeated indeﬁnitely, at the nth stage adding open segments of length 2λn−1 at distance λn−1 from all the previous endpoints. The union of all these open segments is
a graph, with vertices the intersection points of horizontal and vertical segments, and
edges the subsegments between adjacent vertices. We label all the horizontal edges

a , oriented to the right, and all the vertical edges b , oriented upward.

This covering space is called the universal cover of X because, as our general theory will show, it is a covering space of every other connected covering space of X .

The covering spaces (1)–(14) in the table are all nonsimply-connected. Their fundamental groups are free with bases represented by the loops speciﬁed by the listed

words in a and b , starting at the basepoint x0 indicated by the heavily shaded vertex. This can be proved in each case by applying van Kampen’s theorem. One can

also interpret the list of words as generators of the image subgroup p∗ π1(X, x0)

in π1(X, x0) = the induced map

a, b p∗ :

. A general fact
→ π1(X, x0) π1(X

we shall prove about covering spaces , x0) is always injective. Thus we have

is that the at-

ﬁrst-glance paradoxical fact that the free group on two generators can contain as a

subgroup a free group on any ﬁnite number of generators, or even on a countably

inﬁnite set of generators as in examples (10) and (11).

Changing the basepoint vertex changes the subgroup p∗ π1(X, x0) to a conjugate subgroup in π1(X, x0) . The conjugating element of π1(X, x0) is represented by any loop that is the projection of a path in X joining one basepoint to the other. For
example, the covering spaces (3) and (4) differ only in the choice of basepoints, and

the corresponding subgroups of π1(X, x0) differ by conjugation by b .

The main classiﬁcation theorem for covering spaces says that by associating the
subgroup p∗ π1(X, x0) to the covering space p : X→X , we obtain a one-to-one
correspondence between all the different connected covering spaces of X and the

conjugacy classes of subgroups of π1(X, x0) . If one keeps track of the basepoint

vertex p : (X,

xx00)→∈ (XX

, ,

then this is a one-to-one x0) and actual subgroups

correspondence between covering spaces of π1(X, x0) , not just conjugacy classes.

Of course, for these statements to make sense one has to have a precise notion of

when two covering spaces are the same, or ‘isomorphic.’ In the case at hand, an iso-

58 Chapter 1

The Fundamental Group

morphism between covering spaces of X is just a graph isomorphism that preserves the labeling and orientations of edges. Thus the covering spaces in (3) and (4) are isomorphic, but not by an isomorphism preserving basepoints, so the two subgroups of π1(X, x0) corresponding to these covering spaces are distinct but conjugate. On the other hand, the two covering spaces in (5) and (6) are not isomorphic, though the graphs are homeomorphic, so the corresponding subgroups of π1(X, x0) are isomorphic but not conjugate.
Some of the covering spaces (1)–(14) are more symmetric than others, where by a ‘symmetry’ we mean an automorphism of the graph preserving the labeling and orientations. The most symmetric covering spaces are those having symmetries taking any one vertex onto any other. The examples (1), (2), (5)–(8), and (11) are the ones with this property. We shall see that a covering space of X has maximal symmetry exactly when the corresponding subgroup of π1(X, x0) is a normal subgroup, and in this case the symmetries form a group isomorphic to the quotient group of π1(X, x0) by the normal subgroup. Since every group generated by two elements is a quotient group of Z ∗ Z , this implies that every two-generator group is the symmetry group of some covering space of X .

Lifting Properties
Covering spaces are deﬁned in fairly geometric terms, as maps p : X→X that are
local homeomorphisms in a rather strong sense. But from the viewpoint of algebraic topology, the distinctive feature of covering spaces is their behavior with respect to lifting of maps. Recall the terminology from the proof of Theorem 1.7: A lift of a map
f : Y →X is a map f : Y →X such that pf = f . We will describe three special lifting
properties of covering spaces, and derive a few applications of these. First we have the homotopy lifting property, or covering homotopy property,
as it is sometimes called:

Proposition 1.30. Given a covering map f0 : Y →X lifting f0 , then there

space exists

p : X→X , a homotopy
a unique homotopy ft :

Yft→: YX→oXf

, and a f0 that

lifts ft .

Proof: For the covering space p : R→S1 this is property (c) in the proof of Theo-

rem 1.7, and the proof there applies to any covering space.

Taking Y to be a point gives the path lifting property for a covering space

p : X→X , which says that for each path f : I→X and point f (0) = x0 there is a unique path f : I→X lifting f

each lift x0 of the starting starting at x0 . In particular,

the uniqueness of lifts implies that every lift of a constant path is constant, but this

could be deduced more simply from the fact that p−1(x0) has the discrete topology,

by the deﬁnition of a covering space.

Covering Spaces

Section 1.3 59

Taking Y to be I , we see that every homotopy ft of a path f0 in X lifts to a homotopy ft of each lift f0 of f0 . The lifted homotopy ft is a homotopy of paths, ﬁxing the endpoints, since as t varies each endpoint of ft traces out a path lifting a constant path, which must therefore be constant.

Here is a simple application:

Proposition 1.31. The p : (X, x0)→(X, x0) is

map p∗ : injective.

→ π1(X, x0) π1(X, x0)
The image subgroup

induced by a covering space p∗ π1(X, x0) in π1(X, x0)

consists of the homotopy classes of loops in X based at x0 whose lifts to X starting

at x0 are loops.

Proof: An element
homotopy ft : I→X

of of

the kernel of p∗ is represented by a loop f0 f0 = pf0 to the trivial loop f1 . By the remarks

: I→X with a
preceding the

proposition, there is a lifted homotopy of loops ft starting with f0 and ending with a constant loop. Hence [f0] = 0 in π1(X, x0) and p∗ is injective.

For the second certainly represent

eslteamteemntesnot fotfhtehiempargoepoofsipti∗on: π, l1o(oXp,sx0a)t→x0π1li(fXtin, xg0t)o.

loops at x0 Conversely,

a loop representing an element of the image of p∗ is homotopic to a loop having such

a lift, so by homotopy lifting, the loop itself must have such a lift.

If p : X→X is a covering space, then the cardinality of the set p−1(x) is locally
constant over X . Hence if X is connected, this cardinality is constant as x ranges over all of X . It is called the number of sheets of the covering.

Proposition 1.32. The number of sheets of a covering space p : (X, x0)→(X, x0)
with X and X path-connected equals the index of p∗ π1(X, x0) in π1(X, x0) .
Proof: For a loop g in X based at x0 , let g be its lift to X starting at x0 . A product
h g with [h] ∈ H = p∗ π1(X, x0) has the lift h g ending at the same point as g since h is a loop. Thus we may deﬁne a function Φ from cosets H[g] to p−1(x0) by sending H[g] to g(1) . The path-connectedness of X implies that Φ is surjective since x0 can be joined to any point in p−1(x0) by a path g projecting to a loop g at x0 . To see that Φ is injective, observe that Φ(H[g1]) = Φ(H[g2]) implies that g1g2 lifts to a loop in X based at x0 , so [g1][g2]−1 ∈ H and hence H[g1] = H[g2] .

It is important also to know about the existence and uniqueness of lifts of general maps, not just lifts of homotopies. For the existence question an answer is provided by the following lifting criterion:

ffPr:: ((oYYp,,oyysi00t))i→→on((1XX.3,, xx3.00))SuowpfitpfhoseYexigspitavsteihnff-cafon∗conπveec1rt(eiYndg, aysn0pd)acl⊂oecappl∗l:y(Xπp1,ax(tXh0-),c→xo0n)(nXe.c, xte0d).

and a Then

map a lift

60 Chapter 1

The Fundamental Group

When we say a space has a certain property locally, such as being locally pathconnected, we shall mean that each point has arbitrarily small open neighborhoods with this property. Thus for Y to be locally path-connected means that for each point y ∈ Y and each neighborhood U of y there is an open neighborhood V ⊂ U of y that is path-connected. Some authors weaken the requirement that V be pathconnected to the condition that any two points in V be joinable by a path in U . This broader deﬁnition would work just as well for our purposes, necessitating only small adjustments in the proofs, but for simplicity we shall use the more restrictive deﬁnition.

Proof: The ‘only if’ statement is obvious since f∗ = p∗f∗ . For the converse, let
y ∈ Y and let γ be a path in Y from y0 to y . The path f γ in X starting at x0 has a unique lift f γ starting at x0 . Deﬁne f (y) = f γ(1) . To show this is well-

deﬁned, independent of the choice of γ , let γ be another path from y0 to y . Then (f γ ) (f γ) is a loop h0 at x0 with [h0] ∈ f∗ π1(Y , y0) ⊂ p∗ π1(X, x0) . This

means there is a homotopy ht of h0 to a loop h1 that lifts to a loop h1 in X based at x0 . Apply the covering homotopy

fγ

f (y )

property to ht to get a lifting ht . Since h1 is a loop at x0 , so is h0 . By the uniqueness of lifted paths,

x0

fγ

the ﬁrst half of h0 is f γ and the second

f

p

half is f γ traversed backwards, with

the common midpoint f γ(1) =

γ

f γ (1) . This shows that f is

y0

γ

well-deﬁned.

fγ f y

x0

fγ

f (y )

To see that f is continuous, let U ⊂ X be an open neighborhood of f (y) having
a lift U ⊂ X containing f (y) such that p : U→U is a homeomorphism. Choose a

path-connected open neighborhood V of y with f (V ) ⊂ U . For paths from y0 to points y ∈ V we can take a ﬁxed path γ from y0 to y followed by paths η in

V from y to the points y . Then the paths (f γ) (f η) in X have lifts (f γ) (f η)
where f η = p−1f η and p−1 : U→U is the inverse of p : U→U . Thus f (V ) ⊂ U and
f |V = p−1f , hence f is continuous at y .

An example showing the necessity of the local path-connectedness assumption on Y is described in Exercise 7 at the end of this section.
Next we have the unique lifting property:
Proposition 1.34. Given a covering space p : X→X and a map f : Y →X with two lifts f1, f2 : Y →X that agree at one point of Y , then if Y is connected, these two lifts
must agree on all of Y .
Proof: For a point y ∈ Y , let U be an open neighborhood of f (y) in X for which
p−1(U ) is a disjoint union of open sets Uα each mapped homeomorphically to U

Covering Spaces

Section 1.3 61

by p , and let U1 and U2 be the Uα ’s containing f1(y) and f2(y) , respectively. By
continuity of f1 and f2 there is a neighborhood N of y mapped into U1 by f1 and
into U2 by f2 . If f1(y) ≠ f2(y) then U1 ≠ U2 , hence U1 and U2 are disjoint and f1 ≠ f2 throughout the neighborhood N . On the other hand, if f1(y) = f2(y) then U1 = U2 so f1 = f2 on N since pf1 = pf2 and p is injective on U1 = U2 . Thus the
set of points where f1 and f2 agree is both open and closed in Y .

The Classiﬁcation of Covering Spaces

We consider next the problem of classifying all the different covering spaces of

a ﬁxed space X . Since the whole chapter is about paths, it should not be surprising

that we will restrict attention to spaces X that are at least locally path-connected.

Path-components of X are then the same as components, and for the purpose of

classifying the covering spaces of X there is no loss in assuming that X is connected,

or equivalently, path-connected. Local path-connectedness is inherited by covering

spaces, so these too are connected iff they are path-connected. The main thrust of the

classiﬁcation will be the Galois correspondence between connected covering spaces of

X and subgroups of π1(X) , but when this is ﬁnished we will also describe a different method of classiﬁcation that includes disconnected covering spaces as well.

The Galois correspondence arises from the function that assigns to each covering
→ space p : (X, x0) (X, x0) the subgroup p∗ π1(X, x0) of π1(X, x0) . First we con-
sider whether this function is surjective. That is, we ask whether every subgroup of
→ π1(X, x0) is realized as p∗ π1(X, x0) for some covering space p : (X, x0) (X, x0) .
In particular we can ask whether the trivial subgroup is realized. Since p∗ is always injective, this amounts to asking whether X has a simply-connnected covering space.

Answering this will take some work.

A necessary condition for X to have a simply-connected covering space is the

following: Each point x ∈ X has a neighborhood U such that the inclusion-induced

map π1(U , this holds.

x)→π1(X,
To see the

x) is trivial; necessity of

one this

says X is semilocally condition, suppose p

simply-connected if
: X→X is a covering

space with X simply-connected. Every point x ∈ X has a neighborhood U having a

lift U ⊂ X projecting homeomorphically to U by p . Each loop in U lifts to a loop

in U , and the lifted loop is nullhomotopic in X since π1(X) = 0 . So, composing this

nullhomotopy with p , the original loop in U is nullhomotopic in X .

A locally simply-connected space is certainly semilocally simply-connected. For

example, CW complexes have the much stronger property of being locally contractible,

as we show in the Appendix. An example of a space that is not semilocally simplyconnected is the shrinking wedge of circles, the subspace X ⊂ R2 consisting of the
circles of radius 1/n centered at the point (1/n, 0) for n = 1, 2, ··· , introduced in Example 1.25. On the other hand, the cone CX = (X × I)/(X × {0}) is semilocally simply-

connected since it is contractible, but it is not locally simply-connected.

62 Chapter 1

The Fundamental Group

We shall now show how to construct a simply-connected covering space of X if
X is path-connected, locally path-connected, and semilocally simply-connected. To
motivate the construction, suppose p : (X, x0)→(X, x0) is a simply-connected cover-
ing space. Each point x ∈ X can then be joined to x0 by a unique homotopy class of paths, by Proposition 1.6, so we can view points of X as homotopy classes of paths
starting at x0 . The advantage of this is that, by the homotopy lifting property, homotopy classes of paths in X starting at x0 are the same as homotopy classes of paths in X starting at x0 . This gives a way of describing X purely in terms of X .
Given a path-connected, locally path-connected, semilocally simply-connected space X with a basepoint x0 ∈ X , we are therefore led to deﬁne
X = [γ] || γ is a path in X starting at x0

where, as usual, [γ] denotes the homotopy class of γ with respect to homotopies
that ﬁx the endpoints γ(0) and γ(1) . The function p : X→X sending [γ] to γ(1) is

then well-deﬁned. Since X is path-connected, the endpoint γ(1) can be any point of

X , so p is surjective.

Before we deﬁne a topology on X we make a few preliminary observations. Let

U be the collection of
trivial. Note that if the

path-connected open sets U ⊂ X
map π1(U )→π1(X) is trivial for

such that one choice

π1(U )→π1(X) is
of basepoint in U ,

it is trivial for all choices of basepoint since U is path-connected. A path-connected
open subset V ⊂ U ∈ U is also in U since the composition π1(V )→π1(U)→π1(X)

will also be trivial. It follows that U is a basis for the topology on X if X is locally

path-connected and semilocally simply-connected.
Given a set U ∈ U and a path γ in X from x0 to a point in U , let

U[γ] = [γ η] || η is a path in U with η(0) = γ(1)

As the that p

notation
→ : U[γ] U

indicates, U[γ is surjective

] depends since U is

only on the homotopy class [γ] . Observe path-connected and injective since differ-

ent choices of η joining γ(1) to a ﬁxed x ∈ U are all homotopic in X , the map

π1(U )→π1(X) being trivial. Another property is

U[γ] = U[γ ] if [γ ] ∈ U[γ] . For if γ = γ η then elements of U[γ ] have the (∗) form [γ η µ] and hence lie in U[γ] , while elements of U[γ] have the form
[γ µ] = [γ η η µ] = [γ η µ] and hence lie in U[γ ] .

This can be used to show that the sets U[γ] form a basis for a topology on X . For if we are given two such sets U[γ] , V[γ ] and an element [γ ] ∈ U[γ] ∩ V[γ ] , we have
U[γ] = U[γ ] and V[γ ] = V[γ ] by (∗) . So if W ∈ U is contained in U ∩ V and contains
→ γ (1) then W[γ ] ⊂ U[γ ] ∩ V[γ ] and [γ ] ∈ W[γ ] .
The bijection p : U[γ] U is a homeomorphism since it gives a bijection between
the subsets V[γ ] ⊂ U[γ] and the sets V ∈ U contained in U . Namely, in one direction
we have p(V[γ ]) = V and in the other direction we have p−1(V ) ∩ U[γ] = V[γ ] for

Covering Spaces

Section 1.3 63

any [γ ] ∈ U[γ] with endpoint in V , since V[γ ] ⊂ U[γ ] = U[γ] and V[γ ] maps onto V

by the bijection p .
The preceding paragraph implies that p : X→X is continuous. We can also de-

duce that this is a covering space since for ﬁxed U ∈ U , the sets U[γ] for varying [γ]
partition p−1(U ) because if [γ ] ∈ U[γ] ∩ U[γ ] then U[γ] = U[γ ] = U[γ ] by (∗) .

A natural basepoint for X is [x0] , the homotopy class of the constant path at x0 .

Any [γ] ∈ X can be joined to [x0] by a path in X by restricting γ to progressively

shorter segments [0, t] ⊂ [0, 1]
f : I→X be a loop based at [x0

, ]

so X . The

is path-connected. composition pf is

To see then a

that loop

π1(X γ in

) = 0 , let X based

at x0 . Let γt be the path in X obtained by restricting the loop γ to [0, t] . Then [γt]

for t varying from 0 at [γ0] = [x0] = f (0)

to 1 forms a path , so by the unique

liinftiXnglipfrtoinpgertthyeolof othpeγco. vTehrissplaifcte[Xγt→] sXtarwtes

must have [γt] = f (t) for all t . In particular, [γ1] = f (1) = [x0] . Since γ1 = γ , this

says the loop γ = pf is nullhomotopic. Thus p∗([f ]) = 0 in π1(X) . Since p∗ is

injective, [f ] = 0 and hence π1(X) = 0 .
This completes the construction of a simply-connected covering space X→X .

In concrete cases one usually constructs a simply-connected covering space by
more direct methods. For example, suppose X is the union of subspaces A and B for
which simply-connected covering spaces A→A and B→B are already known. Then one can attempt to build a simply-connected covering space X→X by assembling
copies of A and B . For example, for X = S1 ∨ S1 , if we take A and B to be the two
circles, then A and B are each R , and we can build the simply-connected cover X
described earlier in this section by glueing together inﬁnitely many copies of A and
B , the horizontal and vertical lines in X . Here is another illustration of this method:
Example 1.35. For integers m, n ≥ 2 , let Xm,n be the quotient space of a cylinder
S1 × I under the identiﬁcations (z, 0) ∼ (e2πi/mz, 0) and (z, 1) ∼ (e2πi/nz, 1) . Let A ⊂ X and B ⊂ X be the quotients of S1 × [0, 1/2] and S1 × [1/2, 1], so A and B are the mapping cylinders of z zm and z zn , with A ∩ B = S1 . The simplest case is m = n = 2 , when A and B are M¨obius bands and X2,2 is the Klein bottle. We encountered the complexes Xm,n previously in analyzing torus knot complements in Example 1.24.
The ﬁgure for Example 1.29 at the end of the preceding section shows what A looks like in the typical case m = 3 . We have π1(A) ≈ Z , and the universal cover A is homeomorphic to a product Cm × R where Cm is the graph that is a cone on m points, as shown in the ﬁgure to the right. The situation for B is similar, and B is homeomorphic to Cn × R . Now we attempt to build the universal cover Xm,n from copies of A and B . Start with a copy of A . Its boundary, the outer edges of
its ﬁns, consists of m copies of R . Along each of these m boundary

64 Chapter 1

The Fundamental Group

lines we attach a copy of B . Each of these copies of B has one of its boundary lines attached to the initial copy of A , leaving n − 1 boundary lines free, and we attach a new copy of A to each of these free boundary lines. Thus we now have m(n − 1) + 1 copies of A . Each of the newly attached copies of A has m − 1 free boundary lines,
and to each of these lines we attach a new copy of B . The process is now repeated ad
inﬁnitim in the evident way. Let Xm,n be the resulting space. The product structures A = Cm × R and B = Cn × R
give Xm,n the structure of a product Tm,n × R where Tm,n is an inﬁnite graph constructed by an inductive scheme
just like the construction of Xm,n . Thus Tm,n is the union of a sequence of ﬁnite subgraphs, each obtained from the
preceding by attaching new copies of Cm or Cn . Each of these ﬁnite subgraphs deformation retracts onto the
preceding one. The inﬁnite concatenation of these deformation retractions, with the kth graph deformation retracting to the previous one during the time interval [1/2k, 1/2k−1] , gives a deformation retraction of Tm,n onto the initial stage Cm . Since Cm is contractible, this means Tm,n is contractible, hence also Xm,n , which is the product Tm,n × R . In particular, Xm,n is simply-connected.
The map that projects each copy of A in Xm,n to A and each copy of B to B is a covering space. To deﬁne this map precisely, choose a point x0 ∈ S1 , and then the image of the line segment {x0}× I in Xm,n meets A in a line segment whose preimage in A consists of an inﬁnite number of line segments,
appearing in the earlier ﬁgure as the horizontal segments spi-
raling around the central vertical axis. The picture in B is
similar, and when we glue together all the copies of A and B
to form Xm,n , we do so in such a way that these horizontal segments always line up
reexcatcatnlyg.leTihnisandeAcoamnpdoaserescXtamn,gnleinintoainBﬁ. nTihteelycomvaenriyngrepcrtoajnegclteiso,neaXcmh,nf→ormXemd,nfriosmthae
quotient map that identiﬁes all these rectangles.

Now we return to the general theory. The hypotheses for constructing a simplyconnected covering space of X in fact sufﬁce for constructing covering spaces realizing arbitrary subgroups of π1(X) :

Proposition 1.36. Suppose X is path-connected, locally path-connected, and semilo-

cally simply-connected.
space p : XH →X such

Then for every subgroup H ⊂ π1(X, x0) there is a covering that p∗ π1(XH , x0) = H for a suitably chosen basepoint

x0 ∈ XH .

Proof: For points [γ] , [γ ] in the simply-connected covering space X constructed
above, deﬁne [γ] ∼ [γ ] to mean γ(1) = γ (1) and [γγ ] ∈ H . It is easy to see

Covering Spaces

Section 1.3 65

that this is an equivalence relation since H is a subgroup; namely, it is reﬂexive since

H contains the identity element, symmetric since H is closed under inverses, and

transitive since H is closed under multiplication. Let XH be the quotient space of X obtained by identifying [γ] with [γ ] if [γ] ∼ [γ ] . Note that if γ(1) = γ (1) , then
[γ] ∼ [γ ] iff [γη] ∼ [γ η] . This means that if any two points in basic neighborhoods

→ U[γ] and U[γ ] are identiﬁed in XH then the whole
the natural projection XH X induced by [γ]

neighborhoods are identiﬁed. γ(1) is a covering space.

Hence

c

at

If we choose for the x0 , then the image

basepoint of p∗ : π1

(xX0H∈, xX0)H→thπe1e(qXu,ixva0l)enisceexcalacstlsyoHf t.hTe hcoisnisstabnetcpauasthe

for a loop γ in X based at x0 , its lift to X starting at [c] ends at [γ] , so the image of this lifted path in XH is a loop iff [γ] ∼ [c] , or equivalently, [γ] ∈ H .

Having taken care of the existence of covering spaces of X corresponding to all
subgroups of π1(X) , we turn now to the question of uniqueness. More speciﬁcally, we are interested in uniqueness up to isomorphism, where an isomorphism between
covering spaces p1 : X1→X and p2 : X2→X is a homeomorphism f : X1→X2 such
that p1 = p2f . This condition means exactly that f preserves the covering space structures, taking p1−1(x) to p2−1(x) for each x ∈ X . The inverse f −1 is then also an isomorphism, and the composition of two isomorphisms is an isomorphism, so we
have an equivalence relation.
Proposition 1.37. If X is path-connected and locally path-connected, then two path→ → connected covering spaces p1 : X1 X and p2 : X2 X are isomorphic via an isomor→ phism f : X1 X2 taking a basepoint x1 ∈ p1−1(x0) to a basepoint x2 ∈ p2−1(x0) iff
p1∗ π1(X1, x1) = p2∗ π1(X2, x2) .
Proof: If there is an isomorphism f → : (X1, x1) (X2, x2) , then from the two relations
p1 = p2f and p2 = p1f −1 it follows that p1∗ π1(X1, x1) = p2∗ π1(X2, x2) . Conversely, suppose that p1∗ π1(X1, x1) = p2∗ π1(X2, x2) . By the lifting criterion,
→ we may lift p1 to a map p1 : (X1, x1) (X2, x2) with p2p1 = p1 . Symmetrically, we → obtain p2 : (X2, x2) (X1, x1) with p1p2 = p2 . Then by the unique lifting property,
p1p2 = 11 and p2p1 = 11 since these composed lifts ﬁx the basepoints. Thus p1 and p2 are inverse isomorphisms.

We have proved the ﬁrst half of the following classiﬁcation theorem:

Theorem 1.38. Let X be path-connected, locally path-connected, and semilocally

simply-connected. Then there is a bijection between the set of basepoint-preserving
isomorphism classes of path-connected covering spaces p : (X, x0)→(X, x0) and the

set of subgroups of π1(X, x0) , obtained by associating the subgroup p∗ π1(X, x0)

to the covering space (X, x0) . If basepoints are ignored, this correspondence bijection between isomorphism classes of path-connected covering spaces p

gives a
: X→X

and conjugacy classes of subgroups of π1(X, x0) .

66 Chapter 1

The Fundamental Group

Proof: It remains only to prove the last statement. We show that for a covering space
p : (X, x0)→(X, x0) , changing the basepoint x0 within p−1(x0) corresponds exactly
to changing p∗ π1(X, x0) to a conjugate subgroup of π1(X, x0) . Suppose that x1 is another basepoint in p−1(x0) , and let γ be a path from x0 to x1 . Then γ projects to a loop γ in X representing some element g ∈ π1(X, x0) . Set Hi = p∗ π1(X, xi) for i = 0, 1 . We have an inclusion g−1H0g ⊂ H1 since for f a loop at x0 , γf γ is a loop at x1 . Similarly we have gH1g−1 ⊂ H0 . Conjugating the latter relation by g−1 gives H1 ⊂ g−1H0g , so g−1H0g = H1 . Thus, changing the basepoint from x0 to x1 changes H0 to the conjugate subgroup H1 = g−1H0g .
Conversely, to change H0 to a conjugate subgroup H1 = g−1H0g , choose a loop γ representing g , lift this to a path γ starting at x0 , and let x1 = γ(1) . The preceding argument then shows that we have the desired relation H1 = g−1H0g .

A consequence of the lifting criterion is that a simply-connected covering space of a path-connected, locally path-connected space X is a covering space of every other path-connected covering space of X . A simply-connected covering space of X is therefore called a universal cover. It is unique up to isomorphism, so one is justiﬁed in calling it the universal cover.
More generally, there is a partial ordering on the various path-connected covering spaces of X , according to which ones cover which others. This corresponds to the partial ordering by inclusion of the corresponding subgroups of π1(X) , or conjugacy classes of subgroups if basepoints are ignored.

Representing Covering Spaces by Permutations

We wish to describe now another way of classifying the different covering spaces

of a connected, locally path-connected, semilocally simply-connected space X , with-

out restricting just to connected covering spaces. To give the idea, con-

sider the 3 sheeted covering spaces of S1 . There are three of these, X1 ,

X2 , For

and each

X3 of

, with these

the subscript indicating the number
covering spaces p : Xi→S1 the three

of components. different lifts of

a loop in S1 generating π1(S1, x0) determine a permutation of p−1(x0)

sending the starting point of the lift to the ending point of the lift. For

X1 this is a cyclic permutation, for X2 it is a transposition of two points ﬁxing the third point, and for X3 it is the identity permutation. These permutations obviously determine the covering spaces uniquely, up to isomorphism. The same would be true for n sheeted covering spaces of S1 for arbitrary n , even for

n inﬁnite. The covering spaces of S1 ∨ S1 can be encoded using the same idea. Referring

back to the large table of examples near the beginning of this section, we see in the

covering space (1) that the loop a lifts to the identity permutation of the two vertices

and b lifts to the permutation that transposes the two vertices. In (2), both a and b

Covering Spaces

Section 1.3 67

lift to transpositions of the two vertices. In (3) and (4), a and b lift to transpositions of different pairs of the three vertices, while in (5) and (6) they lift to cyclic permutations of the vertices. In (11) the vertices can be labeled by Z , with a lifting to the identity permutation and b lifting to the shift n n + 1 . Indeed, one can see from these examples that a covering space of S1 ∨ S1 is nothing more than an efﬁcient graphical representation of a pair of permutations of a given set.

This idea of lifting loops to permutations generalizes to arbitrary covering spaces.
For a covering space p : X→X , a path γ in X has a unique lift γ starting at a given → point of p−1(γ(0)) , so we obtain a well-deﬁned map Lγ : p−1(γ(0)) p−1(γ(1)) by
sending the starting point γ(0) of each lift γ to its ending point γ(1) . It is evident

that Lγ is a bijection since Lγ is its inverse. For a composition of paths γη we have Lγη = LηLγ , rather than Lγ Lη , since composition of paths is written from left to
right while composition of functions is written from right to left. To compensate for

→ this, let us modify the deﬁnition by replacing Lγ by its inverse. Thus the new Lγ is
a bijection p−1(γ(1)) p−1(γ(0)) , and Lγη = Lγ Lη . Since Lγ depends only on the

homotopy class of γ , this means that if we restrict attention to loops at a basepoint

x0 ∈ X , then the association γ Lγ gives a homomorphism from π1(X, x0) to the group of permutations of p−1(x0) . This is called the action of π1(X, x0) on the ﬁber p−1(x0) .
Let us see how the covering space p : X→X can be reconstructed from the asso-

ciated action of path-connected,

π1(X, x0) on the ﬁber F = p−1(x0) , and semilocally simply-connected, so

assuming that X it has a universal

is connected,
cover X0→X .

We can take the as in the general

points of X0 construction

to of

be homotopy classes of a universal cover. Deﬁne

paths in a map h

X starting
: X0 × F →X

at x0 , send-

ing a pair ([γ], x0) to γ(1) where γ is the lift of γ to X starting at x0 . Then h is

continuous, and in fact a local homeomorphism, since a neighborhood of ([γ], x0) in X0 × F consists of the pairs ([γη], x0) with η a path in a suitable neighborhood of

γ(1) . It is obvious that h is surjective since X is path-connected. If h were injec-

tive as well, it would be a homeomorphism, which is unlikely since X is probably not

homeomorphic to X0 × F . Even if h is not injective, it will induce a homeomorphism

from some quotient space of X0 × F onto X . To see what this quotient space is,

suppose h([γ], x0) = h([γ ], x0) . Then γ and γ are both

γ

paths from x0 to the same endpoint, and from the ﬁgure we see that x0 = Lγ γ (x0) . Letting λ be the loop γ γ , this means that h([γ], x0) = h([λγ], Lλ(x0)) . Conversely, for

x0 x0

γ

any loop λ we have h([γ], x0) = h([λγ], Lλ(x0)) . Thus h

γ

induces a well-deﬁned map to X from the quotient space of x0

X0 × F obtained by identifying ([γ], x0) with ([λγ], Lλ(x0))

γ

for each [λ] ∈ π1(X, x0) . Let this quotient space be denoted Xρ where ρ is the ho-

momorphism from π1(X, x0) to the permutation group of F speciﬁed by the action.

68 Chapter 1

The Fundamental Group

ρ

Notice that the deﬁnition of Xρ of π1(X, x0) on a set F . There is

makes sense whenever we are given an action
a natural projection Xρ→X sending ([γ], x0)

to γ(1) , and this is a covering space since if U ⊂ X is an open set over which the

universal cover X0 is a product U × π1(X, x0) , then the identiﬁcations deﬁning Xρ

simply collapse U × π1(X, x0)× F to U × F .
Returning to our given covering space X→X with associated action ρ , the map

Xloρc→al hXominedoumceodrpbhyishm.isSiancbeijethcitsiohnoamnedomthoerrpefhoisremaXhρo→meXomtaokrepsheisamch

since ﬁber

h of

was a Xρ to

the corresponding ﬁber of X , it is an isomorphism of covering spaces.
If two covering spaces p1 : X1→X and p2 : X2→X are isomorphic, one may ask

how the related.

corresponding actions An isomorphism h : X1

→of Xπ21 (rXes,txri0c)tsotno

tahebiﬁjebcetriosnFF1 1a→ndF2F,2anodveervixd0enatrlye

→ Lγ (h(x0)) = h(Lγ (x0)) . Using the less cumbersome notation γx0 for Lγ (x0) , this
relation can be written more concisely as γh(x0) = h(γx0) . A bijection F1 F2 with

this property is what one would naturally call an isomorphism of sets with π1(X, x0)

action. Thus isomorphic covering spaces have isomorphic actions on ﬁbers. The

converse is also true, and easy to prove. One just observes that for isomorphic actions
→ → ρ1 and ρ2 , an isomorphism h : F1 F2 induces a map Xρ1 Xρ2 and h−1 induces a
similar map in the opposite direction, such that the compositions of these two maps,

in either order, are the identity.

This shows that n sheeted covering spaces of X are classiﬁed by equivalence
classes of homomorphisms → π1(X, x0) Σn , where Σn is the symmetric group on n
symbols and the equivalence relation identiﬁes a homomorphism ρ with each of its conjugates h−1ρh by elements h ∈ Σn . The study of the various homomorphisms from a given group to Σn is a very classical topic in group theory, so we see that this algebraic question has a nice geometric interpretation.

Deck Transformations and Group Actions
For a covering space p : X→X the isomorphisms X→X are called deck transfor-
mations or covering transformations. These form a group G(X) under composition.
For example, for the covering space p : R→S1 projecting a vertical helix onto a circle,
the deck transformations are the vertical translations taking the helix onto itself, so
G(X) ≈ Z in this case. For the n sheeted covering space S1→S1 , z zn , the deck
transformations are the rotations of S1 through angles that are multiples of 2π /n , so G(X) = Zn .
By the unique lifting property, a deck transformation is completely determined
by where it sends a single point, assuming X is path-connected. In particular, only
the identity deck transformation can ﬁx a point of X .
A covering space p : X→X is called normal if for each x ∈ X and each pair of lifts
x, x of x there is a deck transformation taking x to x . For example, the covering

Covering Spaces

Section 1.3 69

space R→S1 and the n sheeted covering spaces S1→S1 are normal. Intuitively, a
normal covering space is one with maximal symmetry. This can be seen in the covering spaces of S1 ∨ S1 shown in the table earlier in this section, where the normal covering
spaces are (1), (2), (5)–(8), and (11). Note that in (7) the group of deck transformations is Z4 while in (8) it is Z2 × Z2 .
Sometimes normal covering spaces are called regular covering spaces. The term
‘normal’ is motivated by the following result.

Proposition 1.39. Let p : (X, x0)→(X, x0) be a path-connected covering space of
the path-connected, locally path-connected space X , and let H be the subgroup p∗ π1(X, x0) ⊂ π1(X, x0) . Then : (a) This covering space is normal iff H is a normal subgroup of π1(X, x0) . (b) G(X) is isomorphic to the quotient N(H)/H where N(H) is the normalizer of

H in π1(X, x0) .

In particular, G(X) is isomorphic
for the universal cover X→X we

to π1(X, x0)/H if X have G(X) ≈ π1(X) .

is

a

normal

covering.

Hence

Proof: We observed earlier in the proof of the classiﬁcation theorem that changing
the basepoint x0 ∈ p−1(x0) to x1 ∈ p−1(x0) corresponds precisely to conjugating H by an element [γ] ∈ π1(X, x0) where γ lifts to a path γ from x0 to x1 . Thus [γ] is in the normalizer N(H) iff p∗ π1(X, x0) = p∗ π1(X, x1) , which by the lifting
criterion is equivalent to the existence of a deck transformation taking x0 to x1 . Hence the covering space is normal iff N(H) = π1(X, x0) , that is, iff H is a normal
subgDroeuﬁpneofϕπ: 1N((XH, x)→0) .G(X) sending [γ] to the deck transformation τ taking x0 to
x1 , in the notation above. Then ϕ is a homomorphism, for if γ is another loop corre-
sponding to the deck transformation τ taking x0 to x1 then γ γ lifts to γ (τ(γ )) , a path from x0 to τ(x1) = ττ (x0) , so ττ is the deck transformation corresponding
to [γ][γ ] . By the preceding paragraph ϕ is surjective. Its kernel consists of classes
[γ] lifting to loops in X . These are exactly the elements of p∗ π1(X, x0) = H .

The group of deck transformations is a special case of the general notion of
‘groups acting on spaces.’ Given a group G and a space Y , then an action of G
on Y is a homomorphism ρ from G to the group Homeo(Y ) of all homeomorphisms
from Y to itself. Thus to each g ∈ G is associated a homeomorphism ρ(g) : Y →Y , which for notational simplicity we write simply as g : Y →Y . For ρ to be a homo-
morphism amounts to requiring that g1(g2(y)) = (g1g2)(y) for all g1, g2 ∈ G and y ∈ Y . If ρ is injective then it identiﬁes G with a subgroup of Homeo(Y ) , and in
practice not much is lost in assuming ρ is an inclusion G Homeo(Y ) since in any case the subgroup ρ(G) ⊂ Homeo(Y ) contains all the topological information about
the action.

70 Chapter 1

The Fundamental Group

We shall be interested in actions satisfying the following condition:

(∗)

Each y ∈ Y has a neighborhood U such that all the images g(U ) for varying g ∈ G are disjoint. In other words, g1(U ) ∩ g2(U ) ≠ ∅ implies g1 = g2 .

The action of the deck transformation group G(X) on X satisﬁes (∗) . To see this,
let U ⊂ X project homeomorphically to U ⊂ X . If g1(U ) ∩ g2(U ) ≠ ∅ for some g1, g2 ∈ G(X) , then g1(x1) = g2(x2) for some x1, x2 ∈ U . Since x1 and x2 must lie in the same set p−1(x) , which intersects U in only one point, we must have x1 = x2 . Then g1−1g2 ﬁxes this point, so g1−1g2 = 11 and g1 = g2 .
Note that in (∗) it sufﬁces to take g1 to be the identity since g1(U ) ∩ g2(U ) ≠ ∅ is equivalent to U ∩ g1−1g2(U ) ≠ ∅ . Thus we have the equivalent condition that U ∩ g(U) ≠ ∅ only when g is the identity.

Given an action of a group G on a space Y , we can form a space Y /G , the quotient

space of Y in which each point y is identiﬁed with all its images g(y) as g ranges over G . The points of Y /G are thus the orbits Gy = { g(y) | g ∈ G } in Y , and

Y /G is called the orbit space of the action. For example, for a normal covering space
X→X , the orbit space X/G(X) is just X .

Proposition 1.40. If an action of a group G on a space Y satisﬁes (∗) , then : (a) The quotient map p : Y →Y /G , p(y) = Gy , is a normal covering space. (b) G is the group of deck transformations of this covering space Y →Y /G if Y is
path-connected. (c) G is isomorphic to π1(Y /G)/p∗ π1(Y ) if Y is path-connected and locally path-
connected.
Proof: Given an open set U ⊂ Y as in condition (∗) , the quotient map p simply
identiﬁes all the disjoint homeomorphic sets { g(U ) | g ∈ G } to a single open set p(U ) in Y /G . By the deﬁnition of the quotient topology on Y /G , p restricts to a homeomorphism from g(U) onto p(U ) for each g ∈ G so we have a covering space. Each element of G acts as a deck transformation, and the covering space is normal since g2g1−1 takes g1(U ) to g2(U ) . The deck transformation group contains G as a subgroup, and equals this subgroup if Y is path-connected, since if f is any deck transformation, then for an arbitrarily chosen point y ∈ Y , y and f (y) are in the same orbit and there is a g ∈ G with g(y) = f (y) , hence f = g since deck transformations of a path-connected covering space are uniquely determined by where they send a point. The ﬁnal statement of the proposition is immediate from part (b) of Proposition 1.39.

In view of the preceding proposition, we shall call an action satisfying (∗) a covering space action. This is not standard terminology, but there does not seem to be a universally accepted name for actions satisfying (∗) . Sometimes these are called ‘properly discontinuous’ actions, but more often this rather unattractive term means

Covering Spaces

Section 1.3 71

something weaker: Every point x ∈ X has a neighborhood U such that U ∩ g(U ) is nonempty for only ﬁnitely many g ∈ G . Many symmetry groups have this proper discontinuity property without satisfying (∗) , for example the group of symmetries of the familiar tiling of R2 by regular hexagons. The reason why the action of this group on R2 fails to satisfy (∗) is that there are ﬁxed points: points y for which there is a nontrivial element g ∈ G with g(y) = y . For example, the vertices of the
hexagons are ﬁxed by the 120 degree rotations about these points, and the midpoints
of edges are ﬁxed by 180 degree rotations. An action without ﬁxed points is called a
free action. Thus for a free action of G on Y , only the identity element of G ﬁxes any point of Y . This is equivalent to requiring that all the images g(y) of each y ∈ Y are distinct, or in other words g1(y) = g2(y) only when g1 = g2 , since g1(y) = g2(y) is equivalent to g1−1g2(y) = y . Though condition (∗) implies freeness, the converse is not always true. An example is the action of Z on S1 in which a generator of Z acts
by rotation through an angle α that is an irrational multiple of 2π . In this case each orbit Zy is dense in S1 , so condition (∗) cannot hold since it implies that orbits are
discrete subspaces. An exercise at the end of the section is to show that for actions on Hausdorff spaces, freeness plus proper discontinuity implies condition (∗) . Note
that proper discontinuity is automatic for actions by a ﬁnite group.

Example 1.41. Let Y be the closed orientable surface of genus 11, an ‘11 hole torus’ as

shown in the ﬁgure. This has a 5 fold rotational symme-

try, generated by a rotation of angle 2π /5 . Thus we have

the cyclic group Z5 acting on Y , and the condition (∗) is

C4

C3

obviously satisﬁed. The quotient space Y /Z5 is a surface

of genus 3, obtained from one of the ﬁve subsurfaces of Y cut off by the circles C1, ··· , C5 by identifying its two
bshoouwndna. rTyhcuisrcwleeshCavi eaandcovCei+ri1ngtospfaocremMth11e→ciMrc3lewCheares
Mg denotes the closed orientable surface of genus g .

C5

C2

C1

p

In particular, we see that π1(M3) contains the ‘larger’

group π1(M11) as a normal subgroup of index 5 , with

C

quotient Z5 . This example obviously generalizes by re-

tnpolafscohilnodgwstybhmyematnweEotruyhl.eorTlehcshisainrgaievcaetceshrais‘actirocmvae’rroginufgmMse1pn1atcbteyhaMmtmifnht+oh1le→erse Maisnmda+ct1oh.veAe5rninefgoxlesdrpcsaiyscmee iMmnge§t→2ry.2Mbihys
then g = mn + 1 and h = m + 1 for some m and n .

As a special case of the ﬁnal statement of the preceding proposition we see that for a covering space action of a group G on a simply-connected locally path-connected space Y , the orbit space Y /G has fundamental group isomorphic to G . Under this isomorphism an element g ∈ G corresponds to a loop in Y /G that is the projection of

72 Chapter 1

The Fundamental Group

a path in Y from a chosen basepoint y0 to g(y0) . Any two such paths are homotopic since Y is simply-connected, so we get a well-deﬁned element of π1(Y /G) associated to g .

This method for computing fundamental groups via group actions on simply-

connected spaces is essentially how
space R→S1 arising from the action

we of

computed Z on R by

π1(S1) in §1.1, via translations. This is

the covering a useful gen-

eral technique for computing fundamental groups, in fact. Here are some examples

illustrating this idea.

Example 1.42. Consider the grid in R2 formed by the horizontal and vertical lines
through points in Z2 . Let us decorate this grid with arrows in either of the two ways shown in the ﬁgure, the difference between the two cases being that in the second case the horizontal arrows in adjacent lines point in opposition directions. The group G consisting of all symmetries of the ﬁrst decorated grid is isomorphic to Z× Z since it consists of all translations (x, y) (x + m, y + n) for m, n ∈ Z . For the second grid the symmetry group G contains a subgroup of translations of the form (x, y) (x + m, y + 2n) for m, n ∈ Z , but there are also glide-reﬂection symmetries consisting of vertical translation by an odd integer distance followed by reﬂection across a vertical line, either a vertical line of the grid or a vertical line halfway between two adjacent grid lines. For both decorated grids there are elements of G taking any square to any other, but only the identity element of G takes a square to itself. The minimum distance any point is moved by a nontrivial element of G is 1 , which easily implies the covering space condition (∗) . The orbit space R2/G is the quotient space of a square in the grid with opposite edges identiﬁed according to the arrows. Thus we see that the fundamental groups of the torus and the Klein bottle are the symmetry groups G in the two cases. In the second case the subgroup of G formed by the translations has index two, and the orbit space for this subgroup is a torus forming a two-sheeted covering space of the Klein bottle.

Example 1.43: RPn . The antipodal map of Sn , x −x , generates an action of Z2
on Sn with orbit space RPn , real projective n space, as deﬁned in Example 0.4. The
action is a covering space action since each open hemisphere in Sn is disjoint from
its antipodal image. As we saw in Proposition 1.14, Sn is simply-connected if n ≥ 2 ,
so from the covering space Sn→RPn we deduce that π1(RPn) ≈ Z2 for n ≥ 2 . A
generator for π1(RPn) is any loop obtained by projecting a path in Sn connecting two antipodal points. One can see explicitly that such a loop γ has order two in π1(RPn) if n ≥ 2 since the composition γ γ lifts to a loop in Sn , and this can be homotoped to
the trivial loop since π1(Sn) = 0 , so the projection of this homotopy into RPn gives
a nullhomotopy of γ γ .

Covering Spaces

Section 1.3 73

One may ask whether there are other ﬁnite groups that act freely on Sn , deﬁning
covering spaces Sn→Sn/G . We will show in Proposition 2.29 that Z2 is the only
possibility when n is even, but for odd n the question is much more difﬁcult. It is easy to construct a free action of any cyclic group Zm on S2k−1 , the action generated by the rotation v e2πi/mv of the unit sphere S2k−1 in Ck = R2k . This action is free since an equation v = e2πi /mv with 0 < < m implies v = 0 , but 0 is not a point of S2k−1 . The orbit space S2k−1/Zm is one of a family of spaces called lens spaces deﬁned in Example 2.43.
There are also noncyclic ﬁnite groups that act freely as rotations of Sn for odd
n > 1 . These actions are classiﬁed quite explicitly in [Wolf 1984]. Examples in the simplest case n = 3 can be produced as follows. View R4 as the quaternion algebra H . Multiplication of quaternions satisﬁes |ab| = |a||b| where |a| denotes the usual Euclidean length of a vector a ∈ R4 . Thus if a and b are unit vectors, so is ab , and
hence quaternion multiplication deﬁnes a map → S3 × S3 S3 . This in fact makes S3
into a group, though associativity is all we need now since associativity implies that any subgroup G of S3 acts on S3 by left-multiplication, g(x) = gx . This action is free since an equation x = gx in the division algebra H implies g = 1 or x = 0 . As a concrete example, G could be the familiar quaternion group Q8 = {±1, ±i, ±j, ±k} from group theory. More generally, for a positive integer m , let Q4m be the subgroup of S3 generated by the two quaternions a = eπi/m and b = j . Thus a has order 2m and b has order 4 . The easily veriﬁed relations am = b2 = −1 and bab−1 = a−1 imply that the subgroup Z2m generated by a is normal and of index 2 in Q4m . Hence Q4m is a group of order 4m , called the generalized quaternion group. Another common name for this group is the binary dihedral group D4∗m since its quotient by the subgroup {±1} is the ordinary dihedral group D2m of order 2m .
Besides the groups Q4m = D4∗m there are just three other noncyclic ﬁnite subgroups of S3 : the binary tetrahedral, octahedral, and icosahedral groups T2∗4 , O4∗8, and I1∗20, of orders indicated by the subscripts. These project two-to-one onto the groups of rotational symmetries of a regular tetrahedron, octahedron (or cube), and
icosahedron (or dodecahedron). In fact, it is not hard to see that the homomorphism
S3→SO(3) sending u ∈ S3 ⊂ H to the isometry v→u−1vu of R3 , viewing R3 as the
‘pure imaginary’ quaternions v = ai + bj + ck , is surjective with kernel {±1} . Then the groups D4∗m , T2∗4 , O4∗8 , I1∗20 are the preimages in S3 of the groups of rotational symmetries of a regular polygon or polyhedron in R3 .
There are two conditions that a ﬁnite group G acting freely on Sn must satisfy:
(a) Every abelian subgroup of G is cyclic. This is equivalent to saying that G contains no subgroup Zp × Zp with p prime.
(b) G contains at most one element of order 2 .
A proof of (a) is sketched in an exercise for §4.2. For a proof of (b) the original
source [Milnor 1957] is recommended reading. The groups satisfying (a) have been

74 Chapter 1

The Fundamental Group

completely classiﬁed; see [Brown 1982], section VI.9, for details. An example of a
group satisfying (a) but not (b) is the dihedral group D2m for odd m > 1 . There is also a much more difﬁcult converse: A ﬁnite group satisfying (a) and (b)
acts freely on Sn for some n . References for this are [Madsen, Thomas, & Wall 1976]
and [Davis & Milgram 1985]. There is also almost complete information about which
n ’s are possible for a given group.
Example 1.44. In Example 1.35 we constructed a contractible 2 complex Xm,n =
→ Tm,n × R as the universal cover of a ﬁnite 2 complex Xm,n that was the union of
the mapping cylinders of the two maps S1 S1 , z zm and z zn . The group
of deck transformations of this covering space is therefore the fundamental group
π1(Xm,n) . From van Kampen’s theorem applied to the decomposition of Xm,n into the two mapping cylinders we have the presentation a, b || amb−n for this group Gm,n = π1(Xm,n) . It is interesting to look at the action of Gm,n on Xm,n more closely. We described a decomposition of Xm,n into rectangles, with Xm,n the quotient of one rectangle. These rectangles in fact deﬁne a cell structure on Xm,n lifting a cell structure on Xm,n with two vertices, three edges, and one 2 cell. The group Gm,n is thus a group of symmetries of this cell structure on Xm,n . If we orient the three edges of Xm,n and lift these orientations to the edges of Xm,n , then Gm,n is the group of all symmetries of Xm,n preserving the orientations of edges. For example, the element a acts as a ‘screw motion’ about an axis that is a vertical line {va}× R with va a vertex of Tm,n , and b acts similarly for a vertex vb .
Since the action of Gm,n on Xm,n preserves the cell structure, it also preserves the product structure Tm,n × R . This means that there are actions of Gm,n on Tm,n and R such that the action on the product Xm,n = Tm,n × R is the diagonal action g(x, y) = g(x), g(y) for g ∈ Gm,n . If we make the rectangles of unit height in the R coordinate, then the element am = bn acts on R as unit translation, while a acts by 1/m translation and b by 1/n translation. The translation actions of a and b on R generate a group of translations of R that is inﬁnite cyclic, generated by translation
by the reciprocal of the least common multiple of m and n .
The action of Gm,n on Tm,n has kernel consisting of the powers of the element am = bn . This inﬁnite cyclic subgroup is precisely the center of Gm,n , as we saw in Example 1.24. There is an induced action of the quotient group Zm ∗ Zn on Tm,n , but this is not a free action since the elements a and b and all their conjugates ﬁx
→ vertices of Tm,n . On the other hand, if we restrict the action of Gm,n on Tm,n to
the kernel K of the map Gm,n Z given by the action of Gm,n on the R factor of Xm,n , then we do obtain a free action of K on Tm,n . Since this action takes vertices to vertices and edges to edges, it is a covering space action, so K is a free group, the
fundamental group of the graph Tm,n/K . An exercise at the end of the section is to determine Tm,n/K explicitly and compute the number of generators of K .

Covering Spaces

Section 1.3 75

Cayley Complexes

Covering spaces can be used to describe a very classical method for viewing

groups geometrically as graphs. Recall from Corollary 1.28 how we associated to each group presentation G = gα || rβ a 2 dimensional cell complex XG with π1(XG) ≈ G
by taking a wedge-sum of circles, one for each generator gα , and then attaching a
2 cell for each relator rβ . We can construct a cell complex XG with a covering space action of G such that XG/G = XG in the following way. Let the vertices of XG be the elements of G themselves. Then, at each vertex g ∈ G , insert an edge joining

g to ggα for each of the chosen generators gα . The resulting graph is known as the Cayley graph of G with respect to the generators gα . This graph is connected since every element of G is a product of gα ’s, so there is a path in the graph joining each vertex to the identity vertex e . Each relation rβ determines a loop in the graph, starting at any vertex g , and we attach a 2 cell for each such loop. The resulting cell

complex XG is the Cayley complex of G . The group G acts on XG by multiplication on the left. Thus, an element g ∈ G sends a vertex g ∈ G to the vertex gg , and the

edge from g to g gα is sent to the edge from gg to gg gα . The action extends to 2 cells in the obvious way. This is clearly a covering space action, and the orbit space

is just XG .

In fact XG is the universal cover of seen by considering the homomorphism

XG ϕ:

since it is simply-connected. This
π1(XG)→G deﬁned in the proof of

can be Propo-

sition 1.39. For an edge eα in XG corresponding to a generator gα of G , it is clear from the deﬁnition of ϕ that ϕ([eα]) = gα , so ϕ is an isomorphism. In particular

the kernel of ϕ , p∗ π1(XG) , is zero, hence also π1(XG) since p∗ is injective.

Let us look at some examples of Cayley complexes.

Example 1.45. When G is the free group on
two generators a and b , XG is S1 ∨ S1 and XG is the Cayley graph of Z ∗ Z pictured at the right. The action of a on this graph is a
rightward shift along the central horizontal

b2 ba 1

b ba

a 1b a1

ab
a e

axis, while b acts by an upward shift along a 2

a2

the central vertical axis. The composition a 1b 1

ab 1

ab of these two shifts then takes the vertex
e to the vertex ab . Similarly, the action of any w ∈ Z ∗ Z takes e to the vertex w .

b 1a 1 b2

b 1a b1

Example 1.46. The group G = Z× Z with presentation x, y || xyx−1y−1 has XG
the torus S1 × S1 , and XG is R2 with vertices the integer lattice Z2 ⊂ R2 and edges
the horizontal and vertical segments between these lattice points. The action of G is by translations (x, y) (x + m, y + n) .

76 Chapter 1

The Fundamental Group

Example 1.47. For G = Z2 = x || x2 , XG is RP2 and XG = S2 . More generally, for
Zn = x || xn , XG is S1 with a disk attached by the map z zn and XG consists of n disks D1, ··· , Dn with their boundary circles identiﬁed. A generator of Zn acts on this union of disks by sending Di to Di+1 via a 2π /n rotation, the subscript i being taken mod n . The common boundary circle of the disks is rotated by 2π /n .
Example 1.48. If G = Z2 ∗ Z2 = a, b || a2, b2 then the Cayley graph is a union of
an inﬁnite sequence of circles each tangent to its two neighbors.

b

a

b

a

b

a

bab b ba a b b e a a b ab a aba

We obtain XG from this graph by making each circle the equator of a 2 sphere, yield-
ing an inﬁnite sequence of tangent 2 spheres. Elements of the index-two normal subgroup Z ⊂ Z2 ∗ Z2 generated by ab act on XG as translations by an even number of units, while each of the remaining elements of Z2 ∗ Z2 acts as the antipodal map on
one of the spheres and ﬂips the whole chain of spheres end-for-end about this sphere. The orbit space XG is RP2 ∨ RP2 .
It is not hard to see the generalization of this example to Zm ∗ Zn with the presentation a, b || am, bn , so that XG consists of an inﬁnite union of copies of the
Cayley complexes for Zm and Zn constructed in Example 1.47, arranged in a tree-like pattern. The case of Z2 ∗ Z3 is pictured below.

ba a

a

a b

ba b

a

b e

a ab2 b

a

b2 b a

ab a b

a

a

Covering Spaces

Section 1.3 77

Exercises

1. For a covering space p : X→X and a subspace A ⊂ X , let A = p−1(A) . Show that the restriction p : A→A is a covering space. → → 2. Show that if p1 : X1 X1 and p2 : X2 X2 are covering spaces, so is their product → p1 × p2 : X1 × X2 X1 × X2 . 3. Let p : X→X be a covering space with p−1(x) ﬁnite for all x ∈ X . Show that X is

compact Hausdorff iff X is compact Hausdorff.

4. Construct a simply-connected covering space of the space X ⊂ R3 that is the union

of a sphere and a diameter. Do the same when X is the union of a sphere and a circle

intersecting it in two points.

5. Let X be the subspace of R2 consisting of the four sides of the square [0, 1]× [0, 1]

together with the segments of the vertical lines x
Show that for every covering space X→X there

= is

1/2, 1/3, 1/4, ··· inside the some neighborhood of

square. the left

edge of X that lifts homeomorphically to X . Deduce that X has no simply-connected

covering space.

6. Let X be the shrinking wedge of circles in Example 1.25, and let X be its covering space shown in the ﬁgure below.

Construct a two-sheeted covering space Y →X such that the composition Y →X→X
of the two covering spaces is not a covering space. Note that a composition of two covering spaces does have the unique path lifting property, however.

7. Let Y be the quasi-circle shown in the ﬁgure, a closed subspace of R2 consisting of a portion of the graph of y = sin(1/x) , the segment [−1, 1] in the y axis, and an arc connecting these two
pieces. Collapsing the segment of Y in the y axis to a point
gives a quotient map f : Y →S1 . Show that f does not lift to the covering space R→S1 , even though π1(Y ) = 0 . Thus local
path-connectedness of Y is a necessary hypothesis in the lifting criterion.

8. Let X and Y be simply-connected covering spaces of the path-connected, locally path-connected spaces X and Y . Show that if X Y then X Y . [Exercise 10 in Chapter 0 may be helpful.]

9. Show that if then every map

a path-connected, locally
X→S1 is nullhomotopic.

path-connected space X [Use the covering space

Rh→as Sπ11.](X

)

ﬁnite,

10. Find all the connected 2 sheeted and 3 sheeted covering spaces of S1 ∨ S1 , up to

isomorphism of covering spaces without basepoints.

78 Chapter 1

The Fundamental Group

11. Construct ﬁnite graphs X1 and X2 having a common ﬁnite-sheeted covering space X1 = X2 , but such that there is no space having both X1 and X2 as covering spaces.
12. Let a and b be the generators of π1(S1 ∨ S1) corresponding to the two S1 summands. Draw a picture of the covering space of S1 ∨ S1 corresponding to the normal subgroup generated by a2 , b2 , and (ab)4 , and prove that this covering space is indeed the correct one.
13. Determine the covering space of S1 ∨ S1 corresponding to the subgroup of π1(S1 ∨ S1) generated by the cubes of all elements. The covering space is 27 sheeted and can be drawn on a torus so that the complementary regions are nine triangles with edges labeled aaa , nine triangles with edges labeled bbb , and nine hexagons with edges labeled ababab . [For the analogous problem with sixth powers instead of cubes, the resulting covering space would have 228325 sheets! And for kth powers with k sufﬁciently large, the covering space would have inﬁnitely many sheets. The underlying group theory question here, whether the quotient of Z ∗ Z obtained by factoring out all kth powers is ﬁnite, is known as Burnside’s problem. It can also be asked for a free group on n generators.]
14. Find all the connected covering spaces of RP2 ∨ RP2 .
15. Let p : X→X be a simply-connected covering space of X and let A ⊂ X be a
path-connected, locally path-connected subspace, with A ⊂ X a path-component of
p−1(A) . Show that p : A→A is the covering space corresponding to the kernel of the map π1(A)→π1(X) . 16. Given maps X→Y →Z such that both Y →Z and the composition X→Z are covering spaces, show that X→Y is a covering space if Z is locally path-connected, and show that this covering space is normal if X→Z is a normal covering space.
17. Given a group G and a normal subgroup N , show that there exists a normal
covering space X→X with π1(X) ≈ G , π1(X) ≈ N , and deck transformation group
G(X) ≈ G/N .
18. For a path-connected, locally path-connected, and semilocally simply-connected
space X , call a path-connected covering space X→X abelian if it is normal and has
abelian deck transformation group. Show that X has an abelian covering space that is a covering space of every other abelian covering space of X , and that such a ‘universal’ abelian covering space is unique up to isomorphism. Describe this covering space explicitly for X = S1 ∨ S1 and X = S1 ∨ S1 ∨ S1 .
19. Use the preceding problem to show that a closed orientable surface Mg of genus g has a connected normal covering space with deck transformation group isomorphic to Zn (the product of n copies of Z ) iff n ≤ 2g . For n = 3 and g ≥ 3 , describe such a covering space explicitly as a subspace of R3 with translations of R3 as deck transformations. Show that such a covering space in R3 exists iff there is an embedding

Covering Spaces

Section 1.3 79

→ of Mg in the 3 torus T 3 = S1 × S1 × S1 such that the induced map π1(Mg) π1(T 3)
is surjective.
20. Construct nonnormal covering spaces of the Klein bottle by a Klein bottle and by a torus.
21. Let X be the space obtained from a torus S1 × S1 by attaching a M¨obius band via a homeomorphism from the boundary circle of the M¨obius band to the circle S1 × {x0} in the torus. Compute π1(X) , describe the universal cover of X , and describe the action of π1(X) on the universal cover. Do the same for the space Y obtained by attaching a M¨obius band to RP2 via a homeomorphism from its boundary circle to a circle in RP2 lifting to the equator in the covering space S2 of RP2 .
22. Given covering space actions of groups G1 on X1 and G2 on X2 , show that the action of G1 × G2 on X1 × X2 deﬁned by (g1, g2)(x1, x2) = (g1(x1), g2(x2)) is a covering space action, and that (X1 × X2)/(G1 × G2) is homeomorphic to X1/G1 × X2/G2 .
23. Show that if a group G acts freely and properly discontinuously on a Hausdorff space X , then the action is a covering space action. (Here ‘properly discontinuously’ means that each x ∈ X has a neighborhood U such that { g ∈ G | U ∩ g(U ) ≠ ∅ } is ﬁnite.) In particular, a free action of a ﬁnite group on a Hausdorff space is a covering space action.
24. Given a covering space action of a group G on a path-connected, locally pathconnected space X , then each subgroup H ⊂ G determines a composition of covering
spaces X→X/H→X/G . Show:
(a) Every path-connected covering space between X and X/G is isomorphic to X/H for some subgroup H ⊂ G .
(b) Two such covering spaces X/H1 and X/H2 of X/G are isomorphic iff H1 and H2 are conjugate subgroups of G .
(c) The covering space X/H→X/G is normal iff H is a normal subgroup of G , in
which case the group of deck transformations of this cover is G/H .
25. Let ϕ : R2→R2 be the linear transformation ϕ(x, y) = (2x, y/2) . This generates
an action of Z on X = R2 − {0} . Show this action is a covering space action and compute π1(X/Z) . Show the orbit space X/Z is non-Hausdorff, and describe how it is a union of four subspaces homeomorphic to S1 × R , coming from the complementary components of the x axis and the y axis.
26. For a covering space p : X→X with X connected, locally path-connected, and
semilocally simply-connected, show:
(a) The components of X are in one-to-one correspondence with the orbits of the action of π1(X, x0) on the ﬁber p−1(x0) .
(b) Under the Galois correspondence between connected covering spaces of X and subgroups of π1(X, x0) , the subgroup corresponding to the component of X

80 Chapter 1

The Fundamental Group

containing a given lift x0 of x0 is the stabilizer of x0 , the subgroup consisting of elements whose action on the ﬁber leaves x0 ﬁxed.
27. For a universal cover p : X→X we have two actions of π1(X, x0) on the ﬁber
p−1(x0) , namely the action given by lifting loops at x0 and the action given by restricting deck transformations to the ﬁber. Are these two actions the same when X = S1 ∨ S1 or X = S1 × S1 ? Do the actions always agree when π1(X, x0) is abelian?
28. Generalize the proof of Theorem 1.7 to show that for a covering space action of a
group G on a simply-connected space Y , π1(Y /G) is isomorphic to G . [If Y is locally path-connected, this is a special case of part (b) of Proposition 1.40.]

29. Let Y be path-connected, locally path-connected, and simply-connected, and let

G1 and G2 be subgroups of Homeo(Y ) deﬁning covering space actions on Y . Show that the orbit spaces Y /G1 and Y /G2 are homeomorphic iff G1 and G2 are conjugate subgroups of Homeo(Y ) .

30. Draw the Cayley graph of the group Z ∗ Z2 = a, b || b2 . 31. Show that the normal covering spaces of S1 ∨ S1 are precisely the graphs that

are Cayley graphs of groups with two generators. More generally, the normal cov-

ering spaces of the wedge sum of n circles are the Cayley graphs of groups with n

generators.
32. Consider covering spaces p : X→X with X and X connected CW complexes,

the cells of X projecting homeomorphically onto cells of X . Restricting p to the

1 skeleton then gives a covering space X1→X1 over the 1 skeleton of X . Show:

(a) (b)

TXXw11→→o XsXuc1ishaacnondvoeXrrm21in→agl scXpo1vaecaerrisengXis1os→pmaocXrepaihfnficdX. X1→2→XX1

are isomorphic iff the restrictions is normal.

(c) The groups of deck transformations of the coverings X→X and X1→X1 are

isomorphic, via the restriction map.

33. In Example 1.44 let d be the greatest common divisor of m and n , and let m = m/d and n = n/d . Show that the graph Tm,n/K consists of m vertices labeled a , n vertices labeled b , together with d edges joining each a vertex to each b vertex. Deduce that the subgroup K ⊂ Gm,n is free on m n − m − n + 1 generators.

Graphs and Free Groups

Section 1.A 81

Since all groups can be realized as fundamental groups of spaces, this opens the way for using topology to study algebraic properties of groups. The topics in this section and the next give some illustrations of this principle, mainly using covering space theory.
We remind the reader that the Additional Topics which form the remainder of this chapter are not to be regarded as an essential part of the basic core of the book. Readers who are eager to move on to new topics should feel free to skip ahead.
By deﬁnition, a graph is a 1 dimensional CW complex, in other words, a space X obtained from a discrete set X0 by attaching a collection of 1 cells eα . Thus X is obtained from the disjoint union of X0 with closed intervals Iα by identifying the two endpoints of each Iα with points of X0 . The points of X0 are the vertices and the 1 cells the edges of X . Note that with this deﬁnition an edge does not include its endpoints, so an edge is an open subset of X . The two endpoints of an edge can be the same vertex, so the closure eα of an edge eα is homeomorphic either to I or S1 .
Since X has the quotient topology from the disjoint union X0 α Iα , a subset of X is open (or closed) iff it intersects the closure eα of each edge eα in an open (or closed) set in eα . One says that X has the weak topology with respect to the subspaces eα . In this topology a sequence of points in the interiors of distinct edges forms a closed subset, hence never converges. This is true in particular if the edges containing the sequence all have a common vertex and one tries to choose the sequence so that it gets ‘closer and closer’ to the vertex. Thus if there is a vertex that is the endpoint of inﬁnitely many edges, then the weak topology cannot be a metric topology. An exercise at the end of this section is to show the converse, that the weak topology is a metric topology if each vertex is an endpoint of only ﬁnitely many edges.
A basis for the topology of X consists of the open intervals in the edges together with the path-connected neighborhoods of the vertices. A neighborhood of the latter sort about a vertex v is the union of connected open neighborhoods Uα of v in eα for all eα containing v . In particular, we see that X is locally path-connected. Hence a graph is connected iff it is path-connected.
If X has only ﬁnitely many vertices and edges, then X is compact, being the continuous image of the compact space X0 α Iα . The converse is also true, and more generally, a compact subset C of a graph X can meet only ﬁnitely many vertices and edges of X . To see this, let the subspace D ⊂ C consist of the vertices in C together with one point in each edge that C meets. Then D is a closed subset of X since it

82 Chapter 1

The Fundamental Group

meets each eα in a closed set. For the same reason, any subset of D is closed, so D has the discrete topology. But D is compact, being a closed subset of the compact space C , so D must be ﬁnite. By the deﬁnition of D this means that C can meet only ﬁnitely many vertices and edges.
A subgraph of a graph X is a subspace Y ⊂ X that is a union of vertices and edges of X , such that eα ⊂ Y implies eα ⊂ Y . The latter condition just says that Y is a closed subspace of X . A tree is a contractible graph. By a tree in a graph X we mean a subgraph that is a tree. We call a tree in X maximal if it contains all the vertices of X . This is equivalent to the more obvious meaning of maximality, as we will see below.

Proposition 1A.1. Every connected graph contains a maximal tree, and in fact any
tree in the graph is contained in a maximal tree.

Proof: Let X be a connected graph. We will describe a construction that embeds

an arbitrary subgraph X0 ⊂ X as a deformation retract of a subgraph Y ⊂ X that

contains all the vertices of X . By choosing X0 to be any subtree of X , for example a

single vertex, this will prove the proposition.

As a preliminary step, we construct a sequence of subgraphs X0 ⊂ X1 ⊂ X2 ⊂ ··· , letting Xi+1 be obtained from Xi by adjoining the closures eα of all edges eα ⊂ X −Xi

having at least one endpoint in Xi . The union i Xi is open in X since a neighborhood

of a point in Xi is contained in Xi+1 . Furthermore, i Xi is closed since it is a union of closed edges and X has the weak topology. So X = i Xi since X is connected.
Now to construct Y we begin by setting Y0 = X0 . Then inductively, assuming that Yi ⊂ Xi has been constructed so as to contain all the vertices of Xi , let Yi+1 be obtained from Yi by adjoining one edge connecting each vertex of Xi+1 −Xi to Yi , and let Y = i Yi . It is evident that Yi+1 deformation retracts to Yi , and we may obtain a deformation retraction of Y to Y0 = X0 by performing the deformation retraction of Yi+1 to Yi during the time interval [1/2i+1, 1/2i] . Thus a point x ∈ Yi+1 − Yi is

stationary ing until it

until this interval, when it reaches Y0 . The resulting

moves into homotopy

Yi ht :

and thereafter continues mov-
Y →Y is continuous since it is

continuous on the closure of each edge and Y has the weak topology.

Given a maximal tree T ⊂ X and a base vertex x0 ∈ T , then each edge eα of X − T determines a loop fα in X that goes ﬁrst from x0 to one endpoint of eα by a path in T , then across eα , then back to x0 by a path in T . Strictly speaking, we should ﬁrst orient the edge eα in order to specify which direction to cross it. Note that the homotopy class of fα is independent of the choice of the paths in T since T is simply-connected.
Proposition 1A.2. For a connected graph X with maximal tree T , π1(X) is a free
group with basis the classes [fα] corresponding to the edges eα of X − T .

Graphs and Free Groups

Section 1.A 83

In particular this implies that a maximal tree is maximal in the sense of not being contained in any larger tree, since adjoining any edge to a maximal tree produces a graph with nontrivial fundamental group. Another consequence is that a graph is a tree iff it is simply-connected.
Proof: The quotient map X→X/T is a homotopy equivalence by Proposition 0.17.
The quotient X/T is a graph with only one vertex, hence is a wedge sum of circles, whose fundamental group we showed in Example 1.21 to be free with basis the loops given by the edges of X/T , which are the images of the loops fα in X .
Here is a very useful fact about graphs:
Lemma 1A.3. Every covering space of a graph is also a graph, with vertices and
edges the lifts of the vertices and edges in the base graph.
Proof: Let p : X→X be the covering space. For the vertices of X we take the discrete goseefttaaXg0uran=piqhpuae−n1ld(ifXta0pI)αp. →lyWinXrgitiptnhagessXpinagtahsthlairfotqiuungoghtipeernaotcphseprptaoycientotoftinhXep0r−e1s(αuxlIt)αi,nagfosmrinaxptsh∈eIeαdα→e.ﬁXnTih,teiwosnee
lifts deﬁne the edges of a graph structure on X . The resulting topology on X is the same as its original topology since both topologies have the same basic open sets, the
covering projection X→X being a local homeomorphism.

We can now apply what we have proved about graphs and their fundamental groups to prove a basic fact of group theory:

Theorem 1A.4. Every subgroup of a free group is free.
Proof: Given a free group F , choose a graph X with π1(X) ≈ F , for example a wedge
of circles corresponding to a basis for F . For each subgroup G of F there is by
Proposition 1.36 a covering space p : X→X with p∗ π1(X) = G , hence π1(X) ≈ G
since p∗ is injective by Proposition 1.31. Since X is a graph by the preceding lemma, the group G ≈ π1(X) is free by Proposition 1A.2.

The structure of trees can be elucidated by looking more closely at the construc-

tions in the proof of Proposition 1A.1. If X is a tree and v0 is any vertex of X , then the construction of a maximal tree Y ⊂ X starting with Y0 = {v0} yields an increasing sequence of subtrees Yn ⊂ X whose union is

all of X since a tree has only one maximal subtree, namely itself.

We can think of the vertices in Yn − Yn−1 as being at ‘height’ n ,

with the edges of of height n − 1 .

Yn − Yn−1 connecting these In this way we get a ‘height

vertices to vertices
function’ h : X→R

assigning to each vertex its height, and monotone on edges.

84 Chapter 1

The Fundamental Group

For each vertex v of X there is exactly one edge leading downward from v , so by following these downward edges we obtain a path from v to the base vertex v0 . This is an example of an edgepath, which is a composition of ﬁnitely many paths each consisting of a single edge traversed monotonically. For any edgepath joining v to v0 other than the downward edgepath, the height function would not be monotone and hence would have local maxima, occurring when the edgepath backtracked, retracing some edge it had just crossed. Thus in a tree there is a unique nonbacktracking edgepath joining any two points. All the vertices and edges along this edgepath are distinct.
A tree can contain no subgraph homeomorphic to a circle, since two vertices in such a subgraph could be joined by more than one nonbacktracking edgepath. Conversely, if a connected graph X contains no circle subgraph, then it must be a tree. For if T is a maximal tree in X that is not equal to X , then the union of an edge of X − T with the nonbacktracking edgepath in T joining the endpoints of this edge is a circle subgraph of X . So if there are no circle subgraphs of X , we must have X = T , a tree.
For an arbitrary connected graph X and a pair of vertices v0 and v1 in X there is a unique nonbacktracking edgepath in each homotopy class of paths from v0 to v1 . This can be seen by lifting to the universal cover X , which is a tree since it is simplyconnected. Choosing a lift v0 of v0 , a homotopy class of paths from v0 to v1 lifts to a homotopy class of paths starting at v0 and ending at a unique lift v1 of v1 . Then the unique nonbacktracking edgepath in X from v0 to v1 projects to the desired nonbacktracking edgepath in X .
Exercises
1. Let X be a graph in which each vertex is an endpoint of only ﬁnitely many edges. Show that the weak topology on X is a metric topology.
2. Show that a connected graph retracts onto any connected subgraph.
3. For a ﬁnite graph X deﬁne the Euler characteristic χ (X) to be the number of vertices minus the number of edges. Show that χ (X) = 1 if X is a tree, and that the rank (number of elements in a basis) of π1(X) is 1 − χ (X) if X is connected. 4. If X is a ﬁnite graph and Y is a subgraph homeomorphic to S1 and containing the basepoint x0 , show that π1(X, x0) has a basis in which one element is represented by the loop Y .
5. Construct a connected graph X and maps f , g : X→X such that f g = 11 but f
and g do not induce isomorphisms on π1 . [Note that f∗g∗ = 11 implies that f∗ is surjective and g∗ is injective.]
6. Let F be the free group on two generators and let F be its commutator subgroup. Find a set of free generators for F by considering the covering space of the graph S1 ∨ S1 corresponding to F .

K(G,1) Spaces and Graphs of Groups

Section 1.B 85

7. If F is a ﬁnitely generated free group and N is a nontrivial normal subgroup of inﬁnite index, show, using covering spaces, that N is not ﬁnitely generated.
8. Show that a ﬁnitely generated group has only a ﬁnite number of subgroups of a given ﬁnite index. [First do the case of free groups, using covering spaces of graphs. The general case then follows since every group is a quotient group of a free group.]
9. Using covering spaces, show that an index n subgroup H of a group G has at most n conjugate subgroups gHg−1 in G . Apply this to show that there exists a normal subgroup K ⊂ G of ﬁnite index with K ⊂ H . [For the latter statement, consider the intersection of all the conjugate subgroups gHg−1 . This is the maximal normal subgroup of G contained in H .]
10. Let X be the wedge sum of n circles, with its natural graph structure, and let
X→X be a covering space with Y ⊂ X a ﬁnite connected subgraph. Show there is a ﬁnite graph Z ⊃ Y having the same vertices as Y , such that the projection Y →X extends to a covering space Z→X .
11. Apply the two preceding problems to show that if F is a ﬁnitely generated free group and x ∈ F is not the identity element, then there is a normal subgroup H ⊂ F of ﬁnite index such that x ∉ H . Hence x has nontrivial image in a ﬁnite quotient group of F . In this situation one says F is residually ﬁnite. 12. Let F be a ﬁnitely generated free group, H ⊂ F a ﬁnitely generated subgroup, and x ∈ F − H . Show there is a subgroup K of ﬁnite index in F such that K ⊃ H and x ∉ K . [Apply Exercise 10.]
13. Let x be a nontrivial element of a ﬁnitely generated free group F . Show there is a ﬁnite-index subgroup H ⊂ F in which x is one element of a basis. [Exercises 4 and 10 may be helpful.]
14. Show that the existence of maximal trees is equivalent to the Axiom of Choice.

In this section we introduce a class of spaces whose homotopy type depends only
on their fundamental group. These spaces arise many places in topology, especially
in its interactions with group theory.
A path-connected space whose fundamental group is isomorphic to a given group
G and which has a contractible universal covering space is called a K ( G , 1) space. The ‘1’ here refers to π1 . More general K(G, n) spaces are studied in §4.2. All these spaces are called Eilenberg–MacLane spaces, though in the case n = 1 they were studied by

86 Chapter 1

The Fundamental Group

Hurewicz before Eilenberg and MacLane took up the general case. Here are some examples:
Example 1B.1. S1 is a K(Z, 1) . More generally, a connected graph is a K(G, 1) with
G a free group, since by the results of §1.A its universal cover is a tree, hence contractible.
Example 1B.2. Closed surfaces with inﬁnite π1 , in other words, closed surfaces other
than S2 and RP2 , are K(G, 1) ’s. This will be shown in Example 1B.14 below. It also follows from the theorem in surface theory that the only simply-connected surfaces without boundary are S2 and R2 , so the universal cover of a closed surface with inﬁnite fundamental group must be R2 since it is noncompact. Nonclosed surfaces deformation retract onto graphs, so such surfaces are K(G, 1) ’s with G free.
Example 1B.3. The inﬁnite-dimensional projective space RP∞ is a K(Z2, 1) since its
universal cover is S∞ , which is contractible. To show the latter fact, a homotopy from the identity map of S∞ to a constant map can be constructed in two stages as follows.
→ First, deﬁne ft : R∞ R∞ by ft(x1, x2, ···) = (1 − t)(x1, x2, ···) + t(0, x1, x2, ···) .
This takes nonzero vectors to nonzero vectors for all t ∈ [0, 1] , so ft/|ft| gives a homotopy from the identity map of S∞ to the map (x1, x2, ···) (0, x1, x2, ···) . Then a homotopy from this map to a constant map is given by gt/|gt| where gt(x1, x2, ···) = (1 − t)(0, x1, x2, ···) + t(1, 0, 0, ···) .
Example 1B.4. Generalizing the preceding example, we can construct a K(Zm, 1) as
an inﬁnite-dimensional lens space S∞/Zm , where Zm acts on S∞ , regarded as the unit sphere in C∞ , by scalar multiplication by mth roots of unity, a generator of this action being the map (z1, z2, ···) e2πi/m(z1, z2, ···) . It is not hard to check that this is a covering space action.
Example 1B.5. A product K(G, 1)× K(H, 1) is a K(G× H, 1) since its universal cover
is the product of the universal covers of K(G, 1) and K(H, 1) . By taking products of circles and inﬁnite-dimensional lens spaces we therefore get K(G, 1) ’s for arbitrary ﬁnitely generated abelian groups G . For example the n dimensional torus T n , the product of n circles, is a K(Zn, 1) .
Example 1B.6. For a closed connected subspace K of S3 that is nonempty, the com-
plement S3 −K is a K(G, 1) . This is a theorem in 3 manifold theory, but in the special case that K is a torus knot the result follows from our study of torus knot complements in Examples 1.24 and 1.35. Namely, we showed that for K the torus knot Km,n there is a deformation retraction of S3 − K onto a certain 2 dimensional complex Xm,n having contractible universal cover. The homotopy lifting property then implies that the universal cover of S3 − K is homotopy equivalent to the universal cover of Xm,n , hence is also contractible.

K(G,1) Spaces and Graphs of Groups

Section 1.B 87

Example 1B.7. It is not hard to construct a K(G, 1) for an arbitrary group G , us-
ing the notion of a ∆ complex deﬁned in §2.1. Let EG be the ∆ complex whose n simplices are the ordered (n + 1) tuples [g0, ··· , gn] of elements of G . Such an n simplex attaches to the (n − 1) simplices [g0, ··· , gi, ··· , gn] in the obvious way, just as a standard simplex attaches to its faces. (The notation gi means that this vertex is deleted.) The complex EG is contractible by the homotopy ht that slides each point x ∈ [g0, ··· , gn] along the line segment in [e, g0, ··· , gn] from x to the vertex [e] , where e is the identity element of G . This is well-deﬁned in EG since when we restrict to a face [g0, ··· , gi, ··· , gn] we have the linear deformation to [e] in [e, g0, ··· , gi, ··· , gn] . Note that ht carries [e] around the loop [e, e] , so ht is not actually a deformation retraction of EG onto [e] .
The group G acts on EG by left multiplication, an element g ∈ G taking the simplex [g0, ··· , gn] linearly onto the simplex [gg0, ··· , ggn] . Only the identity e takes any simplex to itself, so by an exercise at the end of this section, the action
of G on EG is a covering space action. Hence the quotient map EG→EG/G is the
universal cover of the orbit space BG = EG/G , and BG is a K(G, 1) .
Since G acts on EG by freely permuting simplices, BG inherits a ∆ complex
structure from EG . The action of G on EG identiﬁes all the vertices of EG , so BG
has just one vertex. To describe the ∆ complex structure on BG explicitly, note ﬁrst
that every n simplex of EG can be written uniquely in the form

[g0, g0g1, g0g1g2, ··· , g0g1 ··· gn] = g0[e, g1, g1g2, ··· , g1 ··· gn]

The image of this simplex in BG may be denoted unambiguously by the symbol

[g1|g2| ··· |gn] . In this ‘bar’ notation the gi ’s and their ordered products can be

used to label edges, viewing an edge label as the ratio between the two labels on the vertices

g0g1g 2

g0g1g2g3

g3 g2g3

g0g1g2

at the endpoints of the edge,

g1g2

g2

g1g2g3

g2

as indicated in the ﬁgure. With

this notation, the boundary of a simplex [g1| ··· |gn] of BG

g0

g1g2

g1

g0g1

g0

g1

g0g1

consists of the simplices [g2| ··· |gn] , [g1| ··· |gn−1] , and [g1| ··· |gigi+1| ··· |gn]

for i = 1, ··· , n − 1 .

This construction of a K(G, 1) produces a rather large space, since BG is al-
ways inﬁnite-dimensional, and if G is inﬁnite, BG has an inﬁnite number of cells in each positive dimension. For example, BZ is much bigger than S1 , the most efﬁcient
K(Z, 1) . On the other hand, BG has the virtue of being functorial: A homomorphism
f : G→H induces a map Bf : BG→BH sending a simplex [g1| ··· |gn] to the simplex
[f (g1)| ··· |f (gn)] . A different construction of a K(G, 1) is given in §4.2. Here one starts with any 2 dimensional complex having fundamental group G , for example

88 Chapter 1

The Fundamental Group

the complex XG associated to a presentation of G , and then one attaches cells of dimension 3 and higher to make the universal cover contractible without affecting π1 . In general, it is hard to get any control on the number of higher-dimensional cells needed in this construction, so it too can be rather inefﬁcient. Indeed, ﬁnding an efﬁcient K(G, 1) for a given group G is often a difﬁcult problem.
It is a curious and almost paradoxical fact that if G contains any elements of ﬁnite order, then every K(G, 1) CW complex must be inﬁnite-dimensional. This is shown in Proposition 2.45. In particular the inﬁnite-dimensional lens space K(Zm, 1) ’s in Example 1B.4 cannot be replaced by any ﬁnite-dimensional complex.
In spite of the great latitude possible in the construction of K(G, 1) ’s, there is a very nice homotopical uniqueness property that accounts for much of the interest in K(G, 1) ’s:

Theorem 1B.8. The homotopy type of a CW complex K(G, 1) is uniquely determined
by G .

Having a unique homotopy type of K(G, 1) ’s associated to each group G means that algebraic invariants of spaces that depend only on homotopy type, such as homology and cohomology groups, become invariants of groups. This has proved to be a quite fruitful idea, and has been much studied both from the algebraic and topological viewpoints. The discussion following Proposition 2.45 gives a few references.
The preceding theorem will follow easily from:

Proposition 1B.9. Let X be a connected CW complex and let Y be a K(G, 1) . Then every homomorphism → π1(X, x0) π1(Y , y0) is induced by a map (X, x0)→(Y , y0)
that is unique up to homotopy ﬁxing x0 .

To deduce the theorem from this, let X and Y be CW complex K(G, 1) ’s with iso-

morphic fundamental groups. The proposition gives maps f
g : (Y , y0)→(X, x0) inducing inverse isomorphisms π1(X, x0)

: (X, x0)→(Y
≈ π1(Y , y0) .

, y0) and Then f g

and gf induce the identity on π1 and hence are homotopic to the identity maps.

Proof of 1B.9: Let us ﬁrst consider the case that X has a single 0 cell, the base-

point x0 . Given tion of a map f

a homomorphism
: (X, x0)→(Y , y0)

ϕ : π1(X, x0)→π1(Y , y0) , we begin the construc-
with f∗ = ϕ by setting f (x0) = y0 . Each 1 cell

eα1 of X has closure a circle determining an element [eα1 ] ∈ π1(X, x0) , and we let f on the closure of eα1 be a map representing ϕ([eα1 ]) . If i : X1 X denotes

π1( X 1−,−−i−x−−∗−−−0→) −−−f−∗→−−−π−−ϕ→1( Y,y0)

the inclusion, then ϕi∗ = f∗ since π1(X1, x0) is gen-

π1( X, x0)

erated by the elements [eα1 ] . To extend f over a cell eβ2

with

attaching

map

ψβ

: S1→X1 ,

all

we

need

is

for

the

composition f ψβ to be nullhomotopic. Choosing a basepoint s0 ∈ S1 and a path in X1

from ψβ(s0) to x0 , ψβ determines an element [ψβ] ∈ π1(X1, x0) , and the existence

K(G,1) Spaces and Graphs of Groups

Section 1.B 89

of a nullhomotopy of f ψβ is equivalent to f∗([ψβ]) being zero in π1(Y , y0) . We have i∗([ψβ]) = 0 since the cell eβ2 provides a nullhomotopy of ψβ in X . Hence f∗([ψβ]) = ϕi∗([ψβ]) = 0 , and so f can be extended over eβ2 .
→ → Extending f inductively over cells eγn with n > 2 is possible since the attaching
maps ψγ : Sn−1 Xn−1 have nullhomotopic compositions f ψγ : Sn−1 Y . This is

because f ψγ lifts to the universal cover of Y if n > 2 , and this cover is contractible

by

hypothesis, so the lift of f ψγ is nullhomotopic, hence also Turning to the uniqueness statement, if two maps f0, f1

f ψγ : (X,

itself.
x0)→(Y

,

y0)

in-

duce the same homomorphism on to X1 are homotopic, ﬁxing x0 .

π1 , To

then we see extend the

immediately that their restrictions
resulting map X1 × I ∪ X × ∂I→Y

over the remaining cells en × (0, 1) of X × I we can proceed just as in the preceding

paragraph since these cells have dimension n + 1 > 2 . Thus we obtain a homotopy
→ ft : (X, x0) (Y , y0) , ﬁnishing the proof in the case that X has a single 0 cell.

The case that X has more than one 0 cell can be treated by a small elaboration

on this argument. Choose a maximal tree T ⊂ X . To construct a map f realizing a

given ϕ , begin by setting f (T ) = y0 . Then each edge eα1 in X − T determines an element [eα1 ] ∈ π1(X, x0) , and we let f on the closure of eα1 be a map representing ϕ([eα1 ]) . Extending f over higher-dimensional cells then proceeds just as before.
→ Constructing a homotopy ft joining two given maps f0 and f1 with f0∗ = f1∗ also
has an extra step. Let ht : X1 X1 be a homotopy starting with h0 = 11 and restricting

to a deformation retraction of T onto x0 . (It is easy to extend such a deformation retraction to a homotopy deﬁned on all of X1 .) We can construct a homotopy from

f0|X1 to f1|X1 by ﬁrst deforming f0|X1 and f1|X1 to take T to y0 by composing with

ht , then applying the earlier argument to obtain a homotopy between the modiﬁed f0|X1 and f1|X1 . Having a homotopy f0|X1 f1|X1 we extend this over all of X in

the same way as before.

The ﬁrst part of the preceding proof also works for the 2 dimensional complexes

XalGizeadssaoscitahteedintdoupcreedsehnotmatoiomnosrpofhigsrmouopfss. oTmheusmeavperXyGh→omXoHm. oHrpohwiesvmerG, t→heHre

is is

reno

uniqueness statement for this map, and it can easily happen that different presenta-

tions of a group G give XG ’s that are not homotopy equivalent.

Graphs of Groups
As an illustration of how K(G, 1) spaces can be useful in group theory, we shall describe a procedure for assembling a collection of K(G, 1) ’s together into a K(G, 1) for a larger group G . Group-theoretically, this gives a method for assembling smaller groups together to form a larger group, generalizing the notion of free products.
Let Γ be a graph that is connected and oriented, that is, its edges are viewed as arrows, each edge having a speciﬁed direction. Suppose that at each vertex v of Γ we

90 Chapter 1

The Fundamental Group

place a group Gv and along each edge e of Γ we put a homomorphism ϕe from the group at the tail of the edge to the group at the head of the edge. We call this data a graph of groups. Now build a space BΓ by putting the space BGv from Example 1B.7 at each vertex v of Γ and then ﬁlling in a mapping cylinder of the map Bϕe along each edge e of Γ , identifying the two ends of the mapping cylinder with the two BGv ’s at the ends of e . The resulting space BΓ is then a CW complex since the maps Bϕe take n cells homeomorphically onto n cells. In fact, the cell structure on BΓ can be canonically subdivided into a ∆ complex structure using the prism construction from the proof of Theorem 2.10, but we will not need to do this here.
More generally, instead of BGv one could take any CW complex K(Gv , 1) at the vertex v , and then along edges put mapping cylinders of maps realizing the homomorphisms ϕe . We leave it for the reader to check that the resulting space K Γ is homotopy equivalent to the BΓ constructed above.
Example 1B.10. Suppose Γ consists of one central vertex with a number of edges
radiating out from it, and the group Gv at this central vertex is trivial, hence also all the edge homomorphisms. Then van Kampen’s theorem implies that π1(K Γ ) is the free product of the groups at all the outer vertices.
In view of this example, we shall call π1(K Γ ) for a general graph of groups Γ the graph product of the vertex groups Gv with respect to the edge homomorphisms ϕe . The name for π1(K Γ ) that is generally used in the literature is the rather awkward phrase, ‘the fundamental group of the graph of groups.’
Here is the main result we shall prove about graphs of groups:
Theorem 1B.11. If all the edge homomorphisms ϕe are injective, then K Γ is a
K(G, 1) and the inclusions K(Gv , 1) K Γ induce injective maps on π1 .
Before giving the proof, let us look at some interesting special cases:
Example 1B.12: Free Products with Amalgamation. Suppose the graph of groups is
A← C→B , with the two maps monomorphisms. One can regard this data as speci-
fying embeddings of C as subgroups of A and B . Applying van Kampen’s theorem to the decomposition of K Γ into its two mapping cylinders, we see that π1(K Γ ) is the quotient of A ∗ B obtained by identifying the subgroup C ⊂ A with the subgroup C ⊂ B . The standard notation for this group is A ∗C B , the free product of A and B amalgamated along the subgroup C . According to the theorem, A ∗C B contains both A and B as subgroups.
For example, a free product with amalgamation Z ∗Z Z can be realized by map-
ping cylinders of the maps S1 ← S1→S1 that are m sheeted and n sheeted covering
spaces, respectively. We studied this case in Examples 1.24 and 1.35 where we showed that the complex K Γ is a deformation retract of the complement of a torus knot in S3 if m and n are relatively prime. It is a basic result in 3 manifold theory that the