The Classification Theorem for Compact Surfaces
by

Kateryna Tretiakova
a thesis submitted in partial fulfillment
of the requirements for the degree of
Bachelor of Science (Hons.)
in the
Department of Mathematics & Statistics

©Kateryna Tretiakova 2023

We accept this thesis as conforming to the required standards:

Saeed Rahmati
Dept. of Mathematics & Statistics
Thesis Supervisor

Kyle Schlitt
Dept. of Mathematics & Statistics

Shirin Boroushaki
Dept. of Mathematics & Statistics

Dated March 6 2023, Kamloops, British Columbia, Canada

THOMPSON RIVERS UNIVERSITY
DEPARTMENT OF MATHEMATICS & STATISTICS
Permission is herewith granted to Thompson Rivers University to circulate and to have copied
for non-commercial purposes, at its discretion, the above title upon request of individuals or
institutions.

-------------------------------Signature of Author

the author reserves other publication rights, and neither the thesis nor extensive extracts from it may be printed or otherwise reproduced without the
author’s written permission.
the author attests that permission has been obtained for the use of
any copy-righted material appearing in this thesis (other than brief excerpts
requiring only proper acknowledgement in scholarly righting) and that all
such use is clearly acknowledged.

ii

Abstract
The Classification of Surfaces is one of the problems which gave rise to the modern topology. It
has become one of the signature theorems of the area, which now is called algebraic topology.
It states that any closed connected surface is homeomorphic to the sphere, the connected sum
of tori, or the connected sum of projective planes. In this thesis we are going to go over the
geometric, topological, and algebraic tools necessary for understanding, proving and using the
theorem together with some useful examples of surfaces.
Thesis itself consists of three chapters. The first part talks about homotopy theory
and defines the fundamental group, which is an algebraic invariant between topological spaces.
In addition, we learn some basic ways of calculating the fundamental group for some easyto-imagine examples. The second chapter introduces free groups and free products, which
altogether let us calculate the fundamental group in more complex cases. The third and
final chapter introduces the geometric ideas behind the classification theorem, which includes
polygonal regions and labelling schemes together with operations on them. As a result, we
overview the construction of any two-dimensional compact surface and classification theorem
as a main goal.

iii

Acknowledgements
I would like to thank my supervisor Dr. Saeed Rahmati. It has been a new type of adventure
for me and you helped me grow a lot through it.
I would also like to thank all people who once in a while needed to remind me why I love
math and writing this thesis as a result. This includes multiple professors at Thompson Rivers
University, graduate students I have met on the way, and high school kids I have been training
and teaching.
Also a huge thank you to all professors of mathematics in the world who post their notes
online. I cannot reference them but this thesis would have never been finished without all of
you.

iv

Contents
Abstract

iii

Acknowledgements

iv

1 The Fundamental Group
1.1 Homotopy of Paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 The Fundamental Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Covering Spaces and the Fundamental Group of a Circle . . . . . . . . . . . . .
1.4 Retractions and Deformation Retracts . . . . . . . . . . . . . . . . . . . . . . .
1.5 Homotopy Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.6 Fundamental Groups of Other Surfaces . . . . . . . . . . . . . . . . . . . . . . .

1
1
6
10
13
14
16

2 Free Groups
2.1 Free Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Free Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 The Seifert-van Kampen Theorem . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 The Fundamental Group of a Wedge of Circles . . . . . . . . . . . . . . . . . .
2.5 Adjoining a Two-Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20
20
25
26
32
34

3 Classification of Surfaces
3.1 Fundamental Groups of Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Homology of Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Cutting and Pasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 The Classification Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5 Construction of Compact Surfaces . . . . . . . . . . . . . . . . . . . . . . . . .

37
37
43
45
48
54

Appendix A Topology

58

Appendix B Category Theory
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

62
64

v

Chapter 1

The Fundamental Group
1.1

Homotopy of Paths

Two continuous functions from one topological space to another are called homotopic if one can
be “continuously deformed” into the other. Such a deformation is called a homotopy between
the two functions and that is what this chapter is about.
Definition 1.1.1. Continuous maps f0 , f1 : X Ñ Y are said to be homotopic, which is
denoted by f0 » f1 , when there is a continuous map F : X ˆ I Ñ Y , called a homotopy from
f0 to f1 , such that for all points x P X we have F px, 0q “ f0 pxq and F px, 1q “ f1 pxq. With
this, we can define ft0 psq “ F ps, t0 q to be a path from x0 to x1 obtained at t “ t0 .
If f is homotopic to a constant map, i.e., if f » consty for some y P Y , then we say that f is
nulhomotopic.
Theorem 1.1.2. The homotopy relation » is an equivalence relation on the set M appX, Y q
of continuous functions from X to Y .
Proof. Let f, f0 , f1 , f2 : X Ñ Y be continuous maps. We need to check all the conditions of
an equivalence relation:
• Reflexivity (f » f ):
Take the map F : X ˆ I Ñ Y , F px, tq “ f pxq. Since for all t P I and for all x P X we
have F px, tq “ f pxq, F is a homotopy from f to f .
• Symmetry(f0 » f1 ñ f1 » f0 ):
Let F : X ˆI Ñ Y be a homotopy from f0 to f1 , i.e. F px, 0q “ f0 pxq and F px, 1q “ f1 pxq.
Take G : X ˆ I Ñ Y such that Gpx, tq “ F px, 1 ´ tq. This function is continuous since it
is defined as a composition of continuous functions. Moreover, Gpx, 0q “ F px, 1q “ f1 pxq
and Gpx, qq “ F px, 0q “ f0 pxq. Thus, G is a homotopy from f1 to f0 .
• Transitivity(f0 » f1 and f1 » f2 ñ f0 » f2 ):
Let F1 : X ˆ I Ñ Y such that F1 px, 0q “ f0 pxq and F1 px, 1q “ f1 pxq be a homotopy from
1

HOMOTOPY OF PATHS

2

f0 to f1 . Also, let F2 : X ˆ I Ñ Y such that F2 px, 0q “ f1 pxq and F2 px, 1q “ f2 pxq be a
homotopy from f1 to f2 . Take
#
F1 px, 2tq
0 ď t ď 1{2,
F12 px, tq “
F2 px, 2t ´ 1q 1{2 ď t ď 1.
By pasting lemma A.0.28, its components are continuous and since at t “ 1{2 we have
F1 px, 1q “ f1 pxq and F2 px, 1 ´ 1q “ F2 px, 0q “ f1 pxq, F12 is well-defined and a homotopy
from f0 to f2 .

We shall denote the homotopy class of a continuous map f : X Ñ Y by rf s. That is, rf s “
tg P M appX, Y q | g » f u. Moreover, we shall denote the set of homotopy classes of continuous
maps from X to Y by rX, Y s “ M appX, Y q{ ».
Example 1.1.3. Let f, g : R Ñ R be any two continuous, real functions. To show that
f » g, consider a function F px, tq “ p1 ´ tq ¨ f pxq ` t ¨ gpxq. Being a composition of continuous
functions F is continuous. Moreover, F px, 0q “ p1 ´ 0q ¨ f pxq ` 0 ¨ gpxq “ f pxq and F px, 1q “
p1 ´ 1q ¨ f pxq ` 1 ¨ gpxq “ gpxq. Thus, F is a homotopy between f and g. In particular, this
example shows that any continuous map f : R Ñ R is nulhomotopic.
Let’s consider the special case in which f is a path in X. Recall that if f : r0, 1s Ñ X
is a continuous map such that f p0q “ x0 and f p1q “ x1 , we say that f is a path in X from x0
from x1 . We call x0 the initial point and x1 the final point of the path f .
F p0, tq
f1 » f

I ˆI

x0
‚
F p1, tq
‚
x1

F :I ˆI ÑX
f

Figure 1.0.0: the path f can be continuously deformed into f 1
by a continuous deformation F .

HOMOTOPY OF PATHS

3

Definition 1.1.4. Two paths f0 and f1 from I “ r0, 1s to X are path homotopic if they have
the same initial point x0 and the final point x1 , and if there is a continuous map F : I ˆ I Ñ X
such that @s P I and @t P I:
F ps, 0q “ f0 psq,

F p0, tq “ x0 ,

F ps, 1q “ f1 psq,

F p1, tq “ x1 .

F in this case is called a path homotopy between f0 and f1 . If f0 is path homotopic to f1 ,
we write f0 »p f1 . See Figure 1.0.0.
Since the path homotopy is a special case of homotopy, one can conduce the following:
Theorem 1.1.5. The path-homotopy relation »p is an equivalence relation.
Example 1.1.6. Let f0 and f1 be any two maps of a
space X into R2 . Take F px, tq “ p1 ´ tqf0 pxq ` tf1 pxq
as in Example 1.1.3. See Figure 1.1.0. We already know
F is a homotopy map between f0 and f1 . This specific
description of a homotopy is called a straight-line (linear) homotopy since for any p P X it moves the point
f0 ppq to the point f1 ppq along the straight line segment. If
Figure 1.1.0: Straight-line
f0 and f1 are both paths, then F will be a path-homotopy
homotopy between f0 and f1 .
from f0 to f1 .
More generally, let A be any convex subset of R2 . Recall that a set C is convex if the line
segment between any two points in C lies in C, i.e., @x1 , x2 P C, @θ P r0, 1s, θx1 `p1´θqx2 P C.
Then any two paths f0 and f1 between x0 and x1 in A are path-homotopic, since for all s, t P A,
by definition of convex set, we have F ps, tq P A and
F ps, 0q “ f0 psq,

F p0, tq “ p1 ´ tqf0 p0q ` tf1 p0q “ f0 p0q,

F ps, 1q “ f1 psq,

F p1, tq “ p1 ´ tqf0 p1q ` tf1 p1q “ f0 p1q.

Definition 1.1.7. Given paths f and g such that f p1q “ gp0q, the product path f ˚ g (also
called composition or concatenation) is given by
#
f p2sq
0 ď s ď 1{2,
f ˚ gpsq “
gp2s ´ 1q 1{2 ď s ď 1.
Intuitively, the composition law is just given by following one path, and then the other with
twice the speed, as shown in Figure 1.1.1.
Because we need the endpoint of one path to be the beginning point for the other for composition, the set of paths does not form a group. However, if we assume that the paths start and
end at the same point (loops), then they do. We will take a closer look at this group in the
next section!
The operation ˚ can be applied to homotopy classes as well. Once again, let f : I Ñ X be a
path from x0 to x1 and let g : I Ñ X be a path from x1 to x2 . Define rf s ˚ rgs :“ rf ˚ gs.

HOMOTOPY OF PATHS

4

Figure 1.1.1: Construction of a concatenation [Mun00].
Theorem 1.1.8. The operation ˚ between homotopy classes is well-defined.
Proof. Let f 1 P rf s and g 1 P rgs. Because rf 1 s “ rf s and rg 1 s “ rgs, we need to check if
rf 1 s ˚ rg 1 s “ rf s ˚ rgs. Since f and f 1 are path-homotopic, there exists a path homotopy F from
f to f 1 . Likewise, there exists a path homotopy G from g to g 1 . Define H : I ˆ I Ñ X such
that:
#
F p2s, tq
0 ď s ď 1{2,
Hps, tq “
Gp2s ´ 1, tq 1{2 ď s ď 1.
Now we show that H is a path homotopy between f ˚ g and f 1 ˚ g 1 , which are both paths from
x0 to x2 . We know H is continuous by the pasting lemma A.0.28. Let’s check the conditions
from definition of the path-homotopy:
#
F p2s, 0q
0 ď s ď 1{2,
Hps, 0q “
Hp0, tq “ F p0, tq “ x0 ,
Gp2s ´ 1, 0q 1{2 ď s ď 1,
#
F p2s, 1q
0 ď s ď 1{2,
Hps, 1q “
Hp1, tq “ Gp1, tq “ x2 .
F p2s ´ 1, 1q 1{2 ď s ď 1,
The function H is therefore a path homotopy between f ˚ g and f 1 ˚ g 1 . Thus, rf ˚ gs is independent of the class representatives f and g and the operation ˚ is well-defined on equivalence
classes.
Lemma 1.1.9. Let k : X Ñ Y be a continuous function (map) and F be a path homotopy
in X between paths f and f 1 . Then k ˝ F is a path homotopy in Y between paths k ˝ f and
k ˝ f 1.
Proof. The function k ˝ F is continuous by being a composition of continuous functions. Let
x0 and x1 be respectively the initial and the final point of both f and f 1 . Then from the

HOMOTOPY OF PATHS

5

definition, we get the following:
k ˝ F ps, 0q “ kpF ps, 0qq “ kpf psqq “ k ˝ f psq,
k ˝ F ps, 1q “ kpF ps, 1qq “ kpf 1 psqq “ k ˝ f 1 psq,
k ˝ F p1, tq “ kpF p1, tqq “ kpx1 q,
k ˝ F p0, tq “ kpF p0, tqq “ kpx0 q,
which means that k ˝ F is indeed the required homotopy.
Lemma 1.1.10. Let f and g be paths such that f p1q “ gp0q and k : X Ñ Y be a map. Then
k ˝ pf ˚ gq “ pk ˝ f q ˚ pk ˝ gq.
Proof. Let’s start with the right side of the equality and use Definition 1.1.7:
#
k ˝ f p2s, tq
0 ď s ď 1{2,
pk ˝ f q ˚ pk ˝ gqps, tq “
k ˝ gp2s ´ 1, tq 1{2 ď s ď 1.
`
˘
which is equal to k ˝ pf ˚ gq ps, tq.
Theorem 1.1.11. Let f : I Ñ X be a path from x0 to x1 , g : I Ñ X be a path from x1 to x2 ,
and h : I Ñ X be a path from x2 to x3 . Define f¯ : I Ñ X such that f¯psq “ f p1 ´ sq. Given
x P X, let ex denote the constant path ex : I Ñ X carrying all the points of I to one point x
of X. The operation ˚ between homotopy classes has the following properties:
• Identity Elements: rex0 s ˚ rf s “ rf s and rf s ˚ rex1 s “ rf s.
• Inverses: rf s ˚ rf¯s “ rex0 s and rf¯s ˚ rf s “ rex1 s.
• Associativity: prf sq ˚ rgsq ˚ rhs “ rf s ˚ prgs ˚ rhsq.
Proof. Let’s prove separately all the parts of the theorem:
Identity elements: We want to show pex0 ˚ f q »p f . Let e0 : I Ñ I be the constant path at
0 and let i : I Ñ I be the identity path. Because I is convex and both paths e0 ˚ i and i are
paths from 0 to 1 by Example 1.1.6 these two paths are path-homotopic with homotopy F .
By Lemma 1.1.10, we know f ˝ pe0 ˚ iq “ pf ˝ e0 q ˚ pf ˝ iq “ ex0 ˚ f . Then F ˝ f is a homotopy
between f ˝ i and ex0 ˚ f . Similar argument can be done to show the argument towards the
right identity. It follows similarly rf s ˚ rex1 s “ rf s.
Inverses: We want to show pf ˚ f¯q »p ex0 . Given a path f in X from x0 to x1 , let f¯ be a
path from x1 to x0 such that f¯psq “ f p1 ´ sq. We know f ˚ f¯ “ pf ˝ iq ˚ pf ˝ īq “ f ˝ pi ˚ īq.
Similarly, we know ex0 “ f ˚ e0 . Since i ˚ ī and e0 are loops in I based at 0 and I is convex,
there exists the straight line path homotopy F between i ˚ ī and e0 . Then by Lemma 1.1.9,
f ˝ F is a path homotopy between f ˝ pi ˚ īq “ f ˚ f¯ and f ˚ e0 “ ex0 . Therefore, as before, we
find rf s ˚ rf¯s “ rex0 s. By similar reasoning, rf¯s ˚ rf s “ rex1 s.

THE FUNDAMENTAL GROUP

6

Associativity: By the definition of the concatenation one can write
$
’
&f p4sq 0 ď s ď 1{4,
pf ˚ gq ˚ h “ gp4sq 1{4 ď s ď 1{2,
’
%
hp2sq 1{2 ď s ď 1,

$
’
&f p2sq
f ˚ pg ˚ hq “ gp4sq
’
%
hp4sq

0 ď s ď 1{2,
1{2 ď s ď 3{4,
3{4 ď s ď 1.

$
’
0 ď s ď 1{2,
&s{2
Let k : I Ñ I be a map: kpsq “ s ´ 1{4 1{2 ď s ď 3{4,
’
%
2s ´ 1 3{4 ď s ď 1.
Then ppf ˚ gq ˚ hq ˝ k “ f ˚ pg ˚ hq “ pf ˚ pg ˚ hqq ˝ i. Because k and i are both paths from 0
to 1 in the convex space I, k and i are both path-homotopic by Example 1.1.6. Let F be a
path homotopy from k to i spoken of earlier. Then, ppf ˚ gq ˚ hq ˝ F is a path homotopy from
ppf ˚ gq ˚ hq ˝ k “ f ˚ pg ˚ hq to ppf ˚ gq ˚ hq ˝ i “ pf ˚ gq ˚ h. Thus, f ˚ pg ˚ hq »p pf ˚ gq ˚ h and
rf s ˚ prgs ˚ rhsq “ prf s ˚ rgsq ˚ rhs.
Now that we know ˚ is associative, we know rf1 s ˚ rf2 s ˚ ... ˚ rfn s is well-defined. In other words,
no matter how you chop the path, you have the product of all the pieces will give you the same
result. And one can use a very smart chopping in some cases!

1.2

The Fundamental Group

Definition 1.2.1. A path f : r0, 1s Ñ X is called a loop if f p0q “ f p1q. It is said to be based
at x if f p0q “ f p1q “ x. Moreover, a loop is nulhomotopic if it is homotopic to the constant
loop, i.e., the loop f : I Ñ X given by f ptq “ x0 for all t.
With the notion of loops, we can now talk about the group defined under the concatenation:
Definition 1.2.2. The fundamental group or the first homotopy group of X, π1 pX; x0 q,
is the set of equivalence classes of loops f : I Ñ X based at x0 .
Theorem 1.2.3. The fundamental group is a group under composition of loops.
Proof. Certainly composition is an operation taking loops to loops. We first look to see that
composition is well defined on homotopy classes. If f0 » f1 and g0 » g1 , then by composing
the homotopies we get a homotopy of f0 ˚ g0 to f1 ˚ g1 . All other properties of a group come
from Theorem 1.1.11.
Example 1.2.4. The space π1 pRn , x0 q, where x0 P Rn , has trivial fundamental group. To see
this, we have to show every loop is homotopic to the constant loop. For a loop f : I Ñ Rn at
x0 , consider the straight-line homotopy F ps, tq “ t ¨ f psq ` p1 ´ tq ¨ x0 . It defines a homotopy
between f and the trivial loop.
In particular, the unit ball B n in Rn : B n “ tx|x21 ` x22 ` ... ` x2n ď 1u has trivial
fundamental group since all loops at xo in the ball are nulhomotopic.

THE FUNDAMENTAL GROUP

7

Definition 1.2.5. Let α be a path in X from x0 to x1 . Define a “α-hat” map α̂ : π1 pX, x0 q Ñ
π1 pX, x1 q such that α̂prf sq “ rαs ˚ rf s ˚ rαs.
This function is well-defined from Theorem 1.1.8. Then
if f is a loop based at x0 then α̂prf sq is a loop based
at x1 . In other words, we now have a way of “moving” from one points to another using the path between
them, like shown on Figure 1.2.1.
Theorem 1.2.6. The map α̂ is a group isomorphism.
Proof. First, note that α̂ is a homomorphism since
α̂rf s ˚ α̂rgs “ prα̂s ˚ rf s ˚ rαsq ˚ prαs ˚ rgs ˚ rαsq
“ rα̂s ˚ rf s ˚ rgs ˚ rαs “ α̂prf s ˚ rgsq.
Figure 1.2.1: Both loops f1 and f2
We need to show that α̂ is an isomorphism as well.
which are based at x0 can be
Let rhs and rf s be elements of π1 pX, x1 q and π1 pX, x0 q,
transformed to be starting at x1 .
respectively. Then
`
˘
α̂prhsq “ rαs ˚ rhs ˚ rαs “ rαs ˚ rhs ˚ rαs and α̂ α̂qprhsq “ rαs ˚ prαs ˚ rhs ˚ rαsq ˚ rαs “ rhs.
Then, as a result, we get:
`
˘
α̂ α̂rf s “ rαs ˚ rα̂rf ss ˚ rαs “ rαs ˚ prαs ˚ rf s ˚ rαsq ˚ rαs “ rf s.
as needed.
Definition 1.2.7. A space X is path connected if there exists a path joining any two points
(i.e., for all x, y P X there is some path f : I Ñ X with f p0q “ x, f p1q “ y).
The fundamental group of a path connected space does not depend on the choice of base point.
Theorem 1.2.8. Let X be a path connected space with x, y P X. Then, we have an isomorphism of groups π1 pX, xq – π1 pX, yq.
Proof. We can construct the isomorphism π1 pX, xq – π1 pX, yq as follows. Start by choosing a
path f0 from x to y, i.e., f0 : I Ñ X with f0 p0q “ x, f0 p1q “ y. Then, send a loop f1 based at
x to the loop α̂, which is a loop based at y.
Because of the theorem above, it is not particularly important to keep track of the base point
if one is working with a path-connected space. For this reason, base point is usually omitted
in the definition of a fundamental group of a space and we just write π1 pXq.
Note that if a space is not path connected, then for x0 in a component of X, π1 pX, x0 q
provides no information about the other components of X. This is the reason for the study of
fundamental groups being usually restricted to path connected spaces.

THE FUNDAMENTAL GROUP

8

Definition 1.2.9. A space is simply-connected if it is path connected and π1 pX, xq “ 0 for
all points x P X, i.e., every path between two points can be continuously transformed into any
other such path while preserving the two endpoints.
A simply-connected space is a path connected space that has no “holes” that pass through
the entire space. Such a hole would prevent some loops from being shrunk continuously into a
single point.
Theorem 1.2.10. In a simply-connected space X, any two paths that have the same initial
point x0 and endpoint x1 are path homotopic.
Proof. Let α and β be two paths from x0 to x1 . Then α ˚ β is defined and is a loop based at
x0 . Since X is a simply-connected space, this loop is path-homotopic to the constant loop ex0
at x0 . So rαs “ rα ˚ βs ˚ rβs “ rex0 s ˚ rβs “ rβs.
We now develop methods to show that the fundamental group is a topological invariant or, in
other words, a property shared by homeomorphic spaces.
Definition 1.2.11. Let h : X Ñ Y be a continuous map between spaces X and Y with
y0 “ hpx0 q. Then for a loop f in X based at x0 , h ˝ f : I Ñ Y is a loop in Y based at y0 .
We denote this by h : pX, x0 q Ñ pY, y0 q. Define h˚ : π1 pX, x0 q Ñ π1 pY, y0 q by h˚ rf s “ rh ˝ f s.
Then h˚ is the homomorphism induced by h relative to the base point x0 . In the event
that we consider the homomorphism induced by h relative to different base points, we denote
h˚ as phx0 q˚ or phx1 q˚ , etc.
We need to show that in the definition above the map h˚ is well-defined and indeed a
homomorphism. The first condition is true since for f, f 1 P rf s, there is a path homotopy F
between f and f 1 . Then h ˝ F is a path homotopy between h ˝ f and h ˝ f 1 by Lemma 1.1.9.
Moreover, since ph ˝ f q ˚ ph ˝ gq “ h ˝ pf ˚ gq, h is, in fact, a group homomorphism. The induced
homomorphism has two crucial properties.
Theorem 1.2.12. If h : pX, x0 q Ñ pY, y0 q and k : pY, y0 q Ñ pZ, z0 q are continuous, then
pk ˝ hq˚ “ k˚ ˝ h˚ . If i : pX, x0 q Ñ pX, x0 q is the identity map, then i˚ is the identity
homomorphism.
Proof. Since pk ˝ hq˚ prf sq “ rpk ˝ hq ˝ f s, we get pk˚ ˝ h˚ qprf sq “ k˚ ph˚ prf sqq “ k˚ ph ˝ f q “
rk ˝ ph ˝ f qs. Similarly, i˚ prf sq “ ri ˝ f s “ rf s.
Theorem 1.2.13. If h : pX, x0 q Ñ pY, y0 q is a homeomorphism, then h˚ is an isomorphism
of π1 pX, x0 q with π1 pY, y0 q.
Proof. Since h is a homeomorphism, it has an inverse k : pY, y0 q Ñ pX, x0 q. Applying Theorem
1.2.12 we get k˚ ˝ h˚ “ pk ˝ hq˚ “ i˚ , where i is an identity map of pX, x0 q. The same way,
h˚ ˝ k˚ “ ph ˝ kq˚ “ j˚ , where j is an identity map of pY, y0 q. Since both i and j are the identity
homomorphisms of the groups π1 pX, x0 q and π1 pY, y0 q, respectively, k˚ is an inverse of h˚ .

THE FUNDAMENTAL GROUP

9

Theorem 1.2.14. Let h, k : pX, x0 q Ñ pY, y0 q be continuous maps. If h and k are homotopic,
and the image of the base point x0 P X remains fixed at y0 P Y when acted upon by the
homotopy, then the homomorphisms h˚ and k˚ are equal.
Proof. Let H : X ˆ I Ñ Y be the homotopy between h and k such that Hpx0 , tq “ y0 for all
t P I. Then, by definition, Hpx, 0q “ hpxq and Hpx, 1q “ kpxq. Consider a loop f : I Ñ X
based at x0 and the compositions h ˝ f , k ˝ f and H ˝ pf ˆ 1I q : I ˆ I Ñ Y :
I ˆI
Hpf pxq, 0q “ h ˝ f pxq

f ˆ1I

X ˆI

H

Y

Hpf pxq, 1q “ k ˝ f pxq

Hpf p0q, tq “ Hpf p1q, tq “ y0 , @t P I
Then H ˝ pf ˆ 1I q : I ˆ I Ñ Y is a homotopy between h ˝ f and k ˝ f . Moreover, h˚ prf sq “
rh ˝ f s “ rk ˝ f s “ k˚ prf sq, and, thus, k˚ “ h˚ : π1 pX, x0 q Ñ π1 pY, y0 q.
Definition 1.2.15. A space X is contractible if there is a homotopy between the identity
map X Ñ X and a constant map.
Example 1.2.16. Let’s look at a few facts about contractible spaces:
1. I, Rn and the disk Dn are contractible. For the first two spaces, define the homotopy by
F px, tq “ tx. Then f px, 0q “ 0 and f px, 1q “ x, so F is a homotopy from the constant
map 0 to the identity. To see that the disk is contractible it is enough to consider a
straight-line homotopy from the points of the disk to the origin.
2. A contractible space is also path-connected. Let F : X ˆ I Ñ X be a homotopy from
a constant map Cx0 : X Ñ X to the identity, i.e. F px, 0q “ x0 and F px, 1q “ x for all
x P X. For each point x1 P X, the function g : I Ñ X such that gptq “ F px1 , tq gives a
path from x1 to x0 . Thus, all points of X are in the same path components as x0 , so X
itself is path-connected.
3. If Y is contractible, then for any X, the set rX, Y s has a single element. Let F : Y ˆI Ñ Y
be a homotopy from a constant map to the identity, i.e., F py, 0q “ y0 and F py, 1q “ y
for all y P X. Then any map g : X Ñ Y is homotopic to the constant map g 1 pxq “ y0
with a homotopy G : X ˆ I Ñ Y defined by Gpx, tq “ F pgpxq, tq. One can check that
Gpx, 0q “ y0 and Gpx, 1q “ F pgpxq, 1q “ gpxq as needed.
4. If X is contractible and Y is path connected then rX, Y s has a single element. Define F
as a homotopy from part 2 of this example. For any function g : X Ñ Y , the function
f ˝ F is a homotopy between g and a constant map g 1 pxq “ gpx0 q. If Y is path connected,
then any two constant maps are homotopic and, thus, any two maps from X to Y are
homotopic.

COVERING SPACES AND THE FUNDAMENTAL GROUP OF A CIRCLE

10

The fundamental group is a covariant functor from the category Top˚ of pointed topological
spaces and pointed continuous maps to the category Groups of groups and group homomorphisms. For definitions from category theory, the reader is referred to the Appendix B.

1.3

Covering Spaces and the Fundamental Group of a Circle

To explore the fundamental groups of spaces more complex than Rn , consider the following
definition.
Definition 1.3.1. Let p : E Ñ B be a continuous onto map. The open set U of B is evenly
covered by p if the inverse image p´1 pU q can be written as the union of disjoint open sets
Vα Ď E such that for each α, the restriction of p to Vα is a homeomorphism of Vα onto U . The
collection tVα u is a partition of p´1 pU q into slices or fibers.
Definition 1.3.2. Let p : E Ñ B be continuous and onto. If every point b P B has a
neighborhood U , also called trivialized neighbourhood, that is evenly covered by p, then p
is a covering map and E is said to be a covering space of B which is also called the base
space.
A covering map over X is a map that locally
looks like the projection map for some discrete
space as seen on Figure 1.3.1.
Example 1.3.3. The identity map X Ñ X is
always a covering map of X. In this case, we can
take the entire space X to be the neighbourhood
U from the definition. More generally, if F is a
discrete space, then the projection X ˆ F Ñ X
is a covering space of X. We will call the map
X ˆ F Ñ X a trivial cover. Every covering
space looks locally like a trivial cover.

Figure 1.3.1

Note that if p : E Ñ B is a covering map, then for all b P B the subspace p´1 pbq of E has the
discrete topology. One can see that since each “slice” Vα , which is open in E, intersects the
set p´1 pbq in a single point, and so this point must be open in p´1 pbq.
Theorem 1.3.4. Let p : E Ñ B be a covering map. Then p is an open map.
Proof. Let U be an open set in E. If U “ ∅, then ppU q “ pp∅q “ ∅ which is always open.
Therefore, assume that U ‰ ∅. Let x P ppU q. We want to show that x is an interior point
of U . Let V be a neighbourhood of x and let V0 be a path component of p´1 pV q or, in other
words, a slice containing p´1 pxq. Then p restricted to V0 is a homeomorphism onto V . Since
V0 is a path-connected component, it is open in E, and since U is open in, V0 X U is open in
V0 . Since p is a homeomorphism from V0 onto V we have that ppV0 X U q is open in V and is
also open in B. But also x P ppV0 X U q Ď ppU q. So x P IntpppU qq and, thus, ppU q is open.

COVERING SPACES AND THE FUNDAMENTAL GROUP OF A CIRCLE

11

Theorem 1.3.5. The map p : R Ñ S 1 given by the equation ppxq “ pcos 2πx, sin 2πxq is a
covering map.
Proof. One can imagine the real line R getting
wrapped around the circle with the length of a circle
being 1 as shown on Figure 1.3.2. Then each interval
rn, n ` 1s makes exactly one loop around the circle.
Note that p is periodic, so it is enough to discuss in
detail only values of x that lie in (or near) the unit
interval. Consider x P S 1 , and let x0 P R be any point
such that f px0 q “ x, i.e., f ´1 pxq “ tx0 ` k|k P Zu. Let
U Ă S 1 be a small open arc of S 1 such that x P U .
Then the preimage f ´1 pU q consists of a disjoint union
of small intervals surrounding the points x0 ` k for
k P Z. Then U is a trivialized neighbourhood of x.
For construction we use four open sets, U0 , U1 ,
U2 and U3 , described in terms of R2 . Take U0 “
tpx, yq P S 1 : x ą 0u. Since cos 2πx ą 0 in U0 means
that ´π{2 ă 2πx ă π{2, every interval pn´1{4, n`1{4q
Figure 1.3.2: Visualization of the
is getting mapped to U0 by p. To show that every
covering map of S 1 .
such interval is homeomorphic to U0 , note that sin 2πx
is a monotonically increasing continuous function on any of the taken intervals as x is increasing. Therefore, we can provide an inverse continuous function p´1 : S 1 Ñ R where
1
p´1 px, yq “ n ` 2π
arcsin y, which shows that p is homeomorphism on every such interval U0 .
Since for all n P N intervals pn ´ 1{4, n ` 1{4q are disjoint, U0 is evenly covered by p. The
same way it can be shown that U1 “ tpx, yq P S 1 : y ą 0u, U2 “ tpx, yq P S 1 : x ă 0u and
U3 “ tpx, yq P S 1 : y ă 0u are all evenly covered by the intervals pn, n ` 1{2q, pn ` 1{4, n ` 3{4q
and pn ` 1{2, n ` 1q, respectively. Since all of Ui cover S 1 and each of them is evenly covered
by p, p is a covering map.
If p : E Ñ B is a covering map, then p is a local homeomorphism of E with B according
to Definition A.0.10. However, the condition that p is a local homeomorphism is not enough
to claim that p is covering map.
Example 1.3.6. The map p|R` : R` Ñ S 1 given by the equation p|R` pxq “ pcos 2πx, sin 2πxq
is surjective and a local homeomorphism, but not a covering map. One can see that p is not a
covering map because of the behavior of the point b0 “ p|R` p0q “ p1, 0q P S 1 . More specifically,
the point has no neighbourhood which is evenly covered by the map p. From Example 1.3.5
we know that a usual neighbourhood of the point b0 in S 1 can be written as pb0 ´ ϵ, b0 ` ϵq
or depending on p it is pp|R` p0q ´ ϵ, p|R` p0q ` ϵq. The pre-image of these neighbourhoods is
the union of disjoint intervals pn ´ ϵ, n ` ϵq, where n P Z. However, for n “ 0 it becomes the
disjoint union of the interval p0, ϵq and intervals pn ´ ϵ, n ` ϵq for n P N. Each of the intervals
of second kind is evenly covered by the map p as in Example 1.3.5, but the interval p0, ϵq is
not. Therefore, p|R` is not a covering map.

COVERING SPACES AND THE FUNDAMENTAL GROUP OF A CIRCLE

12

The preceding example shows that a restriction of a covering map might not be a covering
map itself. However, in case of an additional condition, we get the following result.
Theorem 1.3.7. Let p : E Ñ B be a covering map. If B0 is a subspace of B and E0 “ p´1 pB0 q,
then the map p0 : E0 Ñ B0 obtained by restricting p is a covering map.
Proof. Let b0 P B0 and U be an open set in B such that U is evenly covered by p and b0 P U .
Let tVα u be a partition of p´1 pU q into slices. Then U X B0 is a neighbourhood of b0 in B0 .
Moreover, sets Vα X E0 are disjoint open sets in E0 whose union is equal to p´1 pU X B0 q, where
each Vα X E0 is mapped homeomorphically onto U X B0 by p. Hence, p0 is a covering map.
Theorem 1.3.8. Let p : E Ñ B and p1 : E 1 Ñ B 1 be covering maps. Then p ˆ p1 : E ˆ E 1 Ñ
B ˆ B 1 is a covering map.
Proof. Take b P B and b1 P B 1 , let U and U 1 be neighbourhoods of b and b1 , respectively, that
are evenly covered by p and p1 . Also, let tVα u and tVα1 u be partitions of p´1 pU q and p1´1 pU 1 q,
respectively, into slices. Then the inverse image under p ˆ p1 of the open set U ˆ U 1 is the union
of all the sets Vα ˆ Vα1 . These are disjoint open sets of E ˆ E 1 where each of them is mapped
homeomorphically onto U ˆ U 1 by the map p ˆ p1 . Therefore, p ˆ p1 is a covering map.
Example 1.3.9. Consider the torus T2 “ S 1 ˆ S 1 . Then the product map p ˆ p : R ˆ R Ñ
S 1 ˆ S 1 , where p is a covering map from Example 1.3.5, is a covering of the torus by the plane
R2 . Since we typically think of S 1 as a subset of R2 , this representation of the torus is the
subset of R4 . Each of unit squares rn, n ` 1s ˆ rm, m ` 1s gets wrapped by p ˆ p entirely around
the torus.
Example 1.3.10. Consider the covering map p ˆ p from Example 1.3.9. Let b0 denote the
point pp0q of S 1 and let B0 denote the subspace B0 “ pS 1 ˆ b0 q Y pb0 ˆ S 1 q Ă S 1 ˆ S 1 . Then B0
is the union of two circles which have the point b0 in common. This is what we call the figureeight space. Considering the space E0 “ p´1 pB0 q which is the infinite grid pR ˆ Zq Y pZ ˆ Rq,
the map p0 : E0 Ñ B0 is a covering map of the figure-eight space by Theorem 1.3.7 since it is
a restriction of the covering map p ˆ p.
The covering spaces are used to prove a classic result in algebraic topology about the fundamental group of a circle. The idea behind it is that the fundamental group of S 1 is generated
by starting at p1, 0q and creating loops that wrap around S 1 a positive integer number of
times (counterclockwise) and loops that wrap around S 1 a negative integer number of times
(clockwise). For the full proof the reader is referred to [Mun00] or [Wil04].
Theorem 1.3.11. The fundamental group of S 1 is isomorphic to the additive group of integers.
Sketch of the proof. Consider a bijection from R to a helix in R3 with a parametrisation defined
by pcos 2πs, sin 2πs, sq. We also identify S 1 as a circle of unit radius inside R2 . Let p : R Ñ S 1
be a map, which is also a covering map, such that ppsq “ pcos 2πx, sin 2πxq. This function can
be thought of as a projection map from R3 to R2 given by px, y, zq ÞÑ px, yq. This means that
R is a covering space of S 1 . Consider the map ϕ : π1 pS 1 , b0 q Ñ Z. One can show that this
map is a group homomorphism, which gives an isomorphism between π1 pS 1 , b0 q and Z.

RETRACTIONS AND DEFORMATION RETRACTS

1.4

13

Retractions and Deformation Retracts

Definition 1.4.1. If A Ă X, a retraction of X onto A is a continuous map r : X Ñ A such
that r|A is the identity map of A. If such a map r exists, we say that A is a retract of X.
Lemma 1.4.2. If a0 P A and r : X Ñ A is a retraction, then r˚ : π1 pX, a0 q Ñ π1 pA, a0 q is
surjective.
Proof. Let ι : A Ñ X be the inclusion map. Then r ˝ ι “ 1A by construction. Then
r˚ ˝ ι˚ “ pr ˝ ιq˚ “ 1A˚ “ 1π1 pAq
by Theorem 1.2.12. Since the right side is an isomorphism r˚ has to be a surjection, while ι˚
has to be an injection.
Theorem 1.4.3 (No-retraction Theorem). There is no retraction of B 2 onto S 1 .
Proof. If S 1 was a retract of B 2 , then the homomorphism induced by the inclusion ι : S 1 Ñ B 2
would be injective. However, the fundamental group of S 1 is non-trivial while the fundamental
group of B 2 is trivial.
Example 1.4.4. There is a retraction r of R2 zt0u onto S 1 given by equation rpxq “ x{||x||.
Therefore, ι˚ , where ι : S 1 Ñ R2 zt0u is the inclusion map, has to be injective, and, hence, nontrivial or, in other words, not nulhomotopic. Similarly, i˚ , where i : S 1 Ñ S 1 is the identity
map, is the identity homomorphism, and hence non-trivial or not nulhomotopic.
Definition 1.4.5. Let A Ă X. We call A a deformation retract of X if the identity map
of X is homotopic to a map that carries X into A. In other words, there exists a continuous
map H : X ˆ I Ñ X such that Hpx, 0q “ x, Hpx, 1q P A for all x P X and Hpa, tq “ a for all
a P A. In this case, we call the homotopy H a deformation retraction of X onto A.
Note that the map r : X Ñ A defined as rpxq “ Hpx, 1q is a retraction of X onto A, and H is a
homotopy between the identity map of X and the map j ˝ r, where j : A Ñ X is the inclusion
map.
Theorem 1.4.6. Let A be a deformation retract of X and let x0 P A. Then the inclusion map
ι : pA, x0 q Ñ pX, x0 q induces an isomorphism of fundamental groups.
Proof. Let r : X Ñ A be the retraction between noted spaces. Then r ˝ ι is the identity map
of A, and by Theorem 1.2.12, r˚ ˝ ι˚ is the identity homomorphism of π1 pA, b0 q, where b0 P A.
Consider the composition ι ˝ r : X Ñ X, which maps X to itself, but is not the
identity map. It is homotopic to the identity map via a homotopy fixing the points of A, i.e.,
a homotopy H : X ˆ I Ñ X with Hpx, 0q “ ι ˝ rpxq, Hpx, 1q “ x, and Hpx0 , tq “ x0 for all
t P I. By Theorem 1.2 since a deformation retraction gives a base-point preserving homotopy
between ι˝r and 1X , we have p1X q˚ “ ι˚ ˝r˚ : π1 pX, x0 q Ñ π1 pX, x0 q. We know ι˚ is injective.
It is also surjective since for any class rf s in π1 pX, x0 q, we have rf s “ ι˚ pr˚ prf sqq. Therefore,
it is an isomorphism.

HOMOTOPY TYPE

14

Example 1.4.7. From the theorem above, one can induce that the inclusion map ι : S n Ñ
Rn`1 zt0u induces an isomorphism of fundamental groups. Thus, π1 pS n q – π1 pRn`1 zt0uq.

Figure 1.4.1: Deformation retractions following R2 ztp, qu above and the punctured torus
below, both resulting in the figure-eight.
Example 1.4.8. Consider R2 ztp, qu, where p, q P R2 , the doubly punctured plane, which
has the figure-eight (recall Example 1.3.10) as a deformation retract. Another space which
has the figure-eight as a deformation retract is the punctured torus, i.e. T 2 zp for some point
p P T 2 . The deformations from this example can be seen in the Figure 1.4.1.

1.5

Homotopy Type

Definition 1.5.1. Let f : X Ñ Y and g : Y Ñ X be continuous maps. Suppose that the map
g ˝ f : X Ñ X is homotopic to the identity map of X, and the map f ˝ g : Y Ñ Y is homotopic
to the identity map of Y . Then maps f and g are called homotopy equivalences, and each
of them is said to be a homotopy inverse of the other. Topological spaces X, Y are said to
be homotopy equivalent or of the same homotopy type, where we denote it by X » Y ,
when there are homotopy equivalences between the spaces.
Note that every homeomorphism f : X Ñ Y is a homotopy equivalence since we can take
g :“ f ´1 . Then if there are spaces X and Y such that X – Y , it would also mean that X » Y .
However, the converse of the statement is not true: consider R and t0u, which are homotopy
equivalent but not homeomorphic. In the Section 2.1.3 we have proved that the relation of
path-homotopy equivalence is an equivalence relation. The same can be done for more general
type of homotopy to show that the relation of homotopy equivalence is an equivalence relation.
Example 1.5.2. If A is a deformation retract of X, then A has the same homotopy type as
X. To show this, take the inclusion map ι : A Ñ X and the retraction map r : X Ñ A. Then

HOMOTOPY TYPE

15

the composition r ˝ι : X Ñ X is the identity map of A, and the composition ι˝r is supposed to
be homotopic to the identity map of X by the definition of the deformation retraction. With
this example, one can think of contractible spaces as spaces that have the homotopy type of a
one-point space.
Example 1.5.3. Consider the figure-eight space X and the
theta space, defined as θ “ S 1 Y p0 ˆ r´1, 1sq. The theta
space is also a deformation retract of R2 ztp, qu, but it is not a
deformation retract of the figure-eight-space. To see that note
that the “bar” p0 ˆ r´1, 1sq in the theta space would need to
remain unchanged during the deformation, but it is not a subspace of the figure eight.
However, we can describe the homotopy equivalences between
them. Consider the figure-eight to be two congruent, tangent Figure 1.5.1: Theta space.
circles and the θ space to be a circle with a diameter drawn.
Then the map g : Y Ñ X can be described as contracting the circle along the diameter to the
center of the circle. Similarly, the map f : X Ñ Y can be described as stretching each tangent
circle to fit into a half of the θ space.
Note that spaces being homotopy equivalent does not mean that they have isomorphic fundamental groups yet. To show this, we need to look at the case when the base point does not
remain the same during the homotopy.
h˚
Lemma 1.5.4. Let h, k : X Ñ Y be continuous maps with
π1 pX, x0 q
π1 pY, y0 q
y0 “ hpx0 q and y1 “ kpx0 q. If h and k are homotopic, then
α̂
k˚
there exists a path α in Y from y0 to y1 such that k˚ “ α̂ ˝ h˚
or, in other words, the following diagram commutes.
π1 pY, y1 q
Proof. Let H : X ˆ I Ñ Y be the homotopy between h and k. Define the required path α
from y0 to y1 as αptq “ Hpx0 , tq. Consider an element f : I Ñ X of π1 pX, x0 q, a path c in
X ˆ I given as cptq “ px0 , tq and loops f0 and f1 in the space X ˆ I given as f0 psq “ pf psq, 0q
and f1 psq “ pf psq, 1q. Then H ˝ f0 “ h ˝ f and H ˝ f1 “ k ˝ f , while H ˝ c “ α.
Consider a map F : I ˆ I Ñ X ˆ I, such that F ps, tq “ pf psq, tq and the following paths in
I ˆ I, which run along the four edges of I ˆ I:
β0 psq “ ps, 0q and β1 psq “ ps, 1q,
γ0 ptq “ p0, tq and γ1 ptq “ p1, tq.
Then F ˝ β0 “ f0 and F ˝ β1 “ f1 , while F ˝ γ0 “ F ˝ γ1 “ c.
The broken-line paths β0 ˚ γ1 and γ0 ˚ β1 are both paths in I ˆ I from p0, 0q to p1, 1q and since
I ˆ I is convex, there is a path homotopy between them by Example 1.1.6. Then F ˝ G is a
path homotopy in X ˆ I between f0 ˚ c and c ˚ f1 . Therefore, H ˝ pF ˝ Gq is a path homotopy
in Y between
pH ˝ f0 q ˚ pH ˝ cqq “ ph ˝ f q ˚ α

and

pH ˝ cq ˚ pH ˝ f1 q “ α ˚ pk ˝ f q,

FUNDAMENTAL GROUPS OF OTHER SURFACES

16

which would mean that
rk ˝ f s “ rαs ˚ rh ˝ f s ˚ rαs
or
k˚ prf sq “ α̂ph˚ prf sqq,
as needed.
The immediate consequence of the Lemma above is that in case of h˚ being injective, surjective
or trivial, k˚ has the same property. Moreover, if h : X Ñ Y is nulhomotopic, then h˚ is the
trivial homomorphism. The most important result of this lemma allows us to extend the idea
of fundamental group to spaces of the same homotopy type.
Theorem 1.5.5. Let f : pX, x0 q Ñ pY, y0 q be a continuous map. If f is a homotopy equivalence
then f˚ : π1 pX, x0 q Ñ π1 pY, y0 q is an isomorphism.
Proof. Consider g : Y Ñ X be a homotopy inverse for f and maps
pX, x0 q

fx 0

pY, y0 q

g

pX, x1 q

fx 1

pY, y1 q,

where x1 “ gpy0 q and y1 “ f px1 q. Then we have induced homomorphisms as follows:
π1 pX, x0 q

pfx0 q˚

π1 pY, y0 q

g˚

π1 pX, x1 q

pfx1 q˚

π1 pY, y1 q.

By assumption, g ˝ f : pX, x0 q Ñ pX, x1 q is homotopic to the identity map, so there is a path
α in X such that pg ˝ f q˚ “ α̂ ˝ p1X q˚ “ α̂. It follows that pg ˝ f q˚ “ g˚ ˝ pfx0 q˚ is an
isomorphism and g˚ is surjective. Similarly, since f ˝ g is homotopic to the identity map 1Y ,
the homomorphism pf ˝ gq˚ “ pfx1 q˚ ˝ g˚ is an isomorphism and g˚ is injective. Therefore, g˚
is an isomorphism. Moreover, we can conclude that pfx0 q˚ “ pg˚ q´1 ˝ α̂ and, thus, pfx0 q˚ is
also an isomorphism.

1.6

Fundamental Groups of Other Surfaces

Theorem 1.6.1. Suppose X “ U YV , where both U and V are open sets of X. Suppose U XV
is path-connected, and that x0 P U X V . Let i and j be the inclusion mappings of U and V ,
respectively, into X. Then the images of the induced homomorphisms
i˚ : π1 pU, x0 q Ñ π1 pX, x0 q and j˚ : π1 pV, x0 q Ñ π1 pX, x0 q
generate π1 pX, x0 q. In other words, given any loop f in X based at x0 , it is path homotopic
to a product of the form g1 ˚ g2 ˚ . . . ˚ gn , where each gi is a loop in X based at x0 which lies
entirely either in U or V .

FUNDAMENTAL GROUPS OF OTHER SURFACES

17

Proof. First, use the Lebesgue number Lemma A.0.30 to choose a subdivision tbi u of I such
that for all i the set f prbi´1 , bi sq is contained in either U or V . If for all i, the set f prbi´1 , bi sq
is contained in U X V , pick this division. Otherwise, let i be an index such that f pbi q R U X V .
Both of the sets f prbi´1 , bi sq and f prbi , bi`1 sq lie fully in either U or V . If f pbi q P U then both
of the sets must lie in U , otherwise, they both must belong to V . In both cases, consider
the same division of I but without bi - let’s call it tci u. This subdivision satisfies the main
condition - for all i the set f prci´1 , ci sq belongs to either U or V - therefore, we can do this
operation until we reach the desired subdivision. Let tai u be the subdivision of I obtained,
i.e., for all i we have f prai´1 , ai sq is either in U or V and f pai q P U X V .
Now, define fi to be the path in X that equals the linear map of I onto rai´1 , ai s followed
by f . Then fi is a path that lies either in U or V , and rf s “ rf1 s ˚ . . . ˚ rfn s. For each i,
since U X V is path-connected, we can choose a path αi in U X V from x0 to f pai q. Since
f pa0 q “ f pan q “ x0 , we can choose α0 and αn to be constant paths at x0 . Now for each i we
have gi “ αi´1 ˚ fi ˚ αi . This means that gi is a loop in X based at x0 whose image lies either
in U or in V . Then we have
rg1 s ˚ . . . ˚ rgn s “ rα0 s ˚ rf0 s ˚ rα1 s ˚ rα1 s ˚ rf1 s ˚ rα2 s ˚ . . . ˚ rαn´1 s ˚ rfn s ˚ rαn s “
rα0 s ˚ rf1 s ˚ . . . ˚ rfn s ˚ rαn s “ rf1 s ˚ . . . ˚ rfn s,
as needed.
Corollary 1.6.2. Suppose X “ U Y V , where both U and V are open sets of X. Suppose
U X V is path-connected and non-empty. If U and V are simply-connected, then X is simplyconnected.
Proof. Since U X V is non-empty, there exists a point x0 P U X V . Both U and V are simplyconnected, so π1 pU, x0 q and π1 pV, x0 q are trivial. Then both of the induced homomorphisms of
the inclusion mappings i˚ and j˚ are trivial homomorphisms, and, thus, π1 pX, x0 q is trivial.
Theorem 1.6.3. The n-sphere S n is simply-connected for n ě 2.
Proof. First, note that for n ě 1, the punctured sphere S n ztpu is homeomorphic to Rn , since
we can define the stereographic projection as a homeomorphism between them. Firstly, let’s
show that for n ě 1, the punctured sphere S n ztpu is homeomorphic to Rn . For a point
p “ p0, . . . , 0, 1q P S n define a map f : pS n ztpuq Ñ Rn as stereographic projection:
f pxq “ f px1 , x2 , . . . , xn`1 q “

1
px1 , . . . , xn q.
1 ´ xn`1

To show that this map is a homeomorphism, we can check that the map g : Rn Ñ pS n ztpuq,
defined by
gpyq “ gpy1 , . . . , yn q “ ptpyqy1 , . . . , tpyqyn , 1 ´ tpyqq,
where tpyq “ 2{p1 ` ||y||2 q, is both right and left inverse for f . Another way of thinking
about it is understanding what it is doing: if we take a line passing through p and the point

FUNDAMENTAL GROUPS OF OTHER SURFACES

18

x P pS n ztpuq, it would intersect the plane Rn ˆ t0u Ă Rn`1 in only one point f pxq ˆ t0u.
Note that the reflection map px1 , . . . , xn`1 q ÞÑ px1 , . . . , xn , ´xn`1 q defines a homeomorphism
of S n zp with S n ztqu, where q “ p0, . . . , 0, ´1q P S n is the south pole of the sphere, so the latter
space is also homeomorphic to Rn .
Now take U “ S n ztpu and V “ S n ztqu be open sets of S n . For n ě 1, the sphere
n
S is path-connected since both U – Rn and V – Rn are path-connected and have the point
p1, 0, . . . , 0q in common. To show that S n is simply-connected, note that U X V “ S n ztp, qu
which is homeomorphic to Rn zt0u. The latter space is path-connected and, thus, U X V is
path-connected. Therefore, by Corollary 1.6.2, U Y V “ S n is simply-connected.
Definition 1.6.4. A topological space M is a topological manifold of dimension n (or
topological n-manifold) if
• M is Hausdorff (recall A.0.24),
• M is second-countable (recall A.0.25), and
• M is locally Euclidean: for all points m P M there exists an open neighbourhood in
which is homeomorphic to an open subset of Rn .
A topological 2-manifold is called a surface.
Theorem 1.6.5. π1 pX ˆ Y, x0 ˆ y0 q is isomorphic with π1 pX, x0 q ˆ π1 pY, y0 q.
Proof. Let p : X ˆ Y Ñ X and q : X ˆ Y Ñ Y be the projection maps. Using the induced
homomorphisms of given maps, define a homomorphism
Φ : π1 pX ˆ Y, x0 ˆ y0 q Ñ π1 pX, x0 q ˆ π1 pY, y0 q
by the equation
`
˘ `
˘
Φprf sq “ p˚ prf sq, q˚ prf sq “ rp ˝ f s, rq ˝ f s .
To show that the map Φ is an isomorphism we need to show that it is bijective. To show that
the map is surjective, let g : U Ñ X be a loop based at x0 and let h : I Ñ Y be a loop based
at y0 . Also, define f : I Ñ X ˆ Y such that f psq “ gpsq ˆ hpsq. Then f is a loop in X ˆ Y
based at x0 ˆ y0 with
Φprf sq “ prp ˝ f s, rq ˝ f sq “ prgs, rhsq,
which means that the element prgs, rhsq lies in the image of Φ. More intuitively, if f is a loop
based at px0 , y0 q, it is nothing more than a pair of loops in X and Y based at x0 and y0 .
Similarly, homotopies of loops are nothing but pairs of homotopies of pairs of loops.
To show that Φ is one-to-one, define f `: I Ñ X ˆ Y˘ as a loop in X ˆ Y based at
x0 ˆ y0 with an identity element being Φprf sq “ rp ˝ f s, rq ˝ f s , which means that p ˝ f »p ex0
and q ˝ f »p ey0 . Let G and H be the respective homotopies in X and Y . Then the map
F : I ˆ I Ñ X ˆ Y defined by F ps, tq “ Gps, tq ˆ Hps, tq is a path homotopy between f and a
constant loop based at x0 ˆ y0 .

FUNDAMENTAL GROUPS OF OTHER SURFACES

19

Note that the preceding theorem can be extended to a finite product of spaces. Moreover, if any of the spaces end up being contractible, they can be dropped from the product.
Example 1.6.6. A natural example to consider, given that π1 pS 1 q – Z, is the torus T 2 “
S 1 ˆ S 1 . Then π1 pT 2 q – Z ˆ Z.

Chapter 2

Free Groups
So far in the previous sections we were able to compute the fundamental group in some basic
cases. For more complicated cases, we need to develop a few more strategies and skills to be
able to describe the structure of the group itself.
Recall the definition of the direct product G “ G1 ˆ G2 ˆ . . . ˆ Gn of a finite number
of groups tGi uni“1 . The elements of G are ordered n-tuples g “ pg1 , . . . , gn q, where gi P Gi ,
with the operation of multiplication denoted by
pg1 , . . . , gn qph1 , . . . , hn q “ pg1 h1 , . . . , gn hn q.
This idea can be extended to a case with infinitely many groups: consider an infinite collection
ś
of groups tGi uiPI , where I is an index set. The direct product in this case is defined as iPI Gi .
Its elements are functions which assign to each index i P I an element gi P Gi with the similar
definition for the multiplication.

2.1

Free Groups

Given a non-empty set X, we would like to construct a free group on this set. There are
different ways to describe free groups and products, and we are going to follow the idea from
[Hun12]. If X “ ∅, then the free group is going to be the trivial group xey. Otherwise, let X ´1
be a set disjoint from X such that |X| “ |X ´1 |. Choose a bijection X Ñ X ´1 and denote an
image of x P X by x´1 . Choose an element I disjoint from X Y X ´1 .
Definition 2.1.1. In this context, a word on X is a sequence pa1 , a2 , . . .q such that aj P X
for j P N and for some n P N, ak “ 1 for all integer k ě n. Define I “ p1, 1, 1, . . .q to be the
empty word.
There are infinitely many such words we can construct on a set, although, some of them seem
to be equivalent. To deal with this problem there are a few reduction operations we can do
to get the reduced word:
1. if we have adjacent x and x´1 , we can delete both,
20

FREE GROUPS

21

2. if ak “ 1 for some k P N, then ai “ 1 for all i ě k.
Notice that every reduced word is of the form pxλ1 1 , xλ2 2 , . . . , xλnn , 1, 1, . . .q, where xi P X and
λi “ ˘1. We will denote such word by xλ1 1 xλ2 2 . . . xλnn . For simplicity, one can also combine the
adjacent identical elements x and x to write x2 and so on for higher powers.
Example 2.1.2. The empty word I is reduced without any reduced operations applied.
Consider a set X “ tx, y, zu. Let w1 “ px, xq and w2 “ px´1 , y, y, y, x´1 , x´1 , x, z, z ´1 q be
words. Their juxtaposition is the sequence w “ px, x, x´1 , y, y, y, x´1 , x´1 , x, z, z ´1 q, which
can be reduced to px, y, y, y, x´1 q “ xy 3 x´1 .
δm with λ , δ “ ˘1 are equal if and only
Two reduced words xλ1 1 xλ2 2 . . . xλnn and y1δ1 , y2δ2 , . . . , ym
i j
if both are I or m “ n and for all 1 ď i ď n we have xi “ yi and λi “ δi . With this definition
denote the set of all reduced words on a set X as F pXq.
To make this a group we need to add an identity and a binary operation to it.

Definition 2.1.3. Consider juxtaposition of two words
δm
δm
q “ pxλ1 1 , xλ2 2 , . . . , xλnn , y1δ1 , y2δ2 , . . . , ym
q,
x ˚ y “ pxλ1 1 , xλ2 2 , . . . , xλnn q ˚ py1δ1 , y2δ2 , . . . , ym

both taken on a given set X.
Intuitively, the empty word I behaves like an identity element, i.e., I ˚ w “ w ˚ I “ w, for any
non-empty word w P F pXq. Also, note that the juxtaposition of two reduced words might not
be reduced, but one can reduce it using the reduction operations.
Theorem 2.1.4. If X is a nonempty set and F “ F pXq is the set of all reduced words on X,
then F is a free group under juxtaposition and it is denoted by F “ xXy instead.
Proof. To verify that F is a group we need to check all properties of a group. We know that the
n , x´λn´1 , . . . , x´λ1 q.
empty word is an identity and a word pxλ1 1 , xλ2 2 , . . . , xλnn q has an inverse px´λ
n
n´1
1
To verify associativity, note that we do not need to reduce the juxtaposition until the very
end. With this, one gets the products of three reduced words x, y, z P X equal to
δm
px ˚ yq ˚ z “ ppxλ1 1 xλ2 2 . . . xλnn q ˚ py1δ1 y2δ2 . . . ym
qq ˚ pz1γ1 z2γ2 . . . zkγk q
δm
“ pxλ1 1 xλ2 2 . . . xλnn y1δ1 y2δ2 . . . ym
q ˚ pz1γ1 z2γ2 . . . zkγk q
δm γ1 γ2
“ xλ1 1 xλ2 2 . . . xλnn y1δ1 y2δ2 . . . ym
z1 z2 . . . zkγk
δm
“ pxλ1 1 xλ2 2 . . . xλnn q ˚ ppy1δ1 y2δ2 . . . ym
q ˚ pz1γ1 z2γ2 . . . zkγk qq “ x ˚ py ˚ zq.

Here, we mention some properties of free groups. If |X| ě 2, then the free group on X is not
abelian since for x, y P X, such that x ‰ y, we have words xy and yx being both reduced but
not equal to each other. Also, every element of such a group except the identity has infinite
order. This being said, if X “ tau, then F is an infinite cyclic group.

FREE GROUPS

22

Theorem 2.1.5 (Universal Mapping Property). Let F be the free
group on a set X and ι : X Ñ F an inclusion map. If G is a
group and ϕ : X Ñ G a map of sets, then there exists a unique
homomorphism of groups ϕ : F Ñ G such that ϕ ˝ ι “ ϕ, i.e., the
following diagram commutes.

F
ι

X

ϕ

ϕ

G

Proof. Define ϕp1q “ e and for a non-empty reduced word on X, define
ϕpxλ1 1 xλ2 2 . . . xλnn q “ ϕpx1 qλ1 ϕpx2 qλ2 . . . ϕpxn qλn .
Since G is a group and λi “ ˘1 for all 1 ď i ď n, the product above is well-defined in G. Such
a definition of ϕ automatically results in ϕ being a homomorphism such that ϕ ˝ ι “ ϕ. To
show that ϕ is indeed unique, consider a homomorphism g : F Ñ G such that g ˝ ι “ ϕ. Then
gpxλ1 1 xλ2 2 . . . xλnn q “ gpx1 qλ1 gpx2 qλ2 . . . gpxn qλn
“ gpιpx1 qλ1 qgpιpx2 qλ2 q . . . gpιpxn qλn q
“ ϕpx1 qλ1 ϕpx2 qλ2 . . . ϕpxn qλn “ ϕpxλ1 1 xλ2 2 . . . xλnn q,
which means ϕ is unique.
The theorem above shows that F is a free object on the set X in the category of groups
according to the Definition B.0.15. This being said, if F 1 is another free object on the same
set X with λ : X Ñ F 1 in the category of groups, then there is an isomorphism ϕ : F Ñ F 1
such that ϕ ˝ ι “ λ.
Corollary 2.1.6. Every group G is the homomorphic image of a free group.
The free group on X is also said to be the freest group generated on a set X. To see why
note that in an arbitrary group there are different products of elements, which give an identity
element as a result. For example,
1. x ˚ x´1 “ e for any element of any group;
2. in a cyclic group of order n, xn “ e.
Any such product is called a relation on a group X. Relations of type (1), which come from
properties of a group, are said to be trivial, while all other ones, like type (2), are said to be
non-trivial. The other way to define a free group on a set is to take a set X with only trivial
relations between its elements.
Definition 2.1.7. Let X be a set and R be a set of reduced words on X. A group G is said to
be the group defined by the generators x P X and relations w “ e for w P R provided
G – F {N , where F is a free group on X and N the normal subgroup of F generated by R. In
this case, we call G “ xX|Ry a presentation of G.

FREE GROUPS

23

These notions also lead to the idea that one can completely describe a group G with
its generating set X and relations set R between them. Note that a presentation of a group
is not unique. To see this, consider a cyclic group Z6 with presentations xa|a6 y and xa, b|a2 “
b3 “ a´1 b´1 aby.
There is a relation between free groups and free abelian groups. For that recall that if x and y
are elements of a group G, then the element rx, ys “ xyx´1 y ´1 P G is called the commutator
of x and y. The notation rG, Gs denotes the subgroup of G generated by all commutators - the
commutator subgroup. Commutators are, in a sense, a measure how much of G fails to be
commutative. In particular, the commutator subgroup is trivial if and only if all commutators
are the identities. We know a few facts about this subgroup:
Theorem 2.1.8. Given a group G, the commutator subgroup rG, Gs is a normal subgroup and
the quotient group G{rG, Gs is abelian. Moreover, if h : G Ñ H is a homomorphism from G to
an abelian group H, then the kernel of h contains rG, Gs, and hence h induces a homomorphism
k : G{rG, Gs Ñ H.
Proof. The theorem consists of 3 different facts, each of which is going to proved in a separate
step.
Step 1. To show that rG, Gs is normal, first, we need to show that any conjugate of a
commutator is in rG, Gs as well:
grx, ysg ´1 “ gpxyx´1 y ´1 qg ´1
“ pgxyx´1 qpy ´1 g ´1 q
“ pgxyx´1 qpg ´1 y ´1 ygqpy ´1 g ´1 q
“ ppgxqypgxq´1 y ´1 qpygy ´1 g ´1 q
“ rgx, ysry, gs,
which is known to be in rG, Gs. Now, consider an arbitrary element z of rG, Gs. This element
is a product of commutators and their inverses. Since
rx, ys´1 “ pxyx´1 y ´1 q´1 “ yxy ´1 x´1 “ ry, xs
or, in other words, every inverse of a commutator is a commutator itself, z is just a product of
commutators z1 , z2 , . . . , zn . Then its conjugate is
gzg ´1 “ gz1 z2 . . . zn g ´1 “ gz1 pg ´1 gqz2 pg ´1 . . . gqxn g ´1 “ pgz1 g ´1 qpgz2 g ´1 q . . . pgxn g ´1 q,
which is a product of commutators’ conjugates, which we know is in rG, Gs, as well. Therefore,
the group rG, Gs is normal.
Step 2. For this step let G1 “ rG, Gs. To show that G{rG, Gs is abelian, we need
paG1 qpbG1 q “ pbG1 qpaG1 q,
which is equivalent to
abG1 “ baG1 ,

FREE GROUPS

24

which is equivalent to
a´1 b´1 abG1 “ G1 .
Since a´1 b´1 ab is a commutator on its own, it belongs to G1 and the last statement follows.
Step 3. Since H is abelian by assumption, h carries each commutator to the identity
element of H. Hence, the kernel of h contains the whole commutator subgroup rG, Gs, so h
induced the desired homomorphism k.
Hence, using this we can transform any free group F to a free abelian group F {rF, F s,
which is called the abelianization, using the natural projection π : F Ñ F {rF, F s.
Definition 2.1.9. If G is a free abelian group, the rank of G is the number of elements in
the generating set of G.
Since for any free group G with n generators the free abelian group G{rG, Gs has rank
n, any system of free generators for G would have n elements.
Theorem 2.1.10. If F and F 1 are free groups on finite sets S and S 1 , then F and F 1 are
isomorphic if and only if S and S 1 have the same rank.
Since now we are dealing with finitely generated abelian groups, we need some properties of such groups. First, recall that the set of all elements of an arbitrary abelian group
A that have finite order is called the torsion subgroup. If we denote the torsion subgroup
by T , then the quotient group A{T is going to be torsion free. In case groups A and A1 are
isomorphic, their torsion subgroups and quotients mod torsion subgroups are also isomorphic.
The converse is true, however, only for finitely generated abelian groups.
Theorem 2.1.11 ([Mas91],[Hun12]). Consider only finitely generated abelian groups. Then
we have the following:
1. Let A be a finitely generated abelian group and let T be its torsion subgroup. Then, T
and A{T are also finitely generated, and A is isomorphic to the direct product T ˆ A{T .
Hence, the structure or A is completely determined by its torsion subgroup T and its
torsion-free subgroup A{T .
2. Every finitely generated torsion-free abelian group is a free abelian group of finite rank.
3. Every finitely generated abelian group G is isomorphic to a product
Zd1 ‘ . . . ‘ Zdk ‘ Z‘n – pZ{d1 Zq ‘ . . . ‘ pZ{dk Zq ‘ Z‘n ,
where Z‘n means the direct product of n copies of the group Z. Moreover, k, n and di
are all uniquely determined and they completely determine the structure of the group G.
However, how do we extract information about a group from its presentation? This
question is answered in, for instance, [D L97]. The reader is welcome to familiarize themselves
with the topic, but here we are going to state the needed theorem and use of it for the finite

FREE PRODUCT

25

X. Consider a presentation P “ xX|Ry of a group G with abelianization Gab . Let us fix the
notation
X “ tx1 , x2 , . . . , xr u, C “ trxi , xj s|1 ď i ă j ď ru, r P N,
where C might be regarded as a subset of any group presented on generators X.
Proposition 2.1.12 ([D L97]). If G “ xX|Ry, then Gab “ xX|R, Cy.
Now, in terms of presentations, part (3) of Theorem 2.1.11 means that every such group
G has a unique presentation of the form
xx1 , . . . , xr |xd11 , . . . , xdkk , Cy,
where k ď r and the di satisfy the conditions of the theorem.

2.2

Free Product

With the idea from the previous section, one can define
Ť the free product of groups. Given a
family of mutually disjoint groups tGi |i P Iu, let X “ iPI Gi and let I be a one-element set
disjoint from X.
Definition 2.2.1. A word is a sequence pa1 , a2 , . . .q such that ai P X Y I and for some n P N,
ai “ 1 for all i ě n.
A word in this case also can get reduced:
1. if ai P X is the identity element of some Gj , then we can delete ai ,
2. if ai and ai`1 belong to the same Gj , we can substitute it with their composition ai ˚Gj
ai`1 , and
3. if ak “ 1, then ai “ 1 for all i ě k.
With this reduction operation, the empty word I, represented by the sequence p1, 1, . . . , 1q, is
already reduced. Every non-empty reduced word can also be written uniquely as a1 a2 . . . an “
pa1 , a2 , . . . , an , 1, 1, . . .q, where ai P X. Considering the same binary operation,śthe juxtaposition, we are able to define the set of all reduced words on X and denote it by ˚iPI Gi .
ś
Theorem 2.2.2. ˚iPI Gi forms a group, free product of the family tGi |i P Iu, under the
juxtaposition.
ś
We can identify Gi with its isomorphic image in ˚iPI Gi .
Theorem 2.2.3 (Characteristic
Property of Free Product). Let
ś˚
ś˚tGi |i P Iu be a family of groups
with free product iPI Gi and family of inclusions ιi : Gi Ñ iPI Gi . If tψi : Gi Ñ H|i P Iu is
a family
ś˚ of group homomorphisms onto a group H, then there exists a unique homomorphism
ψ : iPI Gi Ñ H such that ψ ˝ ιi “ ψi for all i P I.

THE SEIFERT-VAN KAMPEN THEOREM

26

Similarly to the free groups, the free product of groups represents the coproduct in the
category of groups.
ś
Theorem 2.2.4. Consider a group G “ Ş ˚iPI Gi , where all Gi are free groups with taα uαPJi as
respective systems of free generators with iPI Ji “ ∅. Then G is a free group with taα uαPŤiPI Ji
as a system of free generators.
The theorem above can be extended to a free product of any finite number of free groups.
Example 2.2.5. If G is a group defined by generators a, b and relations a2 “ I and b3 “ I,
then G – Z2 ˚ Z3 . Generally, the group defined by the generator c and the relation cm “ I for
some m P N is the cyclic group Zm .

2.3

The Seifert-van Kampen Theorem

In this section, let X “ U Y V be a topological space, where both U and V are open in X.
Moreover, suppose that X, U , V , and U X V are all path-connected and that the fundamental
groups of U and V are known. There are two versions of the main theorem in this section.
Theorem 2.3.1 (Seifert-van Kampen Theorem, modern version). Let x0 P U X V and let
ϕ1 : π1 pU, x0 q Ñ H and ϕ2 : π1 pV, x0 q Ñ H be homomorphisms. Let i1 , i2 , j1 , j2 be the
homomorphisms indicated below, each induced by inclusion.
π1 pU, x0 q
i1

ϕ1

j1

π1 pU X V, x0 q

Φ

π1 pX, x0 q
i2

j2

H

ϕ2

π1 pV, x0 q
If ϕ1 ˝ i1 “ ϕ2 ˝ i2 , then there exists a unique homomorphism Φ : π1 pX, x0 q Ñ H such that
Φ ˝ j1 “ ϕ1 and Φ ˝ j2 “ ϕ2 .
Proof. First, we will show uniqueness of Φ. By Theorem 1.6.1 π1 pX, x0 q is generated by the
images of the induced homomorphisms j1 and j2 . Because Φ is determined by ϕ1 and ϕ2 on
these images, it follows that it is determined on every product of the elements from these
images. However, these products include all of the elements, and so Φ is determined by ϕ1 and
ϕ2 and, therefore, is unique.
To show the existence, consider a path f in X together with its path-homotopy class rf s in
X. If f lies in U , V or U X V , let rf sU , rf sV and rf sU XV denote its path-homotopy class in
U , V and U X V , respectively. The plan is to define several different maps, each building on
the previous, and, for that, consider the steps below.
Step 1. Let’s define a map ρ which assigns an element of the group H to each loop f based

THE SEIFERT-VAN KAMPEN THEOREM

27

at x0 that lies in U or in V . In other words, we want to extend both ϕ1 and ϕ2 to a set map
ρ defined on all loops in X, which are contained in either U or V . Define an element of the
group H by
#
ϕ1 prf sU q if f lies in U,
ρpf q “
ϕ2 prf sV q if f lies in V.
Note that ρ is well-defined because for f lying in both U and V we have ϕ1 prf sU q “ ϕ1 pi1 prf sU XV qq
and ϕ2 prf sV q “ ϕ2 pi2 prf sU XV qq. Moreover, since by assumption ϕ1 ˝ i1 “ ϕ2 ˝ i2 , we also have
ϕ1 prf sU q “ ϕ2 prf sV q. This makes up two facts about our map ρ:
1. If rf sU “ rgsU or rf sV “ rgsV , then ρpf q “ ρpgq (by the initial definition of ρ).
2. If both f and g lie in U or both of them lie in V , then ρpf ˚ gq “ ρpf q ˚H ρpgq since ϕ1
and ϕ2 are homomorphisms.
Step 2. Let’s extend ρ to a map σ, which assigns an element of H to each path f lying in U or
in V such that the map σ also satisfies the condition (1) of ρ and condition (2), when possible,
i.e., when f ˚ g is defined. This makes any path be workable as any other closed loop.
For each x P X, choose a path αx from x0 to x as follows:
• If x “ x0 , let αx be a constant path at x0 .
• If x P U X V , let αx be a path in U X V .
• If x P U or x P V with x R U X V , let αx
be a path in U or V , respectively.
This way for any path f in U or in V from x to
y, we define a loop Lpf q based at x0 such that
Lpf q “ αx ˚ pf ˚ αy q.
Note that because of our choice of αx and αy , if
f was a path in U , then Lpf q would be a loop in
Figure 2.3.0: Construction of a loop.
U as shown on Figure 2.3.0. The same follows
if f was a path in V .
Now, define σpf q “ ρpLpf qq. To show that this map works for us, we need to show that σ is
indeed an extension of ρ and that the properties given hold. If f is a loop based at x0 lying in
either U or V , then we have
Lpf q “ αx0 ˚ pf ˚ αx0 q,
where αx0 is a constant path at x0 . Then Lpf q is path-homotopic to f in either U or V , so
ρpLpf qq “ ρpf q by property (1) of ρ. Hence, σpf q “ ρpf q. To check condition (1), let f and
g be paths which are path homotopic in U or V . If F is a path homotopy in U from f to g,
then the homotopy LpF q is a path homotopy in U from Lpf q to Lpgq. Thus, Lpf q and Lpgq
are path-homotopic in U and so the condition (1) applies. The same can be done for the case

THE SEIFERT-VAN KAMPEN THEOREM

28

when f and g are path homotopic in V . To check condition (2), let f and g be arbitrary paths
in U or V such that f p0q “ x, f p1q “ gp0q “ y and gp1q “ z so f ˚ g is well-defined. Then we
have
Lpf q ˚ Lpgq “ pαx ˚ pf ˚ αy qq ˚ pαy ˚ pg ˚ αz qq »p αx ˚ ppf ˚ gq ˚ αz q,
which means that Lpf q ˚ Lpgq is path homotopic to Lpf ˚ gq. Therefore,
ρpLpf ˚ gqq “ ρpLpf q ˚ Lpgqq “ ρpLpf qq ˚H ρpLpgqq
by condition (2) for ρ. Hence, σpf ˚ gq “ σpf q ˚H σpgq and so property (2) is satisfied.
Step 3. Finally, let’s extend σ to a set map τ which assigns an element of H to an arbitrary
path f of X. Given any path f in X, using the Lebesgue Number Lemma A.0.30, we can
choose a subdivision 0 “ s0 ă s1 ă . . . ă sn “ 1 of the interval I such that f maps each of
the sub-intervals rsi´1 , si s into U or V . Let fi denote the path obtained by restricting f to the
sub-interval rsi´1 , si s. Then fi is a path in U or V with rf s “ rf1 s ˚ . . . ˚ rfn s. Define τ as
τ pf q “ σpf1 q ˚H . . . ˚H σpfn q.
This map will satisfy the similar conditions to ρ and σ:
1. If rf s “ rgs, then τ pf q “ τ pgq.
2. If f ˚ g is well-defined, then τ pf ˚ gq “ τ pf q ˚H τ pgq.
But before we show that the map actually satisfies these claims, let’s show that this definition
of τ is actually independent of the choice of subdivision. For this we need to show that the
value of τ pf q remains the same if we add one additional point p to the subdivision. Let i be
the index such that si´1 ă p ă si with p being a new point. If we compute τ pf q using the new
subdivision, the only change in the the value is the change of σpfi q to σpfi1 q ˚H σpfi2 q, where fi1
and fi2 are paths obtained by restricting f to rsi´1 , ps and rp, si s, respectively. However, since
fi is path homotopic to fi1 ˚ fi2 in U or V , we have τ pfi q “ τ pfi1 q ˚H τ pfi2 q by conditions (1) and
(2) which we know work for τ . Therefore, τ is indeed independent of our choice of subdivision
and hence well-defined.
It immediately follows that τ is an extension of σ: if f is already in U or V , then we can use
the trivial partition r0, 1s “ tt0u, p0, 1q, t1uu to define τ pf q and so τ pf q “ σpf q by definition.
Now, let’s show that τ satisfies the condition (1): if rf s “ rgs, then τ pf q “ τ pgq. Let f and g be
paths in X from x to y and let F be the path homotopy between them. Using the compactness
of r0, 1s2 for the Lebesgue number lemma A.0.30, we can choose subdivisions s0 ă . . . ă sn
and t0 ă . . . ă tm of r0, 1s such that F maps each sub-rectangle rsi´1 , si s ˆ rtj´1 , tj s into U or
V . Let fj be the path fj psq “ F ps, tj q. Then f0 “ f and fm “ g. Note that for all pairs of
paths fj´1 and fj there exists a subdivision s0 , . . . , sn of I such that F carries each rectangle
Ri “ rsi´1 , si s ˆ r0, 1s into either U or V . Given i, consider the linear map of I onto rsi´1 , si s
followed by f or by g - let’s call these maps fi and gi , respectively. The restriction of F to
the rectangle Ri gives a homotopy between fi and gi which is fully happening in either U or
V . However, it is not path-homotopy since the end-points of these restrictions do not have to

THE SEIFERT-VAN KAMPEN THEOREM

29

match. Consider the paths which represent the way of these end points during the homotopy,
i.e., define βi ptq “ F psi , tq. This way βi is the path in X from f psi q to gpsi q with both β0 and
βn being constant paths at x and y, respectively. We would like to show that for all i,
fi ˚ βi »p βi´1 ˚ gi .
For this consider the broken-line path along the bottom and right edges of Ri from si´1 ˆ 0
to si ˆ 1, as shown on Figure 2.3.1. The composition of F with this path is equal to the path

Figure 2.3.1
fi ˚ βi . A similar thing happens when we take the broken-line path along the left and top edges
of Ri and follow it by F - we obtain the path βi´1 ˚ gi . Since Ri is convex, by Example 1.1.6
there is a path homotopy in Ri between two broken-line paths and by Lemma 1.1.9 if we follow
by F , we obtain a path homotopy between fi ˚ βi and βi´1 ˚ gi which takes place in either U
or V . Using the conditions (1) and (2) for σ, we get that
σpfi q ˚h σpβi q “ σpβi´1 q ˚H σpgi q
and, thus, we have
σpfi q “ σpβi´1 q ˚H σpgi q ˚H σpβi q´1 .
Similarly, since β0 and βn are constant maps, and identity elements get mapped to the identity
elements, we have σpβ0 q “ σpβn q “ eH . Now, we can compute using the definition
τ pf q “ σpf1 q ˚H σpf2 q ˚H . . . ˚H σpfn q
“ σpβ0 q ˚H σpg1 q ˚H . . . ˚H σpgn q ˚H σpβn q´1
“ σpg1 q ˚H . . . ˚H σpgn q “ τ pgq.
Therefore, we can deduce that τ pfj´1 q “ τ pf q for each j and so τ pf q “ τ pgq.
Finally, let’s show that τ satisfies the condition (2). Suppose we have a composition of paths

THE SEIFERT-VAN KAMPEN THEOREM

30

f ˚ g in X. Choose a subdivision s0 ă ... ă sn of r0, 1s containing the point 1{2 as a subdivision
point sk such that f ˚ g carries each sub-interval into either U or V . For i “ 1, . . . , k, the
increasing linear map of I to rsi´1 , si s followed by f ˚ g is the same as the increasing linear
map from I to r2si´1 , 2si s followed by f ; let’s call the latter fi . Similarly, for i “ k ` 1, . . . , n,
the linear map of I to rsi´1 , si s followed by f ˚ g is the same as the increasing linear map of I
to r2si´1 ´ 1, 2si ´ 1s followed by g - let’s call this map gi´k . Using the subdivision s0 , . . . , sn
of f ˚ g from before, we have
τ pf ˚ gq “ σpf1 q ˚H . . . ˚H σpfk q ˚H σpg1 q ˚H . . . ˚H σpgn´k q.
Using the subdivision 2s0 , . . . , 2sk of the path f we have
τ pf q “ σpf1 q ˚H . . . ˚H σpfk q.
Similarly, using the subdivision 2sk ´ 1, . . . , 2sn ´ 1 of the path g we have
τ pgq “ σpg1 q ˚H . . . ˚H σpgn´k q.
Therefore, (2) clearly holds since τ pf ˚ gq “ τ pf q ˚H τ pgq.
Step 4. For each loop f in X based at x0 define Φprf sq “ τ pf q. The conditions (1) and (2)
from above show that Φ is a well-defined homomorphism. To show that Φ ˝ j1 “ ϕ1 consider
a loop f in U . Then
Φpj1 prf sU qq “ Φprf sq “ τ pf q “ ρpf q “ ϕ1 prf sU q.
Similarly, for a loop g in V we have
Φpj2 prgsV qq “ Φprgsq “ τ pgq “ ρpgq “ ϕ2 prgsV q.
The classical version of the same theorem assumes the modern version.
Theorem 2.3.2 (Seifert-van Kampen Theorem, classical version). Assume the hypotheses of
the modern version of the theorem. Let x0 P U X V . Consider j : π1 pU, x0 q ˚ π1 pV, x0 q Ñ
π1 pX, x0 q be the homomorphism of the free product that extends the homomorphisms j1 and
j2 . Then, j is surjective, and its kernel is the least normal subgroup N of π1 pU, x0 q ˚ π1 pV, x0 q
that contains all elements represented by words of the form i1 pgq´1 i2 pgq for g P π1 pU X V, x0 q.
The least normal subgroup of the noted product can also be described as a group
generated by all elements of the form i1 pgq´1 i2 pgq for g P π1 pU X V, x0 q and their conjugates.
Proof. Note that by Theorem 1.6.1, π1 pX, x0 q is generated by the images of j1 and j2 and,
thus, j is surjective. For the second part of the theorem, we will firstly show that N Ď kerpjq.
Recall that the kernel of j is a normal subgroup of π1 pU, x0 q ˚ π1 pV, x0 q. Note that it suffices to
show that i1 pgq´1 i2 pgq belongs to the kernel for all g P π1 pU X V, x0 q. To the contrary, if there
was an element of N which does not belong to the kernel, it would still belong to all normal

THE SEIFERT-VAN KAMPEN THEOREM

31

subgroups of π1 pU, x0 q ˚ π1 pV, x0 q, one of which is the kernel itself. Take an inclusion mapping
i : U X V Ñ X, then
ji1 pgq “ j1 i1 pgq “ i˚ pgq “ j2 i2 pgq “ ji2 pgq,
which implies that ji1 pgq “ ji2 pgq and so i´1
1 pgqi2 pgq is getting mapped to the identity element.
´1
Thus, i1 pgqi2 pgq belongs to the kernel of j. Moreover, j induces an epimorphism
k : π1 pU, x0 q ˚ π1 pV, x0 q{N Ñ π1 pX, x0 q
since it is a composition of a homomorphism and an epimorphism. To show that N equals
kerpjq, we need to show that k is injective since N is trivial. For that, it is enough to show
that k has a left inverse.
Let H denote the group π1 pU, x0 q ˚ π1 pV, x0 q{N . Also, let ϕ1 : π1 pU, x0 q Ñ H be the inclusion
map from π1 pU, x0 q to the free product followed by the projection of the free product onto its
quotient by N . Let ϕ2 : π1 pV, x0 q Ñ H be defined similarly. Consider the diagram
π1 pU, x0 q
i1

π1 pU X V, x0 q

j1

π1 pX, x0 q
i2

j2

ϕ1
Φ
k

H

ϕ2

π1 pV, x0 q
Note that from the diagram we can see that ϕ1 ˝i1 “ ϕ2 ˝i2 . Moreover, if g P π1 pU XV, x0 q, then
ϕ1 pi1 pgqq is the coset i1 pgqN in H, and ϕ2 pi2 pGqq is the coset i2 pgqN . Since i1 pgq´1 i2 pgq P N ,
these two cosets are actually equal.
From the modern version of the Seifert-van Kampen Theorem we know that there exists a
homomorphism Φ : π1 pX, x0 q Ñ H such that Φ ˝ j1 “ ϕ1 and Φ ˝ j2 “ ϕ2 . Let’s show
that Φ is the left inverse for k. For this to be true we need Φ ˝ k to act as an identity on
any generator of H, i.e., on any coset of the form gN , where g P π1 pU, x0 q or g P π1 pV, x0 q.
Suppose g P π1 pU, x0 q, then we have
kpgN q “ jpgq “ j1 pgq,
and so it follows that
ΦpkpgN qq “ Φpj1 pgqq “ ϕ1 pgq “ gN,
which is exactly what we need. Similarly, one can show the same thing if g P π1 pV, x0 q.
With Seifert-van Kampen’s Theorem, we can get an exact formula for the fundamental
group of a space X if we know the fundamental groups of a decomposition of X into U , V , and
their intersection U X V . This theorem often is used when “gluing” familiar spaces together
along a common and familiar subspace since instead of U and V we can take the covering
tUα |α P Au of X by path-connected open sets such that the family is closed under finite
intersection and all of its elements include the common point x0 .
Assuming the hypotheses of the Seifert-Van Kampen Theorem.

THE FUNDAMENTAL GROUP OF A WEDGE OF CIRCLES

32

Corollary 2.3.3. If U X V is simply-connected, then there is an isomorphism
k : π1 pU, x0 q ˚ π1 pV, x0 q Ñ π1 pX, x0 q.
Corollary 2.3.4. If V is simply-connected, then there is an isomorphism
k : π1 pU, x0 q{N Ñ π1 pX, x0 q,
where N is the least normal subgroup of π1 pU, x0 q containing the image of the homomorphism
i1 : π1 pU X V, x0 q Ñ π1 pU, x0 q
Definition 2.3.5. The real projective plane RP2 is a quotient space obtained from S 1 by
identifying each point x with its antipode ´x.
Theorem 2.3.6. The projective plane RP2 is a compact surface with fundamental group isomorphic to Z{2Z.
Proof. To show that RP2 is a compact surface, we need to show that it is compact, Hausdorff,
locally Euclidean and second countable. It is second countable since if S 2 has a countable basis
tUn u, the space RP2 would have a countable basis tppUn qu, where p : S 2 Ñ RP2 is a quotient
map. The image space is clearly Hausdorff and locally Euclidean. Moreover, the space RP2 is
compact as an image of a compact space S 2 under a continuous map p.
To calculate the fundamental group, consider S 1 “ U Y V with U “ S 1 ztxu for some point
x R S 1 and V being an open neighbourhood around x. Then U X V is an open disk around
x. Note that the fundamental group of any open neighbourhood is 0, since it is a simplyconnected space, so π1 pV q “ 0. The open disk U is a deformation retraction of S 1 , and,
therefore, π1 pU q – Z. Using Corollary 2.3.4, π1 pRP2 q – xay{xa|a2 y – Z{2Z.

2.4

The Fundamental Group of a Wedge of Circles

Ťn
Definition 2.4.1. Consider a Hausdorff space X “
i“1 Si ,
where each of Si is homeomorphic to the unit circle S 1 . If there
is a point p P X such that Si X Sj “ tpu whenever i ‰ j, then we
call the space X the wedge (bouquet) of the circles tSi uni“1 .
Note that each space Si is compact and, hence, closed
in X. Moreover, since each Si can be imbedded in the plane,
the same can be said about the space X. In other words, if Ci
denotes a circle ofŤradius i in R2 with center at pi, 0q, then X is
homeomorphic to ni“1 Ci .

Figure 2.4.1: Example of
the wedge of five circles.

Theorem 2.4.2. Let X be the wedge of the circles S1 , . . . , Sn with the common point p. Then
π1 pX, pq is a free group. Moreover, if fi is a loop in Si that represents a generator of π1 pSi , pq,
then the loops f1 , . . . , fn represent a system of free generators for π1 pX, pq.

THE FUNDAMENTAL GROUP OF A WEDGE OF CIRCLES

33

Let’s prove a more general result than this for a space X which is a union of infinitely many
circles, which all have a point in common.
Definition 2.4.3. Consider a space X, which is a union of the subspaces Xα for α P J. The
topology of X is said to be coherent with the subspaces Xα provided a subset C of X is
closed in X if C X Xα is closed in Xα for each α. An equivalent definition can be obtained
with open sets instead.
In other words, a topological space is coherent with a family of subspaces if it is a
topological union of those subspaces. In case of a finite collection of circles like before, X is
the union of finitely many closed subspaces tXi uni“1 and so the topology of X is automatically
coherent with these subspaces, since if C X Xi is closed in Xi , it is also closed in X and, thus,
C is the finite union of the sets C X Xi .
Definition 2.4.4. Let X be a space which is a union of the subspaces Sα , α P J, each of which
is homeomorphic to S 1 . If there is a point p P X such that for all α ‰ β we have Sα X Sβ “ tpu
and the topology of X is coherent with the subspaces Sα , then X is called the wedge of the
circles Sα .
Note that the Hausdorff condition, which is included in Definition 2.4.1, is not included
in the infinite case. But it is still required, it just follows from the coherent condition.
Lemma 2.4.5. Let X be the wedge of the circles tSα uαPJ . Then X is normal, and any compact
subspace of X is contained in the union of finitely many circles Sα .
Proof. Firstly note that one-point sets are closed in X. Consider disjoint closed subsets A
and B of the space X such that B does not contain the point p. Choose disjoint subsets Uα
and VŤα of Sα that are open
Ť in Sα and contain tpu Y pA X Sα q and B X Sα , respectively. Let
U “ αPJ Uα and V “ αPJ Vα . Then U and V are disjoint. Now, since all sets Uα contain p
we have that U XSα “ Uα . Similarly, since none of the sets Vα contain p, we have V XSα “ Vα .
Hence, U and V are open in X, and, thus, X is normal.
Consider a compact subspace C of X. Choose a point xα P C XpSα ztpuq if C XpSα ztpuq
is not empty. The set D “ txα u is closed in X, since its intersection with each space Sα is
either empty or a one-point set, which is closed in a Hausdorff space X. For the same reason,
each subset of D is closed in X. Thus, D is a closed discrete subspace of X contained in C
and since C is limit point compact, D must be finite.
Theorem 2.4.6. Let X be the wedge of circles tSα uαPJ with the common point p. Then
π1 pX, pq is a free group. Moreover, if fα is a loop in Sα representing a generator of π1 pSα , pq,
then the loops tfα u represent a system of free generators for π1 pX, pq.
Proof. Let iα : π1 pSα , pq Ñ π1 pX, pq be the homomorphism induced by inclusion and let Gα be
the image of iα . Note that if f is any loop in X based at p, then the image set of f is compact
and so f lies in some finite union of subspaces Sα . Moreover, if f and g are two path-homotopic

ADJOINING A TWO-CELL

34

loops in X, then they are path-homotopic in some finite union of the subspaces of Sα by the
preceding lemma.
To see that the groups tGα u generate π1 pX, pq, consider a loop f in X. It must lie in
Sα1 Y . . . Y Sαn for some finite set of indices. By Theorem 2.4.2, we have rf s as a product of
elements of the groups Gα1 , . . . , Gαn . It follows that iβ is a monomorphism. In the case f is
nulhomotopic in X, f must be path homotopic to a constant in some finite union of spaces
Sα , so by Theorem 2.4.2, f is path homotopic to a constant in Sβ .
Suppose there exists a reduced nonempty word w “ pgα1 . . . gαn q in the elements of
the groups Gα which represents the identity element of π1 pX, pq. Let f be a loop in X whose
path-homotopy class is represented by w. Then f is path homotopic to a constant in X and so
it is path homotopic to a constant in some finite union of subspaces Sα , which is not possible
according to Theorem 2.4.2.
Definition 2.4.7. Given two topological spaces X and Y with points x0 P X and y0 P Y , the
wedge X _ Y of X and Y is defined as the quotient space of their disjoint union where two
copies of the base points (one in X and one in Y ) are identified.
Example 2.4.8. Consider the wedge X of the spaces X1 , . . . , Xn . Let’s show that if for each
i, the common point p is a deformation retract of an open set Wi of Xi , then π1 pX, pq is the
free product of the groups π1 pXi , pq relative to the monomorphisms induced by inclusion.
Consider the problem for the case when X “ X1 _ X2 . We can assume that both X1 and X2
are path-connected since if Ci are the path components containing p in Xi , then π1 pCi , pq “
π1 pXi , pq. Let U “ X1 Y W2 and let V “ X2 Y W1 . Then both U and V are path-connected
since their deformation retracts are X1 and X2 , respectively, and U X V “ W1 Y W2 is simplyconnected since its deformation retract is just the point p. Therefore, by Theorem 2.3.3, there
is an isomorphism π1 pX1 , pq ˚ π1 pX2 , pq – π1 pX, pq.

2.5

Adjoining a Two-Cell

Theorem 2.5.1. Let X be a Hausdorff space and let A be a closed path-connected subspace
of X. Suppose there is a continuous map h : B 2 Ñ X which maps IntpB 2 q bijectively onto
XzA and maps S 1 “ BdpB 2 q onto A. Let p P S 1 , a “ hppq and let k : pS 1 , pq Ñ pA, aq be the
restriction of h. Then the homomorphism
i˚ : π1 pA, aq Ñ π1 pX, aq
induced by the inclusion is surjective, and its kernel is the least normal subgroup of π1 pA, aq
containing the image of k˚ : π1 pS 1 , pq Ñ π1 pA, aq.
Proof. Step 1. Consider the origin 0 of B 2 , its image x0 “ hp0q in X and an open set U “
Xztx0 u of X. Let’s show that A is the deformation retract of U .
Let C “ hpB 2 q and let π : B 2 Ñ C be the restriction of h. Consider the map
π ˆ I : B 2 ˆ I Ñ C ˆ I.

ADJOINING A TWO-CELL

35

Figure 2.5.1: Representation of the construction discussed in Step 1.
Since B 2 ˆ I is compact and C ˆ I is Hausdorff, the map π ˆ I is closed. Since it is closed and
surjective, it is a quotient map by definition. Its restriction
π 1 : pB 2 zt0uq ˆ I Ñ pCztx0 uq ˆ I
is a quotient map as well, since its domain is open in B 2 ˆ I and is saturated with respect
to π ˆ I. It is known that there is a deformation retraction of B 2 zt0u onto S 1 , and so using
the quotient map π 1 it can induce a deformation retraction of Cztx0 u to πpS 1 q. We extend
this deformation retraction to all U ˆ I by letting each point of A remain fixed during the
deformation. Therefore, A is a deformation retract of U .
Then by Theorem 1.4.6 the inclusion of A into U induces an isomorphism of fundamental groups and what we need to prove can be reduced to the following:
Let f be a loop whose class generates π1 pS 1 , pq. Then the inclusion of U into X
induces an epimorphism π1 pU, aq Ñ π1 pX, aq whose kernel is the least normal
subgroup containing the class of the loop g “ h ˝ f .
Step 2. In order to prove the reduced statement, consider the homomorphism π1 pU, bq Ñ
π1 pX, bq induced by inclusion relative to the base point b which does not belong to A.
Let b be any point of U zA. Now, X is the union of the open sets U and V “ XzA “
πpIntpB 2 qq. We know U is path-connected, since A is its deformation retraction. Because π is
a quotient map, its restriction to IntpB 2 q is also a quotient map and hence homeomorphism.
Thus, V is simply-connected. The set U X V “ V ztx0 u is homeomorphic to IntpB 2 qzt0u, so it
is path connected and its fundamental group is infinite cyclic. Since b is a point of U X V , by
Theorem 2.3.4 the homomorphism π1 pU, bq Ñ π1 pX, bq induced by the inclusion is surjective,
and its kernel is the least normal subgroup containing the image of the infinite and cyclic group
π1 pU X V, bq.

ADJOINING A TWO-CELL

36

Step 3. Now, let’s prove the result for a
point a. Let q be the point of B 2 which
is the midpoint of the line segment from 0
to p. Also, let b “ hpqq so b is a point in
U X V . Let f0 be a loop in IntpB 2 qzt0u
based at q that represents a generator of
the fundamental group of this space. Then
g0 “ h˝f0 is a loop in U XV based at b that
represents a generator of the fundamental
group of U X V . By Step 2 we know that
the homomorphism π1 pU, bq Ñ π1 pX, bq induced by the inclusion is surjective and its
kernel is the least normal subgroup conFigure 2.5.2: The situation described in Step 3.
taining the class of the loop g0 “ h ˝ f0 .
To obtain the similar result for the point a, consider γ as the straight line path in B 2 from q
to p and a path δ “ h ˝ γ in U from b to a. The isomorphism induced by the path δ commute
with the homomorphisms, both denoted δ0 , induced by inclusion in the diagram:
π1 pU, bq
δ0

π1 pU, aq

π1 pX, bq
δ0

.

π1 pX, aq

Therefore, the homomorphism of π1 pU, aq into π1 pX, aq induced by inclusion is surjective and
its kernel is the least normal subgroup containing the element δ0 prg0 sq.
The loop f0 represents a generator of the fundamental group of IntpB 2 qzt0u based at
q. Then the loop γ ˚ pf0 ˚ γq represents a generator of the fundamental group of B 2 zt0u based
at p. Therefore, it is path homotopic to either f or its inverse. Suppose the latter: following
the path homotopy by the map h, we note that δ ˚ pg0 ˚ δq »p g in U . Then δ̂prg0 sq “ rgs and
the theorem follows.
Note that the unit ball in the Theorem above can be replaced with any space B which
is homeomorphic to B 2 . We call such space a 2-cell. Then the space X in the Theorem
is obtained by ”adjoining” a 2-cell to A. In other words, the theorem above states that the
fundamental group of X is obtained from the fundamental group of A by killing off the class
k˚ rf s, where rf s generates π1 pS 1 , pq.

Chapter 3

Classification of Surfaces
By now, we have built all the skills we need to be able to classify all the compact surfaces
up to homeomorphism. This problem is more or less trivial for smaller dimensions, i.e., 0
and 1. For the smallest dimension, 0-dimensional connected manifold is just a point, which
means that any 0-dimensional disconnected manifold is just a discrete set. In the case of 1
dimension, we would have a manifold homeomorphic to either a circle or a closed interval in
case of compactness, otherwise, it has to be homeomorphic to the real line R. The reader is
welcome to read more about the one-dimensional case in [Dav87]. In this chapter, though, we
would like to handle the case of compact two-dimensional manifolds.

3.1

Fundamental Groups of Surfaces

We would like to start with some construction. Let’s look at surfaces which can be constructed
as quotient spaces from a polygonal region in a plane.
Consider a point c of R2 and a number a ą 0. Construct a circle in R2 with the center at
c and with the radius a. Given a finite sequence θ0 ă θ1 ă . . . ă θn of real numbers, where
n ě 3 and θn “ θ0 ` 2π, consider the points pi “ c ` apcos θi , sin θi q, which all lie on the circle
described. They also are numbered in counterclockwise order around the circle with pn “ p0 .
The line through pi´1 and pi splits the circle and, as a result, the plane into two closed pieces.
n
Let HŞ
i be the one that contains all the points tpi ui“1 , which we call vertices. Then the space
n
P “ i“1 Hi is what we call the polygonal region determined by the points tpi uni“1 . The
line segments pi pi`1 for all i “ 0, 1, . . . , n ´ 1 with pn “ p0 are called the edges of P . The
union of all edges is what we call the boundary of P and, thus, the region P zBdpP q “ IntpP q
is the interior.
Given a line segment L of R2 , an orientation of L is the ordering of its end points:
initial point a and final point b. In this case, we say that L is oriented from a to b. If
L1 is another line segment, oriented from c to d, then the order-preserving linear map of L
onto L1 is the homeomorphism h that carries the point x “ p1 ´ sqa ` sb of L to the point
hpxq “ p1 ´ sqc ` sd of L1 . Note that h is the straight-line homotopy between straight paths.

37

FUNDAMENTAL GROUPS OF SURFACES

38

Using this, if two polygonal regions P and Q have the same number of vertices, p0 , ..., pn
and q0 , ..., qn , respectively, with p0 “ pn and q0 “ qn , then combining all the separate homeomorphisms using the pasting lemma, we get a homeomorphism h of BdpP q with BdpQq that
carries the line segment from pi´1 to pi by a linear map onto the line segment from qi´1 to qi
as shown on Figure 3.1.1. If p and q are fixed points of IntpP q and IntpQq, respectively, then
this homeomorphism may be extended to a homeomorphism of P with Q which linearly maps
a perpendicular from p to a point x P BdpP q to a perpendicular from q to hpxq.

Figure 3.1.1
Definition 3.1.1. A labelling of the edges of a polygonal region P in the plane is a map
from the set of edges of P to a set S called the set of labels. Given an orientation of each
edge of P , and given a labelling of the edges of P , define an equivalence relation on the points
of P as follows:
#
x,
if x P IntpP q,
x„
hpxq, if x P BdpP q and both x and hpxq belong to edges with the same label.
The quotient space X obtained from this equivalence relation is said to have been obtained by
pasting the edges of P together according to the given orientations and labelling.
Definition 3.1.2. Let n P Nzt1u and let r : S 1 Ñ S 1 be rotation through the angle 2π{n.
Form a quotient space X from the unit ball B 2 by identifying each point x of S 1 with the
points rpxq, r2 pxq, . . . , rn´1 pxq. In this case, X is called the n-fold dunce cap and we will
denote it as Dn .

FUNDAMENTAL GROUPS OF SURFACES

39

Example 3.1.3. Consider the orientations and labelling of the edges of
the triangular region pictured in Figure 3.1. All different orientations and
labellings can give us different quotient spaces. Note that the provided
labellings do not give a full list of all
possible labellings for a triangle.
However, we would like to describe
a method for specifying orientations
and labels for the edges of a polygonal region without drawing a picture.

Figure 3.1: Different labellings for a triangle.

Definition 3.1.4. Let P be a polygonal region with vertices p0 , ..., pn , where p0 “ pn . Given
orientations and a labelling of the edges of P , let a1 , ..., am be the distinct labels that are
assigned to the edges of P . For each k, let aik be the label assigned to the edge pk´1 pk , and
let ϵk be equal `1 or ´1 according to the orientation assigned to this edge, i.e., if it goes from
pk´1 to pk or the reverse. Then the number of edges of P , the orientations of the edges, and
the labelling are completely specified by the symbol
w “ pai1 qϵ1 pai2 qϵ2 . . . pain qϵn ,
which is called a labelling scheme for the edges of P .
We can omit the positive exponents in the labelling scheme to get the scheme to be
looking like words which we have been working with in the previous chapter. Recall the
first figure in Example 3.1.3: the labelling scheme there can be written as a´1 ba if we take
p0 to be a top vertex of the triangle. If we decide to switch p0 we would get the schemes
baa´1 and aa´1 b. It is clear that a cyclic permutation of the terms of the labelling scheme
will change the end space X formed by using the scheme only up to homeomorphism.
Example 3.1.5. A sphere can be constructed by pasting the edges of a square
with the labelling scheme aa´1 bb´1 .
Torus T2 can also be constructed by
pasting the edges of a square, but with
the labelling scheme aba´1 b´1 as shown
on Figure 3.1.2.

Figure 3.1.2: Construction of a torus using a
square labelling.

FUNDAMENTAL GROUPS OF SURFACES

40

Example 3.1.6. Recall that we defined the projective plane to be homeomorphic to the
quotient space of the unit ball B 2 obtained by identifying every point of the boundary with
its antipode. Since the unit square is homeomorphic to the unit ball, we can specify the same
space by the labelling scheme abab.
Theorem 3.1.7. Let X be the space obtained from a finite collection of polygonal regions by
pasting edges together according to some labelling scheme. Then X is a compact Hausdorff
space.
Proof. We will prove the case where X is obtained from a single polygonal region. This can be
extended to an arbitrary collection of polygonal regions. Firstly, note that since the image of
a compact space under a continuous map is compact and the quotient map is continuous, X is
compact. To show that X is also Hausdorff, let’s use the Lemma A.0.27 and instead show that
the quotient map π is a closed map. For this, we need to show that for each closed set C of
P , the set π ´1 pπpCqq is closed in P . Now, the set π ´1 pπpCqq consists of all points of C and all
points of P which are pasted to points of C by the map π. To determine these points consider
the compact subspaces C X e of P for each edge e. If ei is an edge that gets pasted to e, and
if hi : ei Ñ ei is the pasting homeomorphism, then the set De “ π ´1 pπpCqq X e contains the
space hi pCei q. Thus, De equals the union of Ce and the spaces hi pCei q, as ei ranges through
all the edges of P which are pasted to e. Since this union is compact, it is closed in e and in
P . Since π ´1 pπpCqq is the union of the set C and sets De , as e ranges over all edges of P , it
is closed in P , as needed.
Note that if X is obtained by pasting the edges of a polygonal region together, the
quotient map π may map all the vertices of the polygonal region to a single point of X, or it
may not. In the case of the torus, the quotient map does satisfy this condition, while in the
case of the sphere, it does not.
Theorem 3.1.8. Let P be a polygonal region; let
w “ pai1 qϵ1 pai2 qϵ2 . . . pain qϵn
be a labelling scheme for the edges of P . Let X be the resulting quotient space and let π : P Ñ X
be the quotient map. If π maps all the vertices of P to a single point x0 of X, and if a1 , . . . , ak
are the distinct labels that appear in the labelling scheme, then π1 pX, x0 q is isomorphic to the
quotient of the free group on k generators α1 , ..., αk by the least normal subgroup containing
the element
pαi1 qϵ1 pαi2 qϵ2 . . . pαin qϵn .
Proof. The map π sends all vertices of P to a single point of X. Therefore, the space A “
πpBdpP qq is a wedge of k circles. For each i, choose an edge of P which is labelled ai . Consider
the linear map fi of I onto the chosen edge oriented counterclockwise and let gi “ π ˝ fi . Then
the loops g1 , . . . , gk represent the set of free generators for π1 pA, x0 q. The loop f going around
BdpP q once in the clockwise direction generates the fundamental group of BdpP q and, thus,
the loop π ˝ f equals the loop pgi1 qϵ1 . . . pgin qϵn . Now, the needed result follows from Theorem
2.5.1.

FUNDAMENTAL GROUPS OF SURFACES

41

Definition 3.1.9. Consider the space Tn obtained from a 4n-sided polygonal region P with
the labelling scheme
´1
´1 ´1
´1 ´1
pa1 b1 a´1
1 b1 qpa2 b2 a2 b2 q . . . pan bn an bn q.
This space is called the n-fold connected sum of tori or n-torus.
In other words, to construct the 2-fold torus, we can consider the polygonal region P .
See Figure 3.1.3. If we split the polygonal region P along the indicated line c, each of the
resulting pieces represents a torus with an open disc removed. Another way to construct such
surface would be taking two copies of the torus T 2 , deleting a small open disk from each of
them, and pasting the remaining pieces together along their edges. A similar argument for
both of construction techniques shows the construction of the 3-fold torus T #T #T and so on.
See Figure 3.1.4.

Figure 3.1.3: Construction of T #T from a polygonal region [Mun00].

Figure 3.1.4: Construction of T #T #T from a polygonal region.
Theorem 3.1.10. Let X denote the n-fold torus. Then π1 pX, x0 q is isomorphic to the quotient
of the free group on the 2n generators α1 , β1 , . . . , αn , βn by the least normal subgroup containing
the element
rα1 , β1 srα2 , β2 s . . . rαn , βn s,
where rα, βs “ αβα´1 β ´1 .
Proof. To be able to use Theorem 3.1.8 we need to show that the quotient map sends all the
vertices of the space X to a single point x0 of X since all the labels are distinct by definition.
Note that every n-fold torus can be split up into n separated tori, which means that it is

FUNDAMENTAL GROUPS OF SURFACES

42

Figure 3.1.5: Generators for T1 , T2 and T3 , respectively [Hat02].
enough to show that all vertices of a single torus get mapped to a single point. That is true by
construction. For instance, consider a double torus with its generators, all represented in the
middle of Figure 3.1.5. Loops a and b, which are generated from one of the tori, get combined
with loops c and d from the construction of the second tori. All together these four generators
can make up any loop on the double torus.
Definition 3.1.11. Let m ą 1. Consider the space obtained from a 2m-sided polygonal region
P in the plane by means of the labelling scheme
pa1 a1 qpa2 a2 q . . . pam am q.
This space is called the m-fold connected sum of projective planes, or simply the m-fold
projective plane, and denoted by RPm “ RP2 # . . . #RP2 .
Similarly to Theorem 3.1.10, we get the following result for RPm .
Theorem 3.1.12. Let X denote the m-fold projective plane. Then π1 pX, x0 q is isomorphic
to the quotient of the free group on m generators α1 , . . . , αm by the least normal subgroup
containing the element
pα1 q2 pα2 q2 . . . pαm q2 .

Example 3.1.13. The Klein bottle K is
the space obtained from a square by means
of the labelling scheme aba´1 b as shown on
Figure 3.1.6. Moreover, by Theorem 3.1.8
we know the presentation of its fundamental
group: π1 pKq “ xa, b|aba´1 by.
Figure 3.1.6: Construction of the Klein Bottle
[Mun00].

HOMOLOGY OF SURFACES

3.2

43

Homology of Surfaces

By this point we know how to construct a surface or get a presentation of a surface, but we
do not know yet how to compare its fundamental group to a fundamental group of another
surface. This is what we are going to explore in this section.
Definition 3.2.1. Let X be a path-connected space with x0 P X. Define the first homology
group of X as
H1 pX, x0 q “ π1 pX, x0 q{rπ1 pX, x0 q, π1 pX, x0 qs.
We know that if X is a path-connected space, and if α is a path in X from x0 to x1 ,
then there is an isomorphism α̃ of π1 pX, x0 q with π1 pX, x1 q, but the isomorphism depends on
the choice of the path α. We would like to verify a stronger result for the group H1 pXq. In
this case, the isomorphism of the “abelianized fundamental group” based at x0 with one based
at x1 , induced by the path α, is independent of the choice of the path α.
To verify the independence, it suffices to show that if α and β are two paths from x0 to
x1 , then the path α ˚ β induced the identity isomorphism of π1 pX, x0 q{rπ1 pX, x0 q, π1 pX, x0 qs
with itself. Indeed, if rf s P π1 pX, x0 q, then we have
g̃rf s “ rg ˚ f ˚ gs “ rgs´1 ˚ rf s ˚ rgs.
When we pass to the cosets in the abelian group π1 pX, x0 q{rπ1 pX, x0 q, π1 pX, x0 qs, we see that
g̃ induces the identity map.
One can show that the base point is not relevant in the notation of this group similarly
to the fundamental group of a path-connected space. This being said, we will denote it by
H1 pXq instead.
Showed independence of the base point and a path shows that to differentiate between
two surfaces one can compute their homology groups instead of fundamental groups. To do
this we need the following result.
Theorem 3.2.2. Let F be a group with N being a normal subgroup of F . Consider the
projection q : F Ñ F {N . Then the projection homomorphism p : F Ñ F {rF, F s induces an
isomorphism
ϕ : qpF q{rqpF q, qpF qs Ñ ppF q{ppN q.
In other words, if one divides F by N and abelianizes the quotient, they would obtain the same
result as if we abelianize F first and then divide by the image of N in this abelianization.
Proof. Consider the projection homomorphisms p, q, r, s as given in the following diagram,
where qpF q “ F {N and ppF q “ F {rF, F s.
s

qpF q
q

qpF q{rqpF q, qpF qs

u

F

ϕ
p

ψ

v

ppF q

r

ppF q{ppN q

HOMOLOGY OF SURFACES

44

Since r ˝ p maps N to the identity, it induces a homomorphism u : qpF q Ñ ppF q{ppN q.
Now, since the image group is abelian, the homomorphism u induces a homomorphism ϕ :
qpF q{rqpF q, qpF qs Ñ ppF q{ppN q. On the other hand, since s ˝ q maps F onto an abelian
group, it also induces a homomorphism v : ppF q Ñ qpF q{rqpF q, qpF qs. Because s ˝ q maps
N to the identity, the same thing is done by v ˝ p and, thus, v induces a homomorphism
ψ : ppF q{ppN q Ñ qpF q{rqpF q, qpF qs.
We can describe the homomorphisms ϕ and ψ in a similar way such that they are
inverses of each other. For instance, for a given y in qpF q{rqpF q, qpF qs, choose an element x of
F such that s ˝ qpxq “ y. Then ϕpyq “ rpppxqq.
Corollary 3.2.3. Let F be a free group with free generators α1 , α2 , . . . , αn . Let N be the
least normal subgroup of F containing the element x of F and let G “ F {N . Consider the
projection p : F Ñ F {rF, F s. Then G{rG, Gs is isomorphic to the quotient of F {rF, F s, which
is free abelian with basis ppα1 q, . . . , ppαn q, by the subgroup generated by ppxq.
Proof. The group N is generated by x and all of its conjugates. Also, the group ppN q is
generated by ppxq since p is a projection. Therefore, by the preceding theorem, the corollary
follows.
Theorem 3.2.4. If X is the n-fold connected sum of tori, then H1 pXq is a free abelian group
of rank 2n.
Proof. By Theorem 3.1.10, the fundamental group of the n-fold tori is isomorphic to the
quotient of the free group on the 2n generators α1 , β1 , . . . , αn , βn by the least normal subgroup
containing the element
rα1 , β1 srα2 , β2 s . . . rαn , βn s,
where rα, βs “ αβα´1 β ´1 . Now, using Corollary 3.2.3, H1 pXq is isomorphic to the quotient of
the free abelian group F 1 on the set of generators α1 , β1 , . . . , αn , βn by the subgroup generated
by the element rα1 , β1 s . . . rαn , βn s. Since the group F 1 is abelian, the element equals to the
identity element. Using presentations we can write the following:
H1 pTn q “ pπ1 pTn qqab “ xα1 , β1 , . . . , αn , βn |rα1 , β1 s . . . rαn , βn syab “ Z‘2n .

Theorem 3.2.5. If X is the m-fold connected sum of projective planes, then the torsion
subgroup T pXq of H1 pXq has order 2, and H1 pXq{T pXq is a free abelian group of rank m ´ 1.
Proof. By Theorem 3.1.12, π1 pXq is isomorphic to the quotient of the free group F 1 on the set
of generators α1 , . . . , αm by the subgroup generated by pα1 q2 . . . pαm q2 . Let β “ α1 ¨ . . . ¨ αm .
The torsion subgroup T pXq is generated by β 2 and, thus, the order of it is 2. Moreover, the
elements α1 , . . . , αm´1 , β form a basis for F 1 . Now, by Corollary 3.2.3, H1 pXq is isomorphic
to a quotient of a free abelian group generated by m elements by the subgroup generated by
the image of β 2 . Then H1 pXq is isomorphic to the quotient of the m-fold Cartesian product

CUTTING AND PASTING

45

Z ˆ . . . ˆ Z by the subgroup 0 ˆ . . . ˆ 0 ˆ 2Z, which is the same as Zm´1 ‘ Z2 . By Theorem
2.1.11, both H1 pXq{T pXq and T pXq are free abelian. Thus, T pXq is isomorphic to Z2 and so
H1 pXq{T pXq is isomorphic to Zm´1 . As a result, H1 pXq{T pXq is a free abelian group of rank
m ´ 1, as required.
Now, as we have computed the first homology groups for the connected sum of tori and
the connected sum of projective planes, we get the following.
Corollary 3.2.6. Let Tn and RPm denote the n-fold connected sum of tori and the m-fold
connected sum of projective planes, respectively. The the surfaces S 2 , T1 , T2 , . . . , RP1 , RP2 , . . .
are all topologically distinct.

3.3

Cutting and Pasting

Now, as we have developed some algebraic techniques, we also need to catch up on some geometric techniques for computing the fundamental group. These “cut-and-paste” techniques
let us see how a space X can be represented by different collections of polygonal regions and
different labelling schemes.
First, let’s consider cutting. Let P be a polygonal region with successive vertices p0 , p1 , . . . , pn “
p0 . Given k with 1 ă k ă n ´ 1, consider the polygonal regions Q1 with successive vertices
p0 , p1 , . . . , pk , p0 , and Q2 with successive vertices p0 , pk , . . . , pn “ p0 . These regions have the
edge p0 pk in common and the region P is their union.
Now, let’s move (by a translation in R2 ) one of the regions, for instance, Q1 , away
from the region Q2 to obtain two polygonal regions with empty intersection. Call this new
region Q11 . Then the regions Q11 and Q2 are said to have been obtained by cutting P apart
along the line from p0 to pk . See Figure 3.3.1 from right to left. Note that the region P is
homeomorphic to the quotient space of Q11 and Q2 obtained by pasting the edge of Q11 going
from q0 to qk to the edge of Q2 going from p0 to pk , by a linear map of one edge onto the
other. For the reverse operation, suppose we are given two disjoint polygonal regions Q11 with

Figure 3.3.1: Visualization of the pasting operation for a polygonal region P with 7 vertices.
successive vertices q0 , . . . , qk , q0 , and Q2 , with successive vertices p0 , pk , . . . , pn “ p0 . Also,

CUTTING AND PASTING

46

suppose we form a quotient space by pasting the edge of Q11 from q0 to qk onto the edge of Q2
from p0 to pk , by a order-preserving linear map of one edge onto the other.
The points of Q2 lie on a circle and are arranged in counterclockwise fashion. Let us
choose points p1 , . . . , pk´1 on the same circle in such a way that p0 , p1 , . . . , pk´1 , pk are arranged
in counterclockwise order, and let Q1 be the polygonal region with these as successive vertices.
There is a homeomorphism of Q11 onto Q1 that carries qi to pi for each i and maps the edge q0 qk
of Q11 linearly onto the edge p0 pk of Q2 . Therefore, the quotient space before is homeomorphic
to the region P which is the union of Q1 and Q2 . We say that P is obtained by pasting
Q11 and Q2 together along the indicated edge. See Figure 3.3.1 from left to right. We can
summarize this as a theorem.
Theorem 3.3.1. Suppose X is the space obtained by pasting the edges of m polygonal regions
together according to the scheme
y0 y1 , w2 , . . . , wm .
Let c be a label not appearing anywhere in the scheme above. If both y0 and y1 have length at
least two, then X can also be obtained by pasting the edges of m ` 1 polygonal regions according
to the scheme
y0 c´1 , cy1 , w2 , . . . , wm .
Note that the converse of this statement also holds due to the nature of cutting.
We can make a list of elementary scheme operations which we are allowed to perform
without affecting the resulting space X.
1. Cut: replacing the scheme w1 “ y0 y1 with schemes y0 c´1 and cy1 , provided that c does
nor appear elsewhere in the w1 and both y0 and y1 have length at least two.
2. Paste: replacing the scheme y0 c´1 and cy1 by the scheme y0 y1 , provided c does not
appear elsewhere in the total scheme.
3. Relabel: replacing all occurrences of any given label by some other label which does not
appear anywhere in the total scheme. Similarly, one can change the sign of the exponent
of all occurrences of a label.
4. Permute: replacing one of the schemes wi by a cyclic permutation of wi . In other words,
the scheme now begins with a different vertex without changing anything else.
5. Flip: replacing the scheme pai1 qϵ1 . . . pain qϵn with its formal inverse pain q´ϵn . . . pai1 q´ϵ1 .
6. Cancel: deleting pairs aa´1 in the scheme y0 aa´1 y1 given that a does not appear elsewhere in the total scheme and both y0 and y1 have the length at least two. To see the
geometric meaning of this operation, consider Figure 3.3.2.
7. Uncancel: the reverse operation of the previous operation.

CUTTING AND PASTING

47

Figure 3.3.2: Visualization of the cancel operation.
Definition 3.3.2. Two labelling schemes for collections of polygonal regions are equivalent
if one can be obtained from the other by a sequence of elementary scheme operations.
Note that since each elementary operation has its inverse operation also on the list
of the elementary operations, the notion of equivalence, which was introduced above, is an
equivalence relation.
Example 3.3.3. Going back to Example 3.1.13, we already know that the Klein bottle K
is the space obtained from the labelling scheme aba´1 b. It is homeomorphic to the 2-fold
projective plane RP2 #RP2 , which can be seen through the following elementary operations
and visualized as shown on Figure 3.3.3.
aba´1 b „ abc´1 and ca´1 b
„ c´1 ab and b´1 ac´1
´1

„c

´1

aac

„ aacc

cutting
permuting and flipping
pasting
permuting and relabelling

Figure 3.3.3: Klein Bottle transformed into RP2 #RP2 .

THE CLASSIFICATION THEOREM

3.4

48

The Classification Theorem

First we would like to show that every space obtained by pasting the edges of the polygonal
region together in pairs is homeomorphic either to S 2 , Tn or RPm .
Consider polygonal regions P1 , . . . , Pk with labelling schemes w1 , . . . , wk . We call a scheme
proper if each label appears exactly twice in a labelling scheme. Note that a proper scheme
remains proper after any elementary operation from the previous section.
Definition 3.4.1. Let w be a proper labelling scheme for a single polygonal region. We call
w of torus type if each label in it appears exactly twice, each time with a different exponent.
Otherwise, we call w of projective type.
Consider a scheme w of projective type. This being said w has either a label not
appearing twice or a label appearing twice but with the same exponent. Consider the latter
case, i.e., w “ ry0 sary1 sary2 s, where writing ryi s means that yi may be empty.
Lemma 3.4.2. Consider a proper scheme w “ ry0 sary1 sary2 s, where some of the yi for i “
0, 1, 2 may be empty. Then w „ aary0 y1´1 y2 s.

(a) Case where y0 is empty.

(b) General case.

Figure 3.4.1: Labelling schemes operations following the proof of the Lemma 3.4.1.
Proof. Let’s first assume that y0 is empty. In this case we need to show that ary1 sary2 s „
aary1´1 sry2 s. In case of y1 being empty as well, we have an automatic equality. Otherwise, in
case of y2 being empty, one has ary1 sa being equal to a´1 ry1´1 sa´1 , then a´1 a´1 ry1´1 s, which is
also equal to aary1´1 s. Now, if neither of ry1 s or ry2 s is empty, one has to do a similar sequence
of operations to show the same thing. Consider operations shown on Figure 3.4.1a:
ary1 sary2 s „ ary1 sc and c´1 ary2 s

cutting

„ ary1 sc and cry2´1 sa´1
„ ry1 sccry2´1 s
„ aary1´1 sry2 s

permuting and flipping
pasting
permuting and relabelling.

Now we can consider the general case with y0 not being empty. In case of both y1 and y2
being empty, one can permute the scheme and get the wanted result. Otherwise, consider the

THE CLASSIFICATION THEOREM

49

sequence of operations shown on Figure 3.4.1b:
ry0 sary1 sary2 s „ ry0 sab´1 and bry1 sary2 s

cutting

„ b´1 ry0 sa and ry1´1 sb´1 ry2´1 sa´1
„ b ry0 sry1´1 sb´1 ry2´1 s
„ bry2 sbry1 y0´1 s
„ bbry2´1 y1 y0´1 s
„ ry0 y1´1 y2 sb´1 b´1
„ aary0 y1´1 y2 s

permuting and flipping

´1

pasting
permuting and relabelling
by the case prior when y0 is empty
flipping
permuting and relabelling.

Corollary 3.4.3. If w is a scheme of projective type, then it is equivalent to a scheme of the
same length and of the form pa1 a1 qpa2 a2 q . . . pak ak qw1 , where k ě 1 and w1 is either of torus
type or empty.
Proof. The scheme w can be written as ry0 sary1 sary2 s, which by Lemma 3.4.2 is equivalent to
w1 “ aaw1 with the same length as w. If w1 is of torus type, we are done. Otherwise, rewrite
w1 as aarz0 sbrz1 sbrz2 s “ raaz0 sbrz1 sbrz2 s. By the same Lemma, this scheme is equivalent to
w2 “ bbraaz0 z1´1 z2 s “ bbaaw2 with the same length as w. If w2 is of torus type, we are done.
Otherwise, the argument can be continued until we reach the torus type scheme.
Lemma 3.4.4. Consider a proper scheme w “ w0 w1 , where w1 is a scheme of torus type
which does not contain two adjacent terms having the same label. Then w is equivalent to a
scheme w1 “ w0 w2 , where w2 has the same length as w1 and has the form w2 “ aba´1 b´1 w3 ,
where w3 is either of torus type or empty.
Proof. Step 1. First, let’s show that w can be written in the form
w “ w0 ry1 sary2 sbry3 sa´1 ry4 sb´1 ry5 s,

(3.4.1)

where some of the yi might be empty. Let a be the label whose occurrences with opposite
exponents are the close together as possible. Since these occurrences are non-adjacent, there
is at least one other label in between - call this label b. We can, without loss of generality,
assume that b and a appear with positive exponent first. Otherwise, we just need to switch
the labels. Now, since a and a´1 are the closest to each other such labels, the label b´1 cannot
appear before a´1 . Thus, it has to come after a´1 or before a. In the first case, we are finished.
The second scheme is the same if one switch the label a to b´1 and b to a.
Step 2. Consider the form 3.4.1 of w and rewrite it as
w “ w0 ry1 sary2 by3 sa´1 ry4 b´1 y5 s.

THE CLASSIFICATION THEOREM

50

(a) Step 2 operation.

(b) Step 3, second case.

(c) Step 4, second case.

Figure 3.4.2: Labelling schemes operations following the proof of the Lemma 3.4.2.
Now, consider the cutting and pasting operation represented in Figure 3.4.2a. We have the
following result.
w „w0 cry2 by3 sc´1 ry1 y4 b´1 y5 s
„w0 ary2 sbry3 sa´1 ry1 y4 sb´1 ry5 s “ w1

relabelling.

Step 3. If all the schemes y1 , y4 , y5 and w0 are empty, then one gets
w1 “ ary2 sbry3 sa´1 b´1
„ bry3 sa´1 b´1 ary2 s
„ ary3 sba

´1 ´1

b

permuting

ry2 s “ w

2

relabelling.

Otherwise, we can apply the operations represented in
w1 “ w0 ary2 sbry3 sa´1 ry1 y4 sb´1 ry5 s
„ w0 cry1 y4 y3 sa´1 c´1 ary2 y5 s
´1 ´1

„ w0 ary1 y4 y3 sba

b

ry2 y5 s

Figure 3.4.2b
relabelling.

Both times end scheme can be put as w2 “ w0 ary1 y4 y3 sba´1 b´1 ry2 y5 s.
Step 4. Similar to the previous step, if the schemes w0 , y1 and y2 are empty, one gets
w1 “ ary1 y4 y3 sba´1 b´1
„ ba´1 b´1 ary1 y4 y3 s
´1 ´1

„ aba

b

ry1 y4 y3 s “ w

permuting
3

relabelling.

THE CLASSIFICATION THEOREM

51

Otherwise, we can apply the operations
w2 “ w0 ary1 y4 y3 sba´1 b´1 ry2 y5 s
„ w0 ca´1 c´1 ary1 y4 y3 y2 y5 s

Figure 3.4.2c

„ w0 aba´1 b´1 ry1 y4 y3 y2 y5 s

relabelling.

Now, the only thing left to show is that a connected sum of projective planes and tori
is equivalent to a connected sum of projective planes.
Lemma 3.4.5. Any proper scheme w of the form w0 pccqpaba´1 b´1 qw1 is equivalent to the
scheme w1 “ w0 paabbccqw1 .
Proof. Consider the sequence of operations:
w0 pccqpaba´1 b´1 qw1 „ ccrabsra´1 b´1 srw1 w0 s
´1

„ ccrabsrbas

rw1 w0 s

„ rabscrbascrw1 w0 s

permuting
inverse substitution
Lemma 3.4.2

“ rasbrcsbracw1 w0 s
„ bbrac´1 acw1 w0 s

Lemma 3.4.2

“ rbbsarcs´1 arcw1 w0 s
„ aarbbccw1 w0 s

Lemma 3.4.2

„ w0 aabbccw1

permuting.

In particular the theorem above states that the space X “ T1 #RP2 is homeomorphic
to RP3 “ RP2 #RP2 #RP2 . We can visualize it as on Figure 3.4.3.

Figure 3.4.3: Elementary scheme operations showing that T 2 #RP2 – RP3 .

THE CLASSIFICATION THEOREM

52

Theorem 3.4.6 (The Classification Theorem). Let X be the quotient space obtained from a
polygonal region in the plane by pasting its edges together in pairs. Then X is homeomorphic
either to S 2 , to the n-fold torus Tn , or to the m-fold projective plane RPm .
Proof. Let w be a proper labelling scheme of length at least 4 for the quotient space X from
the polygonal region P . We would like to show that w is equivalent to one of the schemes:
1. aa´1 bb´1 , which produces a 2-sphere;
´1
´1 ´1
´1 ´1
2. pa1 b1 a´1
1 b1 qpa2 b2 a2 b2 q . . . pan bn an bn q with n ě 1, which produces Tn ;

3. abab, which produces RP1 ;
4. pa1 a1 qpa2 a2 q . . . pam am q with m ě 2, which produces RPm .
Let’s consider w to be a proper scheme of torus type. Using the method of mathematical
induction, we will show that w is equivalent either to the scheme (1) or to the scheme (2). If
w has length 4, then it has to be either aa´1 bb´1 or aba´1 b´1 , where the first one is a scheme
of type (1) and the second one is of type (3). Assume w has length greater than 4. If w is
equivalent to a shorter scheme of torus type, then by induction hypothesis, it is equivalent
either to the scheme (1) or to the scheme (2). Otherwise, w cannot contain two adjacent
elements having the same label. By Lemma 3.4.4 (taken with empty w0 ), w is equivalent to
a scheme having the same length as w but of the form aba´1 b´1 w3 , where w3 is a non-empty
scheme of torus type. Similarly to w, w3 cannot contain any two adjacent terms having the
same label. Thus, we can use Lemma 3.4.4 again with w0 “ aba´1 b´1 and, as a result, w has
to be equivalent to the scheme of the form
paba´1 b´1 qpcdc´1 d´1 qw4 ,
where w4 is a either empty or of torus type. If it is empty, we are finished and the scheme is
of type (2). Otherwise, we can apply the lemma again until we reach an empty scheme.
Now, let’s consider w to be a proper scheme of projective type. Similarly, we are going to use
induction to show that w is equivalent either to the scheme (3) or to the scheme (4). If w has
length 4, it must be either aabb, which is a scheme of type (4), or aab´1 b, which by Lemma
3.4.2 is equivalent to the scheme abab, which is a scheme of type (3). Now, assume the length
of w is greater than 4. By Corollary 3.4.3, w has to be equivalent to the scheme of the form
w1 “ pa1 a1 q . . . pak ak qw1 ,
where k ě 1 and w1 being either empty or of torus type. In case it is empty, we are done.
Otherwise, if w1 has two adjacent terms with the same label, then w1 is equivalent to a shorter
scheme of projective type and we can apply the induction hypothesis. Otherwise, by Lemma
3.4.4, w1 is known to be equivalent to a scheme of the form
w2 “ pa1 a1 q . . . pak ak qaba´1 b´1 w2 ,

THE CLASSIFICATION THEOREM

53

where w2 is either empty or of torus type. By Lemma 3.4.5, w2 is equivalent to the scheme
pa1 a1 q . . . pak ak qaabbw2 , which is a scheme of type (4). Continuing the same process, we reach
the empty scheme since w is finite, and so it has to be equivalent to the scheme of type (4).
Example 3.4.7. Consider X to be a quotient space obtained from an 8-sided polygonal region
P by means of the labelling scheme abcdad´1 cb´1 . See Figure 3.4.4. Let π : P Ñ X be the
quotient map. We can not use Theorem 3.1.8 since we do not have all vertices mapped to a
single point. Instead, we end up with 2 points x0 and x1 .
In this case we can still calculate the fundamental group of the boundary of X. We can see
that a connects x1 to itself, c connects x0 to itself, while b and d are paths between x0 and
x1 of opposite direction. We now want to calculate its fundamental group. Note that we can
retract the segment d into the point x0 , making the point coincide with x1 . The resulting
deformation retract ends up being the wedge of three circles. Thus, π1 pA, x0 q – Z ˚ Z ˚ Z
by Theorem 2.4.2. One can also show it using labelling schemes, where r s denotes an empty
scheme, with the following operations.
r sarbcdsard´1 cb´1 s „ aad´1 c´1 b´1 d´1 cb´1
´1

„ raasd

´1 ´1

rc

b

´1

sd

´1

rcb

´1 ´1

aabccb

´1

d

s„d
„b

d

´1 ´1

cd

by Lemma 3.4.2

´1

by Lemma 3.4.2

aabc

permuting

„ rb´1 scrd´1 d´1 aabscr s „ ccb´1 b´1 a´1 a´1 d´1 d´1
„ aabbccdd
Therefore, X is homeomorphic to RP3 .

Figure 3.4.4: Polygonal region in Example 3.4.7.

Lemma 3.4.2
relabelling.

CONSTRUCTION OF COMPACT SURFACES

3.5

54

Construction of Compact Surfaces

So far we have proved that every compact connected surface is homeomorphic to a surface
from the list in the Theorem 3.4.6, but we have not clearly showed that every surface can be
obtained by pasting together in pairs the edges of a polygonal region.
Definition 3.5.1. Let X be a compact Hausdorff space. A curved triangle in X is a subspace
A of X and a homeomorphism h : T Ñ A, where T is a closed triangular region in the plane.
If e is an edge of T , we say that hpeq is an edge of A. Similarly, hpvq is a vertex of A if v is a
vertex of T . A triangulation of X is a collection of curved triangles tAi uni“1 in X such that
Ť
n
i“1 Ai “ X and for i ‰ j the intersection Ai X Aj is either empty or a vertex or edge of Ai
and Aj . If X has a triangulation, we say that X is triangulable.
Note that if hi : Ti Ñ Ai is the associated homeomorphism, then if Ai X Aj is an edge
´1
e of both Ai and Aj , then the map h´1
j hi defines a linear homeomorphism hj hi |e of the edge
´1
h´1
i peq of Ti with the edge hj peq of Tj .
Theorem 3.5.2. Every compact surface is triangulable.
The proof of the theorem above is a well-known result of topology. It uses Jordan
curves and the interested reader can find it in [Tho92] or in [AS60]. Prior to proving the main
result, we outline a few propositions to make the main proof easier.
Proposition 3.5.3. If X is a triangular region in the plane and if x is an interior point of one
of the edges of X, then x does not have a neighborhood in X homeomorphic to an open 2-ball.
Proof. Suppose there is a neighbourhood U of x which is homeomorphic to an open ball B in
R2 with the homeomorphism carrying x to 0. Note that the space Xztxu is homeomorphic to
a circle. Let V be an open neighbourhood of 0 contained in B. Choose ϵ such that the open
ball Bϵ of radius ϵ centered at 0 lies in V . Consider the inclusion mappings:
i

Bϵ zt0u
j

Bzt0u
k

V ´0
The inclusion i is homotopic to the homeomorphism hpxq “ x{ϵ, which is scaling the circle,
so by Theorem 1.4.6 it induces an isomorphism of fundamental groups. Therefore, k˚ must be
surjective and so V zt0u cannot be simply-connected. However, a point x on the edge, which
is a part of the boundary of the trianglular region, has arbitrary small neighbourhood W for
which W ztxu is simply-connected. Thus, we reached a contradiction and, as a result, x does
not have a neighborhood in X homeomorphic to an open 2-ball.
Proposition 3.5.4. Let X be the union of k triangles in R3 , each pair of which intersect
in the common edge e. If k ě 3, then a point x of e does not have a neighborhood in X
homeomorphic to an open 2-ball.

CONSTRUCTION OF COMPACT SURFACES

55

Proof. We would like to show that there is no neighbourhood W of x in X such that W ztxu
has abelian fundamental group (since an open 2-ball without an interior point is homotopic
to a circle and has fundamental group Z). Consider a union A of all the edges of triangles
of X which are different from e. The space A is a collection of k “arcs”, each pair of which
intersects in their endpoints. If B is the union of three of the arcs that make up A, then there
is a retraction r of A onto B, obtained by mapping each of the arcs not in B homeomorphically
onto one of the arcs in B, keeping the end points fixed. Then r˚ is an epimorphism by Lemma
1.4.2. Since the fundamental group of B is not abelian, neither is the fundamental group of A
by Theorem 1.4.6. It follows that the fundamental group of Xztxu is not abelian since A is a
deformation retract of Xztxu.
Assume x is the origin in R3 . If W is an arbitrary neighbourhood of 0, we can find a
scaling map f pxq “ ϵx which carries X into W . The image Xϵ “ f pXq is a copy of X lying
inside of W . Consider the inclusion mappings:
i

Xϵ zt0u
j

Xzt0u
k

W ´0
Similarly to the proof of Proposition 3.5.3, k˚ is surjective, and so the fundamental group of
W zt0u cannot be abelian.
Theorem 3.5.5. If X is a compact surface, then X is homeomorphic to the quotient space
obtained from a collection of disjoint triangular regions in the plane by pasting their edges
together in pairs.
Proof. Since the surface is compact, by Theorem 3.5.2, X is triangulable. Let A1 , A2 , . . . , An
be a triangulation of X with corresponding homeomorphisms hi : Ti Ñ Ai . One can get any
triangulation to be disjoint, thus, consider the case when the triangles Ti are already disjoint.
Then the maps hi can be combined to form a map h : T1 Y T2 Y . . . Y Tn Ñ X. Note that this
map is a quotient map since the space E “ T1 Y T2 Y . . . Y Tn is compact and X is Hausdorff.
Moreover, because the map h´1
j ˝ hi is linear when Ai X Aj is an edge, h pastes the edges of Ti
and Tj together by a linear homeomorphism.
First, we need to show that for each edge e, which belongs to the triangulation triangle
Ai , there is exactly one other triangle Aj such that Ai XAj “ e. Note that by Proposition 3.5.3,
there is at least one additional triangle Aj having e as an edge and by Proposition 3.5.4 there
is only one such triangle. Therefore, the quotient map actually pastes the edges of triangles
together in pairs, since each edge appears exactly twice in a scheme.

CONSTRUCTION OF COMPACT SURFACES

56

Now, we would like to show that if the intersection Ai X Aj equals a vertex v, then there is
a sequence, as visualized in Figure 3.5.0, starting with Ai and ending with Aj , of triangles
having v as a vertex such that the intersection
of each triangle of the sequence with its successor is an edge of each. In other words, we
cannot have a case of the wedge of multiple
surfaces. To show that such situation is not
possible, given a common vertex v, define two
triangles Ai and Aj such that v P Ai X Aj to be
equivalent if there exists a sequence of triangles
as mentioned above.

Figure 3.5.0: Visualization of the triangle
sequence with a common vertex v.
Suppose there are two equivalence classes of triangles, and let B and C be the unions
of the triangles in two different equivalence classes. Intersection of the sets B and C consists
of v alone since no triangle in B that has a common edge with a triangle in C. Therefore, for
every sufficiently small neighbourhood W of v in X, the space W ztvu is disconnected, which
contradicts the locally Euclidean property of the surface X.
Theorem 3.5.6. If X is a compact connected triangulable surface, then X is homeomorphic to
the quotient space obtained from a polygonal region in the plane by pasting their edges together
in pairs.
Proof. From the previous theorem there is a collection tTi uni“1 of disjoint triangular regions in
the plane such that X is homeomorphic to the quotient space obtained from the collection by
pasting their edges together in pairs. To extend the previous theorem, we paste the edges of
triangles with the same label together. If two triangular regions have edges with the same label,
we can paste the regions together along these two edges. The result would be one four-sided
region with still proper orientations and labels instead of two triangular regions. Continue
similarly as long as there are two regions having edges bearing the same label. Eventually, one
reaches the situation with either a region with all different labels (exactly what we need) or
with multiple polygonal regions, no two of which have edges bearing the same label. In such
case the space ends up not being connected, which is not possible by the assumption.

Appendices

57

Appendix A

Topology
The following chapter is using materials from the textbooks Topology[Mun00] and General
Topology[Wil04]. As any other axiomatic branch of mathematics, we would like to start with
a set of definitions and axioms, which later would develop theorems and propositions.
Definition A.0.1. A topology on a non-empty set X is a collection T of subsets of X, called
open sets, satisfying
• Both X and ∅ are open, i.e., X P T and ∅ P T .
• The union of any family of open subsets is open.
• The intersection of any finite family of open subsets is open.
A pair pX, T q consisting of a set X together with a topology T on X is called a topological
space.
Example A.0.2. Let X “ ta, b, c, d, e, f u and T1 “ tX, ∅, tau, tc, du, ta, c, du, tb, c, d, e, f uu.
Then T1 is a topology on X as it satisfies all the conditions from the definitions. On the
other hand, the collection T2 “ tX, ∅, tau, ta, c, du, tb, c, d, e, f uu is not a topology on X since
tb, c, d, e, f u X ta, c, du “ tc, du, which does not belong to T2 .
Example A.0.3. Define T as a collection of all subsets of X. It clearly satisfies all the
conditions for a topology. We call this topology discrete on the set X. In this case, we
call pX, T q a discrete space. We can also define the topology with the smallest number of
elements T “ t∅, Xu for a set X. This topology is called indiscrete and the pair pX, T q is
called indiscrete space.
Definition A.0.4. Let pX, T q be a topological space. A subset S of X is said to be a closed
set in pX, T q if its complement XzS is open in pX, T q.
One can define a topology based on closed sets instead of open sets. Then the words
“intersection” and “union” in the definition flip: we would have the intersection of any number
58

59
of closed sets is a closed set together with the union of any finite number of closed sets being
a closed set.
Note that despite the names, open and closed sets are not mutually exclusive. For instance,
in discrete space every set is both open and closed - we call such sets clopen - while in an
indiscrete space pX, T q all subsets of X except X and ∅ are neither open or closed.
Definition A.0.5. A collection B of open subsets of X is a basis for the topology of X if
every open subset of X is the union of some collection of elements of B.
In other words, if B is a basis for a topology T on a set X, then a subset U of X is in
T if and only if it is a union of elements of B.
Definition A.0.6. Let Y be a non-empty subset of a topological space pX, T q. The induced
topology on Y or the subspace topology is defined as T |Y “ tU X Y |U Ă T u.
Example A.0.7. Let X “ ta, b, c, d, e, f u, Y “ tb, c, eu and define T as in Example A.0.2, i.e.,
T “ tX, ∅, tau, tc, du, ta, c, du, tb, c, d, e, f uu. Then the subspace topology on Y is
TY “ tY, ∅, tcuu
Recall from the set theory the notion of equivalence relations.
Definition A.0.8. A relation „ on a set X is said to be an equivalence relation if it is
refrexive (x „ x), symmetric (if x „ y then y „ x) and transitive (if x „ y and y „ z then
x „ z). For an element x P X the equivalence class is defined as all elements that are related
to x:
rxs :“ ty P X|x „ yu.
The set of all equivalence classes of X determines a partition of X.
Definition A.0.9. A map f : X Ñ Y between topological spaces pX, TX q and pY, TY q is
continuous if f ´1 pU q is open in X for every open set U of Y .
Definition A.0.10. Topological spaces pX, TX and pY, TY q are said to be homeomorphic if
there exists a continuous function f : X Ñ Y such that f is bijective and has a continuous
inverse. The map f is said to be a homeomorphism between pX, TX and pY, TY q. We would
write pX, TX q – pY, TY q. One can show that – is an equivalence relation.
A continuous map f : X Ñ Y is said to be a local homeomorphism if every point p P
X has a neighbourhood U Ď X such that f pU q is open in Y and f restricts to a homeomorphism
from U to f pU q.
Definition A.0.11. Let A be a subset of a topological space pX, T q. A point x P X is said
to be a limit point of A if every open set U containing x also contains a point of A different
from x.
Example A.0.12. Consider the topological space pX, T q, where X “ ta, b, c, d, eu and T “
tX, ∅, tau, tc, du, ta, c, du, tb, c, d, euu. Consider A “ ta, b, cu. Then elements b, d and e are
limit points of A, while a and c are not.

60
Proposition A.0.13. Let A be a subset of a topological space pX, T q. Then A is closed in
pX, T q if and only if A contains all of its limit points.
Definition A.0.14. Let A be a subset of a topological space pX, T q. Then the set A Y A1
consisting of A and all its limit points, denoted as a set A1 , is called the closure of A and is
denoted by A.
Definition A.0.15. Let pX1 , T1 q, . . . , pXk , Tk q be topological spaces. The collection of all
subsets of X1 ˆ . . . ˆ Xk of the form U1 , . . . , Uk , where each Uj is open in Xj , forms a basis
for a product topology on X1 ˆ . . . ˆ Xk .
Definition A.0.16. If π : X Ñ Y is a map, a subset U Ď X is said to be saturated with
respect to π if U is the entire preimage of its image: U “ π ´1 pπpU qq.
Definition A.0.17. Let X be a topological space, Y be a set and π : X Ñ Y be a surjective
map. The quotient topology on Y determined by π is defined by the following rule:
U Ď Y is open if and only if π ´1 pU q is open in X. If Y is a topological space itself, the map
π is called the quotient map if it is surjective and continuous and Y has a quotient topology
determined by π.
Here are some useful properties of a quotient map π : X Ñ Y :
• If B is a topological space, a map F : Y Ñ B is continuous if and only if F ˝ π : X Ñ B
is continuous.
• The quotient topology is the unique topology on Y for which the previous property holds.
• A subset K Ď Y is closed if and only if π ´1 pKq is closed in X.
• If π is injective, then it is a homeomorphism.
• If U Ď X is a saturated open or closed subset, then the restriction π|U : U Ñ πpU q is a
quotient map.
• Any composition of π with another quotient map is again a quotient map.
Theorem A.0.18. Let X and Y be topological spaces and let F : X Ñ Y be a continuous map
that is either open or closed.
1. If F is surjective, then it is a quotient map.
2. If F is injective, then it is a topological embedding.
3. If F is bijective, then it is a homeomorphism.
Definition A.0.19. Let X be a topological space, „ be an equivalence relation on X, X{ „ be
the set of all equivalence classes of X and π : X Ñ X{ „ be a natural projection sending each
element x P X to its equivalence class rxs. Endowed with the quotient topology determined
by π, the space X{ „ is called quotient space of X determined by π.

61
Definition A.0.20. A topological space pX, T q is said to be connected if the only clopen
subsets of X are X and ∅. From this, it follows that a topological space pX, T q is not connected
or disconnected if and only if there are non-empty open sets A and B such that A X B “ ∅
and A Y B “ X.
Theorem A.0.21. The union of a collection of connected subspaces of X with a point in
common is connected.
Definition A.0.22.
A topological space X is compact if for every collection
Ť
Ť C of open sets
of X such that APC A “ X there is a finite subcollection F Ď C such that APF A “ X.
Theorem A.0.23. The image of a compact space under a continuous map is compact.
Definition A.0.24. A topological space X is Hausdorff if @ p, q P X such that p ‰ q there
exists a pair of disjoint open subsets U and V in TX such that p P U , q P V and U X V “ ∅.
Note that by this definition singleton sets of a Hausdorff space are closed. In other words, is
X is a Hausdorff space with a point x P X, then Xztxu is open in X. To see that, consider
a point a distinct from
Ť x, By definition A.0.24, there is an open set Ua containing a but not
containing x. Then aPXzx Ua is open as union of open sets, but it also equal to Xztxu.
Definition A.0.25. A topological space X is second-countable if there is a countable basis
for its topology.
Definition A.0.26. Suppose that one-point sets are closed in X. Then X is said to be normal
if for each pair pA, Bq of disjoint closed sets of X, there exists disjoint open sets containing
A and B, respectively. In other words, every two disjoint closed sets of X have disjoint open
neighborhoods.
Note that a normal space is always Hausdorff, but only compact Hausdorff space is normal.
Lemma A.0.27. Let π : E Ñ X be a closed quotient map. If E is normal, so is X.
Theorem A.0.28 (The Pasting Lemma). Let X “ A Y B, where A and B are closed in X.
Let f : A Ñ Y and g : B Ñ Y be continuous. If f pxq “ gpxq for every x P A X B, then f and
g combine to give a continuous function h : X Ñ Y , defined by setting hpxq “ f pxq if x P A
and hpxq “ gpxq if x P B.
Theorem A.0.29 (Extreme Value Theorem). Let f : X Ñ R be a continuous function, where
X is a compact set. Then f is bounded and there exists p, q P X such that f ppq “ supxPX f pxq
and f pqq “ inf xPX f pxq.
Lemma A.0.30 (Lebesgue Number Lemma). For any open cover A of a compact metric space
X, there exists a real number δ ą 0, also called a Lebesgue number for A, such that every
open ball in X of radius δ is contained in some element of A.

Appendix B

Category Theory
Definition B.0.1. A category C consists of a class Ob(C), whose elements are to be called
objects, and a class Mor(C), whose elements are to be called morphisms, satisfying the
following:
• For each morphism f , there are objects A and B, called the source and target of f . In
this case we write f : A Ñ B.
• Given any two morphisms f : A Ñ B and g : B Ñ C, there exists a morphism g ˝ f :
A Ñ C, called the composition of f and g.
• Given any objcet A, there is an identity morphism 1A : A Ñ A such that for any
f : A Ñ B, f ˝ 1A “ f “ 1B ˝ f .
• Morphism composition is associative: given any two morphisms f : A Ñ B, g : B Ñ C
and h : C Ñ D, pf ˝ gq ˝ h “ f ˝ pg ˝ hq.
Definition B.0.2. In any category C, a morphism f : A Ñ B is called an isomorphism if
there is a morphism g : B Ñ A such that f ˝ g “ 1B and g ˝ f “ 1A . In this case, f and g are
called inverses, g is denoted f ´1 , and we say that A is isomorphic to B.
Example B.0.3. Let’s look at some important examples of categories: First, let us define 0
as an empty category (with no objects and no morphisms) and 1 as a category with one object
and the identity morphism.
• Sets is the category of sets and functions between them.
• Setsfin is the category of finite sets and functions between them.
• Groups is the category of groups and group homomorphisms.
• Ab is the category of abelian groups and group homomorphisms.
• Graphs is the category of graphs and graph homomorphisms.
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• VectF is the category of vector spaces over a field F and linear transformations in F .
• Top is the category of topological spaces and continuous mappings.
• Poset is the category of elements of the set and orderings.
Finally, an individual group is itself a category with exactly one object, where all the morphisms
are isomorphisms. For a given group G, this category is called BG.
There are also examples of categories whose objects are sets with distinguished base
points, in addition to possibly other structure.
Definition B.0.4. A pointed set is an ordered pair pX, pq where X is a set and p is an
element of X. Similarly, one can defined objects as pointed topological spaces and so on.
Moreover, if pX, pq and pX 1 , p1 q are both pointed sets, a map F : X Ñ X 1 is called a pointed
map if F ppq “ p1 . In this case, we write F : pX, pq Ñ pX 1 , p1 q.
Example B.0.5. With the definition above, let’s look at categories of pointed objects:
• Set˚ is a category of pointed sets and pointed maps.
• Top˚ is the category of pointed topological spaces and pointed continuous maps.
Definition B.0.6. A subcategory of a category C is a subclass Ob(D)Ď Ob(C) and a subclass
Mor(D) Ď Mor(C) such that any morphism in Mor(D) is between two objects in Ob(D).
For example, 0 is a subcategory of any category.
Definition B.0.7 (Types of morphisms). A morphism f : A Ñ B is called a monomorphism
if it is left cancellative, i.e., f ˝ g “ f ˝ h ñ g “ h. In this case, we say that f is monic.
A morphism f : A Ñ B is called an epimorphism if it is right cancellative, i.e. g ˝f “ h˝f ñ
g “ h. In this case, we say that f is epic.
A morphism is called a bimorphism if it is both epic and monic.
A morphism is called a retraction if it has a left-inverse and a section if it has a right-inverse.
Note that a morphism which is both a retraction and a section is an isomorphism.
Definition B.0.8. An endomorphism is a morphism f : A Ñ A from an object to itself. If
an endomorphism is also an isomorphism, then it is called an automorphism. The class of
endomorphisms of an object A is denoted End(A) and the class of automorphisms is denoted
Aut(A).
Definition B.0.9. A covariant functor (or just a functor) F : C Ñ D between categories
C and D is a mapping Ob C Ñ Ob D and Mor C Ñ Mor D such that:
• F assigns to each object X POb C an object FpXq POb D.
• F assigns to each morphism f P MorC pX, Y q a morphism Fpf q P MorC pFpXq, FpY qq.

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• Fp1A q “ 1F pAq .
• Fpg ˝ f q “ Fpgq ˝ Fpf q.
In short, a functor is a morphism of categories. In particular, every category C has the
identity functor 1C : C Ñ C.
Definition B.0.10. Two categories C and D are isomorphic if there exist functors F : C Ñ D
and G : D Ñ C that are inverses F ˝ G “ 1D and G ˝ F “ 1C . In particular, a functor is
an isomorphism functor if and only if it is bijective on the class of objects and the class of
morphisms.
Definition B.0.11. Let C be a category and tAi |i P Iu be a family of objects of C. A
product for the family tAi |i P Iu is an object P of C together with a family of morphisms
tπi : P Ñ Ai |i P Iu such that for any object B and a family of morphisms tϕi : B Ñ Ai |i P Iu,
there is a unique morphism ϕ : B Ñ P such that πi ˝ ϕ “ ϕi for all i P I.
Definition B.0.12. A coproduct for the family tAi |i P Iu is an object S of C together with a
family of morphisms tιi : Ai Ñ S|i P Iu such that for any object B and a family of morphisms
tψi : Ai Ñ B|i P Iu, there is a unique morphism ψ : S Ñ B such that ψ ˝ ιi “ ψi for all i P I.
Theorem B.0.13. If pP, tπi uq and pQ, tψi uq are both products or both coproducts of the family
tAi |i P Iu of objects of a category C, the P and Q are equivalent.
In many categories, the “objects” are sets or are sets with an added structure (such as
groups). When this is the case, the morphisms can be considered as functions on sets.
Definition B.0.14. A concrete category is a category C together with a function σ that
assigns to each object A of C a set σpAq, called the underlying set of A, such that
1. every morphism mapping A Ñ B of category C is a function on the underlying sets
σpAq Ñ σpBq,
2. the identity morphism of each object A of C is the identity function on the underlying
set σpAq, and
3. composition of morphisms in C agrees with composition of functions on the underlying
sets.
Definition B.0.15. Let F be an object in a concrete category C, X a nonempty set, and
i : X Ñ F a set map. Then object F is free on the set X provided that for any object A of C
and set map f : X Ñ A, there exists a unique morphism of C, f : F Ñ A, such that f ˝ i “ f
as a set map X Ñ A.
Theorem B.0.16. If C is a concrete category, F and F0 are objects of C such that F is free
on the set X and F0 is free on the set X0 and |X| “ |X0 |, then F is equivalent to F0 .

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