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5.15 State and prove the dual of DesarguesвЂ™ theorem. Use the same Figure 5.9a.
5.16 State and prove the dual of PappusвЂ™ theorem. [Hint: with care you can choose notation
exactly dual to that in 5.10, e.g., p : (x = 0), L = p в€© q = (0 : 0 : 1), etc.]
5.17 State and prove the dual of the statement of Exercise 5.6. [Hint: . . . given three 2-planes
of P4 not . . . ]
5.18 Do the same for Exercise 5.5.
Let L , L вЉ‚ P2 be two lines. Prove that a projective linear map П• : L в†’ L can be
5.19
written as the composite of at most 2 perspectivities L в†’ M and M в†’ L from
suitably chosen points of P2 . [Hint: Step 1. If the point of intersection L в€© L = P is
mapped to itself by П•, show that П• is a perspectivity because you can п¬Ѓx the centre
O to deal with 3 points. Step 2. In general, choose a third line M and a centre O so
that П• composed with the perspectivity П€ : M в†’ L is as in Step 1.]
Prove that Pn has a decomposition as a disjoint union of n + 1 subsets
5.20

Pn = {pt} A1 A2 В·В·В· An .

[Hint: Pn = An hyperplane at в€ћ.]
5.21 If k is a п¬Ѓnite п¬Ѓeld with q elements, п¬Ѓnd 2 different proofs of

#(Pn ) = 1 + q + q 2 + В· В· В· + q n .
k

[Hint: the вЂ˜topologicalвЂ™ proof uses the decomposition of the preceding exercise. The
вЂ˜arithmeticвЂ™ method just counts using the deп¬Ѓnition Pn = k n+1 \ 0 /k в€— .]
k
EXERCISES 91

Prove the following statement, announced in 4.5. For n в‰Ґ 2, a bijective map T : An в†’
5.22
An , which preserves the incidence geometry of afп¬Ѓne linear subspaces of An and is
continuous, is afп¬Ѓne linear. [Hint: it is clearly sufп¬Ѓcient to restrict to n = 2. Use the
idea of the sketch proof of HilbertвЂ™s theorem 5.12 to show that any such map is afп¬Ѓne
linear, possibly composed with a continuous п¬Ѓeld automorphism of R. Conclude by
showing that R has no nontrivial continuous п¬Ѓeld automorphisms.]
6 Geometry and group theory

The substance of this chapter can be expressed as the slogan

Group theory is geometry and geometry is group theory.

In other words, every group is a transformation group: the only purpose of being
a group is to act on a space. Conversely, geometry can be discussed in terms of
transformation groups. Given a space X and a group G made up of transformations of
X , the geometric notions are quantities measured on X which are invariant under the
action of G. This chapter formalises the relation between geometry and groups, and
discusses some geometric issues for which group theory is a particularly appropriate
language.
The action of a transformation group on a space is another way of saying symmetry.
To say that an object has symmetry means that it is taken into itself by a group action:
rotational symmetry means symmetry under the group of rotations about an axis.
As a frivolous example, Coventry market pictured in Figure 6.0 has (approximate)
rotational symmetry: if you stand at the centre, all directions outwards are virtually
indistinguishable; you can understand a coordinate frame as a signpost to break the
symmetry, and to enable people to п¬Ѓnd their way around.
Each of the geometries studied in previous chapters had transformations associated
with it: Euclidean motions of E2 , orthogonal transformations as motions of S 2 , Lorentz
transformations as motions of H2 , and afп¬Ѓne and projective linear transformations
of An and Pn . In each case, the transformations form a group. I have already studied
aspects of this setup: for example, several theorems state that transformations are
uniquely determined by their effect on a suitable coordinate frame.
Whenever two branches of mathematics relate in this way, both can beneп¬Ѓt from
the cooperation. The repercussions of symmetry extend into many areas of math and
other sciences. Some examples:

1. The basic idea of the Galois theory of п¬Ѓelds is to view the roots of a polynomial as
permuted amongst themselves by the symmetry group of a п¬Ѓeld extension.
2. Crystallography makes essential use of group theory to understand and classify the
symmetries of lattice structures formed by crystals, and their impurities.

92
6.1 TRANSFORMATIONS FORM A GROUP 93

Figure 6.0 The plan of Coventry market.

3. Requiring the laws of physics to be invariant under a symmetry group has been one
of the most fertile sources of new ideas in math physics:
(a) The assumption in Newtonian dynamics that the laws of motion are invariant under
Euclidean changes of inertial frames leads directly to conservation of momentum
and angular momentum; this will be discussed further in 9.3.1.
(b) The fact that MaxwellвЂ™s equations of electromagnetism are not invariant under the
Galilean group of symmetries of classical Newtonian dynamics, but are invariant
under Lorentzian symmetries, led Einstein to the idea of special relativity.
(c) Modern particle physics classiп¬Ѓes elementary particles in terms of irreducible
representations of symmetry groups. Several particles were п¬Ѓrst predicted from
a knowledge of group representations, before being discovered experimentally.
(See 9.3.3вЂ“9.3.4 for more details.)
(d) In general relativity, EinsteinвЂ™s п¬Ѓeld equation for the curvature tensor of spacetime
was discovered as the only possible partial differential equation invariant under the
pseudo-group of local diffeomorphisms. Einstein himself understood a great deal
more about the principles underlying symmetry in physics than about curvature
in Riemannian geometry.

We divide math up into separate areas (analysis, mechanics, algebra, geometry,
electromagnetism, number theory, quantum mechanics, etc.) to clarify the study of
each part; but the equally valuable activity of integrating the components into a work-
ing whole is all too often neglected. Without it, the stated aim of вЂ˜taking something
apart to see how it ticksвЂ™ degenerates imperceptibly into вЂ˜taking it apart to ensure it
never ticks againвЂ™.

6.1 Transformations form a group
A transformation of a set X is a bijective map T : X в†’ X . (We could equally well
say permutation of X , although this is mainly used for п¬Ѓnite sets.) If T is bijective,
then so is its inverse T в€’1 . If T1 and T2 are maps from X to itself then, as discussed
94 GEOMETRY AND GROUP THEORY

in 2.1, the composite T2 в—¦ T1 means вЂ˜T2 follows T1 вЂ™ or вЂ˜п¬Ѓrst do T1 , then do T2 вЂ™. If T1
and T2 are bijective then so is T2 в—¦ T1 ; thus composition в—¦ is a binary operation

Trans X Г— Trans X в†’ Trans X,

where Trans X is just the set of all transformations of X .

Transformations of a set X form a group Trans X , with composition
Proposition
of maps as the group operation, id X : X в†’ X as the neutral element and T в†’ T в€’1
as the inverse.

This is absolutely content free, but let us check the group axioms anyway.
Proof

As discussed in 2.4, T3 в—¦ T2 в—¦ T1 has no meaning other than the map
Associative
X в†’ X taking x в†’ T3 (T2 (T1 (x))), so that composition of maps is associative.

T в—¦ T в€’1 = T в€’1 в—¦ T = id X . By deп¬Ѓnition T в€’1 (x) = y if and only if
Inverse
T (y) = x. So T (T в€’1 (x)) = T (y) = x and T в€’1 (T (y)) = T в€’1 (x) = y.

id X в—¦ T = T в—¦ id X = T . The left-hand side says вЂ˜п¬Ѓrst do T , then do
Identity
nothingвЂ™. In view of which, you might as well omit the second step. QED

6.2 Transformation groups
A transformation group is a subgroup of Trans X for some set X . In other words, it
is a subset G вЉ‚ Trans X of bijections T : X в†’ X , containing id X , and closed under
composition T1 , T2 в†’ T2 в—¦ T1 and inverse T в†’ T в€’1 .

Usually X has extra structures (for example: distance, algebraic struc-
Discussion
ture, collinearity structure, topology, distinguished elements or subsets), and we take
the set of transformations that preserve these structures:

G = T в€€ Trans X T preserves the given structures of X .

It will usually be obvious that

T preserves structures =в‡’ so does T в€’1 ;
(1)
T1 , T2 preserve structures =в‡’ so does T2 в—¦ T1 ;

so that we get for free that G is a subgroup. This notion includes the symmetry group
of an object, automorphisms in algebra, and many other notions you will meet later
in math and other subjects.

Let X be a п¬Ѓnite set containing n elements labelled
Example 1. вЂ˜No structureвЂ™
{1, . . . , n}. The symmetric group Sn is the group of all permutations of X .
6.3 KLEINвЂ™S ERLANGEN PROGRAM 95

Motions of En form a group Eucl(n). You can
Example 2. Euclidean motions
verify this by using the result that a motion T is of the form T (x) = Ax + b, and
write out the composition and inverse in this form (compare 2.2). However, this is
completely unnecessary: the result is a standard consequence of what I just said,
because motions are deп¬Ѓned explicitly as transformations that preserve distance, so
that (1) holds. The group Eucl(n) has a subgroup consisting of elements T п¬Ѓxing
a chosen point P в€€ En ; if P is the origin, then T (x) = Ax with A an orthogonal
matrix. Hence this subgroup is isomorphic to the orthogonal group O(n) of n Г— n
real orthogonal matrixes. (See 6.5.3 for more on this point.)

Let S be a subset of Euclidean space En , and let
Example 3. Symmetry groups
G be the set of isometries of En which map points of S to points of S. Again, the
general discussion implies that G is a group, since it is the set of transformations of
En preserving the metric and points of S. G is called the symmetry group of S. To
get interesting groups, one chooses special S (see Exercises 6.5вЂ“6.6); for a вЂ˜potato-
shapedвЂ™ set S, there will be no nontrivial symmetries at all.

If V is a vector space over the reals, a transformation
Example 4. Linear maps
T : V в†’ V is linear if and only if T (О»x + Вµy) = what you think; that is, T preserves
the vector space structure. Thus invertible linear transformations form a group GL(V ),
the general linear group of V . If V has п¬Ѓnite dimension, a basis in V gives an
identiп¬Ѓcation V = Rn ; invertible linear maps are then represented by n Г— n invertible
matrixes which form the general linear group GL(n, R). Closely related to the group
GL(n + 1, R) is the projective linear group PGL(n) of projective transformations
discussed in 5.5.
We will see that many of the results of the previous chapters, and many other
questions at the heart of geometry, can be stated as properties of groups such as
Eucl(n), GL(V ) or PGL(V ).

6.3 KleinвЂ™s Erlangen program
Around 1870, Felix Klein formulated the following meta-deп¬Ѓnition:
Geometry is the study of properties invariant under a transformation group.

I have used this principle throughout the previous chapters; for example, distances
and angles are geometric properties in Euclidean geometry exactly because they are
invariant under motions.
In this context, consider the chain

Euclidean geometry En в†’ afп¬Ѓne geometry An в†’ projective geometry Pn . (2)

The corresponding groups of transformations can be expressed as an increasing chain

Eucl(n) вЉ‚ Aff(n) вЉ‚ PGL(n + 1).
96 GEOMETRY AND GROUP THEORY

Here the inclusion of Aff(n) as a subgroup of PGL(n + 1) results from the inclusion
An вЉ‚ Pn as the set of points with x0 = 0: writing T в€€ Aff(n) as usual in the form
T (x1 , . . . , xn ) = Ax + b gives
пЈ«пЈ¶ пЈ«пЈ¶
x1 x1
пЈ¬.пЈ· пЈ¬.пЈ·
пЈ¬.пЈ· b пЈ¬.пЈ·
A
tпЈ¬ . пЈ·= пЈ¬ . пЈ·,
пЈ­xn пЈё 1 пЈ­xn пЈё
0
1 1

so that T в€€ Aff(n) corresponds to A b . It is clear that an element of PGL(n + 1) is
01
in Aff(n) if and only if it takes the hyperplane {x0 = 0} into itself.
The Erlangen program explains the relation between the three geometries in (2) by
saying that as the transformation group gets larger, the invariant properties become
fewer: Euclidean geometry has distances and angles; these are no longer invariants
of afп¬Ѓne geometry, but An has parallels and ratios of parallel vectors; neither of these
notions survives in Pn . As I said in 5.6, the action of the projective group PGL(2) on
P1 is 3-transitive, and it is precisely the size of this symmetry group that says that
there can be no distance function d(P, Q) of two points, and no ratio of distances
d(P, Q) : d(P, R) along lines deп¬Ѓned in projective geometry. The group action was
prominently involved in the deп¬Ѓnition of the cross-ratio in 5.6 and in the deduction
that it is a well deп¬Ѓned function of 4 collinear points.

6.4 Conjugacy in transformation groups
In general, let X be a set and G вЉ‚ Trans X a transformation group of X as in 6.1.
Suppose that T в€€ G is a transformation we want to study, and g в€€ G any element.

What is the conjugate element gT g в€’1 ?
Question

gT g в€’1 is just T viewed from a different angle. We can think of gT g в€’1
as acting on elements gx в€€ g X , rather than x в€€ X , by the rule gx в†’ g(T x). In fact,
the calculation is not very difп¬Ѓcult:

gT g в€’1 (gx) = gT (gg в€’1 )x = g(T (x)). (3)

Thus we can think of g as a вЂ˜change of viewвЂ™, and gT g в€’1 as T expressed in the new
view. In many cases, g will actually be a change of basis in a vector space, and gT g в€’1
the same map T written out in terms of the new basis.

Consider the transposition (12) in the symmetric
Transpositions in Sn
Example 1.
group Sn of all permutations of {1, . . . , n}, the element which transposes 1 and 2 and
leaves everything else п¬Ѓxed. Let g в€€ Sn be any permutation. Then by what I just said,
g(12)g в€’1 should also be a transposition, because it is just (12) viewed from another
6.4 CONJUGACY IN TRANSFORMATION GROUPS 97

g(L)
g(Q)
t(Q) M

Rot(g(P), g(Оё))
Rot(P, Оё)
g(Оё)
P Оё
g(P)
g(M)
L
Q g(t(Q))

The conjugate rotation g Rot(P, Оё )g в€’1 = Rot(g(P ), g(Оё)).
Figure 6.4a

angle. In fact

g(12)g в€’1 = (ab), where a = g(1), b = g(2).

I give the proof, at the risk of spelling out the really obvious:
Proof

g(1) в†’ 1 в†’ 2 в†’ g(2),
g(12)g в€’1 : (4)
g(2) в†’ 2 в†’ 1 в†’ g(1),

and if c = g(1), g(2) then g в€’1 (c) = 1, 2 so that (12) п¬Ѓxes it, and therefore c в†’
g в€’1 (c) в†’ itself в†’ c. QED

Finding the п¬Ѓxed point (or п¬Ѓxed points) of a transforma-
Example 2. Fixed point
tion is an important issue in many geometric contexts. If T п¬Ѓxes P then gT g в€’1 п¬Ѓxes
g(P). The calculation is again really obvious, see (3).

Let T = Rot(P, Оё ) be a rotation of E2 and g в€€ Eucl(2) any
Example 3. Rotation
motion. I determine gT g в€’1 . In order to see the action, consider any line L through P,
and let M be the line such that в€ L P M = Оё. Then T is determined as taking a point
Q в€€ L into the corresponding point of M (that is, T (Q) is the same distance along
M).
Now, as I said, we should view gT g в€’1 as acting on g(E2 ). So draw g(P), g(L)
and g(M). Then gT g в€’1 п¬Ѓxes g(P), and takes points of g(L) into the corresponding
points of g(M) (see Figure 6.4a). This shows that gT g в€’1 = Rot(g(P), g(Оё)), where I
write g(Оё) for the angle в€ g(L)g(P)g(M); in fact g(Оё) = В±Оё (according as g is direct
or opposite).

Let T : An в†’ An be the translation x в†’ x + b and sup-
Example 4. Translation
pose that g в€€ Aff(n) is given by x в†’ Ax + c. By what I said, there is only one thing
gT g в€’1 could possibly be вЂ“ please guess it before reading further.
Now g в€’1 is given by y в†’ Aв€’1 (y в€’ c). So gT g в€’1 is the map

y в†’ Aв€’1 (y в€’ c) в†’ Aв€’1 (y в€’ c) + b в†’ A Aв€’1 (y в€’ c) + b + c. (5)
98 GEOMETRY AND GROUP THEORY

Q g(P)

Q' g(P')
g(Q)
P
g(Q')
P'

Action of Aff(n) on vectors of An .
Figure 6.4b

Multiplying this out gives simply y + Ab. That is, if T is the translation by b then
gT g в€’1 is the translation by Ab.

It is easy to argue that we can write Ab = g(b). In fact g acts on
Remark
в€’в€’ в€’в†’
в€’в€’в€’
в€’в†’ в€’в†’
points of A , so it also acts on based vectors P Q; if b = P Q then Ab = g(P)g(Q)
n

(see Figure 6.4b). With this convention, we can state the conclusion in the form
g(Transl(b))g в€’1 = Transl(g(b)).

I summarise the discussion of this section with the following principle, which is
extremely general in scope.

Let X be a set and g, T : X в†’ X transformations of X . Suppose that
Principle
T has some properties (or is determined by some properties) expressed in terms of
data from X .
Then the conjugate transformation gT g в€’1 : X в†’ X has, or is determined by, the
same properties expressed in terms of g applied to the same data.

Thus T п¬Ѓxes P gives that gT g в€’1 п¬Ѓxes g(P), and T = Rot(P, Оё ) gives gT g в€’1 =
Rot(g(P), g(Оё)).

6.5 Applications of conjugacy

A standard вЂ˜softening upвЂ™ before attacking any kind of geometric object is п¬Ѓrst to
6.5.1
make it as simple as possible by a good choice of coordinates. We have already seen
Normal
this several times in Chapter 1. For example, in 1.14 I expressed any rotation or glide
forms
of the Euclidean plane E2 in the form

cos Оё в€’ sin Оё x +a
x x x
в†’ в†’
or (6)
sin Оё cos Оё в€’y
y y y

with respect to a suitable Euclidean coordinate system. For the glide, you just choose
coordinates so that the reп¬‚ection line is the x-axis. Here the object under study is a
Euclidean motion T в€€ Eucl(2), the change of Euclidean coordinates is also an element
g в€€ Eucl(2) by the discussion in 1.12, and Theorem 1.14 says that gT g в€’1 equals one
of the normal forms (6).
6.5 APPLICATIONS OF CONJUGACY 99

Similar remarks apply to Theorem 1.11. Let T : Rn в†’ Rn be the orthogonal trans-
formation of Rn under study. The result is that in a suitable orthonormal basis, T takes
the block diagonal form of Theorem 1.11. Now T в€€ O(n), and the change of basis is
also given by an orthogonal matrix A в€€ O(n) (because it expresses the standard basis
{e1 , . . . , en } of Rn in terms of the special basis of Theorem 1.11, and both bases are
orthogonal). Thus another way of stating the result is that AT Aв€’1 equals the block
diagonal matrix of Theorem 1.11.
The Jordan normal form of a matrix should be viewed as another example of
conjugation. Consider any linear map Оё : V в†’ V of an n-dimensional complex vector
space V . After a choice of basis, the map Оё is represented by a matrix T в€€ MnГ—n (C).
The theorem is that in a suitable basis, Оё has the diagonal block form
пЈ« пЈ¶
пЈ« пЈ¶ О»i 1
T1 пЈ¬ О»i 1 пЈ·
пЈ¬ пЈ· пЈ¬ пЈ·
T2
Лњ =пЈ¬ пЈ· пЈ¬ пЈ·
..
Ti = пЈ¬ пЈ·.
TпЈ¬ пЈ· with (7)
.
.. пЈ¬ пЈ·
пЈ­ пЈё
. пЈ­ О»i 1 пЈё
Tk
О»i

Recall where this form comes from: the original aim is to choose a basis of V consisting
of eigenvectors, which would reduce the matrix to a diagonal matrix of eigenvalues.
The Jordan normal form is the next best thing if complete diagonalisation turns out
to be impossible.
A coordinate change in Cn changes T into AT Aв€’1 , where A в€€ GL(n) expresses
the change of basis; remember that separate coordinate changes in the domain and
target are not allowed, because they are both the same vector space V . Hence the
theorem on Jordan normal form states that if T is any matrix, for suitable choice
of A the matrix AT Aв€’1 has the shape of (7). If we restrict to a nonsingular matrix
T в€€ GL(n, C), then T в†’ AT Aв€’1 is just conjugacy in GL(n, C).
As a п¬Ѓnal example, consider permutations T в€€ Sn of {1, . . . , n}. Write T as

t = (a1 a2 В· В· В· ak )(ak+1 ak+2 В· В· В· ak+l ) . . .

(recall this means that under T , (a1 в†’ a2 в†’ В· В· В· в†’ ak в†’ a1 ) and so on). If g is the
permutation ai в†’ i then

gT g в€’1 = (12 . . . k)(k + 1 . . . k + l) В· В· В· .

Hence writing a permutation as a product of disjoint cycles can be thought of as
describing conjugacy in the group Sn .

In all the examples discussed here, п¬Ѓnding a normal form of a trans-
Remark
formation T в€€ G is almost the same thing as listing the elements of G modulo the
equivalence relation T в€ј gT g в€’1 . In group theory, the equivalence classes are called
conjugacy classes of G. For example, the above argument gives that the conjugacy
classes of GL(n, C) are exactly the Jordan normal forms (with all О»i = 0). The set of
100 GEOMETRY AND GROUP THEORY

conjugacy classes of a group G is one of the main protagonists in the representation
theory of G.

It happens in lots of problems that we have a subset of elements of a group G, and
6.5.2
вЉ‚ G they generate. I give two quite amusing
we want to know what subgroup
Finding
examples.
generators

The problem of Exercise 2.12 was to prove
Example 1. How to walk a wardrobe
that rotations about any two points P = Q of E2 generate all direct motions of Eucl(2).
I give here a solution based on conjugacy.
How to prove that I can get all the translations? First, I certainly get some transla-
tions, since the composite Rot(P, Оё ) в—¦ Rot(Q, в€’Оё) is a translation in a vector bОё . The
length of bОё is в€љcontinuous function of Оё, and is sometimes nonzero (for example,
a
b90в—¦ has length 2d(P, Q)). It follows by the intermediate value theorem that we can
get a translation by a vector of any fairly short length.
Now I use conjugacy: let T = Transl(bОё ) be a translation, and g = Rot(P, П€)
a rotation. Then the conjugate gT g в€’1 is a translation by the vector g(bОё ) (see 6.4
Example 4):

gT g в€’1 = Transl g(bОё ) .
Thus I can get a translation by any fairly short vector in any direction as a composite
of my generators.

You can buy this puzzle in toy shops, and I am sure
Example 2. The 15-puzzle
you all know it:

HOURS OF FUN

1 2 3 4

5 6 7 8

9 10 11 12

13 14 15

A legal move is to slide the blocks, restoring the blank to the bottom right. As a
result of a legal move you permute the 15 numbered squares, so that clearly

G = legal moves вЉ‚ S15 .

G is the alternating group G = A15 .
Proposition

There exists a 3-cycle T = (11, 12, 15). Just rotate the three
Proof. Step 1
blocks in the bottom right corner.
6.5 APPLICATIONS OF CONJUGACY 101

For any three distinct elements a, b, c в€€ {1, . . . , 15}, there exists a legal
Step 2
move g taking 11 в†’ a, 12 в†’ b, 15 в†’ c (moving the other blocks any-old-how). I
omit the proof, which is not hard: if you have played with the puzzle, you know from
experience that you can put any 6 or 7 of the blocks anywhere you like.

The point of this discussion: by Principle 6.4, gT g в€’1 is the 3-cycle (abc).
Step 3
This is easy, please think it through: a в†’ 11 в†’ 12 в†’ b, . . . .

For any n, the alternating group An is obviously generated by all
End of proof
3-cycles, so that I have proved G вЉѓ A15 . Finally, G вЉ‚ A15 . Indeed, writing 16 for
the blank tile, and removing the restriction that it is always restored to the bottom
right allows us to view G as a subgroup of S16 . But in S16 , every element of G is a
composite of transpositions (AB) where A is the current position of the blank tile,
and you must have evenly many to restore the blank to the bottom right. QED

Note that the Proposition does not immediately explain how to solve the puzzle:
knowing a group up to isomorphism does not tell you how to express its elements as
words in a given set of generators.

The group Aff(n) has two distinguished subgroups:
6.5.3
The
1. the translation subgroup x в†’ x + b, isomorphic to Rn ; and
algebraic
2. the subgroup GL(n)0 of linear maps x в†’ Ax, isomorphic to GL(n) (here linear
structure
means homogeneous linear, that is, п¬Ѓxing 0).
of trans-
formation
Every element of g в€€ Aff(n) can be written in a unique way in the form g : x в†’
groups
Ax + b, that is, g = Tb в—¦ m A , where m A is multiplication by A, and Tb is translation
by b. I write g = (A, b) for short. It follows that

Aff(n) = GL(n) Г— Rn (direct product of sets). (8)

However, (8) is deп¬Ѓnitely not a direct product of groups, because the group law
is not just term by term composition: as we saw in 2.2, the composite g2 в—¦ g1 of
g2 = (A2 , b2 ) and g1 = (A1 , b1 ) is calculated as follows:

x в†’ A1 x + b1 в†’ A2 (A1 x + b1 ) + b2 = (A2 A1 )x + (b2 + A2 b1 ), (9)

so that the group law is

(A2 , b2 ) в—¦ (A1 , b1 ) = (A2 A1 , b2 + A2 b1 ). (10)

This is a bit like a direct product, but the п¬Ѓrst factor A2 interferes with the second
factor b1 before the second factors combine.
102 GEOMETRY AND GROUP THEORY

I summarise the properties of the group given by the product (8) with the group law
(10). Recall п¬Ѓrst that a normal subgroup of a group G is a subgroup H G which is
taken to itself by conjugacy in G; that is, g H g в€’1 = H for all g в€€ G.

This setup has the following properties.
Proposition

The translation subgroup Rn вЉ‚ Aff(n) is a normal subgroup.
(i)
GL(n)0 = (A, 0) A в€€ GL(n) is a subgroup of Aff(n), and is not normal.
(ii)
The п¬Ѓrst projection (A, b) в†’ A of the direct product of sets (8) deп¬Ѓnes a surjective
(iii)
group homomorphism Aff(n) в†’ GL(n), under which the subgroup GL(n)0 maps
isomorphically to GL(n).
The kernel of Aff(n) в†’ GL(n) is Rn .
(iv)
The action of GL(n) on Rn can be described as conjugacy in Aff(n).
(v)

The dramatis personae of the proposition are summarised in the diagram:

Rn в†’
Aff(n) GL(n)
(11)
в€ј
=
GL(n)0

(i) follows from the discussion in 6.4 Example 4: the conjugate of a trans-
Proof
lation by a vector b is another translation, by the vector g(b). (ii) is the same argument,
although the conclusion is different: GL(n)0 preserves 0 в€€ Rn ; therefore by Princi-
ple 6.4, the conjugate subgroup g GL(n)0 g в€’1 preserves g(0). Now in general g(0) = 0,
and therefore g GL(n)0 g в€’1 = GL(n)0 , so that it is not a normal subgroup.
(iii) and (iv) are obvious from the group law. For (v), note that as discussed in
the remark in 6.4 Example 4, the afп¬Ѓne group Aff(n) acts on An , and also acts on
в†’ в€’в€’ в€’в†’
в€’в€’в€’
в€’
vectors of An , taking P Q to g(P)g(Q). This gives a well deп¬Ѓned action of Aff(n) on
в†’ в€’в†’ в€’
в€’
Rn : indeed P Q = P Q means that P Q Q P is a parallelogram; an afп¬Ѓne map takes
в€’ в€’ в€’ в†’ в€’в€’ в€’ в€’ в€’ в€’ в€’в†’
в€’в€’в€’
a parallelogram into another parallelogram, so that also g(P)g(Q) = g(P )g(Q )
(compare Figure 6.4b). Thus the projection ( A, b) в†’ A is just the action of Aff(n)
on Rn (thought of as the free vectors of An ). But this is also the action of Aff(n) by
conjugacy on translations by vectors in Rn . QED

Remarks

1. The same holds for the Euclidean group, with O(n) in place of GL(n). That is, the
same scenario can be replayed word for word with the new cast of players:

Rn в†’
Eucl(n) O(n)
(12)
в€ј
=
O(n)0
6.6 DISCRETE REFLECTION GROUPS 103

2. Philosophy: the groups are contained in the geometry, as transformation groups.
However, the geometry is also contained in the algebra: the vector space Rn and the
action of GL(n) on it are contained in the group structure of Aff(n). To spell this out,
Rn is the subgroup of translations in Aff(n), and the action of GL(n) on Rn is the
conjugacy action of Aff(n) on the translations.
The afп¬Ѓne space An and the action of Aff(n) on it are also buried in the group
structure of Aff(n). Indeed, GL(n)0 is the subgroup of elements preserving 0, and its
conjugates are the subgroups GL(n) P preserving other points P в€€ An . Thus An is in
one-to-one correspondence with these conjugates.
3. This remark is intended for students who know about abstract groups, and what it
means for an abstract group to act on a mathematical structure. (Some details of
what is involved are discussed in Exercise 6.17; see also Section 9.2.) There is a
general notion of semidirect product G H of abstract groups: if a group G acts
on a group H by group homomorphisms, then G H is the set of pairs ( A, b) with
A в€€ G and b в€€ H with the group law ( A2 , b2 ) в—¦ (A1 , b1 ) = (A2 A1 , b2 (A2 b1 )). It is
an easy exercise in abstract groups (Exercise 6.17) to see that this makes G H into
a group, which п¬Ѓts into a diagram like (11).

6.6 Discrete reflection groups
Recall from 2.6 that reп¬‚ections generate all motions of Euclidean space. In general,
a group generated by some set of reп¬‚ections of En is called a reп¬‚ection group. Of
special interest are relatively вЂ˜smallвЂ™ reп¬‚ection groups; in Example 1, the group is
п¬Ѓnite; in Examples 2вЂ“3 it is inп¬Ѓnite but вЂ˜discreteвЂ™ that is, group elements are in a
sense вЂ˜well spacedвЂ™. I do not have space here to elaborate on the theory but I give the
most basic examples.

Two planar reп¬‚ections in Euclidean lines 1 , 2 meet-
Example1. Kaleidoscope
ing at an angle Оё = ПЂ/n generate a п¬Ѓnite group (Figure 6.6a). If s1 and s2 are the two
reп¬‚ections then s2 в—¦ s1 is a rotation through 2ПЂ/n, so (s2 в—¦ s1 )n = id. As an abstract
group this is the dihedral group D2n , containing the cyclic group generated by the
rotation s2 в—¦ s1 as a subgroup of index 2; see Exercise 6.5 for details.
By contrast, to get an idea of what I mean by вЂ˜well spacedвЂ™ group elements, think
of the group generated by reп¬‚ections in two lines that meet at an angle that is an
irrational multiple of ПЂ .

Reп¬‚ections in two parallel mirrors 1 , 2 . This is the
Example 2. BarberвЂ™sshop
inп¬Ѓnite dihedral group D2в€ћ generated by s1 and s2 with s1 = s2 = id, and no other
2 2

relations. It contains the inп¬Ѓnite cyclic group generated by the translation s1 в—¦ s2 as a
subgroup of order 2.

The MusВґ e GrВґ vin is the Paris equivalent of Madame
e e
Example 3. MusВґe GrВґvin
e e
TussaudвЂ™s (the waxworks). They have a spectacular show in which members of the
104 GEOMETRY AND GROUP THEORY

R
R

R

symmetries вЂ˜ вЂ™ us
R

R
R
Figure 6.6a Kaleidoscope.

Figure 6.6b вЂ˜MusВґe GrВґvinвЂ™.
e e

paying public and their children stand inside a kaleidoscope made of mirrors forming a
regular hexagon. At the angles of the hexagon they put exotically decorated columns
(Figure 6.6b). When the lights come on, you have the impression of standing in
an inп¬Ѓnite honeycomb pattern containing periodically arranged family groups with
babies in pushchairs. The reп¬‚ection group here is the group generated by reп¬‚ections
in the six sides of the hexagon. See Exercise 6.6 for details.

Reп¬‚ection groups turn up all over the place in mathematics, from the theory of
Platonic solids through the theory of crystals, Coxeter groups, Lie theory (the Weyl
group), to Riemann surfaces, which are related to Fuchsian groups acting on hyper-
bolic rather than Euclidean space. For a п¬Ѓrst port of call, consult Coxeter .

Exercises
Prove that (n + 1) Г— (n + 1) matrixes with the block form A b where A is n Г— n
6.1 01
and b is n Г— 1 form a group isomorphic to Aff(n). Verify Proposition 6.5.3 in these
terms.
A similarity s : En в†’ En is a transformation which scales distances by a constant
6.2
factor О» > 0 (that is, d(s(x), s(y)) = О»d(x, y) for all x, y). Here О» depends on s only.
(a) Prove that the set of similarities is a transformation group Sim(n) of En .
(b) Sim(n) does not preserve distances in En . Prove that it preserves angles.
EXERCISES 105

(c) Show how to use the scaling factor О» to deп¬Ѓne a group homomorphism Sim(n)
R>0 with Eucl(n) as its kernel.
Prove that the diagonal scalar matrixes diag(О», О», . . . , О») form a subgroup of GL(n),
6.3
equal to the centre (= the set of elements that commute with every matrix). Prove
that PGL(n + 1) is the quotient of GL(n + 1) by its centre (compare 5.5).
Let G be a п¬Ѓnite group of motions of E2 . Prove that there is a point of E2 п¬Ѓxed by
6.4
every element of G. [Hint: take the average.] Deduce a description of every element
of Eucl(n) of п¬Ѓnite order.
Let Sn be the regular n-gon in E2 , for n в‰Ґ 3, and let D2n be the symmetry group of
6.5
Sn . Show that
(a) every element of D2n п¬Ѓxes the centre of S;
(b) D2n contains n rotations (including the identity), which form a subgroup Hn of
D2n isomorphic to the cyclic group of order n;
(c) D2n also contains n reп¬‚ections, and no further elements, hence has order 2n;
(d) D2n is isomorphic to the reп¬‚ection group of 6.6 Example 1.
Denoting by a one of the reп¬‚ections and by b a rotation by a smallest angle, write
out the group elements in terms of a, b. Find the relations holding between a and b.
Deduce from your relations that Hn is a normal subgroup of D2n . [Hint: if you get
stuck, п¬Ѓrst do the case of the square in E2 with vertexes (В±1, В±1); here it is easy
to write out the elements of D8 as a set of matrixes, and doing this case gives you all
the psychological support needed to do the general case.]
The group D2n is called the dihedral group of order 2n, a group which occurs in
many guises in and out of geometry.
6.6 The reп¬‚ection group G corresponding to the MusВґ e GrВґ vin described in 6.6 Exam-
e e
ple 3 and Figure 6.6b is the group generated by reп¬‚ections in the sides of a regular
hexagon H , which acts on E2 preserving the honeycomb tiling by regular hexagons.
Show that
(a) G contains the reп¬‚ections in the 3 diagonals of H , generating a group of symme-
tries of H isomorphic to S3 .
(b) Translations in G form a normal subgroup Z вЉ• Z в€ј T G, with quotient G/T в€ј
= =
S3 .
(c) G is of index 2 in the full group of symmetries of the hexagonal tiling.
[Hint: colour vertexes of the honeycomb tiling alternately black and white.]
Exercises in conjugacy.
Write StabG (x) вЉ‚ G for the set of elements of G that п¬Ѓx x (the stabiliser of x in G);
6.7
prove that StabG (x) is a subgroup.
Let G вЉ‚ Trans X be a transformation group of a set X . For x в€€ X and g, t в€€ G,
prove that t п¬Ѓxes x if and only if gtg в€’1 п¬Ѓxes g(x) (compare 6.4 (3)).
Deduce that StabG (gx) is the conjugate subgroup g StabG (x)g в€’1 .
6.8 Prove that the distinction between direct and opposite motion (Deп¬Ѓnition 1.10) is in-
dependent of the choice of coordinates. [Hint: let T be the motion in question, and g в€€
Eucl(n) a coordinate change. By the principle of 6.4, T is expressed in the new coor-
dinates by gT g в€’1 . It remains to calculate the linear part of gT g в€’1 and its determinant.]
106 GEOMETRY AND GROUP THEORY

6.9 G is a group. Prove that conjugacy is an equivalence relation on G. That is, the relation
g в€ј g if and only if g and g are conjugate in G is an equivalence relation. Determine
all the conjugacy classes in the symmetric group S4 .
6.10 Prove that any two translations Transl(b) by a nonzero vector b are conjugate in
Aff(2). (Compare 6.4 Example 4.) Which translations in Eucl(2) are conjugate?
Prove that two rotations of E2 are conjugate in Eucl(2) if and only if the absolute
6.11
value of the angles are equal.
6.12 Use Principle 6.4 and Theorem 1.14 to list the conjugacy classes of Eucl(2). [Hint:
every motion is conjugate to a standard type. You have to say when two standard types
are conjugate, and to choose exactly one normal form from each conjugacy class.]
Consider the п¬Ѓeld F p = Z/ p with p elements. The projective line P1 p over F p is the
6.13 F
set of 1-dimensional vector subspaces of F p , or equivalently, the set (F2 \ 0)/в€ј. It
2
p
has p + 1 elements, called 0, 1, 2, . . . , p в€’ 1, в€ћ. Use Theorem 5.5 to prove that the
general linear group PGL(2, F p ) has order ( p + 1) В· p В· ( p в€’ 1).
Specialise to p = 5, and the action of PGL(2, F5 ) on the 6 points {0, 1, 2, 3, 4, в€ћ} of
6.14
P1 5 . Write down the 3 maps
F

x в†’ x + 1, x в†’ 2x x в†’ 2 в€’ 2/x
and

(where x is an afп¬Ѓne coordinate) as permutations of these 6 elements.
6.15 Determine the subgroup of S6 (the symmetric group on 6 elements) generated by the
2 elements Пѓ = (abcd) and П„ = (cde f ). [Hint: if you play around for a while with
lots of combinations of the generators, you will notice that it is 3-transitive, but you
only get a few cycle types, so it is probably quite a bit smaller than the whole of S6 .]
(Harder) Determine the subgroup G of the symmetric group S7 generated by Пѓ =
6.16
(1234) and П„ = (34567). [Hint: the answer is S7 . Indeed, G is obviously 3 or 4-
transitive: as with the 15 puzzle (6.5.2 Example 2), you can put any 3 elements
anywhere you like by messing around with the given generators. G also contains an
odd permutation Пѓ , so is not contained in the alternating group A7 . To complete the
proof, you need to п¬Ѓnd a transposition or a 3-cycle; then G must contain A7 by the
same principle as 6.5.2 Example 2.]
6.17 (Assumes abstract group theory) Let G and H be abstract groups. Say what it means
for G to act on H by group homomorphisms ( A, b) в†’ Ab. Under this assumption,
prove that the multiplication

(A2 , b2 ) в—¦ (A1 , b1 ) = (A2 A1 , b2 (A2 b1 )) for Ai в€€ G and bi в€€ H

makes the direct product G Г— H into an abstract group G H , such that the assertions
of Proposition 6.5.3 hold for it.
7 Topology

The word topology in the context of this course has two quite different meanings:

Slogan: a topological space is a вЂ˜metric space without a
вЂ˜Point-set topologyвЂ™
metricвЂ™. In analysis, this idea leads to a fairly minor generalisation of the deп¬Ѓnition of
metric space, but the deп¬Ѓnition of topology has applications in other areas of math,
where it turns out to be logical or algebraic in content. I give the abstract deп¬Ѓnition
and some examples of topological spaces that are deп¬Ѓnitely not metric. This is an
important ingredient in all advanced math (algebra, analysis, arithmetic, geometry,
logic, etc.). Topology has lots of advantages even when the only spaces of interest are
metric spaces. It provides, in particular, a simple rigorous language for вЂ˜sufп¬Ѓciently
nearвЂ™ without epsilons and deltas.

The abstract language gives us tools to study spaces that
вЂ˜Rubber-sheet geometryвЂ™
are geometric in origin, such as the torus and the MВЁ bius strip. Geometric concepts
o
in topology include the winding number and the number of holes of a surface.

Here is a sample of the results proved in this chapter.
If f : S 1 в†’ вЉ‚ R2 is bijective and continuous, then the inverse map f в€’1 : в†’ S 1
1.
is also continuous; that is, f is a homeomorphism. Joke: topology is geometry in
which

в™Ґ = 0.

Imagine trying to prove this from п¬Ѓrst principles! The point is that f can be very
complicated, and f в€’1 might not be given by any simple function.
2. The cylinder is different from the MВЁ bius strip.
o
The winding number: let П• : [0, 1] в†’ R2 \ (0, 0) be a continuous map with П•(0) =
3.
П•(1). Then the number of times П• winds around the origin is not changed by deforming
the loop continuously; in other words, the winding number is a homotopy invariant
of the map П•.

107
108 TOPOLOGY

7.1 Definition of a topological space
Let X be a set. A topology on X is a collection T of subsets of X satisfying the
following three axioms:
r finite intersection U1 , . . . , Un в€€ T =в‡’ U1 в€© В· В· В· в€© Un в€€ T ;
r arbitrary union UО» в€€ T for О» в€€ =в‡’ О»в€€ UО» в€€ T , where is an arbitrary
indexing set;
r conventions on empty set в€…, X в€€ T .

A topological space is a pair X, T consisting of a set X and a topology T = T X on
it. U в€€ T is called an open set of the topology T . We often speak of the topological
space X and its open sets U , omitting T from the notation when it is clear what
topology is intended. V вЉ‚ X is closed if its complement is open; the topology could
be speciп¬Ѓed equally well by the collection of closed sets, which enjoys п¬Ѓnite union
and arbitrary intersection. If Z вЉ‚ X , the closure of Z , denoted Z , is the intersection
of all closed sets containing Z . By the arbitrary intersection property of closed sets,
Z is closed; it clearly contains Z . A neighbourhood of a point x в€€ X is any subset
V вЉ‚ X containing an open set containing x.
We will see presently that if X is a metric space then there is a natural choice of
open sets of X which form a topology. Here are some simpler examples.

Let X = {P1 , P2 , P3 } be a set consisting of 3 points, and
Example 1

T X = {в€…, {P1 }, {P1 , P2 }, X }.

Then {P1 } is open, but every neighbourhood of P2 contains P1 , and every neighbour-
hood of P3 contains both P1 , P2 .

There are two extreme topologies deп¬Ѓned on any set X . The discrete
Example 2
topology has every subset open. The indiscrete topology has no open sets except в€… and
X itself.

The coп¬Ѓnite topology on an inп¬Ѓnite set X is the topology for which the
Example 3
open sets are в€… or the complements of п¬Ѓnite sets; that is, U вЉ‚ X is open if and only if
either U = в€… or X \ U is п¬Ѓnite; it is obvious that this satisп¬Ѓes п¬Ѓnite intersection and
arbitrary union. In this topology, if x в€€ U and y в€€ V are neighbourhoods of any two
points then U в€© V is also the complement of a п¬Ѓnite set, and hence nonempty.

7.2 Motivation from metric spaces
Let (X, d) and (Y, d ) be metric spaces (see Appendix A if you need reminding what
this means) and f : X в†’ Y a map. By deп¬Ѓnition, f is continuous if
7.2 MOTIVATION FROM METRIC SPACES 109

for every x в€€ X and for any given Оµ > 0, there exists Оґ > 0 such that d(x, y) <
Оґ =в‡’ d ( f (x), f (y)) < Оµ.

The intuitive meaning is clear without epsilons and deltas: if x в€€ X is any given point,
I can guarantee that f (y) is arbitrarily close to f (x) by forcing y to be sufп¬Ѓciently
close to x.
The idea of topology on a space is to break up the deп¬Ѓnition of continuity into two
steps. First use the metric to derive the open sets and neighbourhoods of points; then
describe continuity in terms of open sets.

If (X, d) is a metric space, a set U вЉ‚ X is a neighbourhood of x if
Definition
B(x, Оµ) вЉ‚ U for some Оµ. Here B(x, Оµ) is the open ball of radius Оµ centred at x; if you
cannot guess the formal deп¬Ѓnition, look in Appendix A. A set U вЉ‚ X is open if it
is a neighbourhood of every one of its points, that is, for all x в€€ U , B(x, Оµ) вЉ‚ U for
some Оµ. The open sets U of X form a topology on X , the metric topology of (X, d).
(See Exercise 7.1.)

Standard easy result on metric spaces:
Equivalent conditions

в€Ђ x в€€ X and в€Ђ neighbourhood V вЉ‚ Y ,
f is continuous в‡ђв‡’
f в€’1 V вЉ‚ X is a neighbourhood of x

в‡ђв‡’ в€Ђ open V вЉ‚ Y , f в€’1 V вЉ‚ X is open.

In other words, the вЂ˜epsilon-deltaвЂ™ deп¬Ѓnition of continuity for metric spaces can
be replaced by an equivalent condition which involves only open sets of the metric
topology. I will adopt this equivalent condition in 7.3 to deп¬Ѓne continuity for a map
between arbitrary abstract topological spaces.
The idea of a topological space is a natural abstraction and generalisation of the
idea of a metric space. When going from a metric space (X, d) to the corresponding
topological space, we forget the metric, and keep only the notion of neighbourhoods,
or equivalently open sets. There are several advantages. In the context of metric
spaces, closeness means that the distance d(x, y) is small. But just as some things
in life have a value that cannot be expressed as a sum of money, in some contexts
closeness cannot always be expressed as a distance measured as a real number. In
particular, the following three properties are forced on metric spaces by deп¬Ѓnition,
but are optional for topological spaces.

1. Symmetry: in a metric space, x is close to y if and only if y is close to x.
Hausdorff property: given two points x = y в€€ X , there exist disjoint open sets x в€€ U
2.
and y в€€ V (see Figure 7.2a).
Countable neighbourhoods: given a point x в€€ X of a metric space, consider the family
3.
Bn = B(x, n ). Then Bn are neighbourhoods of x; they are countable in number; every
1
110 TOPOLOGY

Figure 7.2a Hausdorff property.

=
identify

S 1 = [0, 1] with the ends identified.
Figure 7.2b

neighbourhood of x contains a Bn ; and Bn = {x}. This can be used in convergence
arguments in analysis (see Exercise 7.4).

The idea of having the open sets speciп¬Ѓed as the basic construction is of course more
abstract and less intuitive than deп¬Ѓnitions in п¬Ѓrst analysis or metric spaces courses,
but abstractness has its own advantages. In many cases, the spaces I am interested in
may actually be metric spaces, but I may not really care about the distances, just in
1. For example, if you think of the circle S 1 вЉ‚ R2 as the
what it means for d(x, y)
identiп¬Ѓcation space obtained by glueing together the ends of the interval [0, 1], then
S 1 is a metric space, with metric

d[0,1] (P, Q), d[0,1] (0, P) + d[0,1] (Q, 1),
d S 1 (P, Q) = min ,
d[0,1] (0, Q) + d[0,1] (P, 1)

which is a fairly tedious expression to work with; but I really do not care about
the metric, only the system of arbitrarily small neighbourhoods of points. A small
neighbourhood of any point other than the вЂ˜seamвЂ™ P0 , the image of the endpoints
0, 1, is given by (x в€’ Оµ, x + Оµ) from the interior of the interval. For P0 , you
glue together small neighbourhoods of the glued endpoints: [0, Оµ) в€Є (1 в€’ Оµ, 1]; see
Figure 7.2b.
As a п¬Ѓnal example, note that the discrete topology on any set X , deп¬Ѓned in 7.1
Example 2, is metric: just set d(x, y) = 1 for every x = y. On the other hand, the
indiscrete topology is not metric.
7.3 CONTINUOUS MAPS AND HOMEOMORPHISMS 111

7.3 Continuous maps and homeomorphisms
If X and Y are topological spaces, a map f : X в†’ Y is continuous if
7.3.1
Definition of
f в€’1 (U ) вЉ‚ X is open for every open U вЉ‚ Y .
a contin-
uous map
Notice that I am already omitting mention of the topologies T X and TY . To use the
language literally, I should have said the following: let X, T X and Y, TY be topological
spaces, then f is continuous if

U в€€ TY =в‡’ f в€’1 (U ) в€€ T X .

If X is any set with the discrete topology of 7.1 Example 2, then every
Example 1
map X в†’ Y from X to any topological space is continuous. If X has the indiscrete
topology, then every map Y в†’ X from any topological space to X is continuous.

Consider an inп¬Ѓnite п¬Ѓeld k with the coп¬Ѓnite topology on it (see 7.1
Example 2
Example 3). Let f : k в†’ k be a polynomial map given by a в†’ f (a), where f is a
polynomial in one variable. Then f is continuous. For U вЉ‚ k is open if and only
if U = в€… or U is the complement of a п¬Ѓnite set, say U = k \ {b1 , . . . , bn }; then
f (x) = bi has at most deg f solutions, so that f в€’1 (U ) is also the complement of a
п¬Ѓnite set.

A map f : X в†’ Y is a homeomorphism if f is bijective, and both f and f в€’1 are
7.3.2
continuous. This means that
Definition of
a homeo-
f: X в†”Y T X в†” TY ,
and
morphism

or in other words, f is an isomorphism of all the structure there is. X and Y are
homeomorphic, written X Y , if there exists a homeomorphism f : X в†’ Y .

R.
An open interval (a, b) is homeomorphic to the real line, (a, b)
Example 3
For example, the map

в€’1 2x в€’ 1
1
f : (0, 1) в†’ R f (x) = + =
deп¬Ѓned by
1в€’x x(1 в€’ x)
x
is a homeomorphism, illustrated in Figure 7.3a.

The square is homeomorphic to the circle in R2 . To see this, put the
Example 4
square inside the circle and project out from an interior point (see Figure 7.3b). A
similar radial projection argument shows also that the full square is homeomorphic to
the closed disc {x 2 + y 2 в‰¤ 1} вЉ‚ R2 . In Theorem 7.14 below I show that if f : S 1 в†’
R2 is any one-to-one and continuous map (that is, a simple closed curve) then f is a
homeomorphism.
112 TOPOLOGY

y
xв€’
y
x в€’x

x

R.
Figure 7.3a (0, 1)

O

Figure 7.3b Squaring the circle.

If (X, d) and (Y, d ) are metric spaces and f : X в†’ Y is an isometry,
Example 5
then f is a homeomorphism. Note however a map f can set up a homeomorphism
between (the metric topologies of) metric spaces without being an isometry, as in
Examples 3 and 4 above. Being homeomorphic is a much coarser relation on met-
ric spaces than being isometric. Example 7.4.2 discusses this issue from a slightly
different point of view.

7.3.3 The group Homeo(X ) of self-homeomorphisms is a transformation group of the
Homeomor- topological space X (compare 6.1). In the framework of the Erlangen program
phisms and of Section 6.3, topology can be viewed as the study of properties invariant under
the Erlangen Homeo(X ).
The homeomorphism group of X = R is already an uncomfortably large inп¬Ѓnite
program
group, and its action mixes up the points of R like anything, so at п¬Ѓrst sight it seems
hard to imagine how any invariant properties can survive. However, such properties
do exist; one example is between-ness, or separation, derived from the order relation
of R: a homeomorphism f takes three real numbers x, y, z with y between x and z
into f (x), f (y), f (z) with f (y) between f (x) and f (z); this follows at once from the
intermediate value theorem.
7.4 TOPOLOGICAL PROPERTIES 113

If a geometry has lines which are homeomorphic copies of the real line R, then the
separation property can be formulated in the geometry: a point cuts a line into two
disconnected subsets, and hence it makes sense to ask whether a point Q on a line
lies between two other points P, R. Euclidean and hyperbolic geometry are examples
where this property holds. In contrast, the lines (great circles) of spherical geometry
have the topology of the circle S 1 , so they have the вЂ˜no separationвЂ™ property: cutting a
point leaves behind a set which is still connected. See 9.1 for the historic signiп¬Ѓcance
of this issue.

The following 5 spaces are not homeomorphic (for proofs, please be patient
7.3.4
until 7.4.4):
The homeo-
morphism
(1) the closed interval [a, b];
problem
the open interval (a, b) R;
(2)
the circle S 1 ;
(3)
the plane R2 ;
(4)
the sphere S 2 вЉ‚ R3 .
(5)
The examples here and in 7.3.2 illustrate an important general point. If you want to
prove that two given topological spaces X and Y are homeomorphic, then it is your job
to supply a homeomorphism f : X в†’ Y , for example by a geometric construction;
or at least, to prove that one exists. On the other hand, to prove that X and Y are not
homeomorphic, you need to п¬Ѓnd some property of spaces that is the same for
homeomorphic spaces, but different for X and Y . This is called the вЂ˜homeomorphism
problemвЂ™.
The next few sections introduce some basic notions of topology and use them to
prove assertions of this type. Algebraic topology has as one of its main aims to develop
systematic invariants of topological spaces that can be used to prove that spaces are
not homeomorphic, notably the fundamental group ПЂ1 (X, x0 ) and homology groups
Hi (X, Z); but in this book I work only with very simple ideas.

7.4 Topological properties
Some properties of a topological space depend only on the topology. A topological
property of topological spaces is a property that can be expressed in terms of points
and open sets only. Homeomorphisms preserve topological properties.
For example, if X is a metric space, then bounded is not a topological property: it
depends on distance (d(x, y) в‰¤ K for some K ), and not just on the topology. Thus
(a, b) R (see Figure 7.3a), but the left-hand side is bounded, while the right-hand
side is not.

A topological space X is connected, if it cannot be written as a disjoint union of two
7.4.1
nonempty open subsets; that is, there does not exist any decomposition
Connected
space
X = U1 with U1 , U2 open,
U2

where denotes disjoint union.
114 TOPOLOGY

x2

x1

Figure 7.4a Path connected set.

A path in a space X is a continuous map П• : [0, 1] в†’ X ; X is path connected if for
any two points x1 , x2 в€€ X , there exists a path П• with П•(0) = x1 and П•(1) = x2 (that
is, any two points can be joined by a path). (See Figure 7.4a.)
Connected and path connected are both topological properties, since only open
sets and continuous maps appear in their deп¬Ѓnitions. Thus given two spaces X, Y , if
X Y then X and Y are either both (path) connected or both (path) disconnected.

Lemma

(1) The interval [0, 1] is connected.
(2) A path connected set is connected.

For (1), suppose [0, 1] = U1 U2 with opens U1 , U2 [0, 1]. Say 0 в€€ U1 ,
Proof
and consider

z = sup x [0, x] вЉ‚ U1 ,

where sup is least upper bound from your п¬Ѓrst analysis course. The sup exists by
the completeness axiom of the reals. If z в€€ U1 , then because U1 is open, there is a
neighbourhood of z in U1 , that is, [z, z + Оµ) вЉ‚ U1 for some Оµ > 0, so z is not an upper
bound. If z в€€ U2 , there is a neighbourhood of z in U2 , so an interval (z в€’ Оµ, z] disjoint
from U1 and so z в€’ Оµ is a strictly smaller lower bound, which also contradicts the
deп¬Ѓnition of z as sup. (The proof is the same as that of the intermediate value theorem
in a п¬Ѓrst analysis course.)
To show (2), suppose X is path connected and X = U1 U2 with opens U1 , U2
X . Then choose x в€€ U1 and y в€€ U2 and apply the deп¬Ѓnition of path connected, so
that there is a continuous map П• : [0, 1] в†’ X with П•(0) = x and П•(1) = y. Then
[0, 1] = П• в€’1 (U1 ) П• в€’1 (U2 ) is a disjoint union, with both П• в€’1 (U1 ) and П• в€’1 (U2 ) open
and nonempty, which contradicts (1). QED

If X is any topological space, deп¬Ѓne a relation on X by setting x в€ј y if and only if
there is a connected subset U of X containing x, y. It is clear that в€ј is symmetric and
reп¬‚exive, and a bit of thought tells you that it is also transitive, hence it is an equivalence
relation. Equivalence classes of в€ј are called components of the topological space X .
7.4 TOPOLOGICAL PROPERTIES 115

The property used to deп¬Ѓne a path connected space corresponds to our
Remark
usual perception of вЂ˜connectednessвЂ™: you can get from point A to point B using
an unbroken вЂ˜pathвЂ™. In the context of surface travel, mainland Eurasia forms a path
connected space but the United States does not: you cannot get from New York to
Alaska without crossing Canada or going by air or sea. However, in the context
of general topological spaces, connectedness as deп¬Ѓned above, without reference to
paths, is preferable. By the Lemma, path connectedness implies this more general
form of connectedness. Similar remarks apply to components: the deп¬Ѓnition is a
natural extension of the obvious notion under which the connected components of
the United Kingdom include mainland Britain and mainland Northern Ireland, along
with any number of smaller islands around the coast.

7.4.2 The space X is compact if
Compact
for every cover X = О»в€€ UО» of X by an arbitrary collection of opens
space
UО» , there exists a п¬Ѓnite number of indexes О»1 , . . . , О»n в€€ such that X =
n
i=1 UО»i .

(Slogan: every open cover has a п¬Ѓnite subcover.) This property manifestly depends
only on open sets.
A sequence of points a1 , a2 , . . . in a topological space X converges to a limit
l в€€ X , written ain в†’ l, if for any neighbourhood U of l, the ai are eventually all in
U. In other words,

for every open set U of X with l в€€ U , there exists n 0 such that ai в€€ U for
all i в‰Ґ n 0 .

In other words, a1 , a2 , . . . tend to l в€€ X if, for any measure of closeness, the ai are
eventually all close to l.
The space X is sequentially compact if every sequence has a convergent sub-
sequence, that is, for every inп¬Ѓnite sequence a1 , a2 , . . . of points of X , there exists a
point x в€€ X and a sequence i 1 , i 2 , . . . of indexes such that ain в†’ x. (Slogan: every
sequence has a convergent subsequence.)
The following statement relates these notions to each other and to more familiar
ones in metric spaces.

Proposition

For V a subset of Rn with its usual (Euclidean) metric,
(1)

V is closed and bounded в‡ђв‡’ V is sequentially compact.

For X any metric space and V вЉ‚ X a subset,
(2)

V is sequentially compact в‡ђв‡’ V is compact.
116 TOPOLOGY

Here is a brief discussion of where you can п¬Ѓnd this in the literature. Compactness is
the subject of Sutherland , Chapter 5. The statement that a closed bounded subset
of Rn is compact is the HeineвЂ“Borel theorem, proved in , Theorem 5.3.1 for n = 1,
and in general (by reducing to the case n = 1) in Theorem 5.7.1. Compact implies se-
quentially compact (in a metric space) is proved in , Theorem 7.2.6. The other way
round, sequentially compact implies compact (in a metric space), is proved in ,
Chapter 7. The proof is a bit tricky, but Sutherland breaks it up into 3 self-contained
steps, each of which takes a half-page. (See also, for example, Rudin , 2.31вЂ“2.40.)
This is not primarily a course on foundational stuff in metric spaces, and I take
a common sense approach: when I am working in a metric space, I use compact or
sequentially compact more-or-less interchangeably. With general topological spaces,
the language of compactness is more natural and more convenient.

Consider the n-sphere
Example

S n = {(x1 , . . . , xn ) в€€ Rn | x1 + В· В· В· + xn = 1}.
2 2

You have already seen two different metrics on S n : one is the Euclidean distance of
points on S n вЉ‚ Rn , and the other one is the spherical distance d(x, y) = arccos(x В· y)
(see 3.1 and compare Exercise 3.10). However, points are close to each other in one
of the metrics if and only if they are close in the other; said differently, the metric
topologies given by the two metrics are the same. Under the Euclidean metric inherited
from Rn , the set S n is bounded (distance 1 from the origin) and closed (clearly) so S n
is compact by (1) of the Proposition.

Let X , Y be topological spaces and f : X в†’ Y a surjective contin-
7.4.3 Proposition
Continuous uous map. Then if X is compact, so is Y .
image of a
compact
You just have to write out the deп¬Ѓnitions: if Y = VО» , an arbitrary union
Proof
space is
of open sets, let UО» = f в€’1 (VО» ). Then UО» is open, and X = UО» . Therefore there
compact n
exists a п¬Ѓnite set of indexes О»1 , . . . , О»n such that X = i=1 UО»i . Finally,
n n
Y = f (X ) = f (UО»i ) = VО»i . QED
i=1 i=1

Pretty easy wasnвЂ™t it? This shows what a convenient property compactness is.
Compare the result in analysis: a continuous function f : [a, b] в†’ R is bounded and
attains its bound. This is hard to prove from п¬Ѓrst principles, but is really easy once
you have established the deп¬Ѓnition of compactness, and proved Proposition 7.4.2.
The notion of compactness is a powerful tool, and you should learn to use it,
even if you put off studying the proofs until later. A typical use is the kind of вЂ˜con-
tinuity implies uniform continuityвЂ™ argument used all over the place in analysis. If
f : [a, b] в†’ R is continuous, then given Оµ > 0, for all x в€€ [a, b], you can force f (x )
that close to f (x) by squeezing x within Оґ of x; here Оґ depends on x, but compactness
allows you to choose one Оґ that works uniformly for all x в€€ [0, 1].
7.5 SUBSPACE AND QUOTIENT TOPOLOGY 117

There is a famous Bertrand Russell quotation about the advantages of the axiomatic
method: they are the advantages of theft over honest labour. You must either under-
stand a proof of the HeineвЂ“Borel theorem (e.g. Sutherland , Theorem 5.7.1), or
take it on trust as an axiom and accept the advantages.

7.4.4 The notions set up so far are already enough to give a proof of the statement in 7.3.4.
An For example, the topological nature of connectedness implies that (a, b) [a, b],
because any point disconnects the left-hand side. In more detail, if x в€€ (a, b) is any
application
of topo- point then it disconnects (a, b) into two disjoint open intervals (a, x) and (x, b);
if П• : [a, b] в†’ (a, b) were a homeomorphism, then П•(a) = x в€€ (a, b) would be an
logical
interior point, so (a, b) \ x would be disconnected, whereas [a, b] \ {a} = (a, b] is
properties
connected. For exactly similar reasons,

[a, b], R2 , S 2
S1 (any 2 points disconnect the left-hand side)
R R2 (any point disconnects the left-hand side)
R2 or S 2
[a, b] (any 3 points disconnect the left-hand side).

To complete the argument, note that

[a, b], S 1 , S 2 (a, b), R, R2 ;

because all the spaces on the left-hand side are compact, and all those on the right are
not.

7.5 Subspace and quotient topology
If X is a topological space and Z вЉ‚ X a subset, write i : Z в†’ X for the inclusion
map, that is i(z) = z в€€ X for every z в€€ Z . Then the subspace topology of Z is the
topology whose open sets are of the form U в€© Z , where U is an open of X . If X is a
metric space with the topology deп¬Ѓned by the metric d, then the subspace topology
of Z is also metric, deп¬Ѓned by the same metric restricted to Z . This deп¬Ѓnition of
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