ńņš. 5 |

5.16 State and prove the dual of Pappusā™ theorem. [Hint: with care you can choose notation

exactly dual to that in 5.10, e.g., p : (x = 0), L = p ā© 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 ā‚ P2 be two lines. Prove that a projective linear map Ļ• : L ā’ L can be

5.19

written as the composite of at most 2 perspectivities L ā’ M and M ā’ L from

suitably chosen points of P2 . [Hint: Step 1. If the point of intersection L ā© L = P is

mapped to itself by Ļ•, show that Ļ• is a perspectivity because you can ļ¬x the centre

O to deal with 3 points. Step 2. In general, choose a third line M and a centre O so

that Ļ• composed with the perspectivity Ļ : M ā’ 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 Ā·Ā·Ā· An .

[Hint: Pn = An hyperplane at ā.]

5.21 If k is a ļ¬nite ļ¬eld with q elements, ļ¬nd 2 different proofs of

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

k

[Hint: the ā˜topologicalā™ proof uses the decomposition of the preceding exercise. The

ā˜arithmeticā™ method just counts using the deļ¬nition Pn = k n+1 \ 0 /k ā— .]

k

EXERCISES 91

Prove the following statement, announced in 4.5. For n ā„ 2, a bijective map T : An ā’

5.22

An , which preserves the incidence geometry of afļ¬ne linear subspaces of An and is

continuous, is afļ¬ne linear. [Hint: it is clearly sufļ¬cient to restrict to n = 2. Use the

idea of the sketch proof of Hilbertā™s theorem 5.12 to show that any such map is afļ¬ne

linear, possibly composed with a continuous ļ¬eld automorphism of R. Conclude by

showing that R has no nontrivial continuous ļ¬eld 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 ļ¬nd 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 afļ¬ne 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 beneļ¬t 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 ļ¬elds is to view the roots of a polynomial as

permuted amongst themselves by the symmetry group of a ļ¬eld 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 Maxwellā™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 classiļ¬es elementary particles in terms of irreducible

representations of symmetry groups. Several particles were ļ¬rst predicted from

a knowledge of group representations, before being discovered experimentally.

(See 9.3.3ā“9.3.4 for more details.)

(d) In general relativity, Einsteinā™s ļ¬eld 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 ā˜taking something

apart to see how it ticksā™ degenerates imperceptibly into ā˜taking it apart to ensure it

never ticks againā™.

6.1 Transformations form a group

A transformation of a set X is a bijective map T : X ā’ X . (We could equally well

say permutation of X , although this is mainly used for ļ¬nite sets.) If T is bijective,

then so is its inverse T ā’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 ā—¦ T1 means ā˜T2 follows T1 ā™ or ā˜ļ¬rst do T1 , then do T2 ā™. If T1

and T2 are bijective then so is T2 ā—¦ T1 ; thus composition ā—¦ is a binary operation

Trans X Ć— Trans X ā’ 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 ā’ X as the neutral element and T ā’ T ā’1

as the inverse.

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

Proof

As discussed in 2.4, T3 ā—¦ T2 ā—¦ T1 has no meaning other than the map

Associative

X ā’ X taking x ā’ T3 (T2 (T1 (x))), so that composition of maps is associative.

T ā—¦ T ā’1 = T ā’1 ā—¦ T = id X . By deļ¬nition T ā’1 (x) = y if and only if

Inverse

T (y) = x. So T (T ā’1 (x)) = T (y) = x and T ā’1 (T (y)) = T ā’1 (x) = y.

id X ā—¦ T = T ā—¦ id X = T . The left-hand side says ā˜ļ¬rst do T , then do

Identity

nothingā™. 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 ā‚ Trans X of bijections T : X ā’ X , containing id X , and closed under

composition T1 , T2 ā’ T2 ā—¦ T1 and inverse T ā’ T ā’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 ā Trans X T preserves the given structures of X .

It will usually be obvious that

T preserves structures =ā’ so does T ā’1 ;

(1)

T1 , T2 preserve structures =ā’ so does T2 ā—¦ 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 ļ¬nite set containing n elements labelled

Example 1. ā˜No structureā™

{1, . . . , n}. The symmetric group Sn is the group of all permutations of X .

6.3 KLEINā™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 deļ¬ned explicitly as transformations that preserve distance, so

that (1) holds. The group Eucl(n) has a subgroup consisting of elements T ļ¬xing

a chosen point P ā 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 Ć— 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 6.5ā“6.6); for a ā˜potato-

shapedā™ 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 ā’ V is linear if and only if T (Ī»x + Āµy) = 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 ļ¬nite dimension, a basis in V gives an

identiļ¬cation V = Rn ; invertible linear maps are then represented by n Ć— 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 Kleinā™s Erlangen program

Around 1870, Felix Klein formulated the following meta-deļ¬nition:

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 ā’ afļ¬ne geometry An ā’ projective geometry Pn . (2)

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

Eucl(n) ā‚ Aff(n) ā‚ 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 ā‚ Pn as the set of points with x0 = 0: writing T ā Aff(n) as usual in the form

T (x1 , . . . , xn ) = Ax + b gives

ļ£«ļ£¶ ļ£«ļ£¶

x1 x1

ļ£¬.ļ£· ļ£¬.ļ£·

ļ£¬.ļ£· b ļ£¬.ļ£·

A

tļ£¬ . ļ£·= ļ£¬ . ļ£·,

ļ£xn ļ£ø 1 ļ£xn ļ£ø

0

1 1

so that T ā 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 afļ¬ne 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 deļ¬ned in projective geometry. The group action was

prominently involved in the deļ¬nition of the cross-ratio in 5.6 and in the deduction

that it is a well deļ¬ned function of 4 collinear points.

6.4 Conjugacy in transformation groups

In general, let X be a set and G ā‚ Trans X a transformation group of X as in 6.1.

Suppose that T ā G is a transformation we want to study, and g ā G any element.

What is the conjugate element gT g ā’1 ?

Question

gT g ā’1 is just T viewed from a different angle. We can think of gT g ā’1

Answer

as acting on elements gx ā g X , rather than x ā X , by the rule gx ā’ g(T x). In fact,

the calculation is not very difļ¬cult:

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

Thus we can think of g as a ā˜change of viewā™, and gT g ā’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 ā’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 ļ¬xed. Let g ā Sn be any permutation. Then by what I just said,

g(12)g ā’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(Īø))

Rot(P, Īø)

g(Īø)

P Īø

g(P)

g(M)

L

Q g(t(Q))

The conjugate rotation g Rot(P, Īø )g ā’1 = Rot(g(P ), g(Īø)).

Figure 6.4a

angle. In fact

g(12)g ā’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) ā’ 1 ā’ 2 ā’ g(2),

g(12)g ā’1 : (4)

g(2) ā’ 2 ā’ 1 ā’ g(1),

and if c = g(1), g(2) then g ā’1 (c) = 1, 2 so that (12) ļ¬xes it, and therefore c ā’

g ā’1 (c) ā’ itself ā’ c. QED

Finding the ļ¬xed point (or ļ¬xed points) of a transforma-

Example 2. Fixed point

tion is an important issue in many geometric contexts. If T ļ¬xes P then gT g ā’1 ļ¬xes

g(P). The calculation is again really obvious, see (3).

Let T = Rot(P, Īø ) be a rotation of E2 and g ā Eucl(2) any

Example 3. Rotation

motion. I determine gT g ā’1 . In order to see the action, consider any line L through P,

and let M be the line such that ā L P M = Īø. Then T is determined as taking a point

Q ā 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 ā’1 as acting on g(E2 ). So draw g(P), g(L)

and g(M). Then gT g ā’1 ļ¬xes g(P), and takes points of g(L) into the corresponding

points of g(M) (see Figure 6.4a). This shows that gT g ā’1 = Rot(g(P), g(Īø)), where I

write g(Īø) for the angle ā g(L)g(P)g(M); in fact g(Īø) = Ā±Īø (according as g is direct

or opposite).

Let T : An ā’ An be the translation x ā’ x + b and sup-

Example 4. Translation

pose that g ā Aff(n) is given by x ā’ Ax + c. By what I said, there is only one thing

gT g ā’1 could possibly be ā“ please guess it before reading further.

Now g ā’1 is given by y ā’ Aā’1 (y ā’ c). So gT g ā’1 is the map

y ā’ Aā’1 (y ā’ c) ā’ Aā’1 (y ā’ c) + b ā’ A Aā’1 (y ā’ 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 ā’1 is the translation by Ab.

It is easy to argue that we can write Ab = g(b). In fact g acts on

Remark

ā’ā’ ā’ā’

ā’ā’ā’

ā’ā’ ā’ā’

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 ā’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 ā’ 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 ā’1 : X ā’ X has, or is determined by, the

same properties expressed in terms of g applied to the same data.

Thus T ļ¬xes P gives that gT g ā’1 ļ¬xes g(P), and T = Rot(P, Īø ) gives gT g ā’1 =

Rot(g(P), g(Īø)).

6.5 Applications of conjugacy

A standard ā˜softening upā™ before attacking any kind of geometric object is ļ¬rst 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 Īø ā’ sin Īø x +a

x x x

ā’ ā’

or (6)

sin Īø cos Īø ā’y

y y y

with respect to a suitable Euclidean coordinate system. For the glide, you just choose

coordinates so that the reļ¬‚ection line is the x-axis. Here the object under study is a

Euclidean motion T ā Eucl(2), the change of Euclidean coordinates is also an element

g ā Eucl(2) by the discussion in 1.12, and Theorem 1.14 says that gT g ā’1 equals one

of the normal forms (6).

6.5 APPLICATIONS OF CONJUGACY 99

Similar remarks apply to Theorem 1.11. Let T : Rn ā’ 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 ā O(n), and the change of basis is

also given by an orthogonal matrix A ā 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 Aā’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 Īø : V ā’ V of an n-dimensional complex vector

space V . After a choice of basis, the map Īø is represented by a matrix T ā MnĆ—n (C).

The theorem is that in a suitable basis, Īø has the diagonal block form

ļ£« ļ£¶

ļ£« ļ£¶ Ī»i 1

T1 ļ£¬ Ī»i 1 ļ£·

ļ£¬ ļ£· ļ£¬ ļ£·

T2

Ė =ļ£¬ ļ£· ļ£¬ ļ£·

..

Ti = ļ£¬ ļ£·.

Tļ£¬ ļ£· with (7)

.

.. ļ£¬ ļ£·

ļ£ ļ£ø

. ļ£ Ī»i 1 ļ£ø

Tk

Ī»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 Aā’1 , where A ā 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 Aā’1 has the shape of (7). If we restrict to a nonsingular matrix

T ā GL(n, C), then T ā’ AT Aā’1 is just conjugacy in GL(n, C).

As a ļ¬nal example, consider permutations T ā Sn of {1, . . . , n}. Write T as

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

(recall this means that under T , (a1 ā’ a2 ā’ Ā· Ā· Ā· ā’ ak ā’ a1 ) and so on). If g is the

permutation ai ā’ i then

gT g ā’1 = (12 . . . k)(k + 1 . . . k + l) Ā· Ā· Ā· .

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, ļ¬nding a normal form of a trans-

Remark

formation T ā G is almost the same thing as listing the elements of G modulo the

equivalence relation T ā¼ gT g ā’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 Ī»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

ā‚ 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, Īø ) ā—¦ Rot(Q, ā’Īø) is a translation in a vector bĪø . The

length of bĪø is ācontinuous function of Īø, and is sometimes nonzero (for example,

a

b90ā—¦ 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(bĪø ) be a translation, and g = Rot(P, Ļ)

a rotation. Then the conjugate gT g ā’1 is a translation by the vector g(bĪø ) (see 6.4

Example 4):

gT g ā’1 = Transl g(bĪø ) .

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 ā‚ 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 ā {1, . . . , 15}, there exists a legal

Step 2

move g taking 11 ā’ a, 12 ā’ b, 15 ā’ 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 ā’1 is the 3-cycle (abc).

Step 3

This is easy, please think it through: a ā’ 11 ā’ 12 ā’ b, . . . .

For any n, the alternating group An is obviously generated by all

End of proof

3-cycles, so that I have proved G ā A15 . Finally, G ā‚ 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 ā’ x + b, isomorphic to Rn ; and

algebraic

2. the subgroup GL(n)0 of linear maps x ā’ Ax, isomorphic to GL(n) (here linear

structure

means homogeneous linear, that is, ļ¬xing 0).

of trans-

formation

Every element of g ā Aff(n) can be written in a unique way in the form g : x ā’

groups

Ax + b, that is, g = Tb ā—¦ 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) Ć— Rn (direct product of sets). (8)

However, (8) is deļ¬nitely 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 ā—¦ g1 of

g2 = (A2 , b2 ) and g1 = (A1 , b1 ) is calculated as follows:

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

so that the group law is

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

This is a bit like a direct product, but the ļ¬rst 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 ļ¬rst 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 ā’1 = H for all g ā G.

This setup has the following properties.

Proposition

The translation subgroup Rn ā‚ Aff(n) is a normal subgroup.

(i)

GL(n)0 = (A, 0) A ā GL(n) is a subgroup of Aff(n), and is not normal.

(ii)

The ļ¬rst projection (A, b) ā’ A of the direct product of sets (8) deļ¬nes a surjective

(iii)

group homomorphism Aff(n) ā’ GL(n), under which the subgroup GL(n)0 maps

isomorphically to GL(n).

The kernel of Aff(n) ā’ 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 ā’

Aff(n) GL(n)

(11)

ā¼

=

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 ā Rn ; therefore by Princi-

ple 6.4, the conjugate subgroup g GL(n)0 g ā’1 preserves g(0). Now in general g(0) = 0,

and therefore g GL(n)0 g ā’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 afļ¬ne group Aff(n) acts on An , and also acts on

ā’ ā’ā’ ā’ā’

ā’ā’ā’

ā’

vectors of An , taking P Q to g(P)g(Q). This gives a well deļ¬ned action of Aff(n) on

ā’ ā’ā’ ā’

ā’

Rn : indeed P Q = P Q means that P Q Q P is a parallelogram; an afļ¬ne map takes

ā’ ā’ ā’ ā’ ā’ā’ ā’ ā’ ā’ ā’ ā’ā’

ā’ā’ā’

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) ā’ 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 ā’

Eucl(n) O(n)

(12)

ā¼

=

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 afļ¬ne 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 ā 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 ā G and b ā H with the group law ( A2 , b2 ) ā—¦ (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 ļ¬ts into a diagram like (11).

6.6 Discrete reflection groups

Recall from 2.6 that reļ¬‚ections generate all motions of Euclidean space. In general,

a group generated by some set of reļ¬‚ections of En is called a reļ¬‚ection group. Of

special interest are relatively ā˜smallā™ reļ¬‚ection groups; in Example 1, the group is

ļ¬nite; in Examples 2ā“3 it is inļ¬nite but ā˜discreteā™ that is, group elements are in a

sense ā˜well spacedā™. I do not have space here to elaborate on the theory but I give the

most basic examples.

Two planar reļ¬‚ections in Euclidean lines 1 , 2 meet-

Example1. Kaleidoscope

ing at an angle Īø = Ļ/n generate a ļ¬nite group (Figure 6.6a). If s1 and s2 are the two

reļ¬‚ections then s2 ā—¦ s1 is a rotation through 2Ļ/n, so (s2 ā—¦ s1 )n = id. As an abstract

group this is the dihedral group D2n , containing the cyclic group generated by the

rotation s2 ā—¦ s1 as a subgroup of index 2; see Exercise 6.5 for details.

By contrast, to get an idea of what I mean by ā˜well spacedā™ group elements, think

of the group generated by reļ¬‚ections in two lines that meet at an angle that is an

irrational multiple of Ļ .

Reļ¬‚ections in two parallel mirrors 1 , 2 . This is the

Example 2. Barberā™sshop

inļ¬nite dihedral group D2ā generated by s1 and s2 with s1 = s2 = id, and no other

2 2

relations. It contains the inļ¬nite cyclic group generated by the translation s1 ā—¦ s2 as a

subgroup of order 2.

The MusĀ“ e GrĀ“ vin is the Paris equivalent of Madame

e e

Example 3. MusĀ“e GrĀ“vin

e e

Tussaudā™s (the waxworks). They have a spectacular show in which members of the

104 GEOMETRY AND GROUP THEORY

R

R

R

symmetries ā˜ ā™ us

R

R

R

Figure 6.6a Kaleidoscope.

Figure 6.6b ā˜MusĀ“e GrĀ“vinā™.

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 inļ¬nite honeycomb pattern containing periodically arranged family groups with

babies in pushchairs. The reļ¬‚ection group here is the group generated by reļ¬‚ections

in the six sides of the hexagon. See Exercise 6.6 for details.

Reļ¬‚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 ļ¬rst port of call, consult Coxeter [5].

Exercises

Prove that (n + 1) Ć— (n + 1) matrixes with the block form A b where A is n Ć— n

6.1 01

and b is n Ć— 1 form a group isomorphic to Aff(n). Verify Proposition 6.5.3 in these

terms.

A similarity s : En ā’ En is a transformation which scales distances by a constant

6.2

factor Ī» > 0 (that is, d(s(x), s(y)) = Ī»d(x, y) for all x, y). Here Ī» 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 Ī» to deļ¬ne a group homomorphism Sim(n)

R>0 with Eucl(n) as its kernel.

Prove that the diagonal scalar matrixes diag(Ī», Ī», . . . , Ī») 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 ļ¬nite group of motions of E2 . Prove that there is a point of E2 ļ¬xed by

6.4

every element of G. [Hint: take the average.] Deduce a description of every element

of Eucl(n) of ļ¬nite order.

Let Sn be the regular n-gon in E2 , for n ā„ 3, and let D2n be the symmetry group of

6.5

Sn . Show that

(a) every element of D2n ļ¬xes 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 reļ¬‚ections, and no further elements, hence has order 2n;

(d) D2n is isomorphic to the reļ¬‚ection group of 6.6 Example 1.

Denoting by a one of the reļ¬‚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, ļ¬rst do the case of the square in E2 with vertexes (Ā±1, Ā±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 reļ¬‚ection group G corresponding to the MusĀ“ e GrĀ“ vin described in 6.6 Exam-

e e

ple 3 and Figure 6.6b is the group generated by reļ¬‚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 reļ¬‚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 ā• Z ā¼ T G, with quotient G/T ā¼

= =

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) ā‚ G for the set of elements of G that ļ¬x x (the stabiliser of x in G);

6.7

prove that StabG (x) is a subgroup.

Let G ā‚ Trans X be a transformation group of a set X . For x ā X and g, t ā G,

prove that t ļ¬xes x if and only if gtg ā’1 ļ¬xes g(x) (compare 6.4 (3)).

Deduce that StabG (gx) is the conjugate subgroup g StabG (x)g ā’1 .

6.8 Prove that the distinction between direct and opposite motion (Deļ¬nition 1.10) is in-

dependent of the choice of coordinates. [Hint: let T be the motion in question, and g ā

Eucl(n) a coordinate change. By the principle of 6.4, T is expressed in the new coor-

dinates by gT g ā’1 . It remains to calculate the linear part of gT g ā’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 ā¼ 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 ļ¬eld 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)/ā¼. It

2

p

has p + 1 elements, called 0, 1, 2, . . . , p ā’ 1, ā. Use Theorem 5.5 to prove that the

general linear group PGL(2, F p ) has order ( p + 1) Ā· p Ā· ( p ā’ 1).

Specialise to p = 5, and the action of PGL(2, F5 ) on the 6 points {0, 1, 2, 3, 4, ā} of

6.14

P1 5 . Write down the 3 maps

F

x ā’ x + 1, x ā’ 2x x ā’ 2 ā’ 2/x

and

(where x is an afļ¬ne 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 Ļ = (abcd) and Ļ„ = (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 Ļ =

6.16

(1234) and Ļ„ = (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 Ļ , so is not contained in the alternating group A7 . To complete the

proof, you need to ļ¬nd 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) ā’ Ab. Under this assumption,

prove that the multiplication

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

makes the direct product G Ć— 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 ā˜metric space without a

ā˜Point-set topologyā™

metricā™. In analysis, this idea leads to a fairly minor generalisation of the deļ¬nition of

metric space, but the deļ¬nition of topology has applications in other areas of math,

where it turns out to be logical or algebraic in content. I give the abstract deļ¬nition

and some examples of topological spaces that are deļ¬nitely 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 ā˜sufļ¬ciently

nearā™ without epsilons and deltas.

The abstract language gives us tools to study spaces that

ā˜Rubber-sheet geometryā™

are geometric in origin, such as the torus and the MĀØ 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 ā’ ā‚ R2 is bijective and continuous, then the inverse map f ā’1 : ā’ S 1

1.

is also continuous; that is, f is a homeomorphism. Joke: topology is geometry in

which

ā™„ = 0.

Imagine trying to prove this from ļ¬rst principles! The point is that f can be very

complicated, and f ā’1 might not be given by any simple function.

2. The cylinder is different from the MĀØ bius strip.

o

The winding number: let Ļ• : [0, 1] ā’ R2 \ (0, 0) be a continuous map with Ļ•(0) =

3.

Ļ•(1). Then the number of times Ļ• 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 Ļ•.

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 ā T =ā’ U1 ā© Ā· Ā· Ā· ā© Un ā T ;

r arbitrary union UĪ» ā T for Ī» ā =ā’ Ī»ā UĪ» ā T , where is an arbitrary

indexing set;

r conventions on empty set ā…, X ā T .

A topological space is a pair X, T consisting of a set X and a topology T = T X on

it. U ā 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 ā‚ X is closed if its complement is open; the topology could

be speciļ¬ed equally well by the collection of closed sets, which enjoys ļ¬nite union

and arbitrary intersection. If Z ā‚ 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 ā X is any subset

V ā‚ 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 = {ā…, {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 deļ¬ned on any set X . The discrete

Example 2

topology has every subset open. The indiscrete topology has no open sets except ā… and

X itself.

The coļ¬nite topology on an inļ¬nite set X is the topology for which the

Example 3

open sets are ā… or the complements of ļ¬nite sets; that is, U ā‚ X is open if and only if

either U = ā… or X \ U is ļ¬nite; it is obvious that this satisļ¬es ļ¬nite intersection and

arbitrary union. In this topology, if x ā U and y ā V are neighbourhoods of any two

points then U ā© V is also the complement of a ļ¬nite 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 ā’ Y a map. By deļ¬nition, f is continuous if

7.2 MOTIVATION FROM METRIC SPACES 109

for every x ā X and for any given Īµ > 0, there exists Ī“ > 0 such that d(x, y) <

Ī“ =ā’ d ( f (x), f (y)) < Īµ.

The intuitive meaning is clear without epsilons and deltas: if x ā X is any given point,

I can guarantee that f (y) is arbitrarily close to f (x) by forcing y to be sufļ¬ciently

close to x.

The idea of topology on a space is to break up the deļ¬nition 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 ā‚ X is a neighbourhood of x if

Definition

B(x, Īµ) ā‚ U for some Īµ. Here B(x, Īµ) is the open ball of radius Īµ centred at x; if you

cannot guess the formal deļ¬nition, look in Appendix A. A set U ā‚ X is open if it

is a neighbourhood of every one of its points, that is, for all x ā U , B(x, Īµ) ā‚ U for

some Īµ. 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

ā x ā X and ā neighbourhood V ā‚ Y ,

f is continuous āā’

f ā’1 V ā‚ X is a neighbourhood of x

āā’ ā open V ā‚ Y , f ā’1 V ā‚ X is open.

In other words, the ā˜epsilon-deltaā™ deļ¬nition 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 deļ¬ne 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 deļ¬nition,

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 ā X , there exist disjoint open sets x ā U

2.

and y ā V (see Figure 7.2a).

Countable neighbourhoods: given a point x ā 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 speciļ¬ed as the basic construction is of course more

abstract and less intuitive than deļ¬nitions in ļ¬rst 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 ā‚ R2 as the

what it means for d(x, y)

identiļ¬cation 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 ā˜seamā™ P0 , the image of the endpoints

0, 1, is given by (x ā’ Īµ, x + Īµ) from the interior of the interval. For P0 , you

glue together small neighbourhoods of the glued endpoints: [0, Īµ) āŖ (1 ā’ Īµ, 1]; see

Figure 7.2b.

As a ļ¬nal example, note that the discrete topology on any set X , deļ¬ned 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 ā’ Y is continuous if

7.3.1

Definition of

f ā’1 (U ) ā‚ X is open for every open U ā‚ 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 ā TY =ā’ f ā’1 (U ) ā T X .

If X is any set with the discrete topology of 7.1 Example 2, then every

Example 1

map X ā’ Y from X to any topological space is continuous. If X has the indiscrete

topology, then every map Y ā’ X from any topological space to X is continuous.

Consider an inļ¬nite ļ¬eld k with the coļ¬nite topology on it (see 7.1

Example 2

Example 3). Let f : k ā’ k be a polynomial map given by a ā’ f (a), where f is a

polynomial in one variable. Then f is continuous. For U ā‚ k is open if and only

if U = ā… or U is the complement of a ļ¬nite set, say U = k \ {b1 , . . . , bn }; then

f (x) = bi has at most deg f solutions, so that f ā’1 (U ) is also the complement of a

ļ¬nite set.

A map f : X ā’ Y is a homeomorphism if f is bijective, and both f and f ā’1 are

7.3.2

continuous. This means that

Definition of

a homeo-

f: X ā”Y T X ā” 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 ā’ Y .

R.

An open interval (a, b) is homeomorphic to the real line, (a, b)

Example 3

For example, the map

ā’1 2x ā’ 1

1

f : (0, 1) ā’ R f (x) = + =

deļ¬ned by

1ā’x x(1 ā’ 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 ā¤ 1} ā‚ R2 . In Theorem 7.14 below I show that if f : S 1 ā’

R2 is any one-to-one and continuous map (that is, a simple closed curve) then f is a

homeomorphism.

112 TOPOLOGY

y

xā’

y

x ā’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 ā’ 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 inļ¬nite

program

group, and its action mixes up the points of R like anything, so at ļ¬rst 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 ā˜no separationā™ property: cutting a

point leaves behind a set which is still connected. See 9.1 for the historic signiļ¬cance

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 ā‚ 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 ā’ 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 ļ¬nd some property of spaces that is the same for

homeomorphic spaces, but different for X and Y . This is called the ā˜homeomorphism

problemā™.

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 Ļ1 (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) ā¤ 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 Ļ• : [0, 1] ā’ X ; X is path connected if for

any two points x1 , x2 ā X , there exists a path Ļ• with Ļ•(0) = x1 and Ļ•(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 deļ¬nitions. 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 ā U1 ,

Proof

and consider

z = sup x [0, x] ā‚ U1 ,

where sup is least upper bound from your ļ¬rst analysis course. The sup exists by

the completeness axiom of the reals. If z ā U1 , then because U1 is open, there is a

neighbourhood of z in U1 , that is, [z, z + Īµ) ā‚ U1 for some Īµ > 0, so z is not an upper

bound. If z ā U2 , there is a neighbourhood of z in U2 , so an interval (z ā’ Īµ, z] disjoint

from U1 and so z ā’ Īµ is a strictly smaller lower bound, which also contradicts the

deļ¬nition of z as sup. (The proof is the same as that of the intermediate value theorem

in a ļ¬rst analysis course.)

To show (2), suppose X is path connected and X = U1 U2 with opens U1 , U2

X . Then choose x ā U1 and y ā U2 and apply the deļ¬nition of path connected, so

that there is a continuous map Ļ• : [0, 1] ā’ X with Ļ•(0) = x and Ļ•(1) = y. Then

[0, 1] = Ļ• ā’1 (U1 ) Ļ• ā’1 (U2 ) is a disjoint union, with both Ļ• ā’1 (U1 ) and Ļ• ā’1 (U2 ) open

and nonempty, which contradicts (1). QED

If X is any topological space, deļ¬ne a relation on X by setting x ā¼ y if and only if

there is a connected subset U of X containing x, y. It is clear that ā¼ is symmetric and

reļ¬‚exive, and a bit of thought tells you that it is also transitive, hence it is an equivalence

relation. Equivalence classes of ā¼ are called components of the topological space X .

7.4 TOPOLOGICAL PROPERTIES 115

The property used to deļ¬ne a path connected space corresponds to our

Remark

usual perception of ā˜connectednessā™: you can get from point A to point B using

an unbroken ā˜pathā™. 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 deļ¬ned 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 deļ¬nition 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 = Ī»ā UĪ» of X by an arbitrary collection of opens

space

UĪ» , there exists a ļ¬nite number of indexes Ī»1 , . . . , Ī»n ā such that X =

n

i=1 UĪ»i .

(Slogan: every open cover has a ļ¬nite 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 ā X , written ain ā’ 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 ā U , there exists n 0 such that ai ā U for

all i ā„ n 0 .

In other words, a1 , a2 , . . . tend to l ā 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 inļ¬nite sequence a1 , a2 , . . . of points of X , there exists a

point x ā X and a sequence i 1 , i 2 , . . . of indexes such that ain ā’ 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 āā’ V is sequentially compact.

For X any metric space and V ā‚ X a subset,

(2)

V is sequentially compact āā’ V is compact.

116 TOPOLOGY

Here is a brief discussion of where you can ļ¬nd this in the literature. Compactness is

the subject of Sutherland [24], Chapter 5. The statement that a closed bounded subset

of Rn is compact is the Heineā“Borel theorem, proved in [24], 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 [24], Theorem 7.2.6. The other way

round, sequentially compact implies compact (in a metric space), is proved in [24],

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 [21], 2.31ā“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 ) ā Rn | x1 + Ā· Ā· Ā· + xn = 1}.

2 2

You have already seen two different metrics on S n : one is the Euclidean distance of

points on S n ā‚ Rn , and the other one is the spherical distance d(x, y) = arccos(x Ā· 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 ā’ 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 deļ¬nitions: if Y = VĪ» , an arbitrary union

Proof

space is

of open sets, let UĪ» = f ā’1 (VĪ» ). Then UĪ» is open, and X = UĪ» . Therefore there

compact n

exists a ļ¬nite set of indexes Ī»1 , . . . , Ī»n such that X = i=1 UĪ»i . Finally,

n n

Y = f (X ) = f (UĪ»i ) = VĪ»i . QED

i=1 i=1

Pretty easy wasnā™t it? This shows what a convenient property compactness is.

Compare the result in analysis: a continuous function f : [a, b] ā’ R is bounded and

attains its bound. This is hard to prove from ļ¬rst principles, but is really easy once

you have established the deļ¬nition 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 ā˜con-

tinuity implies uniform continuityā™ argument used all over the place in analysis. If

f : [a, b] ā’ R is continuous, then given Īµ > 0, for all x ā [a, b], you can force f (x )

that close to f (x) by squeezing x within Ī“ of x; here Ī“ depends on x, but compactness

allows you to choose one Ī“ that works uniformly for all x ā [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 Heineā“Borel theorem (e.g. Sutherland [24], 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 ā (a, b) is any

application

of topo- point then it disconnects (a, b) into two disjoint open intervals (a, x) and (x, b);

if Ļ• : [a, b] ā’ (a, b) were a homeomorphism, then Ļ•(a) = x ā (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 ā‚ X a subset, write i : Z ā’ X for the inclusion

map, that is i(z) = z ā X for every z ā Z . Then the subspace topology of Z is the

topology whose open sets are of the form U ā© Z , where U is an open of X . If X is a

metric space with the topology deļ¬ned by the metric d, then the subspace topology

of Z is also metric, deļ¬ned by the same metric restricted to Z . This deļ¬nition of

ńņš. 5 |