Alder - An Introduction to Algebraic Topology

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An Introduction toAlgebraic Topology

Michael Alder

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An Introduction toAlgebraic

Topology

by

Michael D Alder


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This Edition Michael D. Alder, 2001

Warning: This edition is not to becopied, transmitted, excerpted or printed

except on terms authorised by the publisher


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Contents

1 Preliminaries 9

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.1.1 What is Topology? . . . . . . . . . . . . . . . . . . . . 9

1.1.2 Why do we care? . . . . . . . . . . . . . . . . . . . . . 12

1.1.3 Invariants . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.1.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.2 Back to Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.2.1 Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . 20

1.2.2 Topological Spaces . . . . . . . . . . . . . . . . . . . . 25

1.2.3 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

1.2.4 Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . 45

1.2.5 Odd . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

1.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

2 Categories and Functors and 53

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CONTENTS

2.1 Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

2.2 The Fundamental Group Functor . . . . . . . . . . . . . . . . 58

2.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3 Singular Homology 75

3.1 The Homology Groups . . . . . . . . . . . . . . . . . . . . . . 75

3.2 Hn is a functor . . . . . . . . . . . . . . . . . . . . . . . . . . 85

3.3 Homotopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

3.4 Homology of Pairs . . . . . . . . . . . . . . . . . . . . . . . . 96

3.5 Excision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

3.5.1 Subdivision . . . . . . . . . . . . . . . . . . . . . . . . 112

3.5.2 Small Cubes . . . . . . . . . . . . . . . . . . . . . . . . 119

3.5.3 Excision at Last . . . . . . . . . . . . . . . . . . . . . . 122

3.6 The Mayer-Vietoris Theorem . . . . . . . . . . . . . . . . . . . 124

3.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

3.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

Bibliography 141

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Introduction to Algebraic Topology 5

PrefaceThis book was used for the University of Western Australias MathematicsDepartments Course in Algebraic Topology taken by Honours (Fourth) year

students in 1996. Views expressed herein are those of the author, not thoseof the University nor the Department insofar as either can be said to have aview.

The material covered represents a short (one semester, two lectures a week)and therefore minimalist sketch of Algebraic Topology. I had briefly consid-ered subtitling the book Algebraic Topology for Dummies but it would beunfair to dummies to let them find out about Topology or indeed any otherbranch of Pure Mathematics. There are, after all, plenty of other subjectsfor them to pass their time with.

The book was much influenced by notes by Patrick Hew of lectures given in1995 by Lyle Noakes, by Lyles own notes, and contain various borrowings,notably from Edwin Spaniers Algebraic Topology [1] and most extensivelyfrom William Masseys A basic Course in Algebraic Topology [2]. I alsofound that old favourite of mine, G.F. Simmons Introduction to Topologyand Modern Analysis [3] useful for jogging my memory, after a rather longperiod of time. The treatment may also have been influenced by CTC Wallscourse given over quarter of a century ago in Cambridge and Liverpool. Thisis a suitable place to thank Terry for giving me an inferiority complex whichpromises to last a lifetime1.

The use of computers is having a profound effect on Mathematics and Science,turning parts of Pure Mathematics into an experimental science (it has alwayshad an element of this, we all have to do sums, but it is vastly increasedin recent times). There is an aspect however which has been made muchmore important, and that is qualitative theories. If you want to study thekinds of things that can happen, a computer simulation may be quite uselessor positively misleading. If the orbits of a planet are elliptical and close

1All human beings are ignorant and stupid, it is part of the design specification. Thesooner they find this out the better. I didnt find out until relatively late, which is regret-table. I realise that this view is completely out of touch with modern educational thought,

which believes in giving everybody a high self-esteem by never letting them do anythingdifficult. There are other good things about it, too.

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6 MICHAEL D. ALDER

to circular, a numerical approximation, as is produced by a computer, willusually produce orbits which spiral outwards. The finer the approximation,the smaller the effect and the tighter the spiral, but never, ever, do you getre-entrant curves. There is a rather considerable difference between these andreal orbits. So in order to use a computer intelligently (which is, admittedly,

an uncommon and decidedly eccentric thing to do), the machine needs to besupplemented by some heavy thought about the nature of the possibilities.Do we really get outward spirals, or inward spirals (much more plausiblephysically) or do we get re-entrant curves? These qualitative considerationsare quintessentially topological, and topology is essential to understandingthem.

As a geometer, by training and temperament, I want to be able to go fromphysical and geometric intuitions and make them respectable parts of mathe-matics. This means that a lot of topology consists of definitions, intended to

articulate some intuitive idea. There are a lot of subtle complications whichcan arise in topology, and much experiment has gone into finding definitionswhich make your life easy, or at least as easy as possible. Unfortunately,the resulting definitions are not always obvious and intuitively natural tothe beginner. So you have the choice of either starting off with natural andobvious definitions and then having a horrible mess later on, or taking someless than obvious definitions as a starting point, which take a while to getused to but save you effort in the longer run. The definition of a topologicalspace is a case in point. Some trust in your teachers, although generallyundesirable and often difficult to attain, is useful. Alternatively, a consider-able quantity of patience is essential, because the construction of the heavy

machinery needed to do the calculations takes a lot of preliminary work. Ihave tried to mitigate the awfulness of the abstraction by briefly discussingmotivational considerations, but it has to be faced that there is no RoyalRoad to Geometry.

A mathematician has to be able to use both hemispheres of his brain inconcert: he has to be able to think about his subject in three languagessimultaneously, and to hop between them like a kangaroo on a pointed pogostick. He must be able to simultaneously translate between Images, Englishand Algebra. The student who uses this book must beware of confusing the

English or the pictorial parts with a new sort of Mathematics which is done bywaving the arms. It is much harder than that. So although I may be chatty

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in part of the notes, I am stern and precise where it matters. Vagueness andsloppiness are useful as heuristics for developing the intuitions part of theway, but real mathematics culminates in a crystaline purity, seldom achievedbut always aimed at.

A look at any of the books on algebraic topology will make it clear that thiscourse barely scratches the surface of the subject. Populist and commercialnotions of what a University is for, have ensured that the preparation ofstudents in these decadent times is such that more would be unattainable.Even so, it is, as warned, tough stuff that will take all your efforts. Onthe other hand, the self-esteem of those who have mastered the subject islegendary. From the heights of algebraic topology, it is easy to look down onrather a lot. With which ambiguous comment, I close.

The symbol denotes the end of a proof.

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Chapter 1

Preliminaries

1.1 Motivation

1.1.1 What is Topology?

Algebraic Topology takes problems of a more or less geometric sort, andtranslates them into problems of a clearly algebraic sort. The geometricsort needs some clarification. Everyone has heard that Topology is rubbersheet geometry, where we can deform the objects. In the kind of geometryone used to meet at school, one thought of a triangle as existing independent

of its coordinates. Two triangles which differed by a shift or a rotation were,pretty much, the same triangle. We said they were congruent, and made noessential distinction between them. Any property of a rigid geometric objectwas shared by any congruent object. Such geometry can be thought of as astudy of objects in Rn which are invariant under the Euclidean Group.

Projective Geometry, not much now studied, extended the study to the Pro-jective Group. Two objects are as near as dammit the same in ProjectiveGeometry if one is a perspective view of the other from some angle. If ABCis a triangle in the plane, and abc is another triangle in a possibly different

plane, both planes sitting inR3

, so that the lines through Aa, Bb and Ccmeet in a point P, then the two triangles look the same from the point P.

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Some of the theory of Projective Geometry has practical implications forrecognising objects in images and computer graphics.

In Topology, we go nearly all the way. The intuitive idea is, roughly, that wedo not distinguish two objects which can be deformed into each other. We

imagine a world in which objects are not rigid as in Euclidean geometry butare made of plasticene or chewing gum or something which can be deformedby stretching and compressing and twisting, but not tearing or gluing. Wearticulate this by saying that we deal with spaces which are regarded asequivalent if there is a homeomorphism between them:

Definition 1.1.1 If U and V are topological spaces,

f : U V

is a homeomorphism iff

1. it has an inverse map g : V U, such that f g is the the identitymap on V and g f is the identity map on U.

2. Both f and g are continuous

Thus the unit interval [0, 1] and that of double the length [0, 2] are obviouslyhomeomorphic by the function y = 2x and its inverse x = y/2.

For the present purposes, if you do not know what a topological space is,take it to be a metric space. If you do not know what a metric space is, youhave problems, but I shall give definitions shortly.

Exercise 1.1.1.1 Show that homeomorphism is an equivalence relation onany set of spaces.

Show that a homeomorphism is 1-1 and onto.

Hence the definition of a topologist as a man who cant tell a coffee cup from adoughnut. This applies to the toroidal doughnut invented by Americans who

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cleverly charge us for the hole: if such a doughnut were made of plasticene,you could deform it progressively and continuously into a coffee cup, thehandle of the cup being the bit around the hole in the doughnut.

It is plausible to the intuitions that you cannot however deform a solid ball

into a doughnut. Just digging a hole through the ball would give a doughnut,but there would have to be some tearing to get the hole. This would make therequisite maps discontinuous. Similarly, if we take the real line, we can showthat it is homeomorphic to the open interval (1, 1) by explicitly constructingmaps, but it is not, one might feel intinctively, homeomorphic either to theclosed interval [1, 1] or to R2. These claims of what is intuitively plausibleof course refer to ones intuitive feelings about what continuous maps can andcannot do. They need to be proved carefully, not just left to ones woollyfeelings.

Exercise 1.1.1.2 Construct an explicit map from (1, 1) to R which is ahomeomorphism. Show that (1, 1) is not homeomorphic to [1, 1].

Note that if we relax the continuity conditions on the maps between objects,we have the condition that two objects are equivalent iff they have the samecardinality, there are the same number of points in the two objects. Allthe above objects, the doughnut, the solid ball, R, (1, 1) and [1, 1] areequivalent in this sense. We are doing set theory if we take this approach,which is another and much less interesting subject.

Exercise 1.1.1.3 Show that there is a bijection (1-1 and onto map) betweenR andR3.

Show that there is a bijection between [1, 1] and (1, 1).

Show that the exponential map

exp : [0, 1) Cgiven by

exp(t) = cos(2t) + i sin(2t)

is a continuous, 1-1 map from a half open interval onto the unit circle. Doesthis mean that the unit circle and the half open interval are homeomorphic?

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1.1.2 Why do we care?

Why do we care about such things? Who gives a damn if spaces are home-omorphic or not? Why play with rubber sheet geometry? There are two

distinct answers.

The first is that it is useful, it has applications.

For example, it is important when one comes to study differential equationson spheres and tori. This is perfectly intelligible at the intuitive level: Youcan imagine lots of little arrows defining a vector field on a sphere or thesurface of a doughnut, and you can probably credit that there is sense inhaving a smooth curve which is a solution to such a system. Moreover, thesethings can arise physically with no trouble at all: If we are studying vibrationsin a crystal, we can take a unit cube of the crystal which is repeated regularly,

and regard the whole crystal as, approximately, an infinite replication ofthis cube. Thus we are looking at functions defined on the cube which areperiodic. If we did this for a function on the real line which was periodicover [, ], we can treat this as a function defined on the circle. If itwere a square in two dimensions, we would get a torus, and if a cube, weget a three dimensional version of a torus. Or consider functions or vectorfields defined on R2 which have the property that they vanish at infinity-a condition common in Physical systems. Then we can extend the vectorfield to one on a sphere in the way which you may have met in ComplexFunction Theory. Now suppose we perturb the vector field a little bit. Thismay correspond to deforming the space it is on. Two vector fields may be

deformed into each other continuously, or they may be of a different type.The qualitative type of a vector field or flow on a sphere is determined, moreor less, by the zeros of the vector field and what type of zeros they are. Itis the topological type of the space it is on which determines what kind offlows can exist. For example, it is easy to have a flow on a torus which hasno fixed points at all, but there is no such flow on a sphere. Deforming thesphere, taking any space homeomorphic to the standard sphere, does notchange that statement.

Definition 1.1.2 A flow on a space U is a continuous map

v : R U U

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with the following properties:

u U, v(0, u) = u (1.1)t R, v(t, ) : U U is a homeomorphism, and (1.2)

s, t

R,

u

U, v(s, v(t, u)) = v(s + t, u) (1.3)

Definition 1.1.3 A flow v on a space U has u U a fixed point of v ifft R, v(t, u) = u

Exercise 1.1.2.1 Construct a flow on the unit circle which has no fixedpoint. It might be simpler to construct a flow on R2 which takes the unitcircle into itself so that the only fixed point of the flow onR2 is the origin.

Construct a flow on the torus which has no fixed point.

Construct a flow on the sphere which has precisely one fixed point.

So if we want to study qualitative properties of flows on such surfaces asspheres and tori, we need to know something of the properties of the under-lying spaces which are topological properties. Since computers enable us tostudy numerically flows on all sorts of spaces, but since the qualitative fea-tures get thoroughly buggered up by such things as truncation and roundingerrors, the qualitative theory of flows, also known as Dynamical Systems, israther important. It can save you trusting your computer blindly, and helptackle issues where the computer behaves like a rather convoluted randomnumber generator.

Another example of an application of Topology which may appeal to someoccurs in image analysis, when we want to find straight lines in images. Afavourite method used by Computer Scientists and Engineers is called theHough Transform. What we do is to consider an image as a collection ofpoints in R2, and then consider the space of all lines in the plane. This spaceis a new space as far as you are concerned, one of the Grassmannian Spaces.It is called Real Projective 2-Space, or RP2. You can think of it as follows:

Take the space of lines in the plane and write them as ax + by + c = 0.Now the triple (a,b,c) determines a line, which might lead you to the idea

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Figure 1.1: Construction of Real Projective 2-Space

that the space of lines in the plane is justR3

. This would be quite wrong,because if we take two such triples (a,b,c) and (A,B,C) such that one is anon-zero multiple of the other, (A,B,C) = (a,b,c) for some = 0, thenthey determine the same line. So we fix up that a2 + b2 + c2 = 1. This mightlead you to the belief that the space of lines in the plane is just the 2-sphere.This is still wrong. Opposite points on the 2-sphere correspond to the sameline. So we throw away the southern hemisphere, c < 0. We still need toglue together opposite points on the equator. If we glue together one pair ofopposite points, we get a peculiar looking bent disk with lips as in figure 1.1

The final business of gluing together the two bounding circles is complicated

by the fact that we cannot just stretch them around. The wrong pointsget glued. We have to twist one of the circles before gluing. This meansthat it is not possible to build the object in three dimensional space. It is,however, a perfectly respectable two dimensional surface, just like a sphereor a torus. Or a Klein Bottle. The somewhat woolly idea of gluing edgestogether sounds as though it needs to be made precise, but a woolly idea isoften a good starting point1.

1This is still wrong as a representation of the lines in the plane. It contains a line(0, 0, 1), the North pole of the hemisphere. This, does not correspond to any actual linein the plane, the equation 0x+ 0y + 1 = 0 is not the equation of any line unless you think

the empty set is a line. If you look to see what happens as a, b get very close to zero andc compensates, you discover that we are getting lines which are further and further away

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The following is an excerpt from a paper of mine on image analysis:

The Hough Transform

It is possible to obtain a maximum likelihood estimate from a set of points

drawn from a line or lines inR2, by turning to the space, M, of all such linesinR2. It is usual to adjoin to the space of all lines in the plane a line atinfinity, which produces the spaceRP2, real projective 2-space. This may beregarded as unrealistic, since the line at infinity is not going to appear in anyimage, but neither are lines sufficiently remote, since images are bounded.We may more plausibly remove a contractible neighbourhood of the line atinfinity and obtain the space M which is homeomorphic to RP2 with a diskremoved from it. By parametrising this space, homeomorphic to a Mobiusstrip, and observing that a point in the original image (by which is meantthe compact rectangle inR2 in which our line is sought) generates a curvein the space of lines, the line in the image space is transformed into a setof curves intersecting in the point which parametrises the given line. If thepoints of the image are subject to noise, then the transform of them will yielda cluster of points the centroid of which may be construed as a maximumlikelihood estimate of the parametrisation of the underlying line. This isgenerally known in the image processing literature as the Hough Transform.It transforms the problem of finding a line or lines in the image space intothe problem of finding a cluster or clusters, or maxima of a density function,in the line space.

Engineers try to parametrise the space of lines in various ways; if you were

to write each line as y = mx + c, then the mc space is the transform space.This has a problem, namely that you miss out on the vertical lines. You canget around this by using two parametrisations, one with y = mx + c andone with x = my + c. This has some complications because most lines areobtained twice. So engineers have sought a parametrisation which gets every-thing and gets it only once. Unfortunately there is no such parametrisation,because it would be a homeomorphism between a disk or rectangle in R2 anda Mobius strip. And there isnt one. Note that parametrisations are localhomeomorphisms. So Topology occurs quite naturally in understanding thelimitations of algorithms in image analysis. It is rather useful, if you are anengineer searching hard for something, to be told it isnt there to be found.

from the origin. So the point is said to define a line at infinity.

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You give up sooner, so you save time.

These are some examples, but there are heaps of others. Topology comes upin Theoretical Physics, in many areas. It is central to modern mathematics.

So Topology is useful.

The second reason for studying the geometry of spaces up to homeomor-phism, that is, only caring about properties which are preserved by everyhomeomorphism, is that it is fun. This is, of course, rather a subjective as-sessment, and maybe as fun goes it is pretty highbrow. Still, if it wasnt fun,people wouldnt do it unless they got paid much more money than academicsget.

1.1.3 Invariants

It is clear then that there are some spaces which look different to the classicalgeometer, but which look to be equivalent to the topologist; but that notall spaces look equivalent to a topologist, or there would be little interest instudying them. One of the natural things to try to do when sorting out whichobjects are equivalent is to look for a simple thing like a number which cannotbe different for two objects if they are homeomorphic. I claimed that R andR2 are not homeomorphic, and I assure you as a scholar and a gentleman2

that Rn and Rm are not homeomorphic unless n = m. The dimension,

then is an example of an invariant, something which gives us a test to seeif things are homeomorphic. If they are not, it may be the case that thedimensions are the same, as with [1, 1] and (1, 1). But if the dimensionsare different, the spaces cannot be homeomorphic. Or to put it another way,

2I hope you are not naive enough to trust me on this. A scholar, just maybe, but agentleman only for short periods and in a bad light. As a mathematician, your job isnot to trust anybody except on a purely temporary basis. You will therefore, as a goodstudent, spend a lot of your spare time trying to prove that

Rn = Rm n = m

where = denotes the relation of being homeomorphic. It looks fairly easy, does it not?Heh, heh, heh.

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a homeomorphism must preserve the dimension, or the dimension is invariantunder the transformation which is a homeomorphism.

So one way to tell, sometimes, if things are not homeomorphic, is to computethe dimension. If the same, could be, if different, no way.

Of course, a sphere and a torus have the same dimension, so it is worthlooking for other, more subtle invariants. Let us look at one which allows usto tell that a sphere and a torus are not homeomorphic.

Definition 1.1.4 The unit 2-sphere is defined to be the set

x R3 : x = 1

More generally, the unit n-sphere is defined to be

x Rn+1 : x = 1

Definition 1.1.5 The unit 2-torus is defined to be the cartesian productspace

S1 S1

A triangulationof a surface is, intuitively, a process of chopping up the surfaceinto triangles, so that the triangles intersect not at all or along their edges

or vertices. This is not very precise, but if we take a tetrahedron regardedas four triangles sitting in R3, we get something which is homeomorphic toa 2-sphere and is triangulated.

Exercise 1.1.3.1 Prove that a tetrahedron surface consisting of four trian-gles is in fact a sphere despite the pointy bits (known, technically, as thevertices).

Now count the number of vertices (V), Edges (E) and faces (F) in the trian-

gulation. For the sphere/tetrahedron we have V = 4, E= 6 F= 4. The EulerCharacteristic is the number = V F + E = 2. Suppose we subdivide a

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18 MICHAEL D. ALDER

face of the tetrahedron. Then we add one new vertex, lose one face and gainthree, so gain two faces, and also gain another three edges. Counting theedges as negative, this sums to zero, so the Euler characteristic is unchanged(invariant) under the operation of refining the triangulation. Actually, anytriangulation of the sphere will produce an Euler characteristic of 2. This

needs proof, but some experimenting will give you a conviction that theclaim is defensible. A little thought shows that if we could show that anytwo triangulations of a space have a common refinement, we would be done.

Exercise 1.1.3.2 Construct a formal definition of a triangulation of a sur-face. At least, make it as formal as you know how; this may require you todefine a surface formally, too. Construct a formal definition of a refine-ment of a triangulation. Can you prove that any two triangulations have acommon refinement3?

Triangulating a torus is a little more complicated. Think of it as a squarewhich has had two opposite edges glued together to give a tube, and then thetwo circular ends glued together to give the torus. Now you can triangulatethe square, carefully, so that when you do the gluing, it stays a triangulation,but some of the edges have been glued together. There is a triangulationshown in Masseys book [2] with V=9, E=27 and F= 18, giving = 0.There is a somewhat more extremal triangulation with two triangular facesto give the original square, and gluings to give V=1, E= 3 and F = 2, again

giving = 0. Again, the same argument as before shows that refining thetriangulation doesnt change , and a rather careful analysis can establishthat every triangulation of a torus (or indeed any surface) has the sameEuler Characteristic. And since 2 = 0, the Torus and the 2-sphere are nothomeomorphic.

Exercise 1.1.3.3 Triangulate the spaceRP2 and calculate its Euler charac-teristic.

3The correct answer to this question is probably No. It first requires you to define a

triangulation carefully, of course, and then to grapple with the possibility of some ratherfrightful ones.

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This argument may not altogether convince you, I hope it doesnt, becausethe details which I left out are quite difficult. Nevertheless. the idea oftriangulating the space is an important one, because it turns it into a muchmore algebraic sort of object, consisting of vertices and edges and faces, allwith certain incidence relationships between them, and the idea of being able

to compute invariants for spaces is an important one.

1.1.4 Summary

This completes the motivational part of the chapter; I have indicated thatTopology is about spaces and that there are important spaces which aretopologically distinct, but where the problem of showing that they are nothomeomorphic can be quite hard. I have indicated briefly how there is anapproach through invariants, one of which is the dimension, and one of whichis the Euler Characteristic, , which can be obtained by triangulating thespaces, and then computing numbers which can be combined to give . Ihave been disgustingly woolly about details, and the next section is designedto establish the basic machinery that will be needed to make what I havesketched out intellectually respectable. It will also allow us to build farmore powerful apparatus for investigating the highly non-trivial problem ofdeciding if two spaces are homeomorphic.

We noted that some surfaces can be embedded in R3 and some cannot, andthere is clearly a problem of deciding when a space can be embedded in

another. This is another problem tackled by algebraic topology. There aremany others, such as the question of which kinds of vector fields can exist onwhich kinds of manifolds, but even describing most of the problems whichcan be solved can be a lot of work, so I stop here.

1.2 Back to Basics

The following is material which is logically a prerequisite for this course, and

I expect readers to know it thoroughly. It is possible to learn it here but noteasy.

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1.2.1 Metric Spaces

Definition 1.2.1 A Metric Space, (X,d), is an ordered pair the first compo-nent of which is a set and the second of which is a map

d : X X Rsatisfying the following conditions:

x, y X, d(x, y) 0 (1.4)x X, d(x, x) = 0, and x, y X, d(x, y) = 0 x = y (1.5)

x,y,z X, d(x, z) d(x, y) + d(y, z) (1.6)

The map d is called the metric; it is sometimes known as a distance function.

Definition 1.2.2 If (X, d) is a metric space and U X, then (U, d) is asubspace with the inherited metric obtained by restricting d to U U.

Definition 1.2.3 For every Metric Space (X, d), for every positive real rand for every x X, the open ball B(x, r) X is the set

{y X : d(x, y) < r}

Definition 1.2.4 For any metric space (X, d) and for any subset U X,U is open in X iff U is a union of open balls.

Exercise 1.2.1.1 Show that if(X, d) is a metric space, so is (X,

d), wherex, y X, d(x, y) =

d(x, y). Find a case where (X, d) is a metric but

(X, d2) is not.

Definition 1.2.5 If (x, d) and (Y, e) are two metric spaces, and if there isa map f : X Y, then we say that f is continuous at a X iff

R+ R+x X : d(x, a) < d(f(x), f(a)) <

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Definition 1.2.6 If (x, d) and (Y, e) are two metric spaces, and if there isa map f : X Y, then we say that f is continuous iff a X, f is continuous at a.

This must look familiar! It generalises to arbitrary metric spaces the essentialidea of continuity on R.

We often take the metrics for granted and write f : X Y is a continuousmap between metric spaces X and Y.

Definition 1.2.7 For any map f : X Y, and for any V Y,

f1V = {x X : f(x) V}

Proposition 1.2.1 A map f : X Y between metric spaces (X, d) and(Y, e) is continuous iff V Y is open in Y f1V is open in X.

Proof:

Let V be open in Y. Then a f1V, there exists an open ball B in Vcontaining f(a), since f(a) V and V is a union of open balls. If the centreof B is at c and the radius of B is r1, then a B d(a, c) < r1, so there isan open ball B

of radius(r1 d(a, c)) centred on f(a) contained in V. Sincef is continuous it is continuous at a, so there is an open ball A on a, theimage by f of which is inside B. Hence f1V is a union of open balls and isopen in X.

a X, R+, let V be the open ball on f(a) in Y of radius . Thenf1V is an open set in X and contains a. Hence it contains an open ballcontaining a, and hence it contains an open ball centred on a. Its radius isthe required .

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Corollary 1.2.1.1 The composite of continuous maps is continuous.

One of the important things that has come out of Algebraic Topology is arealisation that it is the maps which are allowed between objects which are

important, since there is some sort of structure preserved by the maps, andthis is how you define that structure. Hence:

Definition 1.2.8 If d, d are two metrics on a spaceX, the metrics are saidto be equivalent iff the identity map IX : (X, d) (X, e) and the mapIX : (X, e) (X, d) are both continuous.

If two metrics on X are equivalent, then it is easy to see that all the mapsout of X which are continuous with respect to one metric are continuouswith respect to the other. This also holds for maps into X.

Exercise 1.2.1.2 Prove the above assertion.

Proposition 1.2.2 Two metrics d, e on a set X are equivalent iff every setwhich is open in (X, d) is open in (X, e) and vice versa.

Proof.

The identity map on X, IX, takes every subset to itself; if U is open in(X, e)then I1X U = U is open in (X, d) since IX : (X, d)

(X, e) is continuous.

Similarly the other way round.

The map dogma which claims that it is the maps which determine thestructure, tells us that if we are interested in the underlying structures onmetric spaces, we should be looking at the open sets on the space.

Proposition 1.2.3 In any metric space (X, d), the empty set is open inX, and X is open in X. The intersection of two open setsU, V is open, andfor any set of open sets, the union of them all is open in X.

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Proof.

That X is open in X is rather trivial: take every open ball on every pointof X. That the empty set, , is open is surprising, since on the face of itit isnt a union of open balls. Given a collection C of open balls of X, the

union over C is the set of points which are in at least one ball in the set Cof balls. If C is empty, this is also empty.

If U, V are open in X, they are both unions of open balls, and if x U V,then there is an open ball BU containing x and contained in U. Without lossof generality, we may take BU to be centred on x. Similarly there is an openball centred on x, BV which is contained in V. The smaller of the two ballsis in both, i.e. is in U V. Hence U V is open in X.

If

{U

j: j

J}

is a set of open sets of X, then

x X, x

jJ

{Uj} j, x Uj

Hence there is an open ball containing x which is contained in Uj and hencecontained in the union. Consequently, the union is a union of open balls, andis therefore open. Note that this holds (vacuously) if J = .

Exercise 1.2.1.3 What is the intersection of an empty collection of sets?

The above theorem leads, like many theorems, fairly naturally to a definition:

Definition 1.2.9 A Topological Space is a set X and a non-empty collection

T = {Uj : j J}

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24 MICHAEL D. ALDER

of subsets of X satisfying the following conditions:

T (1.7)X T (1.8)

i, j

J, Ui, Uj

T Ui

Uj

T(1.9)

K, {Uj T : j K} ,

jK

Uj T (1.10)

We shall call T the collection of open sets ofX in what follows.

We say that the open sets are closed under pairwise intersections and arbi-trary unions, and that and the whole space are always open. Then the lastproposition asserts that every metric space is a topological space, with theopen sets of the topology being the open sets of the metric. The converse isnot true.

Exercise 1.2.1.4 Take a two point set X = {x, y}. List all the topologieson X. Show that one, at least, cannot be derived from a metric.

It is possible to go to even more primitive objects than topologies, but thereare not many reasons for doing so. On the other hand, there are topologicalspaces which arise in the study of sets of functions where there is no goodmetric. So topological spaces are a Good Thing. On yet another hand, mostof the spaces studied by algebraic topologists are embedded safely in Rn forsome n, and consequently are metric spaces. On the fourth hand, life is oftencleaner, simpler and altogether less hassle if we go to the basics and workwith least machinery. Sometimes, all the extra machinery just gets in theway of seeing what is happening.

For reasons of simplicity and austerity I shall therefore use topological spacesas the basic framework. The abstraction frightens some people to death,and charms others. I suggest that you see it as a kind of game and tryto enjoy it. The next advice, superfluous to the intelligent and wasted onthe rest, is to make up an example of every new object defined. In factseveral examples, some straightforward and some bizarre. This is (a) good

fun, requiring ingenuity and perhaps creativity and (b) extremely useful incoming to grips with the abstract ideas.

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1.2.2 Topological Spaces

Repeating ourselves for completeness:

Definition 1.2.10 If X is any set, a topology for X is a collection T ofsubsets of X satisfying:

T and X T (1.11)U, V X, U,V T U V T (1.12)J, {Uj T : j J} ,

jJ

Uj T (1.13)

Definition 1.2.11 A topological space is a set X together with a topologyT for X.

Definition 1.2.12 If (X, T) and (Y, S) are topological spaces and if f :X Y is a map, then f is continuous iff

V Y, V S f1V T

Exercise 1.2.2.1 Show that the composite of continuous maps is continu-ous.

Definition 1.2.13 For any set X, the map IX : X X is the identity,x X, IX(x) = x

Exercise 1.2.2.2 Show that the identity map IX : (X, T) (X, S) iscontinuous iff the topologies S, T are the same.

Definition 1.2.14 A map f : X Y between topological spaces (X, T)and (Y, S) is a homeomorphism iff it is continuous and has a continuousinverse, that is, iff

g : Y

X : f

g = IY, g

f = IX

and both f and g are continuous.

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Definition 1.2.15 A map f : X Y between topological spaces (X, T)and (Y, S) is an embedding of X in Y iff it is a homeomorphism onto itsimage.

Thus S1

normally comes embedded in C or R2

.

Definition 1.2.16 The subsets of X which are elements of T are calledthe open sets of the topological space. A closed set is one which is the setcomplement of an open set.

Definition 1.2.17 If(X, T) is a topological space, and ifV X is a subset,then we define

T |V = {Vj : j J} where {Uj : j J} = T and j J, Vj = V Uj

Proposition 1.2.4 T |V is a topology for V.

Proof.

By definition, T |V, and V T |V.

It is clear that for any two sets Vi, Vj T |V,Vi Vj = (Ui Uj ) V

soT |

V is closed under intersections.

By distributivity, it is also closed under arbitrary unions.

Definition 1.2.18 The topologyT |V on the subsetV of the topological space(X, T) is called the relative or subspace topology.

Exercise 1.2.2.3 If(X, T) is a topological space derived from a metric space(X, d) and V X, show that the relative topology on V is derived from themetric restricted to V.

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You will notice that there are such things as linear spaces, linear maps,linear subspaces and cartesian product linear spaces; there are groups, ho-momorphisms, subgroups and cartesian product groups. There are also met-ric spaces, continuous maps, metric subspaces and cartesian product metricspaces, and there are topological spaces, continuous maps and topological

subspaces. You might also note that sometimes, there are quotient linearspaces, and quotient groups. The question comes up, are there quotient met-rics? Are there quotient topologies? The answers are sometimes and yes,respectively. The question are there cartesian product metrics? is of courseyes, with a choice of several, but there is a unique product topology on thecartesian product of two topological spaces.

Definition 1.2.19 Let denote an equivalence relation on a set X, and letT be a topology on X. LetX/ denote the equivalence classes, and let

: X X/ denote the projection which takes each point x ofX to the class [x] containingx.

Then let T/ denote the collection of subsets of X/ :V X/ T/ iff 1V is open in X

This collection of subsets of X/ is called the quotient topology on X/ .

Calling something a topology doesnt make it one:

Proposition 1.2.5 The quotient topology is a topology on X/ .

Proof.

T/ since 1 = . Similarly, X/ T/ .U, V X/ , U, V T/ 1U T and 1V T

1U 1V T

1(U

V)

T U V T/

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28 MICHAEL D. ALDER

Similarly for unions.

Note that this ensures that the projection map is continuous, just. If thetopology on X had any fewer open sets, or the topology on the quotient setany more, it wouldnt be.

Exercise 1.2.2.4 Construct an equivalence relation on a finite set, constructa topology on the set and the construct the quotient topology.

When we talked rather vaguely of gluing points together to get a new space,we are implicitly defining a rather obvious equivalence relation on the pointsof the first space, then taking the quotient space with the quotient topology.

For example, take the topology induced from the standard metric on the unitsquare in R2. Define an equivalence relation on the points contained in thesquare as follows: a point in the interior is equivalent only to itself. A pointon the bottom boundary is equivalent to itself and also to the point with thesame x-coordinate on the top boundary, and the points on the left and rightboundaries are paired if their y-coordinate is the same. Then the quotientspace, with the quotient topology, is homeomorphic to a torus.

Exercise 1.2.2.5 Show that there exists a metric space and an equivalenceon it so that, regarding the metric space as having the induced topology, the

quotient space is not a metric space.

Exercise 1.2.2.6 If X is a space and A is a subspace, we write X/A todenote the quotient space obtained when every point of A is equivalent toevery other point of A, and every point of X A is equivalent only to itself.

If X = [0, 1] and A = {0, 1}, describe X/A.

If X is a 2-sphere and A is the equator, describe X/A.

If X is the two dimensional disk{x R2

: x 1}, and A is the boundary,S1, what is X/A?

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If X = [0, 1] and A is the sequence (1, 1/2, 1/4, , 1/2n, , 0), draw apicture of X/A. What difference does it make if 0 is left out of A?

Definition 1.2.20 If (X,

T) and (Y,

S) are topological spaces, and if X

Y

is the cartesian product of the underlying sets, with X : X Y X andY : X Y Y the two projection maps, and if U T and V S, thenI define a topology (T S) on X Y by stipulating:

1X U 1Y V (T S)

and every (open) set in T Sis a union of such sets.

This makes the projection maps just continuous. It is also an open map,

that is, it takes open sets to open sets. Again, just. If the topology on theproduct had any more open sets, it would not be open. Since the set oftopologies on a base set is partially ordered by inclusion, we could use theobservations on just continuous and just open to characterise the quotienttopology.

Again, pious hopes are not enough; we have to show that there really is atopology here.

Proposition 1.2.6 The product topology is a topology on the cartesian prod-

uct.

Proof:

Do it as an exercise.

Exercise 1.2.2.7 Define the product topology for the cartesian product of afinite collection of topological spaces.

Find maps which are open and others which are not.

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A map between topological spaces is closed iff the image of every closed setis closed. Show that the projection fromR2 to R which projects onto the firstcomponent is a map which is (a) continuous (b) open and (c) not closed.

Show that the space X

Y and the spaceY

X, with the product topologies,

are homeomorphic.

Exercise 1.2.2.8 Construct a definition to make precise the fluffy notionof maps being just continuous. You may need to look at the family of alltopologies and their obvious partial ordering. Give a definition of the producttopology via this idea. There are several approaches.

Exercise 1.2.2.9 Construct a definition of a product topology for any col-lection of topological spaces. Do so in such a way that the projection maps

are continuous and open. Be careful!

S1 was defined earlier, and T2 was defined via the cartesian product.

Exercise 1.2.2.10 Let the space defined by gluing together opposite sidesof the unit square as above be called T21 . Observe that it is defined as atopological space after all these definitions. Show that it is homeomorphic toT2.

Note that the constructions we have got for making topological spaces fromold ones are quite powerful, and easily extensible. For example, suppose Ihave two topological spaces which are surfaces, and I embed a disk in each.Now I chop out the interior of the disks, and glue the two circular boundariestogether. This gives me a new object, called the connected sum of the twosurfaces. There is a bit of a problem of ensuring that the homeomorphismclass of resulting objects will not depend on the details of the embeddingsand the gluing, so the idea needs a bit of work, but it is plausible that thiscan be done.

Exercise 1.2.2.11 Take two copies of the real lineR

andR

and let bethe unique field isomorphism between them, with x = (x). Identify x R

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with x R for everyx except zero. The result is essentially a Real line withtwo distinct zeros. Show that there is a well defined topology, specify the opensets. Show the space is not metrisable, i.e. there is no metric on it whichgives the topology.

We can place additional requirements on topological spaces in order to makethem at least a little more like the metric spaces we know and love. One is:

Definition 1.2.21 A topological space (X, T) is said to be hausdorff or T2iff for every pair of distinct points x, y X there exist disjoint open setsU, V T with x U and y V.

Exercise 1.2.2.12 Show that every metrisable space is hausdorff.

Exercise 1.2.2.13 Show that in a non-empty hausdorff space, the points areclosed. That is,

x X, X {x} is open

From now on, we shall simply take the topology for granted and simply referto the open sets on X, and write that X is a topological space. Also, unlessotherwise stated, we assume that all the spaces we shall be concerned withare hausdorff.

Many notions which you are used to from studying Rn make sense and are

much simpler in the setting of topological spaces.

Definition 1.2.22 LetS0 denote the topological space with two points, andevery subset open.

You can probably see why the above space is so called. The topology whereevery subset is open, or alternatively every singleton set is open, is is calledthe discrete topology. Such spaces are usually disconnected in an obviousintuitive sense, and S0 is the simplest of them, except for the empty set. The

topology at the other end of the partially ordered set of topologies on a setX is the chaotic topology which has precisely two open sets if X = .

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Definition 1.2.23 For any topological space X, we say that a map f : X S0 which is onto is a disconnection of X. If there exists a disconnection of X,we say that X is disconnected. If X is not disconnected because there existsno disconnection, we say X is connected.

Exercise 1.2.2.14 Show that a topological space X is disconnected iff thereexist two non-empty open sets U,V which are disjoint and contain X in theunion. Prove thatR {0} is disconnected, but thatR is connected. Hencededuce that removing any point forR gives something not homeomorphic toR. Show thatR {p} is homeomorphic to R {q} for any points p, q R.

Definition 1.2.24 A path from a to b in X, for any two points a, b X,is a continuous map f : [0, 1] X such that f(0) = a, f(1) = b. Thetopological space [0, 1] has the obvious topology derived from the usual metricas a subspace ofR.

Exercise 1.2.2.15 Show that the open sets onR are the sets which are aunion of open intervals. Show that [0, 1) is open in [0, 1] but not inR.

Definition 1.2.25 A space X is path connected iff a, b X, there is apath from a to b.

Exercise 1.2.2.16 Show thatR is path connected.

Show that S1

is path connected.

Show that if X and Y are path connected, so is X Y.

Exercise 1.2.2.17 Show that if a topological spaceX is path connected, thenit is connected.

Show that the converse is false.

One of the useful properties that a subset ofRn could have was to be closed

and bounded. This comes up in lots of approximation theorems. The gener-alisation of closed presents no problems:

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Definition 1.2.26 A subset U of a topological space X isclosed iff the com-plement, X U is open in X.

Definition 1.2.27 A cover of a spaceX by open sets is a family of open sets

the union of which contains X. A subcover is a selection of the sets from thefamily.

The notion of boundedness seems to require a metric. We can however trans-late the Heine-Borel theorem intro a definition. Recall:

Theorem 1.2.1 (Heine-Borel)

A subset X of R which is closed and bounded has the property that everycover of X by open sets has a finite subcover.

Proof.

See any good textbook on point set topology or elementary analysis, e.g. [3].

This leads to the definition:

Definition 1.2.28 A topological space X iscompact iff every cover of X byopen sets has a finite subcover.

Exercise 1.2.2.18 Show that every closed subspace of a compact space iscompact.

Show that the image of a compact space by a continuous map is compact.

Exercise 1.2.2.19 Show that the product of hausdorff spaces is hausdorff.

Exercise 1.2.2.20 Show that if a point x is in the complement X A of a

compact subspace A of a hausdorff space X, then there are disjoint open setsU and V such that A U and x V.

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Exercise 1.2.2.21 Show that every compact subspace of a hausdorff spaceis closed, but that compact subspaces need not be closed in general.

Deduce that if we have a 1-1 continuous map of a compact space into ahausdorff space, the inverse is also continuous.

The following result is at first sight surprising, but follows from a suitablechoice of definition of the product topology:

Theorem 1.2.2 (Tychonov)

Any non-empty product of compact spaces is compact.

Proof.

See any good book on analytic topology. For example, G.F. Simmons Intro-duction to Topology and Modern Analysis [3].

Exercise 1.2.2.22 Prove the Tychonov theorem for the case of a finite prod-uct of spaces.

We cannot leave the subject of topological spaces without saying a littleabout limit points. There are two big areas of mathematics, algebra andtopology. Algebra is about algorithms for doing sums. Topology is aboutnotions of approximation or closeness. So ultimately the point of studyinganalytic or point set topology is to understand what Newton was on aboutwhen he took limits.

Definition 1.2.29 If A X for a topological space X, then a X is saidto be a closure point of A iff every open set containing a contains points ofA.

Definition 1.2.30 If A

X for a topological space X, then the set of allclosure points of A is written A and is called the closure of A.

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Definition 1.2.31 If A X for a topological space X, then a X is saidto be a limit point of A iff every open set containing a contains points of Aother than a.

Definition 1.2.32 A subset A X of a topological space, X, is said to bediscrete iff no point of X is a limit point of A.

Exercise 1.2.2.23 Show that if X is a compact hausdorff space, there canbe no infinite discrete subset. Alternatively, every infinite subset of a compacthausdorff space has a limit point.

Exercise 1.2.2.24 Show that 0 is a limit point ofR+.

Definition 1.2.33 For any space X, and any subspace A X, the bound-ary of A in X is the set A of points which are in the closure of A and alsoin the closure of the complement of A.

Definition 1.2.34 IfA is a subset of a topological space X, then theinteriorof A is the union of all open subsets of X contained in A.

The interior and closure operators give us (generally) new subspaces, andmost of the foundations of analysis can be obtained by taking these opera-tors as primitives instead of going through open sets. That is to say, I candefine a space by giving operators on its subspaces with certain axiomaticproperties, and then derive a definition of an open set, instead of startingwith an axiomatic definition of open sets and deriving a definition of theclosure and interior operators.

Exercise 1.2.2.25 Find a subset ofR2 which is closed and has empty inte-rior. Find another which is open and has empty closure. Find another whichhas the closure of the interior empty while the set is not empty. Find anotherwhich has the interior of the closure empty while the set is non-empty. Findanother which has the closure of the interior the same as the interior of theclosure. Let all your examples be different.

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36 MICHAEL D. ALDER

1.2.3 Groups

We are doing algebraic topology, which means we have to know the elementsof algebra as well as of topology. Fortunately, algebra is easy (CTC Wall)

and anyway the amount I shall assume you know is very small.

Definition 1.2.35 A group is a set G together with a binary operation, , : G G G

such that:

a,b,c G : (a b) c = a (b c) (1.14)e G a G : a e = e a = a (1.15)

a G a1 G : a a1 = e = a1 a (1.16)

Definition 1.2.36 A group G is abelian iff

a, b G : a b = b a (1.17)

We usually suppress the and so write ab for a b, except when the groupis abelian when we write a + b.

A group may be given or presented via generators and relations. In thesimplest case there are no (non-trivial) relations:

Definition 1.2.37 The free group on a non-empty set X is the set of (finitelength) strings

xni1i1

xni2i2

xnikikunder concatenation, where the xi are any elements of X but consecutive xiare different, and the ni are integers which may be zero, and where x

0 is thegroup identity, e for any x X. When concatenating, the rule of indices

xixj = xi+j

is followed, in particular x X : xx1 = e. This is a trivial relation, notto be compared with the non-trivial ones we consider subsequently. We shallwrite F(X) for the free group on X.

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Example 1.2.3.1 The free group on a singleton set X = {x} has terms ofthe form xn for any integer n, and hence is isomorphic to Z, the group ofintegers. It is, of course, abelian.

The free group on the set X =

{x, y

}having two elements has strings of the

form:xn1yn2xn3yn4 yn

each of which is an element ofF(X), which is not abelian.

A relation on a group is an equation of the form

xni1i1

xni2i2

ynikik = ewhere e is the identity of the group, and where the terms xi in the relationare elements of the group.

Example 1.2.3.2 On the free group on one element x, take the relationx2 = e. Then the group satisfying this condition is justZ2.

Definition 1.2.38 S is a set of generators for the group G iff S G andevery element of G is in the free group on S

It is rather taken for granted here that a string on the free group on S isinterpreted as a product in the group G of the elements of G that are in S.This can be expressed rather easily using the idea of a homomorphism.

Definition 1.2.39 If G, H are groups, a map

f : G His a (group) homomorphism iff

x, y G, f(xy) = f(x)f(y)

where we have used concatenation of symbols to denote the (generally differ-ent) operations in each group.

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Definition 1.2.40 An isomorphism between two groups G, H is a map f :G H which has an inverseg : H G both maps being homomorphisms.

Exercise 1.2.3.1 Show that if f is a homomorphism with inverse h. Then

h is also a homomorphism.

Definition 1.2.41 The kernel of a homomorphism of groups, f : G His the set

ker(f) = {g G : f(g) = eH}where eH is the identity in H.

Exercise 1.2.3.2 Show that the kernel of a homomorphism is a subgroup ofthe domain.

Definition 1.2.42 The image of a homomorphism of groups, f : G His the set

im(f) = {h H, g G : f(g) = h}

Exercise 1.2.3.3 Show that the image of a homomorphism is a subgroup ofthe codomain.

Definition 1.2.43 If G, H are groups, we can make G H into a group bydefining the binary operation on the cartesian product by the rule

(g1, h1) (g2, h2) = (g1g2, h1h2)

where I have used juxtaposition for the two operations in G, H.

Now observe that ifS is any set and iff : S G is a map of the set into agroup G, there is an obvious and unique extension of f to the free group onthe set S, f : F(S) G, which is a group homomorphism. All we have todo is to define

f on a string in the obvious way given that we have f definedon the basic terms. In particular, f(xn) = (f(x))n.

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Exercise 1.2.3.4 Define the extension f rigorously. Show it is unique, i.e.show that there is only one homomorphism of the free group F(S) into Gwhich agrees with f on the elements of S.

We may now give a more intelligible definition of a generating set:

Definition 1.2.44 A non-empty subset S G of a group G is said to be aset of generators for G iff i : F(S) G is onto, where i is the inclusionmap of S in G.

In the case of the above definition, the kernel of the map i is a normalsubgroup of the free group F(S) and equating any element of this to theidentity gives a relation on G. When this is done for a set of generators ofthe kernel, we say that G is presented in terms of generators and relations.

If G is an abelian group, we often signify this by using additive notation;thus we write g1 + g2 instead of g1g2, and 0 instead ofe for the identity, and2g for g2.

Example 1.2.3.3 For the case of a singleton set {1} which generatesZ andthe relation 2x = 0, we note that we have 2Z is the kernel, which is a freegroup on {2}, and that Z2 is the resulting group. Evidently all the finitecyclic groups can be presented in an analogous way.

Exercise 1.2.3.5 Translate the above into multiplicative notation.

It is a fact, although we do not give a proof, that:

Proposition 1.2.7 Every subgroup of a free group is free. The trivial sub-group is, by a not unreasonable convention, generated by the empty set ofgenerators.

Any group can be presented in terms of generators and relations; more pre-cisely:

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40 MICHAEL D. ALDER

Proposition 1.2.8 Any group G is the image by a homomorphism of a freegroup. That is, any group is a quotient group of a free group.

Proof.

Take as a set of generators of G, G itself. Then the unique extension of theidentity map gives a homomorphism from F(G) to G which is onto.

Definition 1.2.45 The commutator subgroup of a group G is the smallestnormal subgroup containing all elements of the form xyx1y1, for every pairof x, y G. The expression xyx1y1 is denoted [x, y] and the commutatorsubgroup written [G, G].

Proposition 1.2.9 The quotient group G/[G, G] is abelian. It is sometimescalled the abelianisation of G.

Proof.

An exercise for the diligent student.

Just as we have defined the free group on a set S, we can define the FreeAbelian Group on S as the abelianisation ofF(S). An easier way to think ofit is to take it that the strings of symbols from S which are elements of theFree Abelian Group on S have the symbols themselves commuting. Henceeverything commutes.

Example 1.2.3.4 The free abelian group on a singleton set {x} is still, upto isomorphism, Z. The free abelian group on two generators isZ Z. Theusual expression is in terms of the direct sum Z

Z. The direct sum is the

same as the direct product for a finite collection of abelian groups.

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Definition 1.2.46 If{Gj : j J} is a set of abelian groups, the direct sum

jJ

Gj

is the subset of the cartesian productjJ

Gj

which contains only those prod-uct terms for which all but a finite number of the components are the identity0j Gj. It is common to use direct sum notation for finite sets of abeliangroups rather than the direct or cartesian product.

We can see that there is a trivial abelian analogue of the result that anygroup can be presented in terms of generators and relations:

Proposition 1.2.10 Every abelian group is the homomorphic image of afree abelian group.

Proof.

Take the free abelian group on the group G itself, let I : G G be the iden-tity, and observe that, just as for the free group, there is a unique extensionof I to I, a homomorphism from the free abelian group on G to G, which israther trivially onto.

A particularly important case for us is when an abelian group is given as a

homomorphic image of a free abelian group, and when the free abelian groupis the free abelian group on some finite set. The examples ofZ,Z Z andZ2 given above obviously satisfy this condition.

Definition 1.2.47 An abelian group is said to be finitely generated when ithas a finite generating set, alternatively when it is the homomorphic imageof a free abelian group on a finite set.

There are a number of intuitively obvious results which can be proved, and

should be proved carefully for those unused to elementary group theory. Inparticular:

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Proposition 1.2.11 The free abelian group on a set of k elements is iso-morphic to Zk, andZk is isomorphic to Z iff k = .

Definition 1.2.48 An element g of a group G has order n iff gn = e and

gk

= e n|kWe take the order to be positive.

Proposition 1.2.12 The elements of an abelian group G having finite orderform a subgroup of G.

Definition 1.2.49 The subgroup of elements of an abelian group G of finiteorder is called the torsion subgroup of G. If the torsion subgroup of G is theidentity element alone, then G is said to be torsion free. If every element ofG has finite order, G is said to be a torsion group

Proposition 1.2.13 If an abelian group G has torsion subgroup T, then thegroup G/T is torsion free.

Proposition 1.2.14 If abelian groupsG, H are isomorphic, then so are theirtorsion subgroups, T, U and the quotient free abelian groups G/T, H/U.

None of the above results are particularly difficult. The following result is

known as the classification theorem for finitely generated abelian groups andis relatively complicated. It need not be proved if you are not an algebraist,but it needs to be known. Fortunately it is easy to remember:

Theorem 1.2.3 (Classification of finitely generated abelian groups)

Every finitely generated abelian group G is isomorphic to the direct sum of theTorsion subgroup T with the quotient group G/T. The latter is isomorphicto the direct sum of k copies ofZ, for non-negative integer k. The formermay be written, up to isomorphism, as a sum C1 C2 Cn, where eachCi is a finite cyclic group of order mi, and each mi is a divisor of mi+1, andthese orders are uniquely determined.

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Definition 1.2.50 The order of G/T for a finitely generated abelian groupG with torsion group T is called the rank of G. The orders of the torsionsubgroup cyclic components are called the torsion coefficients of G.

For a proof of the above theorem, see any reasonable book on group theory.(A reasonable book on group theory will contain a proof by definition ofreasonable.)

Exercise 1.2.3.6 Show thatZ2Z3Z3 is a finitely generated abelian group,and find the torsion coefficients.

Finally, a result we shall need later.

Definition 1.2.51 If G is an abelian group and A, B are subgroups, A + Bis the set

{a + b}, a A, b B

Proposition 1.2.15 IfA, B are subgroups of an abelian group G, thenA+Band A B are subgroups of G.

Proof:

It sufffices to prove closure, which is obvious.

Theorem 1.2.4 (First Isomorphism Theorem)

If A, B are subgroups of an abelian group G, and if i : A A + B is theobvious inclusion, the map

i :A

A

B

A + BB

is an isomorphism.

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44 MICHAEL D. ALDER

Proof.

It is obvious that i(A B) is a subgroup of B, so

a

A, a + (A

B)

A

A B i(a) = (a + 0) + B

A + B

B

So there really is an induced homomorphism

i :A

A B A + B

B

a A, [a] = a + (A B) ker(i)

i(a)

B

a A B [a] = 0 A

A B

Hence i is 1-1.

a A, b B, (a + b) + B A + BB

a + (b + B) A + BB

a + B A + BB

i(a + A B) = a + (b + B) A + BB

So i is onto.

Hence i is an isomorphism.

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This concludes the algebra with which I shall assume familiarity. There is abit more algebra to come, but if you are fluent with the above material, itshould not be difficult. If the above material looks deep and mysterious, doall the proofs of the propositions.

1.2.4 Manifolds

First, as for earlier subjects, a superficial, chatty, discursive outline of thegeneral ideas with examples and no proofs. If this were a course on Zoology,this is where I should be showing you photographs of elephants prior todefining one. (This gives you some idea of how much I know bout Zoology.)

It should be apparent that all the analytic or point set topology that hasbeen discussed above, is technical machinery for studying something else. It

is true that some mathematicians live their whole lives worrying about themachinery, but it was developed for a reason, in fact two separate reasons.The two major applications are, first to spaces of functions so as to geta precise grip on differential and integral operators, and second to realtopology, the study of manifolds. It is the second that motivates us here.

By way of warning, it should be said that other sorts of spaces are important;there are spaces which arise in order to get a grip on manifolds, and thenatural spaces for algebraic topology are probably the complexes. There areseveral varieties of these, too, and we shall meet some of them later. Still, itwas studying manifolds that led to the origins of real topology, and so it isdesirable to have a good feeling for what they are.

The simplest manifolds are the one dimensional curves, and the two dimen-sional surfaces. One of the important things about a manifold is that it hasa dimension. Once we have defined them properly, the curves and surfacesare just particular examples.

The one dimensional manifolds are pretty simple: up to homeomorphismthere are just two: one is R and the other is S1. This is too simple and afalse statement as it stands, since a collection of manifolds of the same di-

mension can be a manifold, so I need to say that there are only two connectedmanifolds, R and S1. Of these, only S1 is compact. The two dimensional

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compact, connected manifolds, otherwise known as compact, connected sur-faces, are a bit more complicated; RP2 is one, S2 is another, and T2, thetorus is a third. We can regard the torus as obtained by sticking a handle ona sphere. We can stick any finite number of handles on a sphere, the numberof handles is called the genus, and when we do this we get different compact,

connected surfaces. Or we can glue handles on to RP2 to get another lotof compact connected surfaces. And this is all the compact, connected twodimensional manifolds there are.

The above assertion is is a very strong claim indeed, and a claim whichrequires proof; and a long and complicated proof it is, if done carefully.Of course we cant even begin to tackle it because proofs are relative todefinitions and we havent got any. Yet.

The three dimensional compact connected manifolds are too horrible to be

discussed in this course, and for dimensions higher than three there is nouseful way to describe them all. Still, much can be done with particularcases, and saying things about particular cases is both possible and highlydesirable. Some reasons were discussed in the first part of this chapter:manifolds and flows on them arise in nature.

One of the ways manifolds tend to arise is by constraints on measurementvariables, and these occur in various ways, some of them very physical. Thesimplest mathematical example is probably a linear relation between twovariables as in ax + by = 0. These give linear manifolds, in general. Moreinteresting is when we have non-linear constraints on two variables, as in

x2 + y2 1 = 0. This, of course, gives the unit circle. In general, youmight conjecture that there is a generalisation of the rank-nullity theoremto the effect that if you have n variables and k independent differentiableconditions on them, you get a smooth n k dimensional solution space,which is a smoooth manifold sitting in Rn. This, when stated correctly, isthe implicit function theorem, and isnt as strong as the conjecture.

We saw also that manifolds can arise differently: we got RP2 by a quotientprocess of a rather different sort. The manifolds which arise by constrainingn variables are, when they exist, embedded in Rn. But RP2 came defined

in an intrinsic way which said nothing about an embedding. So it would beessential to have a definition which does not depend on an embedding. This

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would allow us to define a manifold without respect to where it is sitting.

To give an example of this, I can define S1 by saying that it is obtainedby gluing two copies of [1, 1] together, joining the two pairs of end points.Alternatively, I can say it is [0, 1]/

{0, 1

}, the quotient of the unit interval by

identifying its end points. These are an intrinsic definitions which do notsay where, if anywhere, the space is embedded. Note, also, that the universewe live in is three dimensional: there are grounds for thinking that the entireuniverse is compact. We do not know whether we live in S3, or S1 S2 orS1S1S1, or conceivably something much nastier. And we have no reasonto suppose that there is anything outside the Universe in which our Universeis embedded. So intrinsic definitions of manifolds are essential.

To get a grasp of what can be done here, throw your mind back to anearlier stage of your mathematical development and the way in which youparametrised surfaces and curves in order to do integration of vector fields(or differential forms if you did it properly). Basically, a parametrisation ofa surface involved mapping some nice region in R or R2 onto the surface, ora part of it. You may have had to do this with several nice regions, as whenyou want to integrate over a sphere. Basically, you took squares or rectanglesin the plane, and set one to the northern half and one to the southern half,or at least, this is one way which comes out of the spherical polar coordinateframework.

There are other ways of tackling this, but a possible way is to start out bysaying that we get a manifold by gluing cubes together by mapping the cubes

into the manifold. We shall use this approach when dealing with HomologyTheory, the primary tool of algebraic topology. In the case of one dimension,we glue intervals, in the case of surfaces we glue squares onto the manifold.In general the problem with this is that we can get rather a lot of other thingsbesides manifolds.

As well as having to talk about manifolds, we shall be interested in manifoldswith boundary. The unit square in R2

I2 = {(x, y) R2 : 0 x 1&0 y 1}has a boundary to it, and we want to be able to describe these things as well.

After that rather extensive bit of chat about what sort of things manifolds

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48 MICHAEL D. ALDER

are, why we love them and are curious about them, why we need an intrinsicdefinition rather than one which gives us the object as a subset ofRn, let usget to the nitty-gritty and give a definition which although imperfect will dofor a first pass:

Definition 1.2.52 Let X be a hausdorff space, and let { Uj : j J} be aset of open subsets ofRn. Let{fj : Uj X} be a set of continuous maps

from the open sets such that each fj is 1-1 and the images of the Uj coverX, that is

x X, j J, u Uj : fj (u) = x

Then if whenever Ui Uj = we have that the map f1i fj is a homeo-morphism on its domain, we say that the family of maps {fj : j J} is a

family of charts, and thatX together with this family of charts is a manifoldof dimension n.

The family of charts is called an atlas. It is formally convenient to extend theatlas by sticking in all possible charts which are compatible with the atlas, butthis is a technicality which must not distract you from the essentials. Again,we do not stipulate here that the domains of the charts, the Uj should beopen disks or open cubes, but we dont say that they cant be either.

Exercise 1.2.4.1 Let U1 be the open interval (, ) which is an open setinR. Letf1 : U1 S1 R2 be the map which sends t to (cos t sin t) R2.Let U2 be the same as U1, but let f2 : U2 R2 be defined by f2(t) =( cos t, sin t). Show explicitly that this defines an atlas for S1 and hencethat S1 really is a manifold of dimension 1.

Note that our construction makes a manifold locally a metric space, indeedlocally a linear space, Rn. This is cheering, but it does not necessarily makeit a proper metric space, since distances in one chart may be different from

distances in another. Still, we can hope for a good many properties of thenice spaces which are open subsets ofRn to be carried over.

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Exercise 1.2.4.2 Note that we started off with a hausdorff space. Supposewe had dropped this condition, making it, say, any old set. Find a thoroughlynasty quasi-manifold which isnt metrisable.

Exercise 1.2.4.3 Show that S2 is a manifold, likewise T2

Exercise 1.2.4.4 Show that the product of two manifolds is a manifold in anatural way.

Exercise 1.2.4.5 Show that the quotient space of a manifold need not be amanifold.

If we want to extend the idea to a manifold with boundary, which, be it

noted is not a manifold, we need to distinguish between the interior pointsof the manifold which are treated as above, and the boundary points whichlook rather special. Take a half ofRn,

Hn = {(x1, x2, xn) Rn : x1 0}and require that some of the points of the space X should be covered by theboundary of Hn instead of an open set in Rn.

Exercise 1.2.4.6 Define the term manifold with boundary using the above

idea.

Show that [0, 1] is a one dimensional manifold with boundary, and that D2 ={x R2 : x 1} is a two dimensional manifold with boundary.

Show that the product of a manifold with a manifold with boundary is always amanifold with boundary, and that the product of two manifolds with boundaryis also a manifold with boundary.

Show that the boundary of a manifold with boundary is a manifold, and showthat the boundary of a compact connected manifold with boundary is a com-

pact, but not generally connected, manifold. (This will require you to definethe boundary of a manifold with boundary)

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The connected sum operation on surfaces was sketched earlier: we take anembedding of D2, the unit disk in R2 into one surface (i.e. two dimensionalmanifold) and another of the same unit disk into another surface. We thenremove the interior of the disks from both surfaces, and identify the matchingpoints on the circle. It should be clear to those of a nervous disposition that

there are some perfectly horrible maps of D2 into R2, and that this isntnecessarily a straightforward job. For higher dimensions, it can only getworse- meditate upon the Alexander Wild Horned sphere, for example. Ifyou feel foolhardy, try the next exercise:

Exercise 1.2.4.7 Show that if we do this operation on the two manifoldswhich are both copies ofR2 with single charts in the atlas in both cases, weget a manifold. Similarly, show carefully that attaching a handle to R2 givesa manifold.

1.2.5 Odd

There is one result for compact metric spaces which is needed in the sequeland which doesnt logically belong anywhere we have been so far. First afew definitions.

Definition 1.2.53 If U

X is a subset of a metric space (X, d), the diam-

eter of U is sup{d(u, v) : u, v U} when this exists.

Definition 1.2.54 Let{Uj : j J} be a cover of a metric space (X, d) byopen sets. We say that the real number > 0 is a Lebesgue number for thecover, if for every subset A X of diameter less than , there is at least one

j J such that A Uj.

It is clear that we can cover some spaces with open sets which get progres-

sively smaller, and where the amount of intersection of the sets also getssmaller, so no Lebesgue number exists. On the other hand:

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Theorem 1.2.5 (Lebegues Covering Lemma)

In a compact metric space, every open cover has a Lebesgue number.

There is a proof in [3].

1.3 Summary

The preceding back to basics section summarised all the material on met-ric spaces, topological spaces and groups which you are assumed to know.Whether you actually do know it is not altogether clear, but if not, the rem-edy is in your own hands. By now you can be assumed to be sufficientlyadult to take whatever steps are necessary to make up for the deficiencies of

your education so far, when they become too painfully apparent.

This summary isnt going to summarise the preceding section, because thepreceding section is a summary. Sort that one out.

We are now ready to start on the new stuff which I shall assume you dontknow about.

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Chapter 2

Categories and Functors and

2.1 Categories

You will by now have noticed certain basic similarities which keep on comingback to haunt us.

There are things called sets. Between most pairs of them, there may beseveral things called maps. Given any set, X, there are usually some subsetsof the set X. Given two sets,X, Y there is a product set, X Y. If there isa map f : X

Y and another g : Y

Z, then there is a map called the

compositeand written g f : X Z. Particularly important kinds ofmapsoccur between sets; for instance every set X is associated with an identitymap, IX. If f : X Y is any map between sets, then f IX = IY f = f.Ifg : Y X is another map, and iff g = IY and g f = IX, then we saythat f, g are inverses. In this case, there is an equivalence between some ofthe sets determined by the invertible maps.

If I went through every occurrence of the word set and every occurrenceof the word map and replaced them by group and group homomorphism,the paragraph would still be true. If I went through and replaced every

occurrence of set by linear space and map became linear map, the samething would hold. Other possibilities will doubtless occur to you. This leads

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us to a new stage of abstraction:

Definition 2.1.1 A category is a collection of two sorts of things, Objectsand maps. The collection of objects is generally a class which is not a set.

Each object, X, is associated with a particular map IX. Each map f, isassociated with a pair of objects, dom(f) and ran(f). In the case of the mapIX, both of these are the object X. IfX,Y ,Z are three objects (not necessarilydistinct), and iff, g are two maps (not necessarily distinct), and if dom(g) =ran(f) = Y anddom(f) = X, ran(g) = Z, then there is a map written gf, withdom(gf) = X, ran(gf) = Z. For composites of three or more, the compositionis associative whenever it is defined.

Exercise 2.1.0.1 List as many distinct categories as you can think of. Abouta dozen is average.

Any sensible abstraction like this is intended to save effort by doing thingsin abstract categories rather than doing them one damn time after anotherin several different categories. Naturally, we write

f : X Ywhenever f is a map in the category under discussion and dom(f) = X, ran(f)= Y. Sometimes we write:

X Yf

or

X Yf

for this. The composite of maps f, g will be written gf:

X Yf

Zg

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Now we can define a monomorphism in an abstract category as follows:

Definition 2.1.2 A map f : X Y is said to be a monomorphism iffwhenever g, h : A

X are maps,

fg = fh g = h

It is clear that for the category of sets, if f is not 1-1, then it may be thatf g = f h but g = h. g and h may differ, sending a point a to g(a) = h(a).But iff(g(a)) = f(h(a)) we may still have fg = fh. Conversely, if f is 1-1,then if g and h differ on some point a A, then so will fg and fh. So wehave shown that in the category of sets, the 1-1 maps are monomorphisms.

Dually:

Definition 2.1.3 A map f : X Y is said to be an epimorphism iffwhenever g, h : Y Z are maps,

gf = hf g = h

It is obvious that if, in the category of sets, the map f is not onto, then wecould have g = h but gf = hf. Conversely, if f is onto, and if g(f(x)) =h(f(x)) for every x X, then g = h.

Exercise 2.1.0.2 Show that monomorphisms and epimorphisms are whatyoud expect them to be in

(a) the categories of Groups and group homomorphisms and

(b) the category of topological spaces and continuous maps.

Likewise, w

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