The Chichilnisky–Heal Approach toArrow’s Impossibility Theorem

Anton Vernersson

Master Thesis, 30hpMaster in Mathematics, 120hp

Spring 2018Department of Mathematics and Mathematical Statistics

AbstractThis essay explores a surprising intersection between economics and algebraic topology.

On the economical side social choice is studied, a field which includes topics such as votingtheory. Within standard economics this is often approached by discrete methods, suchas Arrow’s impossibility theorem. This essay will instead follow a continuous proof byChichilnisky and Heal, which is based on a considerable amount of algebraic topology.Therefore, this essay will cover a great deal of algebraic topology. In particular, the subjectof obstruction theory, which provides clear conditions for the extension of a function to alarger space, is studied.

SammanfattningDenna uppsats utforskar en oväntad blandning av nationalekonomi och algebraisk

topologi. På den ekonomiska sidan så studeras "social choice". Ett ämne som bland annatinkluderar valteori. Inom den rådande ekonomiska teorin så används ofta diskret matem-atik för att lösa problem i ämnet. Denna uppsats följer istället en kontinuerlig metodkonstruerad av Chichilnisky och Heal. Vilken bygger på algebraisk topologi och specielltpå "obstruction theory" som syftar till att utdvidga funktioner till större mängder.

Contents

1. Introduction 12. Economic Framework 72.1. Introductory Economics 72.2. The Chichilnisky-Heal setting 123. Algebraic Topology 153.1. Deformation of Paths 153.2. Category Theory 183.3. H-groups 213.4. Exact Sequences of Sets of Homotopy Classes 243.5. Higher Homotopy Groups 323.6. CW-Complexes 353.7. Homotopy Functors 414. Obstruction Theory 454.1. Eilenberg-Maclane Spaces and Cohomology 454.2. The Tower of Postnikov 504.3. The Final Obstruction 585. Proof of Theorem 2.16 636. Intriguing Equivalences 657. References 69

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1. Introduction

All voting systems are fundamentally flawed! Already in 1785, Marie Jean An-toine Nicolas de Caritat, Marquis of Condorcet, a French political scientist pointedout this in his “Essai sur l’application de l’ analyse à la probabilité des décisionsrendues à la pluralité des voix ”. Despite its age, the insights still have impact onmodern day elections. Consider the following short example of the work done byNicolas de Condorcet. Assume that there is a remarkably small country consistingonly of three persons. They are aptly named A, B, and C. Despite the size of theircountry they decide that it needs a president. Since there are only three individualsthey decide to elect the most agreeable candidate. The following are the preferencesheld by A, B, and C, respectively.

A : A B C.B : B C A.C : C A B.

From this it follows that C is preferred to A by both B and C. Therefore it wouldseem reasonable begin by considering C as president. However, both A and Bprefers B instead of C. It might then seem like B is the most suitable president.Then again, both A and C would rather prefer to have A as president. Thus, nomatter who is elected, an unambiguously more popular candidate always exists.This is what is known as the Condorcet paradox.

Centuries later Kenneth Arrow a legendary American economist, and Nobellaureate, proved in 1950 what is known today as Arrow’s impossibility theorem.Informally, it says that no rank-order electoral system can be designed that alwayssatisfies the following fairness-type properties:

(1) if every voter prefers alternative A over alternativeB, then the group prefersA over B;

(2) if every voter’s preference between A and B remains unchanged, then thegroup’s preference between A and B will remain unchanged;

(3) there is no single voter that possess the power to always determine thegroup’s preference, i.e. there is no dictator

(see [4] for the formal statement and proof). This theorem has been argued among,generalized by, and inspired many (see e.g. [23] or [6]). From the work done byArrow an entire field known as social choice theory has sprung. The statementof Arrow’s impossibility theorem is discrete in its nature, and the proof is notsurprisingly solely combinatorial.

Contrary to the discrete origin of social choice theory, Chichilnisky introduced in1980 a, continuous, topological approach [10]. It is this approach we are interestedin this essay, especially a topological version of Arrow’s impossibility theorem. Weshall present here a fairly self-contained proof of the following theorem proved byChichilnisky and Heal [11], see also [9].

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Theorem 2.16. Let P be the space of preference profiles. A continuous functionφ :∏nk=1 P → P , which satisfies anonymity and unanimity, exists if and only if P

is contractible.

There are benefits using this continuous model. First, the framework used byChichilnisky and Heal agrees with many common assumptions from modern microe-conomics. Secondly, we can determine whether or not a social choice rule existsbased solely on if the preference space P is contractible. Finally, the topologicalconstruction also allows for an application to a wider section of economics, specifi-cally to the issue of aggregating preferences held by individuals.

Given the benefits of continuous social choice it is somewhat surprising that it notwidely used. One could argue that this perhaps is because of its position somewherebetween mathematics and economics. On one hand the methods and tools used arenot commonly known among economists, and on the other hand there are no newmathematical results for a topologist. This combined with the great success ofthe discrete model by Arrow does at least provide a partial answer for why thecontinuous model is not more well-known. Here it should also be mentioned thatthere have been attempts to unify and justify the interplay between the topologicaland traditional combinatorial approaches of the social choice theory, we restate onesuch result in Section 6.

The method used by Chichilnisky and Heal [11] in proving the sufficiency makesuse of obstruction theory from algebraic topology. This is quite a general methodof solving problems involving the extension of maps to larger domains. While onecould go for a more modern approach such as [25] we will instead follow the classicaltextbook [24]. A benefit of this is that our treatment of algebraic topology, andobstruction theory, will closely follow what was available to Chichilnisky and Heal.The specific theorem we will use is the following.

Theorem 4.35. Let ι ∈ Hn(Y, y0;π) be n-characteristic for a simple (n − 1)-connected pointed space Y where n ≥ 1, and let (X,A) be a relative CW complexsuch that Hq+1(X,A;πq(Y, y0)) = 0 for q > n. A map f : A→ Y can be extendedover X if and only if δf∗(ι) = 0 in Hn+1(X,A;π).

As one can see this is quite a technical theorem and we will spend the entirety ofSection 3 and Section 4 working towards a proof. The division between the sectionsabout algebraic topology is made so that Section 3 introduce and addresses thefundamental parts of the subject and Section 4 deals with the topics specific to theobstruction theory. Within Sections 3 and Section 4 the theory, unless otherwisenoted, is from the monograph [24] written by Edward Spanier.

Section 2 gives an introduction to economics. It starts with a broad presentationof modern microeconomics, with a focus on the mathematics. Following that is apresentation of the construction specific to the ideas used in the proof of Theo-rem 2.16, which is presented in Section 5. Following the proof of our main resultin Section 5 we will continue with Section 6 and an exploration of some interestinginterconnections between Brouwer’s Fixed Point Theorem, Arrow’s Theorem andthe continuous social choice rule, as given by Chichilnisky and Heal. Specifically,

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we will rely on [9] and [26] to show that both the economical theorems are equiv-alent to Brouwer’s Fixed Point Theorem, Theorem 6.1. This is quite remarkable,as the theorem by Arrow is entirely combinatorial while the work by Chichilniskyand Heal is continuous.

Even though this essay is, up to a point, self-contained the reader is assumed tohave mathematical maturity and basic knowledge of general topology and abstractalgebra.

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AcknowledgementThere are a number of people without whose support this essay would never havematured into what it is today. I would like to thank, in alphabetical order, JasperGräflich, Christoffer Olsson, and Aron Persson for reading and commenting on mywork. I only hope that my last days of writing have not undone what you so kindlyhelped me build. I would also like to thank Per Åhag, my supervisor, for your timeand advise but especially for never doubting that this essay would be completed.Finally I would like to thank my fiancée, Clara Nygren, for supporting me in whathas undoubtedly been some of my most difficult times.

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2. Economic Framework

To understand social choice theory it is not necessary to have a solid foundationin economics. However, in order to see why the approach by Chichilnisky and Healis appealing it is important to know the basics. This is what we shall begin to doin Section 2.1, where we introduce ordered spaces and the order isomorphisms thatcharacterize modern economical theory. we continue by looking at differentiabilityof these morphisms. This will then allow us to formulate the specific setting ofChhichilnisky and Heal and state the main theorem, Theorem 2.16, in Section 2.2.

2.1. Introductory Economics. The first point of interest in any treatment ofmodern economics is the concept of preference relations and their associated or-der isomorphisms. Within economics these are used to describe and model thebehaviour of economical agents. The agents are of course not some kind of cheapversions of James Bond but an individual or firm acting within an economy. Thesetools are the tools of the trade for modern economists. To a mathematician a pref-erence relation is just a binary relation and the order isomorphism is a monotonetransformation.

In this section we will look at the mathematical details of how preference relationsare used in economics. This treatment will be somewhat informal and a reader wellversed in mathematics or economy will find that more than a few details are omitted.Ideally there would be no trade off between quality and quantity but this is notthe case. Instead of a complete treatment this section aims at a brief introductiontrying to get the reader up to speed. Perhaps breaking with what was just statedwe begin by properly defining binary relations in general and the preference relationin particular.

Definition 2.1. A binary relation R of a setX is a subset R ⊂ X×X. If (x, y) ∈ R,we write xRy.

Definition 2.2. Let Y be a set and let y1, y2 and y3 be elements in Y . If < is abinary relation on Y such that it is:

(1) reflexive, y1 < y1 for all y ∈ Y ;

(2) transitive, if y1 < y2 and y2 < y3 then y1 < y3;

(3) complete, for all y1 and y2 ∈ Y either y1 < y2 or y2 < y1.Then we say that < is a preference relation on Y and that (Y,<) is an ordered set.

A preference relation can be subdivided into a symmetric relation, i.e. one wherey1 < y2 is equivalent to y2 < y1. And an assymetric, i.e. on where y1 < y2 impliesthat y2 < y1 does not hold. On the real line, with its usual ordering, we have thesymmetric relation = and the asymmetric relation >.

Dealing with orderings in a economical setting the symmetric relation is usuallydenoted by ∼ and is said to be an indifference relation. The asymmetric relation isdenoted (or sometimes ≺ when appropriate). We say that is a strict preferencerelation. In economics we use (order) preferences to represent preferences held byindividuals. Consider the following example on how preferences held by an inidivualis captured by preference relations.

Example 2.3. Let Alice be an individual who consumes bread and wine and enjoysspending time away from work. Let b, w, l ∈ R+, then the triple (b, w, l) ∈ R3

+

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of bread, wine and leisure represents a possible consumption choice that Alicecould make. Lets say that Alice currently consumes ten breads, drinks two bottlesof wine and has eighty hours of free time a week, i.e. her consumption choiceis: (10, 2, 80). In this case she would certainly prefer that bundle to the bundle:(10, 2, 70), because it is identical except that less work is necessary. But more thaneither of those she would prefer the bundle: (12, 2, 80). Therefore (10, 2, 70) <(10, 2, 80) < (12, 2, 80).

There is an issue in how this framework is used in practice and we will addressthat before proceeding. Commonly it is assumed that an increase in one good,while the other remains the same, is always a good thing. Within the context ofour previous example this means that there is no amount of bread Alice could eatsuch that a small increase would not make her happier. This is of course absurd:no one would prefer to eat a hundred loafs of bread for lunch over having just one.This is, however, not due to a fault of the theory but of the practicioner. All suchcases, perhaps with the exception to some “off the wall” cases, are due to a modelmisspecification. In our example we have neglected to include effects to Alice health(and her presumed utility of it).

Whether this short exposition has convinced the reader that more is alwaysbetter, or not, it is a fundamental assumption of economics. In the literature thisproperty is commonly known as strong monotonicity. Technically we say that, for achoice set in Rn such as the previous quantities, then < satisfies strong monotonicityif and only if, for a relation x < y, x 6= y implies x y. For more on the economicalaspects of this please see Chapter 7 of [27].

We shall now get a bit more specific, or perhaps a bit less, dependent on thereader’s point of view. In the previous example of bread, wine and leisure weassumed that Alice could make her choice of bread, wine and leisure on a subset ofR3. But that is not something we have to do. The next few paragraphs will showhow the model works at a higher level of abstraction. A level at which it is notnecessary to assume that Y ⊂ Rn. Instead we shall work with ordered topologicalspaces. I will assume that the reader is already familiar with the basics of topology.However, we will not use that much general topology and the treatment will againbe quite informal. With this said let us formulate what we wish to study.

Let Y be a topological space (the topology is not important at this point) to-gether with the binary relation <. We will write this pair as (Y,<). Now, (Y,<)is where agents make their consumption choices. What we wish to do is to expressthe order on Y in terms of the order on another topological space, X. To simplifythe exposition we will denote the ordering on a set Y by <Y , and the order on aset X by <X . A preference relation, <, does naturally define equivalence classeson its choice space Y by its symmetric part ∼. Because of this we could equallywell work with Y and <Y or the quotient space Y/ ∼ with Y . In the future wewill often not mention this and use Y for both spaces.

We will now begin to look directly at maps f : (Y,Y ) → (X,X) and inparticular maps f : (Y,) → (R, >). The first step will be to show that there is amap f such that the orders on the two spaces are isomorphic.

Definition 2.4. Let (Y,<Y ) and (X,<X) be ordered sets with y′, y′′ ∈ Y , and letf : Y → X be a map. If y′ <Y y′′ ↔ f(y′) <X f(y′′), then we say that f is anorder isomorphism.

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To get an order isomorphism we will have to assume two things about the order:that it is a total order and it is order separable. We define these in the followingtwo definitions. A total order is actually what most people would already assumethat a order is. For instance the standard order ≥ on R is a total order.

Definition 2.5. A binary relation < is a total order if it is a preference relation,as by Definition 2.2, and is also antisymmertic, i.e. for all y1, y2 ∈ Y then (y1 <y2 and y2 < y1)⇒ y1 ∼ y2.

Reflexivity and antisymmetry are quite straightforward and do not carry anydeeper economical meaning. On the other hand, transitivity two do have someeconomic implications. First of all, transitivity removes cyclic behaviour. If forinstance Alice exchanges A for B and B for C, then she is unwilling to trade C forA again. Thereby it will be no possibility for a looping behaviour. This is in mostcases a quite rational assumption. Perhaps there are persons who would do so butthat would break the minimum level of rationality often assumed in economics.

Definition 2.6. Let X be a set ordered by the preference relation <. If thereexists a countable set Z ⊂ X such that, for all x, y ∈ X\Z, x y implies thatthere exists a z ∈ Z such that x z y, then we say that (X,<) is order separable(in the sense of Birkhoff).

There are a number of equivalent formulation of Definition 2.6 (see Proposi-tion 1.4.4 of [7]). But the core of order separability is that there are no open gapsin the order on X. That is, on R with the standard topology a preference relation< is Birkhoff separable if there is no open set, (a, b), such that a b. With thespecific formulation of Birkhoff separability we have that there exists a dense, withrespect to <, subset Z ⊂ X. This can be used to explicitly construct a order iso-morphism f : X → R. We shall partially show how this construction is carried out.While we will leave out some details the general idea will remain the same and theinterested reader could look at Chapter 1.6 of [7].

Example 2.7 (An explicit order isomorphism f : X → R). First consider thepiecewise map r : X ×X → 0, 1 defined by

r(x, y) =

1 if x y0 otherwise

Now, since by definition Z is a countable set ordered by we can create the sum

(1) f(x) =∑

2−nr(zn, x),

where zn∞n=1 = Z. Certainly, f(x) is convergent and f(x) ∈ [0, 1]. Since Z isdense in X it follows that if x y, then there is a zn ∈ Z for some n such thatx zn y. Hence f(x) > f(y).

Similarly, if f(x) > f(y), then∑2−nr(zn, x) >

∑2−nr(zn, y),

which can only happen if r(zn, x) > r(zn, y) for some n. But that is to say thatthere is a zn ∈ Z such that x zn y. Since x and y are separated it follows thatx y.

Given the ease by with an order isomorphism can be constructed one could belead to believe that the implications of it are quite trivial. This is sadly true and in

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almost all settings additional assumptions are placed on f . One useful assumption isthat of continuity. For if the order isomorphism is continuous then the intermediatevalue theorem applies. Furthermore, the “level curves” of a continuous map are alsocontinuous.

Continuity is often a minimum requirement on the maps an economist studies.Thankfully, this is not a severe assumption on the preference relation. All thatis required is second countability, see e.g. [21], of the choice space in addition tothe assumptions already made. See [7] for a number of related proofs, including aconstructive proof similar to that of the plain order isomorphism.

So far we have not touched upon a very important part of order isomorphisms:uniqueness of representation. To this end let f1 : Y → R and f2 : Y → R betwo order isomorphisms. If there is a continuous and monotonically increasingfunction g : R → R, then g must be an order preserving mapping on R. Since gis an order isomorphism on R such that g(f1) = f2, then the orders induced byf1 and f2 must be the same. Therefore order isomorphisms are only unique upto monotone and continuous transformations. This view is within economics oftencalled “ordinal utility”. Note that order isomorphisms are often known as utilityfunctions in economics. However, this view is seemingly unable to explain the listof common isomorphisms in Example 2.8, as all these should be equivalent underthe ordinal assumption.

Example 2.8. The following are common economical examples of order isomor-phisms.

(1) Cobb-Douglas: f(x1, x2) = axα1x1−α2 . A very common and easy function

that can be used in a variety of situations where there are two distinctgoods.

(2) Constant elasticity of substitution: f(x1, x2) = a(xρ1 + xρ2)−ρ. A general-ization of the Cobb-Douglas function. For an account of elasticity, pleasesee [27].

(3) The Khaneman-Tversky utility function: f(x) = λxxα , where λx|x<0 is

smaller than λx|x≥0. A common, and revolutionary, function used in con-junction with decision under risk. Please see [19] for further information.

2.1.1. Differentiability. The difference between the isomorphisms of Example 2.8lies in differentiability. One cannot understate the usefulness of this property withineconomics. While continuity is important, many results and theories of moderneconomics relies heavily on differentiability. For instance, the optimization problemsinvolved in the behaviour of agents is almost exclusively done using derivatives. Wewill work with two examples that illustrates the use of the derivative. The first oneis a simple, bare bones, approach to finding the optimal consumption of goodswith a budget restriction. And the second is a slightly more intricate one, whichdescribes consumption within a larger economy.

Example 2.9. Let Bob be an agent in an economy with two goods, which are easilyquantifiable and infinitely divisible. That is, Bob’s choice space Y is a subset of R2.Furthermore, assume that Bob is endowed at the start with a vector ω of 2 goods.Certainly, the endowment of a good could be zero and possibly even negative. Letthe best trade opportunity in this economy be one unit of good one for two units

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of good two. If Bob has an initial wealth of ω = (2, 3), then the possible pointsBob can obtain are on the line between (3.5, 0) and (0, 7), we call this the budgetconstraint.

Say that Bob’s preferences are represented by a Cobb-Douglas of the formf(x1, x2) = x0.2

1 x0.82 . Since the slope of the budget constraint is −2, it follows

that the optimal consumption bundle is when∂x1

∂x2= −2.

This example is economically quite simple. In an economy one would expectthat the individual, Bob, would interact with other individuals and firms. The stepwe will take in the next example is to look at a more involved market example.

Example 2.10. Let X ⊂ R be a consumption set we wish to study and let Y ⊂ Rbe a consumption set we are not interested in. For instance, if we want to studythe consumption of cars, then this is our set X and Y would be a representation ofour other consumption, such as food and housing. Also, let f : X → R be an orderisomorphism and let the order on Y be represented by idY . Thus, the combinedorder isomorphism g : X × Y → R is given by g(x, y) = f(x) + y. Now, assumethat all individuals are equivalent and endowed with goods at a value of ω. In thissetting the consumption of, y, must be the difference between the endowment, ω,and cost of x, c(x). Hence the problem can be formulated as

maxx,y

(u(x) + y)

subject to y = ω − c(x).

This can be rewritten asmaxx

(u(x) + ω − c(x)) ,

which has the first-order condition∂f

∂x= c′(x).

This gives an indication as to how optimization is used in economics.

As can be discerned from the previous examples it is not mathematically difficultto work with this type of economics. The challenge, in this case, is to properlydefine differentiability in a satisfactory way. One could introduce differentiabilityby assuming that X is a subset of R and then only look at differentiable functions asthe order isomorphisms. Another possibility is to view Y as a topological manifoldand introduce a differentiable structure on it. We will take the former path but thereader may, if so inclined, find an explanation of the latter method in Chapter 8 of[7].

What we will do is to use hyperplanes as done by [13]. To simplify we assumethat Y = Rn+, which is the positive cone of Rn. This is a common assumption butit may seem severe. However, much of what economics study can be made to fitnicely into the positive part of Rn. All goods, for instance, are traded in positivequantities as are financial commodities and time. If we also assume that thingssuch as health is quantifiable then there is really no loss in generality to assumethat Y = Rn+. By using this assumption we can make the following definition.

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Figure 1. Order isomorphisms with some normals represented as arrows

Definition 2.11. Let S ⊂ Rn be of codimension 1 then we say that S is a hyper-plane.

Since < is satisfies monotonicity and is continuous it follows that the level curves,or indifference surfaces, are continuous surfaces in Rn of codimension 1. We takea moment here and consider what we are doing. Our order isomorphic mapping ofRn+ to R will take a path beginning at 0 ∈ Rn+ and map this to R. This will take onedimension of Rn+ while the remaining points will belong to the equivalence classesof ∼. Transitivity and strong monotinicity of < will assure that the equivalenceclasses do not intersect and are uniformly of dimension n − 1 (except possibly for0).

Now, we will need normals of the hypersurfaces. By definition, a normal onthe equivalence classes is the direction of the biggest increase of f : Rn+. This isapparent if we assume that f is differentiable and then use the gradient as the nor-mal. Economically the normal gives the preferred change in consumption, given aninitial consumption bundle. In Figure 1 we can see the normals of one-dimensionalhypersurfaces.

Definition 2.12. Let f : Rn+ → R be a differentiable map and let ∇ be thestandard gradient on Rn. We say that for any point x ∈ Rn+ the vector ∇f(x), inthe direction of increasing f , is the normal at x. We shall often call the normalspreferences.

As we going for the ordinal approach to isomorphisms we are not interestedin the magnitude of the normal. But unlike the continuous case we do attachimportance to the shape of the hyperplanes. Consequentially the direction of thenormal does matter. This is what explains the difference between the isomorphismsof Example 2.8.

2.2. The Chichilnisky-Heal setting. One of the biggest issues in economics isthat of aggregation: is there any way to combine preferences held by the individ-uals of a group into a single preference for the entire group. As an example wehave elections, which were mentioned in the introduction and studied by Arrow.This is also solved by Arrow: under some plausible restrictions there is no way toconsistently aggregate individual preferences into a group preference. While thisresult is a great one it not founded on the same base as the usual economics we

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have discussed here. Instead it is a discrete model. In this section we will workwith the Chichilnisky-Heal framework, which sets out to answer the same questionbut based on the theory we have discussed so far.

We will now explore the Chichilnisky-Heal model and state the main theoremof this essay: Theorem 2.16. Fundamental to the model are the preferences, ornormals, of Definition 2.12. To an individual with an order isomorphism f : Rn+ →R the normal at a point x ∈ Rn+ represents the change to x that gives the largestincrease in R.

Definition 2.13. Let P = p|p is a preference. Then a social choice rule (for nindividuals) is a map

φ :

n∏i=1

P → P.

In addition we say that∏ni=1 P is the space of profiles.

The social choice rule thus takes preferences pini=1 ∈∏P and map them into

a single preference p ∈ P . Less formally the social choice rule takes the collectionof order preserving maps of each of the individuals and produce a new, aggregated,order preserving map.

We would like φ to behave nicely with respect to the following conditions. Firstof all, φ should not discriminate between individuals, each one should have an equalamount of power in the final preference. Also, we would like the final preferencenot to be arbitrary in the following sense: if all preferences are P then the resultingpreference should also be P . However, this is not to say that if y <i y′ for all i,1 ≤ i ≤ n, implies φ(y) < φ(y′).

Lastly, we shall require that the social choice map is continuous with respect tothe product topology it inherits from P . Of all requirements this last one is thefurthest from the usual assumptions. Still, it is mathematically, the most familiarwhile the other two require some defining.

Definition 2.14. Let P be a set of preferences, let pini=1 ∈∏ni=1 P , and let

φ :∏ni=1 P → P . If pjnj=1 is an arbitrary permutation of pni=1, then we say

that φ is anonymous if, and only if, φ(pjnj=1) = φ(pini=1).

Clearly if we require φ to be anonymous we will by definition exclude solutionssuch as φ(p1, . . . , pn) = p1. Those kinds of solutions are known as dictatorialsolutions, which would not be acceptable in most practices. Another type of solutionthat we want to exclude are those of the kind φ(pi, . . . , pi) = pj where pi 6= pj . Orin other words: when all individual preferences agree so should φ.

Definition 2.15. Let P be a set of preferences, let pi ∈ P and let φ be a socialchoice rule. We say that φ respects unanimity (or is unanimous) if the followingholds

φ(pi, . . . , pi) = pi.

Both anonymity and unanimity can be motivated by what can be considered“fair” in a choice situation. However, continuity is not as readily motivated from justthe act of choosing. Instead one would have to rely on the usefulness of continuity inother applications, such as discussed in the previous section. Finally, as continuityis the defining feature of the idea by Chichilnisky and Heal, as compared to Arrow’s,one could argue that this is the most interesting property of the model.

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This is nearly all we need to state the main theorem of the essay. Notablycontractible spaces will not be defined even though they are instrumental in thetheorem. Instead we shall define them in the next section. For now one may thinkof a contractible space as a space, which can be deformed into a single point. Whilethis is enough to understand the theorem we shall see that the devil, as per usual,is in the details. Therefore, in order to prove Theorem 2.16 we must work troughsome algebraic topology, which we do after a brief discussion of the theorem.

Theorem 2.16. Let P be the space of preference profiles. A continuous functionφ :∏i∈I P → P , which satisfies anonymity and unanimity, exists if and only if P

is contractible.

This is a striking result. At first glance it is somewhat surprising that thesolution to a economical problem lies within algebraic topology, a subject rarelyused in economy. If we were to dig deeper it might be expected that contractibilityis sufficient for the existence of a well behaved social choice rule. Intuitively if Pis deformable to a point then all p ∈ P are, in a sense, equivalent to each other.Then their order does not matter and therefore φ satisfies anonymity. By similarreasoning would it be impossible for the social choice rule not to be unanimous sinceall preferences are equal. Finally, continuity can also be motivated by the idea thatwe are only working with a single point, which implies a trivial continuity. Thereforeit seems natural that contractibility is a sufficient condition for φ. But that it isnecessary for

∏ni=1 P to be contractible – that is quite remarkable since it is a

somewhat severe restriction.In order to prove Theorem 2.16 we have to develop quite a bit of algebraic

topology. Specifically we are interested in extending maps f : A → Y to a mapf : X → Y , where A ⊂ X.

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3. Algebraic Topology

The aim of this section is to provide a coherent exposure to the fundamentalsof algebraic topology. We will go as far as to Obstruction Theory (Section 4) in afairly rigorous way. Still, the exposition of (for instance) Homology and Cohomology(Section 4.1) will be kept very short. Ultimately, the goal with the use of algebraictopology is to prove Theorem 4.35, which we will then use in the proof of the maintheorem (Theorem 2.16). As a note of caution: we will use the shorthand notationfg for the composition of two maps throughout the essay.

The reader is expected to know the basics of general topology. However, we willstate three definitions for reference and as a reminder.

Definition 3.1. Let X be a topological space and let Xjj∈J be a sequence oftopological spaces, for some indexing set J . Given functions gi : Xi → X, wesay that the finest topology on X such that each gi is continuous is the coinducedtopology (on X by gi).

Definition 3.2. Let X be a topological space and let X ′ ⊂ X. We say that thequotient space is X/X ′ together with the topology coinduced by the projection mapX → X ′.

Definition 3.3. Let X be a topological space and let

A = A|A is a subspace of X

be a collection of subspaces of X. If the topology on X is coinduced by the inclusionmaps i : A → X, then we say that X has a topology coherent with A . This is alsoknown as the weak topology.

Historically Homotopy has its roots in the study of integration on C by Cauchy[8]. But the field is also partially rooted in the connectedness of spaces studiedby Riemann [22]. Naturally, neither of them had access to what we now know ashomotopy. To lay down the basics of algebraic topology we will begin with goingback to the roots and the deformation of paths. In Section 3.2 we will move intomore modern methods as we look at category theory, which perhaps is the definingfeature of modern algebraic topology. Chronologically, category theory is a quiterecent addition to mathematics. It was only in the mid twentieth century whenthe field was explored by Samuel Eilenberg and Saunders MacLane. The homotopyintroduction will then be ended by looking at H-groups in Section 3.3, which willbe important as we move into Section 3.5.

3.1. Deformation of Paths. Homotopy is the study of deformations of paths,i.e. curves in topological spaces. In general, homotopy theory asks the question:how do loops behave on a space? For instance, if every loop on a space X can becontinuously deformed into a point, then we say that the space is contractible. If aspace is not contractible, then the way in which it fails to be so conveys informationabout it. We will go deeper into this later. Especially into contractibility, whichis defined in Definition 3.7. But first, we will make precise what kind of spaces weare going to study.

One could work with simple topological spaces: a set X together with a topologyon it. That would be sufficient for most of the elementary topics presented hereand it is the approach of [16]. However, we will stay closer to [24] and use so called

16

topological pairs. An advantage is that the ordinary topological spaces are specialcases of topological pairs.

Definition 3.4. Let X be a topological space and let A be a subspace of X. Thenwe say that the pair (X,A) is a topological pair. The special pair (X, ∅) is commonlydenoted X rather than (X, ∅).

We can now use topological pairs to define the continuous deformation of a map.Since the interval [0, 1] will be used frequently throughout the essay, we will useI = [0, 1]. Furthermore, we use I to denote the set 0, 1.

Definition 3.5. Let (X,A) and (Y,B) be topological pairs and letX ′ be a subspaceof X. Furthermore let f0 and f1 be functions (X,A) → (Y,B) such that f0|X′ =f1|X′ . If there exists a continuous map

F : (X,A)× I → (Y,B)

such that F (x, 0) = f0(x), F (x, 1) = f1(x) for all x ∈ X, and F (x, t) = f1(x) =f2(x) for all x ∈ X ′ and all t ∈ I, then we say that f0 is homotopic to f1 relativeto X ′.

In Definition 3.5 the map F : (X,A)× I → (Y,B) is so central to the point thatit has its own name. We say that any map such as F is a homotopy relative to X ′from f0 to f1, and write F : f0 ' f1 rel X ′. The special case X ′ = ∅ would implythat nowhere on X is f1 = f0. On the other hand if we have a homotopy relativeto X for f0 and f1, then the maps f0 and f1 must be equivalent everywhere onX. In fact, for X ′′ ⊂ X ′ and a homotopy F : f0 ' f1 rel X ′, there is a homotopyG : f0 ' f1 rel X ′′. Figure 2 shows two maps that are homotopic relative to theirendpoints.

When it comes to the spaces we have the convex spaces as a special case. Re-gardless of how the space is otherwise defined: a convex space will always possess anative homotopy F . We prove this in Corollary 3.6 and it is illustrated in Figure 2.

Corollary 3.6. Let X be an arbitrary space and let Y be a convex space. For anytwo functions f0, f1 : X → Y there is a natural homotopy F (x, t) = tf1 + (1− t)f0.

Proof. Because, if f1(x) and f2(x) are in Y , then for t ∈ I, tf1 + (1− t)f0 must bein the Y . Also, F (x, 0) = f0(x) and F (x, 1) = f1(x).

f ′

f

Figure 2. The continuous deformation of f to f ′ by F (x, t) = tf1+(1−t)f0. The functions f and f ′ are homotopic relative to their endpoints.

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A path is said to be contractible if it is homotopic to a point, that is: if it iscontinuously deformable to a constant map. Surely the types of maps that arecontractible on a pair (X,A) says something about the pair. To formalize this ideawe will use identity maps a lot, so let idX will be the identity map on X. For atopological pair, (X,A), then id(X,A) = idX since A ⊂ X so that idA = idX |A.Definition 3.7. Let (X,A) be a topological pair and let c : (X,A) → (X,A) bea constant map of (X,A) to (X,A). If there is a homotopy F : (X,A) × I →(X,A) such that idX ' c under F , then we say that F is a contraction of (X,A).Furthermore we say that (X,A) is contractible.

Based on the existence of a natural homotopy on every convex space we couldexpect them to have some special relation to contractibility, which they do.

Proposition 3.8. Every convex space is contractible.

Proof. This follows easily from Corollary 3.6 since on a convex space we can al-ways use the homotopy F (x, t) = tidX + (1 − t)c. Thus a convex space must becontractible.

We have now discussed the absolute basics of homotopy theory. It would be pos-sible to continue from here and not mention categories but we would still implicitlyrely on them. Therefore, we are going to properly introduce category theory. Toput homotopy in a categorical context we will need two theorems (Theorem 3.9 andTheorem 3.11). In addition, the perhaps most useful piece of notation used in thisessay is introduced in Definition 3.10.

Theorem 3.9. Homotopy relative to X ′ is an equivalence relation in the set ofmaps from (X,A) to (Y,B).

Proof. We prove reflexivity, symmetry and transitivity.(1) Reflexivity. For f : (X,A) → (Y,B) define F : f ' f rel X by F (x, t) =

f(x).

(2) Symmetry. Given F : f0 ' f1 rel X ′, define F ′ : f1 ' f0 rel X ′ byF (x, t) = F (x, 1− t).

(3) Transitivity. Given F : f0 ' f1 rel X ′ and G : f1 ' f2 rel X ′, defineH : f0 ' f2 rel X ′ by

H(x, t) =

F (x, 2t) 0 ≤ t ≤ 1

2 ,G(x, 2t− 1) 1

2 ≤ t ≤ 1.

Note that H is continuous because its restriction to each of the closed sets X× [0, 12 ]

and X × [ 12 , 1] is continuous.

Since homotopy yields a equivalence relation on maps between two homotopypairs one would expect there to be equivalence classes as well. Not only do we haveand use such classes but they also turn out to be extremely useful. The followingdefinition and the notation will be very important in the remainder of the essay.

Definition 3.10. Let (X,A) and (Y,B) be topological pairs and let X ′ be a subsetof X. Furthermore, let F be a homotopy relative to X ′ on the set of maps between(X,A) and (Y,B). Then the homotopy classes relative to X ′ are the disjoint equiv-alence classes on the set of maps between (X,A) and (Y,B) induced by any F . Wedenote the collection of homotopy classes relative to X ′ by [X,A;Y,B]X′ .

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Let us expend a few words on notation. For a map f : (X,A) → (Y,B) we willuse [fX′ ] to denote its class in [X,A;Y,B]X′ . If X ′ = ∅ then we write [f ] insteadof [f ]∅ and [X,A;Y,B] instead of [X,A;Y,B]∅.

We now continue with the final result we shall need to move into categories: thatcomposition preserves homotopy. This is specifically needed in the very definitionof a category, see (3) in Definition 3.12.

Theorem 3.11. Composites of homotopic maps are homotopic.

Proof. Let f0, f1 : (X,A) → (Y,B) be homotopic relative to X ′ and let g0, g1 :(Y,B) → (Z,C) be homotopic relative to Y ′, where f1(X ′) ⊂ Y ′. To show thatg0, f0 g1, f1 : (X,A) → (Z,C) are homotopic relative to X ′, let F : f0 ' f1 rel X ′and G : g0 ' g1 rel Y ′. Then the composite

(X,A)× I F−→ (Y,B)g0−→ (Z,C)

is a homotopy relative to X ′ from g0f0 to g0f1 and the composite

(X,A)× I f1×idI−−−−→ (Y,B)× I G−→ (Z,C)

is a homotopy relative to f−11 (Y ′) from g0f1 to g1f1. Since X ′ ⊂ f−1

1 (Y ′), we haveshown that g0f0 ' g0f1 rel X ′ and g0f1 ' g1f1 rel X ′. The result now follows sinceby Theorem 3.9 homotopy relative to X ′ is transitive.

3.2. Category Theory. The reader is surely excited to see how categories can beused to work with homotopy, or perhaps even eager to see what a category is. Onecould say that a category is like a set together with a family of maps between sets.But this comparison is lacking as we shall see: a category is a far more interestingconcept. What category theory gives us both in connection with homotopy and ingeneral is the ability to solve problems in new settings. For instance, we are goingto solve the extension problem, see Definition 3.18, by moving the problem intothe setting of groups. This move is conducted by what is known as a functor, seeDefinition 3.15.

As a quick explanation of notation: we are going to use hom (X,Y ) to denotethe set of morphisms between the objects, as defined in the following definition.For two spaces X and Y the set hom (X,Y ) would be all functions from X to Y .

Definition 3.12. A category C consists of(1) a class of objects,

(2) for each ordered pair of objects X and Y in C a set hom (X,Y ) of mor-phisms with domain X and range Y ,

(3) for every ordered triple of objects X, Y , Z, a function associating to a pairof morphisms f : X → Y and g : Y → Z their composite

gf = g f : X → Z.

These satisfy associativity : if f : X → Y , g : Y → Z and h : Z →W , then

h(gf) = (hg)f : X →W.

As well as identity : for every object Y there is a morphism idY : Y → Y such that,if f : X → Y , then idY f = f and if h : Z →W , then hidY = h.

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Given the terminology used at the end of Definition 3.12 one could be led tobelieve that a category is some kind of group. That is certainly not true. A groupon the other hand is, however, a special kind of category with just one object withisomorphisms as morphisms on it. One should not confuse this with the categoryof abelian groups, abG which has abelian groups as objects and homomorphismsbetween them as morphisms. Another (important) category is T∗: the category ofpointed spaces. The objects of T∗ are spaces with distinguished points and mapspreserving these distinguished points as morphisms. When we use spaces in theremainder of the essay these are assumed to belong to T∗

More specifically, we say that a topological spaceX is a pointed space if, and onlyif, it has a distinguished point x0 ∈ X. Commonly a pointed space is denoted by(X,x0) and a pointed topological pair by (X,A, x0), where x0 ∈ A. Similarly, a basispoint preserving map is a map f : (X,A, x0)→ (Y,B, y0). Pointed spaces togetherwith basis point preserving maps do form a category, with spaces as objects andmaps as morphisms. In the future we will assume that all spaces are pointed andall maps are basis point preserving. However, as said previously, we will suppressthe (X,A, x0) notation in favour of the shorter (X,A).

Even tough it was just stated that categories are not groups, some of the thingscommonly associated with the latter exists on the former as well. For instance givena morphism f in a category C if there is a morphism g in C such that g f = id,we say that g is a left inverse of f . Similarly if f g = id, then g is a right inverseof f . If f has both a left and right inverse then these are equal (see Spanier [24]Lemma 1.1) and we say that g is the inverse.

Definition 3.13. Let C be a category with objects X and Y and a morphism f .If f : X → Y is such that there is an inverse g : Y → X, then we say that f is anequivalence, denoted f : X ≈ Y .

With the basics of categories in place we will now show that there is a homotopycategory, as defined in Proposition 3.14.

Proposition 3.14. There is a homotopy category with topological pairs as objectsand homotopic maps as morphisms.

Proof. All that has to be done is to verify the conditions of Definition 3.12. First,there is a class of objects: the topological pairs. Since the continuous functionssatisfies condition two then the homotopy classes must also satisfy it. Finally,from Theorem 3.11 we know that composites of homotopic maps are homotopic.Therefore we do indeed have a homotopy category and we say that it is the homotopycategory of pairs. We will use C0 to denote this category.

While categories may surely be exciting, there is not much use in a single cate-gory. Instead, the usefulness of category theory lies in the ability to move betweencategories. Because sometimes a difficult problem in one category has a quite simplesolution in another. However, this requires the transition between two categoriesto be rigorously defined.

Definition 3.15. Let C and D be categories, let X and Y be objects in C . Acovariant functor T then consists of two things. First, an object function whichassigns to each object X in C an object T (X) in D . Second, a morphism functionassigning to each morphism f of C a morphism T (f) : T (X) → T (Y ) in D suchthat:

20

(1) T (idX) = idT (X)

(2) T (gf) = T (g)T (f)

If instead T (f) : T (X) ← T (Y ) and T (gf) = T (f)T (g), we say that T is a con-travariant functor.

To use categories and functor as tools for solving problems it is immensely helpfulto have diagrams. We define these in Definition 3.16.

Definition 3.16. Let X, Y and Z be objects of a category and let f : X → Y ,g : Y → Z and h : X → Z be morphisms of this category. We can represent this ina diagram where arrows indicate direction of morphisms.

Y

X Z

gf

h

If the diagram is such that g f = h then we say that the diagram commutes.

Diagrams can also be used in reference to functors. We will, for instance, usethem in the definition of the natural transformation next. In this definition we willuse a morphism between functors.

Definition 3.17. Let C and D be categories and let F : C → D and G : C → D betwo contravariant (or covariant) functors. Furthermore, let X and Y be objects ofC and let f be a morphism on C . If there is a morphism, Φ, between F and G suchthat the following diagram exists then we say that Φ is a natural transformation.

F (X) G(X)

F (Y ) G(Y )

F (f)

ΦX

G(f)

ΦY

If Φ makes the diagram commute then Φ is said to be a natural isomorphism.

The natural transformations are important within category theory. We will,however, not make much use of them and the reader is referred to Chapter 1.4 of[25] for more details. Instead, we shall look at the use of functors (and diagrams)in stating and solving problems.

Definition 3.18. Let X and Y be topological spaces and let A be a subspace ofX such that there is a continuous function f : A→ Y . Then the extension problemis the determination of whether or not f can be extended to a continuous functionf∗ : X → Y . Diagrammatically this is equivalent to the existence of the dashed inthe following diagram.

A Y

X

f

f∗

As a brief side note: in category theory one can often take a problem or a proofand construct its dual by flipping the direction of all morphisms. For instance, theextension problem of Definition 3.18 has a dual in the lifting problem, which is thequestion of existence of a map λ making the following diagram commute.

21

A Y

Xλ

f

Furthermore, if one proves something for a covariant functor, say, then thatresult has a dual result for a contravariant functor. This will be of importancelater.

Aside from being a good example the extension problem is also important forthe main theorem of the essay. In the proof of Theorem 2.16 we will show thata function exists on a subset and then extend it to a larger set. We will now usefunctors to derive a necessary condition for the existence of f∗ in Theorem 3.19.

Theorem 3.19. Let T be a covariant functor from the category of topological spacesand continuous maps to a category C . A necessary condition that a map f : A→ Yis extendible to X is that there exist a morphism ϕ : T (X) → T (Y ) such thatϕ T (idX) = T (f).

Proof. Assume that f ′ : X → Y is an extension of f . Then f ′i = f . ThereforeT (f ′) T (i) = T (f), and T (f ′) can be taken as the morphism ϕ.

In Theorem 3.19 we do not limit ourselves in the choice of functor. We coulduse the forgetful functor, which strips the objects of some of their structure. Forinstance topological spaces could be stripped of their topological structures sendingthem to the category of sets.

As a tangential note there are functor categories where the objects are func-tors and the morphisms, which map functors to each other, are known as naturaltransformations. It is surprising, but every category can actually be embedded ina functor category by the Yoneda embedding. But this is not what we are going touse. Instead we will look at two very special functors in Example 3.20

Example 3.20. Let X, Y and Z be objects of C and let f : X → Z be a morphismof C . Then there is a covariant functor πY such that πY (X) = hom(Y,X) andπY (f) = f# : hom(Y,X) → hom(Y,Z). There is also a contravariant functor πY

such that πY (X) = hom(X,Y ) and πY (f) = f# : hom(Z, Y )→ hom(X,Y ).

The functors of Example 3.20 are the fundamental blocks of the type of algebraictopology that we are going to use: the homotopy groups. We will use these functorsto study the extension problem in the category of groups. However, a glaring issueis that there is nothing group-like about either πY nor πY , yet. In the next sectionwe will work out a remedy for this.

3.3. H-groups. To make πY take values in the category of groups and homomor-phisms we will endow the topological spaces with an additional structure. This willyield new spaces known as H-spaces and H-cospaces in honour of the work done byHeinz Hopf, see for instance [17]. With some additional assumptions on the spaceswe get H-groups and H-cogroups. In conjunction with these spaces we will definethe loop space ΩX and the suspension SX of a topological space X. While thesuspension is to us the far more important one, both will come to use in Section 3.4.Unfortunately, our treatment of what is quite an interesting subject will be briefand the reader is referred to Chapter 2 of [3] or Section 9.2 of [25] for more. For abrief historical overview please see Chapter 26 of [18].

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Technically, the topological spaces will be endowed with a multiplication. Withjust the multiplication and nothing else we have the H-space. To create a properH-group we shall need some additional assumptions on the multiplication.

Definition 3.21. Let P be a topological space and let µ : P × P → P . We saythat P is an H-space, if the constant map c : P → P is such that µ (c, idP ) : P →P × P → P and µ (idP × c) : P → P × P → P are both homotopic to idP .

We will now impose associativity on µ. To do so on a topological space ratherthan a group will require us to use diagrams. By associativity in a homotopysense we mean that the following diagram is homotopy commutative, i.e. that itcommutes in the homotopy category.

P × P × P P × P

P × P P

idP ×µ

µ×idP

µ

µ

In the definition we are only interested in the behaviour in the homotopy categoryand not on T∗. In other words it is just required that

µ (c, idP ) ' µ (idP , c),

rather than an equality. This should not be surprising since we look at πP , whichjust contains the homotopy equivalence classes.

In addition to associativity we need inverses to create a group. Again it is in thehomotopy category the work will be done. We say that a map, ψ : P → P , is ahomotopy inverse for P and µ if, and only if

µ (id, ψ) ' c ' µ (ψ, id),

where c : P → P is the constant map on P . By adding associativity and the inverseto a H-space we get an H-group. This is formalized in the following definition.

Definition 3.22. Let P be a H-space with multiplication µ : P × P → P . Ifµ (µ × id) ' µ (id × µ) and there is a homotopy inverse ψ of µ on P , then wesay that P is an H-group.

The reader interested in reading more about topological groups is referred to [2].For that reader it may be useful to know that our H-groups (as by Definition 3.22)are technically paratopological groups. That is, however, not a concern to us. Weshall instead focus on a few key properties ofH-groups. In particular, Theorem 3.23make it clear why we are interested in those groups.

Theorem 3.23. If P is an H-group, then πP is a contravariant functor from thecategory of topological spaces and continuous functions to the category of groups andhomomorphisms. If P is an abelian H-group, then πP takes values in the categoryof abelian groups.

Proof. See the discussion preceding Theorem 1.5.1 of [24].

The perhaps most useful example of an H-group is known as the loop space.If (Y, y0) is a space with distinguished point y0, then the loop space, ΩY , of Y isthe set of continuous maps ω : (I, I) → (Y, y0). That is, ΩY are all loops in Ybased at y0. We shall return to Ω later on, both when discussing suspensions and

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in Section 4 while addressing obstructions. Now we will instead move to a perhapseven more interesting space.

As said earlier: many things in category theory have useful duals. In this case,there is a dual construction to H-groups, which dualizes Theorem 3.23. To arriveat that result we need to define a new kind of multiplication on topological spaces.Let Q be a topological space then we want a multiplication akin to

ν : Q→ Q×Q.However, since Q × Q is not a pointed space such a multiplication will not work.Instead we shall introduce the wedge, which produces a pointed product of spaces.

Definition 3.24. Let X and Y be pointed spaces with basis points x0 and y0,respectively. Then the wedge X ∨ Y is defined as the quotient space of the disjointunion of X and Y with x0 identified with y0. Specifically

X ∨ Y = (X t Y )/(x0 ∼ y0).

Definition 3.25. Let Q be a pointed topological space and let ν : Q→ Q ∨Q bea continuous multiplication. Then we say that πQ is an H cogroup if for a constantmap c : Q→ Q both ν (c, idQ) : Q→ Q∨Q→ Q and ν (idQ, c) : Q→ Q∨Q→ Qare homotopic to idQ. And also the square

Q Q ∨Q

Q ∨Q Q ∨Q ∨Q

ν

ν

idQ×ν

ν∨idQ

commutes homotopically.

With the H-cogroups we have a dual result to that of Theorem 3.23.

Theorem 3.26. If Q is an H cogroup, πQ is a covariant functor from the homotopycategory of pointed spaces with values in the category of groups and homomorphisms.If Q is an abelian group then this functor takes values in the category of abeliangroups and homomorphisms.

Proof. Please see the discussion preceding Theorem 1.5.5 of [24].

Similar to how loop spaces are the quintessential example of an H-group we havesuspensions as a prominent example of an H-cogroup.

Definition 3.27. Let X be a pointed space with x0 as basis point. Then thereduced suspension of X, SX is the quotient space

(X × I)/(X × 0)(∪x0 × I) ∪ (X × 1)

Thus, the suspension is an extrusion of the space X with the top and bottomcollapsed to a point. Given this, our definition may seem strange. One wouldperhaps expect the suspension to be (X × I)/(X × 0 ∪ X × 1) and this isactually the (non-reduced) suspension. While the latter suspension could makemore sense geometrically, it does not preserve the basis point of X.

We can now apply what we have done so far to the extension problem. Thisis done on some very specific spaces: the n + 1 dimensional ball, En+1, and then-dimensional sphere, Sn. The use of S both here and in Definition 3.27 is not amistake. One can view the n-dimensional sphere as n successive suspensions of theunit interval. We will go deeper into this in Definition 3.41. For now we will look

24

at the extension of a map f : Sn → Y to a map f ′ : En+1, which is a special caseof what we want to achieve. This is Theorem 1.3.12 of [24].

Example 3.28. Let p0 be any point of Sn and let f : Sn → Y , then f can becontinuously extended to En+1 if and only if f is null homotopic, i.e. homotopicto the constant map. We first show that being null homotopic implies that f canbe extended to En+1. Let F : f ' c, where c is the constant map of Sn to y0 ∈ Y .Define an extension f ′ of f to En+1 by

f ′(x) =

y0 0 ≤ ‖x‖ ≤ 1

2

F (x/‖x‖, 2− 2‖x‖) 12 ≤ ‖x‖ ≤ 1

Since F (x, 1) = y0 for all x ∈ Sn, the map f ′ is well-defined. Furthermore, f ′ iscontinuous because its restrictions to each of the closed sets x ∈ En+1|0 ≤ ‖x‖ ≤12 and x ∈ E

n+1| 12 ≤ ‖x‖ ≤ 1 are continuous. Since F (x, 0) = f(x) for x ∈ Sn,f |Sn = f and f ′ is a continuous extension of f to En+1.

It now remains to show that if the map f can be extended to En, then f ' c. Iff has the continuous extension f ′ : En+1 → Y , define F : Sn × I → Y by

F (x, t) = f ′((1− t)x+ tp0).

Then F (x, 0) = f ′(x) = f(x) and F (x, 1) = f ′(p0) for x ∈ Sn. Since F (p0, t) =f ′(p0) for t ∈ I, F is a homotopy relative to p0 from f to the constant map tof ′(p0). This shows that f is null homotopic relative to p0, which in turn impliesthat f is null homotopic.

3.4. Exact Sequences of Sets of Homotopy Classes. We will now get deeperinto algebraic topology. While the goal is to be thorough, the amount of materialto be covered is vast. The layout is structured in the following way. The presentsection looks into the algebraic property of exactness, with the goal of creating anexact sequence of spaces. This is then combined with H-spaces in Section 3.5 todefine the higher homotopy groups. On its own an exact sequence is easy to define,we do this in definitions 3.31 and 3.32. However, more work is necessary beforeexact sequences can be used on topological spaces.

In the category of groups and homomorphisms we have the concepts of kernelsand images. For two groups A and B and a morphism f : A→ B, we say that thekernel of f are precisely the elements of A which maps to 0 ∈ B under f and wedenote those by ker f . In other words ker f = f−10. The image are the elementsof B given by f(A). We will denote the image of f by Im f . The most importantkernel and image, to us, is covered in Example 3.29.

Example 3.29. Let (X,A), (X ′, A′) and (Y,B) be topological pairs, and let 0 ∈[X,A;Y,B] be the class of null homotopic maps (X,A) → (Y,B). For a map,f : (X ′, A′)→ (X,A), we say that the kernel of the induced map f#, ker f#, is thehomotopy classes of maps g : (X ′, A′)→ (Y,B) such that f#g = 0.

Much in the same spirit we can define the image in a homotopy sense.

Definition 3.30. Let (X,A), (X ′, A′) and (Y,B) be topological pairs and let fbe a map (X ′, A′) → (X,A). The image of the induced map f#, Im f#, is thehomotopy classes of compositions f# g, where [g] ∈ [X ′, A′;Y,B].

25

In abstract algebra we say that a series of the form

0→ Af−→ B

g−→ C → 0,

where f and g are morphisms and A, B and C are groups, is a short sequence.Furthermore, a short sequence is said to be exact if ker g = Im f . We will nowuse the image and kernel of induced maps to create a corresponding definition fortopological spaces.

Definition 3.31. Let (X ′, A′), (X,A), (X ′′, A′′) and (Y,B) be topological pairs.A short exact sequence of (pairs and maps) is a three-term sequence of pairs andmaps is a sequence

· · · → (X ′, A′)f−→ (X,A)

g−→ (X ′′, A′′)→ . . .

(possibly terminating at either end) such that for any pair (Y,B) the associatedsequence of pointed sets

· · · → [Y,B;X ′, A′]f#−−→ [Y,B;X,A]

g#−−→ [Y,B;X ′′, A′′]→ . . .

is exact.A coexact three-term sequence of pairs and maps is a sequence

· · · → (X ′, A′)f−→ (X,A)

g−→ (X ′′, A′′)→ . . .

such that for any pair (Y,B) the associated sequence of pointed sets

· · · → [X ′′, A′′;Y,B]g#−−→ [X,A;Y,B]

f#

−−→ [X ′, A′;Y,B]→ . . .

is exact.

However, short exact sequences are of limited interest. In order to prove anythinginteresting the sequences has to be longer than three spaces.

Definition 3.32. Let there be a sequence

(2) · · · → (Xn+1, An+1)fn−→ (Xn, An)

fn−1−−−→ (Xn−1, An−1)→ · · ·of topological pairs and functions, possibly terminating with a 0 at both, one orneither end. If ker fk ⊂ Im fk+1 for k + 1 ≤ n, then we say that the sequence isexact at k. Should the sequence be exact at every n, then we say that it is a longexact sequence.

The previous definition connects spaces with exactness in the homotopy category.We will now work to rewrite the sequence

(X ′, A′)f−→ (X,A)

g−→ (X ′′, A′′)

in terms of H-cogroups. The first step is to introduce the reduced cone, whichis the lower half of a suspension. From this the mapping cone is created, and inTheorem 3.35 the coexactness of a sequence including the mapping cone is proven.

Definition 3.33. Let X be a pointed space with basis point x0, then the absolutereduced cone is the quotient space

CX = X × I/(X × 0 ∪ I × x0).Similarly for a topological pair (X,A) the relative reduced cone is the quotient spaceC(X,A) = (CX,CA).

26

Just as the case were with the suspension, the reduced cone is a construction topreserve the basis point. Indeed, to see that (X × I)/((X × 0) ∪ (I × x0)) is astructure that preserves the basis point consider the following. Let X be the diskE2, then X × I is a regular cylinder. Identifying the entire top of the cylinder witha point, i.e. X × I/(X × 0), yields a regular cone. Comparing this to the reducedcone it is clear that the latter is a regular cone with the line above x0 collapsed toa single point.

Definition 3.34. Let X ′ and X be pointed spaces with basis points x′0 and x0,respectively. Furthermore, let f be a map between X ′ and X, then the mappingcone of f , Cf is defined to be the space CX ′ ∨X with the identification (x′, 1) 'f(x′) for all x′ ∈ X ′.

We will now show that there is a coexact sequence (X ′, A′)f−→ (X,A) →

(Cf ′ , Cf ′′), with f ′ and f ′′ defined later. By definition this is true if

[Cf ′ , Cf ′′ ;Y,B]→ [X,A, Y,B]→ [X ′, A′;Y,B]

is exact. We utilise this in the next theorem. Later on, the reverse implication,that coexactness in C implies exactness in the homotopy category, is used to proveTheorem 3.48. But at that point the sequence will have been heavily modified,while keeping its coexactness.

Theorem 3.35. For any map f : (X ′, A′)→ (X,A) the sequence

(X ′, A′)f−→ (X,A)

i−→ (Cf ′ , Cf ′′)

is coexact.

Proof. Let (Y,B) be arbitrary (with B not necessarily closed in Y ) and considerthe sequence

[Cf ′ , Cf ′′ ;Y,B]i#−→ [X,A;Y,B]

f#

−−→ [X ′, A′;Y,B].

We now show that Im i# ⊂ ker f#. The composite i f : (X ′, A′) → (Cf ′ , Cf ′)equals the composite

(X ′, A′) → C(X ′, A′) → C(X ′, A′) ∨ (X,A)k−→ (Cf ′ , Cf ′),

where k is the canonical map to the quotient. However, the inclusion map (X ′, A′) →C(X ′, A′) is null homotopic (see Lemma 7.1.1 of [24]). Therefore, i f is null ho-motopic, and so Im(f# i#) = 0, proving that Im i# ⊂ ker f#.

Assume that g : (X,A) → (Y,B) is such that f#[g] = 0 (that is, g f isnull homotopic). There is a map G′ : C(X ′, A′) ∨ (X,A) → (Y,B) such thatG′|C(X′,A′) = G and G′|(X,A) = g (see Lemma 7.1.1 of [24]). Since

G′[x′, 1] = G[x′, 1] = g(f(x′)) = G′(f(x′)) for all x′ ∈ X ′,there is a map h : (Cf ′ , Cf ′) → (Y,B) such that G′ = h k. Then h|(X,A) = g,showing that h i = g or [g] = i#[h]. Therefore ker f# ⊂ Im i#.

Theorem 3.35 is quite interesting as it places no condition on the maps andspaces that goes into it. Hence, we can take a map j : (Cf ′ , Cf ′′)→ (Ci′ , Ci′′) andTheorem 3.35 then assures that the sequence

(Cf ′ , Cf ′′)j−→ (Ci′ , Ci′′)

k−→ (Cj′ , Cj′′)

27

CX ′

f

XCf

CX

i

CX ′

X

Figure 3. The left figure depicts the composition of Cf with the iden-tification of CX ′ and X by f . Similarly, the centre figure shows thesame relation between CX and Cf by i for Ci. From this it is easy tosee that Ci/CX = Cf/X, which is illustrated in the right figure.

is coexact. By extension, the following sequence must be coexact as well.

(3) (X ′, A′)f−→ (X,A)

i−→ (Cf ′ , Cf ′′)j−→ (Ci′ , Ci′′)

k−→ (Cj′ , Cj′′).

Equation 3 is a fundamental part in the definition of the homotopy groups (inDefinition 3.43), especially in showing that the homotopy sequence of pairs is exact.In the end we like the corresponding homotopy sequence to be groups. As we know,πQ takes on values in the category of groups whenever Q is an H-cogroup, whichthe suspension is. We would therefore like to transform the cones into suspensions.

The following part will show how to transform a mapping cone into a suspensionby collapsing its base. To do this we will use a collapsing map, which is a mapk : (Y,B)→ (Y,B)/Y ′ for a topological pair (Y,B) and a subspace Y ′ of Y . First,we will show that the collapsing map k is a homotopy equivalence. In Figure 3 areduced mapping and collapsed mapping cones can be seen.

To help the reader navigate the remainder of the section and all the way toTheorem 4.11, a schematic of the connections within the theory is included inFigure 4.

Lemma 3.36. Let (Y,B) be a pair and let Y ′ be a closed subset of Y . Assume thatthere is a homotopy H : (Y,B)× I → (Y ′, B) such that:

(1) H(y, 0) = y for y ∈ Y ,

(2) H(Y ′ × I) ⊂ Y ′,

(3) H(Y ′ × 1) = y0.Then the collapsing map, k : (Y,B)→ (Y,B)/Y ′, is a homotopy equivalence.

Proof. Define a map, f : (Y,B)/Y ′ → (Y,B), by the equation

f(k(y)) = H(y, 1), y ∈ Y,which is well defined, because H(Y ′ × 1) = y0. We show that f is a homotopyinverse of k. By definition of f , we see that H is a homotopy from 1(Y,B) to f k.On the other hand, because H(Y ′ × 1) ⊂ Y ′, there is a homotopy

H ′ : (Y,B)/Y ′ → (Y,B)/Y ′,

such that H ′(k(y), t) = k(H(y, t)) for y ∈ Y and t ∈ I. Then,k(f(k(y))) = k(H(y, 1)) = H ′(k(y), 1), y ∈ Y.

28

4.11p. 49

3.74p. 43

3.73p. 43

3.66p. 40

3.50p. 35

3.48p. 34

3.42p. 31

4.10p. 48

4.9p. 48

3.39p. 30

3.38p. 29

3.36p. 27

3.72p. 42

3.69p. 41

3.65p. 40

3.63p. 39

3.62p. 39

3.64p. 39

3.46p. 34

3.49p. 35

3.44p. 33

3.40p. 31

Figure 4. A schematic of the interdependencies between the lemmas,corollaries and theorems necessary to prove Theorem 4.11.

Therefore, H ′ is a homotopy from the identity map of (Y,B)/Y ′ to k f , and f isa homotopy inverse of k. Since k has a homotopy inverse, it must be a homotopyequivalence.

We will now extend the result of Lemma 3.36 to a map k : (Ci′ , Ci′′) →(Ci′ , Ci′′)/CX. This is done in two steps: first we show the existence of a wellbehaved homotopy F : C(X,A)× I → (Y,B), and then use it to find the extendedk.

29

Lemma 3.37. Given a map f : C(X,A)→ (Y,B) and a homotopy G : (X,A)×I →(Y,B) of f |(X,A). Then, there is a homotopy F : C(X,A) × I → (Y,B) of f suchthat F |(X,A) × I = G.

Proof. An explicit formula for F is

F ([x, t], t′) =

f [x, t(1 + t′)] t(1 + t′) ≤ 1,G(x, t(1 + t′)− 1) 1 ≤ t(1 + t′).

Theorem 3.38. Let f : (X ′, A′) → (X,A) be a map and let i : (X,A) →(Cf ′ , Cf ′′). Then, CX ⊂ Ci′ , (Ci′ , Ci′′)/CX = (Cf ′ , Cf ′′)/X, and the collapsingmap

k : (Ci′ , Ci′′)→ (Ci′ , Ci′′)/CX

is a homotopy equivalence.

Proof. Ci′ is the quotient space of CX ′ ∨ CX with the identifications [x′, 1] =[f(x′), 1] for all x′ ∈ X ′, hence CX ⊂ Ci′ . It follows that

Ci′/CX = Cf ′/(Cf ′ ∩ CX) = Cf ′/X.

Similarly, Ci′′/CA = Cf ′′/A, and because Ci′′ ∩ CX = CA,

(Ci′ , Ci′′)/CX = (Cf ′ , Cf ′′)/X.

This proves the first two parts of the corollary. An illustration of this can be seenin Figure 3 where the absolute space case is depicted.

To prove the remaining statement: that k : (Ci′ , Ci′′) → (Ci′ , Ci′′)/CX is ahomotopy equivalence; we make the following definitions. Let F : C(X,A) ×I → C(X,A) be the contraction defined by F ([x, t], t′) = [x, (1 − t′)t], and letg : C(X ′, A′)→ (Ci′ , Ci′′) be the composite

C(X ′, A′) ⊂ C(X ′, A′) ∨ C(X,A)→ (Ci′ , Ci′′),

where the second map is the canonical map. The composite

(X ′, A′)× I f×1−−−→ (X,A)× I ⊂ C(X,A)× I F−→ C(X,A) ⊂ (Ci′ , Ci′′)

is a homotopy G : (X ′, A′)×I → (Cf ′ , Cf ′′) such that G(x′, 0) = [f(x′), 1] = g[x′, 1].By Lemma 3.37, there is a homotopy F ′ : C(X ′, A′) × I → (Ci′ , Ci′′) such thatF ′|(X ′, A′)× I = G and F ′([x′, t], 0) = g[x′, t]). Then, there is a homotopy

H : (Ci′ , Ci′′)× I → (Ci′ , Ci′′),

defined by the equationsH([x′, t]t′) = F ′([x′, t], t′) x′ ∈ X ′; t, t′ ∈ I,

H([x, t], t′) = F ([x, t], t′) x ∈ X; t, t′ ∈ I,which is well-defined since F ′([x′, 1], t′) = G(x′, t′) = F ([f(x′), 1], t′). Then Hsatisfies (1), (2) and (3) of Lemma 3.36 with (Y,B) = (Ci′ , Ci′′) and Y ′ = CX.Therefore k : (Ci′ , Ci′′)→ (Ci′ , Ci′′)/CX is a homotopy equivalence.

Consulting the right figure of Figure 3, it is obvious by inspection that Ci/CXand Cf/X are equivalent but also suspensions. Formally, as Cf = CX ′ ∨X, withthe identification x′ ∈ X ′ × 0 = f(x′), it follows that Cf/X = SX ′. By the samereasoning Ci/CX = SX ′. Finally, since A′ ⊂ X ′ the two previous observationshold for A′ as well. This shows that (Ci′ , Ci′′)/CX = (Cf ′ , Cf ′′)/X = S(X ′, A′) in

30

the homotopy category. Another implication is that k : (Ci′ , Ci′′)→ (Ci′ , Ci′′)/CXis a homotopy equivalence between Ci′ , Ci′′ and S(X ′, A′).

All of the preceding discussion is indicative of the way forward. The natural pathis to replace both of the latter mapping cones of Equation 3 with their homotopyequivalent suspensions. This then yields a sequence of H-cogroups on which we canwork. Formally, we have the following lemma.

Lemma 3.39. For any map f : (X ′, A′)→ (X,A) the sequence

(X ′, A′)f−→ (X,A)

i−→ (Cf ′ , Cf ′′)k−→ S(X ′, A′)

Sf−−→ S(X,A)

is coexact.

Proof. By Corollary 3.38, there is a homotopy equivalence

k′ : Ci′ , Ci′′ → (Cf ′ , Cf ′′)/X = S(X ′, A′).

Defining k = k′ j, for j : (Cf ′ , Cf ′′) → (Ci′ , Ci′′), yields a collapsing map

(Cf ′ , Cf ′′)j−→ (Ci′ , Ci′′)

k′−→ S(X ′, A′). Also by Corollary 3.38, there is a homo-topy equivalence

k′′ : (Cj′ , Cj′′)→ (Cj′ , Cj′′)/CCi′ = (Ci′ , Ci′′)/Cf ′ = S(X,A).

Defining the collapsing map (Ci′ , Ci′′) → (Ci′ , Ci′′)/Cf ′ as the composition l k′ :

(Ci′ , Ci′′)l−→ (Cj′ , Cj′′)

k′−→ S(X ′, A′) gives the following diagram.

(Ci′ , Ci′′)

S(X ′, A′) S(X,A)

kh′

g

Here g : S(X ′, A′) → S(X,A) is given by g([x′, t]) = [f(x′), 1 − t]. Defining ahomotopy H : (Ci′ , Ci′′)× I → S(X,A) by

H([x′, t], t′) = [f(x′), 1− tt′] x′ ∈ X ′; t, t′ ∈ I,

H([x, t], t′) = [x, (l − t′)t] x ∈ X; t, t′ ∈ I,

shows that k ' g k′. Therefore there is a homotopy-commutative diagram,

(Cf ′ , Cf ′′) (Ci′ , Ci′′) (Cj′ , Cj′′)

S(X ′, A′) S(X,A)

j

kk′

l

k′′

g

in which k′ and k′′ are homotopy equivalences. Since Equation 3 is coexact andboth k′ and k′′ are homotopy equivalences, the sequence

(X ′, A′)f−→ (X,A)

i−→ (Cf ′ , Cf ′′)k−→ S(X ′, A′)

Sf−−→ S(X,A)

must be coexact as well.

Now, Lemma 3.39 hints at the possibility to, from any exact sequence, createexact sequences of suspensions. This possibility will give us a way to create anarbitrary number of sequences. In the proof we shall use the fact that exactnessof a sequence in (X,A) is a property that holds against any (Y,B). It is thereforefeasible to try to use the dual property of Ω and S.

31

Lemma 3.40. If the sequence

(X ′, A′)f−→ (X,A)

g−→ (X ′′, A′′)

is coexact, so is the suspended sequence

S(X ′, A′)Sf−−→ S(X,A)

Sg−−→ S(X ′′, A′′).

Proof. For any pair (Y,B) let Ω(Y,B) = (ΩY,ΩB). By Theorem 2.8 of [24], thereis a commutative diagram in which the vertical maps are equivalences of pointedsets.

[S(X ′′, A′′);Y,B] [S(X,A);Y,B] [S(X ′, A′); y,B]

[X ′′, A′′; Ω(Y,B)] [X,A; Ω(Y,B)] [X ′, A′; Ω(Y,B)]

(Sg)# (Sf)#

g# f#

Hence, Im(Sg)# = ker(Sf)# in the top sequence is equivalent to Im g# = ker f#

in the bottom sequence.

Therefore, if we suspend (X ′, A′)f−→ (X,A)

i−→ (Cf ′ , Cf ′′), we get a coexactsequence

S(X ′, A′)Sf−−→ S(X,A)

Si−→ S(Cf ′ , Cf ′′).

However, from Lemma 3.39, we know that

(X ′, A′)f−→ (X,A)

i−→ (Cf ′ , Cf ′′)k−→ S(X ′, A′)

Sf−−→ S(X,A)

is coexact. By an argument similar to that preceeding Equation (3), we can guessthat, since Im k = kerSf , the two sequences can be joined into a longer coexactsequence. Continuing this process for more spaces would then result in an arbi-trarily long coexact sequence. This is proven formally in the following theorem,Theorem 3.42. But as we are going to make use of iterated suspensions, i.e. supen-sions of suspensions of suspensions and so on, we make the following definition tosimlify the notation.

Definition 3.41. If (X,A) is a topological pair, then the n-th suspension of (X,A),Sn(X,A), is recursively defined by

Sn(X,A) = S(Sn−1(X,A)) n ≥ 1

withS0(X,A) = (X,A).

With this notation and the previous lemmas, the next theorem is trivial.

Theorem 3.42. For any map f : (X ′, A′)→ (X,A) the sequence

(4) (X ′, A′)f−→ (X,A)

i−→ · · · Snf−−−→ Sn(X,A)

Sni−−→ Sn(Cf ′ , Cf ′′)Snk−−→ · · ·

· · · Snk−−→ Sn+1(X ′, A′)

Sn+1f−−−−→ · · ·

is coexact.

32

(X,A)

π1 (X

,A)

π2(X,A)

π 3(X,A

)

Figure 5. Homotopy functors built as the equivalences of maps fromn-dimensional spheres to (X,A).

Proof. From Lemma 3.39 and Lemma 3.40, for n ≥ 0 there is a coexact sequence

(5) Sn(X ′, A′)Snf−−−→ Sn(X,A)

Sni−−→ Sn(Cf ′ , Cf ′′)Snk−−→ Sn+1 · · ·

· · · Snk−−→ (X ′, A′)

Sn+1f−−−−→ Sn+1(X,A).

Since every three-term subsequence of the sequence in the theorem is contained inone of these five-term coexact sequences, the result follows.

3.5. Higher Homotopy Groups. This section will be all about two things. Wefirst define the relative homotopy groups and using the results of the previoussection we show in Theorem 3.48 that the homotopy sequence of pairs is exact.This will be important for the next topic of the section: n-connectedness. Ourbig result of this subsection is Corollary 3.50, which shows that (En, Sn−1), whereEn is the n-dimensional ball, is n-connected. This last result holds an importantapplication to CW-complexes, which are treated in the next section.

Definition 3.43. Let (X,A) be a pair with base point x0 ∈ A. For n ≥ 1 then-th relative homotopy group, denoted by πn(X,A, x0), is defined to be equal to[Sn−1(I, I);X,A]. The absolute homotopy group, πn(X,A, x0) is similarly definedas [Sn−1(I);X].

In other words, the n-th homotopy group of (X,A), πn(X,A, x0) is the equiv-alence class of maps from the nth suspension of (I, I) to the pair (X,A). Thesuspension Sn(I, I) is the topological pair of the n-dimensional ball and the n-dimensional sphere as a subspace. See Figure 5 for a rough depiction.

Since πn(X,A) has a group structure it must therefore have a trivial element.Following the standard practice of abstract algebra we shall denote this elementby 0. In the case of the absolute homotopy group, πn(X), a map α : Sn →X represents the trivial element of πn(X) for n ≥ 1 if, and only if, α can becontinuously extended to En+1. We are going to prove this for the topologicalspace (X,A) in Theorem 3.44. In order to do that a minor, but technical, rewriting

33

of the suspension map must be undertaken. The point is to show that Sn(I, I) ishomeomorphic to the relative hypercube: (In, In, z0).

First, Sn−1(I, I) is homeomorphic to (I× In−1, I× In−1)/(I× In−1 ∪ 0× In−1).Because of this homeomorphism it is true that the homotopy classes of

(In, In, I × In−1)/(I × In−1 ∪ 0× In−1)→ (X,A, x0)

are isomorphic to πn(X,A, x0) for all n ≥ 1. Furthermore, for z ∈ (I× In−1∪0×In−1), we have

(In, In, z) ⊂ (I × In−1 ∪ 0 × In−1).

If we let z0 =∏n

0 then since (I × In−1 ∪ 0 × In−1) is contractible we have

(In, In, z0) ⊃ (I × In−1 ∪ 0 × In−1).

Hence,(In, In, z0) = (I × In−1 ∪ 0 × In−1).

Therefore the suspension Sn(I, I) is homeomorphic to the set (In, In, z0).It is known that the n-cube In is homeomorphic to the ball En and that the

border of the n-cube, In, is homeomorphic to the sphere Sn. Assuming that thehomeomorphism sends z0 to p0 ∈ Sn. Then for n ≥ 1, πn(X,A, x0) is in a one toone correspondence with the equivalence classes of

(En, Sn, p0)→ (In, In, z0).

It is by using (En, Sn−1, p0), in place of the suspension, we will approach theclassification of maps representing 0 in πn. The proof itself is fairly straightforward,involving only the construction and checking of homotopies.

Theorem 3.44. Given a map α : (En, Sn−1, p0) → (X,A, x0), then [α] = 0 inπn(X,A, x0) if, and only if, α is homeotopic relative to Sn−1 to some map of Ento A.

Proof. Assume [α] = 0 in πn(X,A, x0). Then there is a homotopy

H : (En, Sn−1, p0)× I → (X,A, x0)

from α to the constant map En → x0. The issue here is to find a homotopy relativeto Sn−1. However, a homotopy H ′ relative to Sn−1 from α to some map En to Acan be constructed from H as follows

H ′(z, t) =

H(

z1−t/2

), t 0 ≤ ‖z‖ ≤ 1− t

2 ,

H(

z‖z‖ , 2− 2‖z‖

)1− t

2 ≤ ‖z‖ ≤ 1.

Conversely, if α is homotopic relative to Sn−1 to some map α′, such thatα′(En) ⊂ A, then [α] = [α′] in πn(X,A, x0), and it suffices to show that [α′] = 0in πn(X,A, x0). A homotopy H : (En, Sn−1, p0) × I → (X,A, x0) from α′ to theconstant map c : En → x0 is defined by

H(z, t) = α′((1− t)z + tp0).

34

Consider the case where every map α : (En, Sn−1, p0) → (X,A) satisfies thecondition of Theorem 3.44. This would mean that every map is the zero elementof πn(X,A). As an example consider the case when n = 1. Then we would look athomotopies from the bounded interval relative to the endpoints. The requirementis then that every map α(I, I, p0) → (X,A, x0) is homotopic relative to I to somemap I → A. In other words it has to be possible to connect each pair of points in(X,A) by a path. This property is what we will call n-connectedness. It is definedas follows.

Definition 3.45. A pair (X,A) is said to be n-connected for n ≥ 0 if, for 0 ≤ k ≤ n,every map α : (Ek, Sk−1) → (X,A) is homotopic relative to Sk−1 to some map ofEk to A.

By Theorem 3.44 and Definition 3.45, an n-connected space has trivial homotopygroups up to πn. Conversely, a space with trivial homotopy groups to dimension nis obviously n-connected. For future reference we state this as a corollary.

Corollary 3.46. A pair (X,A) is n-connected for n ≥ 0 if and only if every pathcomponent of X intersects A, and for every point a ∈ A and every 1 ≤ k ≤ n,πk(X,A, a) = 0.

As we proceed, we make use of a map ∂ : π(X,A, x0)→ πn−1(A, x0), defined by

∂[α] = [α|Sn−1(I)],

for α : Sn−1(I, I)→ (X,A).

Definition 3.47. Let (X,A) be a pair of pointed spaces with inclusion maps i :A → X and j : (X, x0) → (X,A). Then the homotopy sequence of (X,A) is thesequence of pointed sets

· · · → πn+1(X,A)∂−→ πn(A)

i#−→ πn(X)j#−−→ πn(X,A)

∂−→ · · · i#−→ π0(X).

Recall that in Theorem 3.42 we proved that

(X ′, A′)f−→ (X,A)

i−→ · · · Snf−−−→ Sn(X,A)

Sni−−→ Sn(Cf ′ , Cf ′′)Snk−−→ · · ·

· · · Snk−−→ Sn+1(X ′, A′)

Sn+1f−−−−→ · · ·

is coexact. By using this, one can show that the homotopy sequence is exact.This exactness will play an important part when we are to show that (En, Sn) isn-connected.

Theorem 3.48. The homotopy sequence of a pair is exact.

Proof. Let f : (I , 0) ⊂ (I , I) and let f ′ : I ⊂ I and f ′′ : 0 → I. By Theo-rem 3.42, there is a coexact sequence

(I , 0) f−→ (I , I)i−→ (Cf ′ , Cf ′′)

k−→ S(I , 0) Sf−−→ S(I , I)→ · · ·

Let g : (Cf ′ , Cf ′′) → (I, I) be the homeomorphism defined by g([0, t]) = 0 andg([1, t]) = t. Then the composite g i is the inclusion map i′ : (I , I) → (I, I), andthe composite k g−1 equals the composite

(I, I)k′−→ (I/I),

h−→ (S(I), 0)

35

where k′ is the collapsing map and h is the homeomorphism used in identifyingπn(X, x0) with πn(X). Therefore, there is a coexact sequence

(I , 0) f−→ (I , I)i′−→ (I, I)

hh′−−−→ S(I , 0) Sf−−→ · · · ,This yields an exact sequence

· · · → πn+1(X,A)(Sni′)#−−−−−→ πn(A)

(Snf)#−−−−−→ πn(X)(Sn−1(hk′))#−−−−−−−−−→ · · ·

· · · (Sn−1(hk′))#−−−−−−−−−→ πn(X,A)→ · · · → π0(X).

The proof is completed by the trivial verification that

(Sn, i′I#) = ∂, (Snf)# = i#, and (Sn−1(h k′))# = j#.

Now, since the homotopy sequence of (X,A) is exact it follows by definitionthat Im ∂ = ker i#. Should it be the case that πk(A) = 0, for some k, thenker ∂ = πk+1(X,A). This follows since the kernel of ∂ : πk+1(X,A) → πk(A)must be the entire space πk+1(X,A), if ∂ maps everything to 0. Furthermore, ifπk+1(X) = 0, then Im j# must necessarily be 0. Since if this weren’t the case,then j could not possibly be an inclusion. Given the preceding reasoning it isclear that if both πk+1(X) and πk(A) are trivial, then πk+1(X,A) = 0. Referringback to Corollary 3.46 one can see a connection between what was just discussedand n-connectedness. It is this connection that provides a method to prove theconnectedness of (En+1, Sn). Of course, given that Sn is (n − 1)-connected forn ≥ 1.

Theorem 3.49. Sn is (n− 1)-connected for n ≥ 1.

Proof. For a proof please see Theorem 3.4.1 of [24].

Corollary 3.50. For n ≥ 0, (En+1, Sn) is n-connected.

Proof. For n ≥ 0, En+1 is path connected and Sn is nonempty; therefore everypath component of En+1 meets Sn. If x ∈ Sn, then πk(En+1, x) = 0 for 0 ≤ k,because En+1 is contractible. By Theorem 3.49, πk(Sn, x) = 0 if 0 ≤ k < n. Itfollows from Theorem 3.48 that πk(En+1, Sn, x) = 0 for 1 ≤ k ≤ n. The resultfollows from Corollary 3.46.

Thus we have a pair that we know is n-connected. Now, the idea is to use thispair to construct a more general space which is n-connected. The most appealingway to do is that is to intruduce the CW-complex.

3.6. CW-Complexes. In this section we introduce what J. H. C. Whitehead in[28] named “closure finite complexes with weak topology” or CW-complexes forshort. These spaces satisfy a particular kind of n-connectedness, which we addressin Lemma 3.66. The remainder of the section will be quite technical. While this re-sult is important in our context it is still only scratching the surface of the theory ofCW-complexes. The reader who wants to explore the versatility of CW-complexesmay read Section 10.5 and Section 10.6 of [20], for instance. A nice exposition onthe topological aspects of the closure finite topology can be found in [29, p.89].

Before working with CW-complexes, it is necessary to talk about their construc-tion. A CW-complex is “built” by adding together topological spaces of successively

36

higher dimensions. These spaces are known as n-cells. Formally we have the fol-lowing definition.

Definition 3.51. Let X be a topological space. If X is homeomorphic to then-dimensional ball, En, we say that X is an n-cell and denote it by en.

Homomorphically there is just one of each n-dimensional cell. For instance, apoint is the 0-cell, e0, and the unit interval I is the 1-cell, e1. If we attach bothends of the 1-cell to the 0-cell then we get the pointed circle S1. In a similar waywe could use cells to construct many of the topological spaces that we normallyuse. For instance S2 can be constructed by attaching the edge of a 2-cell to a 0-cell.However, the manner in which cells are attached is important. We therefore makethe following definition regarding the attachment process.

Definition 3.52. Let A be a topological space and let enj be a sequence of n-cells.Then we say that X is obtained from A by adjoining n-cells if:

(1) for each j, enj ⊂ X;

(2) if enj = enj ∩A, then for j 6= i, enj − enj and eni − eni are disjoint;

(3) X has a topology coherent with A, enj and X = A ∪ ujenj ;

(4) for each j there is a map

fj : (En, Sn−1)→ (enj , enj )

such that fj(En) = enj , fj maps En−Sn−1 homeomorphically onto enj − enj ,and enj has the topology coinduced by fj and the inclusion map enj → enj .

We say that a map such as in (4) is a characteristic map.

This process of constructing X from A will constitute our definition of a CW-complex. While this will suffice it does not clarify either the “weak topology” orthe “closure finiteness”. Unfortunately, we shall not spend time on those aspects ofCW-complexes. After the definition we shall instead work with another propertyof the complexes.

Definition 3.53. A space X obtained from A by adjoining n-cells is a relativeCW-complex. Since it is a topological pair we will often just write it as (X,A).

This process of creating spaces is remarkably powerful. Some of its power stemsfrom the possibility to continuously retract the complex back to A. In fact, all theresults, which we are to use, are consequences of this. However, there are a lot tobe said about CW-complexes and an interested reader could consult [12] or [1]. Wewill move on to retractions instead.

Definition 3.54. Let A be a subspace of the topological space X with the inclusionmap i : A → X. If there is a continuous map r : X → A such that ri = idA, thenwe say that A is a retract of X.

In Section 5 we will return to the retract. To continue with the CW-complexeswe will define the strong deformation retract.

Definition 3.55. If there is a retraction r of X to A such that if i : A → X, thenidX ' ir rel A, then we say that A is a strong deformation retract.

37

The notion of a retract is very similar to this strong deformation retract. Thedifference between these two concepts is the order in which the inclusion and re-traction are applied. Specifically, the latter definition requires the retraction ofi : A → X to be homotopic to idX relative to A. While the former definitionrequired i r to be homotopic to idA. Formally, a strong deformation retract is aleft homotopy inverse while a retract is a right homotopy inverse.

For every CW-complex (X,A) the space A is a strong deformation retract of X.This will follow from the way that (X,A) is constructed. To prove it one must firstlook into the structure of spaces created by adjoining n-cells. Such a study couldwell constitute its own section and for that the reader may wish to read Chapter 3of [24]. Our exposition will, as usual, be brief.

We begin by defining a simplicial complex. From our perspective this is a abstractform of a CW-complex, almost combinatorial in its nature. For instance, it is notapparent that there is a non trivial topology on a simplicial complex. However, weshall later construct one.

Definition 3.56. Let K be a set of vertices v together with a collection s ofsubsets of s. If s contains all singletons of K and all nonempty subsets of a setin s is again in s, then we say that K is a simplical complex. Furthermore, wesay that each s ∈ s is a simplex.

The dimension (of a simplex) is defined as the number of vertices that are insidethe simplex. Building on that, the dimension of a simplicial complex is the largestdimension of simplexes contained within it. It should be apparent that this way ofthinking of dimension corresponds well with what we usually think that it shouldbe.

Let us now discuss the faces of a simplex. Let s′ be a subset of s, then wesay that s′ is a face of s. A word of warning: the faces of a simplex does notnecessarily correspond with what a face is usually thought to be. In particular,faces are not only on what would geometrically be “the outside” of the simplex buton all combination of vertices. Furthermore, if s′ 6= s, then we say that s′ is aproper face of s.

Since they are not defined with a topology, it follows that simplicial complexare not topological spaces but sets. However, there exists a covariant functor thatwill take simplicial complexes into the category of topological spaces. We will notdescribe this in any detail and the interested reader is referred to page 111 in [24].In broad strokes, one would begin with a family of functions of the form α : K → I.If we assume that each function α satisfies the following two conditions:

(1) for any α, v ∈ K|α(v) 6= 0 is a simplex of K;

(2) for any α,∑v∈K α(v) = 1.

Then, we denote this set by |K|. One can then use |K| to construct a metrictopology, denoted by |K|d.

It is also possible to define a similar structure on the simplexes of K. Let s bea simplex of K, then we adopt the following notation

|s| = α ∈ |K| : α(v) 6= 0⇒ v ∈ s.

From |K|d, |s| inherits a topology, denoted |s|d. In turn, by the relation between sand K, this creates a weak topology on |K|. For details please, again, see [24]. wecan now use this in the following lemma.

38

Lemma 3.57. For any simplex s, |s| × 0 ∪ |s| × I is a strong deformation retractof |s| × I.

Proof. For a proof, please see Lemma 3.2.3 of [24]

For a simplicial complex there is an analogous construction to the subspace of aspace, known as a subcomplex.

Definition 3.58. Let K and L be simplicial complexes. If each simplex s of L isalso a simplex of K, then we say that L is a subcomplex of K.

As a sidenote, this provides a way to relate a topological pair (X,A) to a simpli-cial complex, specifically to a simplicial pair (K,L). Technically, one would denotethis by ((K,L), f), where f is a homeomorphism (|K|, |L|) 7→ (X,A). We say that((K,L), f) is a triangulation and that (X,A) is a polyhedral pair, if it has a tri-angulation. This will be used more in Section 4.1, when we look at homology andcohomology.

At the moment we will use the subcomplex L of K to give a generalization ofLemma 3.57.

Lemma 3.59. For any subcomplex L ⊂ K the subspace |K|×0∪|L|×I is a strongdeformation retract of |K| × I.

Proof. Please see Lemma 3.2.4 of [24]

Now, Lemma 3.59 can be used to prove the existence of a similar strong defor-mation retract on a CW-complex, (X,A). The key is that Lemma 3.59 provides astrong deformation retract D : (En × I)× I → En × I.

Lemma 3.60. If X is obtained from A by adjoining n-cells, then X × 0∪A× I isa strong deformation retract of X × I.

Proof. For each n-cell enj of X −A let

fj : (En, Sn−1)→ (enj , enj )

be a characteristic map. Let D : (En × I) × I → En × I be a strong deformationretraction of En × I to En × 0 ∪ Sn−1 × I (which exists, by Lemma 3.59). Thereis a well-defined map Dj : (en × I)× I → enj × I characterized by the equation

Dj((fj(z), t), t′) = (fj × 1I)(D(z, t, t′)) z ∈ En; t, t′ ∈ I.

Then there is a map D′ : (X × I) × I → X × I such that D′|(ej×I)×I = Dj andD′(a, t, t) = (a, t) for a ∈ A, and D′ is a strong deformation retraction of X × I toX × 0 ∪A× I.

We will now look at an interesting topic in algebraic topology: cofibrations.Later, in Section 4 we will work with fibrations, which are closer to the fibers thatone would normally meet in for instance differential geometry.

Definition 3.61. Let f : X ′ → X be a map and let Y be an arbitrary space.For maps g : X → Y and G : X ′ × I → Y such that g(f(x′)) = G(x′, 0) forx ∈ X. If there is a map F : X × I → Y such that F (x, 0) = g(x) for x ∈ X andF (f(x′), t) = G(x′, t) for x′ ∈ X ′ and t ∈ I, then f is a cofibration. Consider thefollowing diagram.

39

X ′ × 0 X ′ × I

Y

X × 0 X × I

f×id0 f×idI

G

g

Clearly X × 1∪A× I ⊂ X × I and also A× I ⊂ X × 0∪A× I. Therefore, givenmaps g : X → Y and G : A× I → Y such that G(x, 0) = g(x) for x ∈ A, there is asimple way to find a map F : X × I → Y . Let f : X × 0 ∪ A × I → Y be definedby f(x, 0) = g(x) for x ∈ X and f(x, t) = G(x, t) for x ∈ A and t ∈ I. Then, byLemma 3.60, there is a continuous map F : X × 0 ∪ A× I → X × I → Y . This isdisplayed in the following diagram and stated in the next corollary.

A× 0 A× I

Y X × 0 ∪A× I

X × 0 X × I

G

fg

F

Corollary 3.62. If X is obtained from A by adjoining n-cells, then the inclusionmap i : A → X is a cofibration.

Given Corollary 3.62 one may expect that the inclusion map i : A → X for arelative CW-complex should be a cofibration. This is indeed true and almost animmediate consequence of the corollary. Since if X can not be obtained from Aby attaching just cells of dimension n, then by the definition of CW-complexes onecan attach cells of successively higher dimension. At each step one then appliesCorollary 3.62.

Theorem 3.63. If (X,A) is a relative CW-complex, then the inclusion map A →X is a cofibration.

Proof. This follows from Corollary 3.62, using induction.

Because of Theorem 3.63, there is a simple way to extend homotopies on a CW-complex (X,A). For any map f : X → Y , if there is a homotopy F : A × I →Y , then there is a homotopy F ′ : X × I → Y . We will use this to prove thatthere is a homotopy, relative to A, from (X,A) to any n-connected space (Y,B) inTheorem 3.65. However, in order to do that a technical lemma is necessary.

Lemma 3.64. Let X be obtained from A by adjoining n-cells and let (Y,B) be apair such that πn(Y,B, b) = 0 if n ≥ 1, for all b ∈ B. Furthermore let (Y,B) besuch that every point of Y can be joined to B by a path if n = 0. Then, any mapfrom (X,A) to (Y,B) is homotopic relative to A to a map from X to B.

Proof. See Lemma 7.6.3 of [24].

Similar to how we had to generalize Corollary 3.62 to Theorem 3.63 the same hasto be done here. It is still possible to rely on an iterative process to show that eachstage taken in the construction of the CW-complex satisfies Lemma 3.64. However,the proof will be trickier if the number of dimensions of the CW-complex can be

40

infinite. To circumvent that we will assume that dim(X − A) ≤ n, for some n.But this is purely an act of convenience: the property holds for any CW-complexregardless of dimension.

Theorem 3.65. Let (X,A) be a relative CW-complex, with dimension (X−A) ≤ n,and let (Y,B) be n-connected. Then any map from (X,A) to (Y,B) is homotopic,relative to A, to a map from X to B.

Proof. This follows, using induction, from Corollary 3.46, Lemma 3.64 and Theo-rem 3.63.

The preceding proof leaves us in a position to prove that a CW-complex is (n−1)-connected. It would actually be possible to prove a stronger result: that (X, (X,A))is (n− 1)-connected. But again, this is not necessary. Instead we shall use the factthat (X,A) is (n− 1)-connected and therefore πq(X,A) = 0 for q < n to prove animportant result in Section 4.

Lemma 3.66. For n ≥ 1 if X is obtained from A by adjoining n-cells, then (X,A)is (n− 1)-connected.

Proof. For k ≤ n − 1, let f : (Ek, Sk−1) → (X,A) be a map. We will now createtwo spaces. Because f(Ek) is compact, there exist a finite number, say e1, . . . , em,of n-cells of X−A such that f(Ek) ⊂ A∪mi=1 ei. For 1 ≤ i ≤ m, let xi be a point ofei − ei. From this we can create the spaces Y = A ∪mi=1 (ei − xi) and ei − ei. Bothof these spaces intersect f(Ek) in a set open in f(Ek) for 1 ≤ i ≤ m .

There is a simplicial triangulation of Ek, say K, such that (identifying |K| withEk) for every simplex s ∈ K either f(|s|) ⊂ Y or, for some 1 ≤ i ≤ m, f(|s|) ⊂ei− ei. Let A′ be the subpolyhedron of Ek which is the space of all simplexes s ∈ Ksuch that f(|s|) ⊂ Y , and for 1 ≤ i ≤ m, let Bi be the subpolyhedron which isthe space of all simplexes s of K such that f(|s|) ⊂ ei − ei. Then Sk−1 ⊂ A′ andEk = A′ ∪mi=1 Bi. If i 6= j, then Bi −A′ is disjoint from Bj −A′. Let Bi = Bi ∩A′and observe that (Bi, Bi) is a relative CW-complex, with dim(Bi−Bi) ≤ k ≤ n−1.

For 1 ≤ i ≤ m the pair ((ei − ei), (ei − ei) − xi) is homeomorphic to (En −Sn−1, (En − Sn−1 − 0)) and has the same homotopy groups as (En, Sn−1). ByCorollary 3.50, (En, Sn−1) is (n− 1)-connected. It follows from Theorem 3.65 thatf |(Bi− Bi) is homotopic relative to Bi to a map from Bi to (ei− ei)−xi. BecauseBi − Bi is disjoint from Bj − Bj for i 6= j, these homotopies fit together to definea homotopy relative to A′ of f to some map f ′ such that f ′(Ek) ⊂ Y . Clearly, Ais a strong deformation retract of Y . Therefore f ′ is homotopic relative to Sk−1 toa map f ′′ : (En, Sn−1) → (X,A) such that f ′′(Ek) ⊂ A. Then f ' f ′ ' f ′′, allhomotopies relative to Sk−1. Therefore (X,A) is (n− 1)-connected.

Theorem 3.67. For n finite or infinite let f : X → Y be an n-equivalence and let(P,Q) be a relative CW-complex with the dim (P −Q) ≤ n. Given maps g : Q→ Xand h : P → Y such that h|Q = f g, there exists a map g′ : P → X such thatg′|Q = g and f g′ ' h relative to Q.

Proof. Let Zf ve the mapping cylinder of f , with inclusion maps i : X → Zf andj : Y → Zf , and retraction r : Zf → Y a homotopy inverse of j. Then, in thefollowing diagram

41

Q P

X Zf

g jh

i

a homotopy i g ' j h|Q can be found whose composite with r is constant. ByTheorem 3.63, there is a map h′ : P → Zf such that h′|Q = i g, and such thatr h′ ' r j h relative to Q. We regard h′ as a map from (P,Q) to (Zf , X). Since(Zf , X) is n-connected and dim (P −Q) ≤ n, it follows from Theorem 3.65 that h′is homotopic relative to Q to some map g′ : P → X. Then g′|Q = g and

f g′ = r i g′ ' r j h = h

all the homotopies being relative to Q. Hence g′ has the desired properties.

3.7. Homotopy Functors. The functor πn is sometimes known as the homotopyfunctor. However, there is in actuality a wide class of functors, called the homotopyfunctors. For example we do not only have πn but also the cohomology functorHn, which we will look at in Section 4.1. Regardless of the kind of functor theyall have to respect equivalence between homotopic maps. We shall now define thisequivalence in a category theoretic sense.

Definition 3.68. Let C be a category with objects X, Y , Z, and Z ′ as well asmorphisms f0 : X → Y and f1 : X → Y , an coequalizer of f0 and f1 is a morphismj : X → Z in C such that:

(1) j f0 = j f1,

(2) if j′ : X → Z ′ is a morphism in C such that j′ f0 = j′ f1, there is amorphism g : Z → Z ′ such that j′ = g j.

The coequalizer can be represented in a diagram in the following way.

X Y Z

Z ′

f0

f1

j

j′g

As a remark for reference, the map j is in [24] called the equalizer. We have,however, gone with the greater literature (such as [25]) in naming it the coequalizer.We now show that there are equalizers on C0.

Lemma 3.69. Let Ynn≥0 be objects of C0 that are subspaces of a space Y in C0

such that Yn → Yn+1 is a cofibration for all n ≥ 0, Y = ∪nYn, and Y has thetopology coherent with Yn. Let in : Yn → Yn+1, idn : Yn → Yn, and jn : Yn → Ybe the inclusion maps. Then, the homotopy class [jn] :

∨Yn → Y is an equalizer

in C0 of the homotopy classes[∨in

]:∨Yn →

∨Yn and

[∨idn

]:∨Yn →

∨Yn.

Proof. Since jn+1 in = jn idn, it follows that jn ∨in = jn

∨idn. Given a

map j′ :∨Yn → Z ′ such that j′

∨in ' j′

∨idn, let j′n : Yn → Z ′ be defined by

j′n = j′|Yn. Then, jn+1 in ' j′n and using the fact that Yn → Yn+1 is a cofibrationand by induction on n, there is a sequence of maps gn : Yn → Z ′ such that gn ' j′nand gn+1i = gn. Let g : Y → Z ′ be the map such that g|Yn = gn. Then, gj ' j′,completing the proof.

42

Definition 3.70. A homotopy functor is a contravariant functor H from C0 to T∗such that the following two conditions hold.

(1) If [j] : X → Z is an coequalizer of [f0], [f1] : A→ X and if u ∈ H(X) is suchthat H([f0])(u) = H([f1])(u), there is a v ∈ H(Z) such that H([j])(v) = u.

(2) If Xλλ is an indexed family of objects in C0 and iλ : Xλ →∨Xλ, there

is an equivalence

H[iλ]λ : H(∨

Xλ

)≈∏

H(Xλ).

All homotopy functors have something known as coefficient groups. The q-thcoefficient group is the functor taken on the q-th suspension of I, Sq. For thehomotopy functor πY this means that πq(Y ) is the q-th coefficient group. In generalthe q-th coefficient group is H(Sq). It is natural to wonder whether two homotopyfunctors share the same group structure on their coefficient groups. And even moreinteresting: is H(Sq) ≈ πq(Y )? This would surely depend in some way on the spaceY and in the next definition we formalize this dependence.

Definition 3.71. An element u ∈ H(Y ) is said to be n-universal for H, wheren ≥ 1, if the homomorphism

Tu : πY (Sq)→ H(Sq),

where Tu is defined by

Tu([f ]) = H([f ])(u), [f ] ∈ πY ,is an isomorphism for 1 ≤ q < n and an epimorphism for q = n. If u is n-universalfor all n ≥ 1 we say that Y is a classifying space for H.

Thus, if Y is a classifying space for H then all coefficient groups of H and πY areisomorphic.

The use we will make of the classifying spaces is somewhat technical in its nature.Recall that the ultimate goal is to extend a map on a subset of P onto P in itsentirety. In our case we could perhaps do this using only the functor πq(P ) sincethe space and function we try to extend is particularly simple. Nonetheless, weshall try and do this more generally, which results in a problem: πq(Y ) is seldomeasy to calculate. It would therefore be convenient to use another functor and thisis what classifying spaces will allow us to do.

We now continue with the classifying spaces and prove Theorem 3.74. This isdone over three theorems in an inductive fashion. First, in Lemma 3.72, we willshow that we can in a sense extend an element of H(Y ) to a larger set by adding1-cells to Y . We then build on this in Lemma 3.73 and allow attachment of (n+1)-cells.

As a small note on notation, we shall, given an inclusion i : X → X ′ andu ∈ H(X ′), write u|X instead of H(i)(u). This is also the equivalent to a restrictionin this case.

Lemma 3.72. Let H be a homotopy functor, Y an object in C0, and u ∈ H(Y ).There exist an object Y ′ in C0, obtained from Y by attaching 1-cells, and a 1-universal element u′ ∈ H(Y ′) such that u′|Y = u.

Proof. For each λ ∈ H(S1) let S1λ be a 1-sphere and define Y = Y

∨(∨λ S

1λ

). Then

Y ′ is an object of C0 obtained from Y by attaching 1-cells. If gλ is the composite

43

S1 ≈−→ S1λ ⊂ Y ′, it follows from condition (2) of Definition 3.70 that there is an

element u′ ∈ H(Y ′) such that u′|Y = u and H(gλ)(u′) = λ for λ ∈ H(S1). SinceTu′([gλ]) = λ, Tu′([S1;Y ′]) = H(S1), and u′ is 1-universal.

Lemma 3.73. Let H be a homotopy functor and u ∈ H(Y ) an n-universal elementfor H, with n ≥ 1. There exist an object Y ′ in C0, obtained from Y by attaching(n+ 1)-cells, and an (n+ 1)-universal element u′ ∈ H(Y ′) such that u′|Y = u.

Proof. For each λ ∈ H(Sn+1) let Sn+1λ be an (n + 1)-sphere, and for each map

α : Sn → Y such that H(α)(u) = 0 attach an (n + 1)-cell en+1a to Y by α. Let

Y ′ be the space obtained from Y∨∨

λ Sn+1λ by attaching the (n+ 1)-cells en+1

α .Then Y ′ is an object of C0 obtained from Y by attaching (n+ 1)-cells.

If gλ : Sn+1 → Y∨∨

λ Sn+1λ is the composite

Sn+1 −→≈Sn+1λ → Y

∨(∨λ

Sn+1λ

),

it follows from (2) of Definition 3.70 that there is an element u ∈ H(Y∨(∨

λ Sn+1λ

))such that u|Y = u and H(α)u = 0. Let Snα be an n-sphere and let f0 :

∨α S

nα →

Y∨(∨

λ Sn+1λ

)be the constant map. Furthermore, let the map f1 :

∨α S

nα →

Y∨(∨

λ Sn+1λ

)be such that Snα is mapped by α. Then the inclusion

j : Y∨(∨

λ

Sn+1λ

)→ Y ′

is a map such that [j] is an equalizer of [f0] and [f1]. Since H(f0)u = 0 = H(f1)u,by condition (1) of Definition 3.70, there is an element u′ ∈ H(Y ′) such thatH(j)u′ = u. Thus, u′|Y = u and to complete the proof we need only show that u′is (n+ 1)-universal.

There is a commutative diagram

πq+1(Y ′, Y ) πq(Y ) πq(Y′) πq(Y

′, Y )

H(Sq)

∂

Tu

i#

Tu′

with the top row exact. Since Y ′ is obtained from Y by attaching (n + 1)-cells,it follows from Lemma 3.66 that πq(Y ′, Y ) = 0 for q ≤ n. Therefore, i# is anisomorphism for q < n and an epimorphism for q = n. Since u is n-universal,Tu is an isomorphism for q < n and an epimorphism for q = n. Furthermore, ifa ∈ [Sn;Y ] is in the kernel of Tu, then a is represented by a map α : Sn → Y and

a = [α] ∈ ∂(πn+1(en+1α , en+1

α )) ⊂ ∂(πn+1(Y ′, Y )) = ker i#.

Therefore, for q = n, ker Tu = ker i#, and so Tu′ is an isomorphism from πn(Y ′)to H(Sn). For any λ ∈ H(Sn+1) the map j gλ : Sn+1 → Y ′ has the property that

Tu′([j gλ]) = H(gλ)(u) = λ,

showing that Tu′ is an epimorphism for q = n+1, and so u′ is (n+1)-universal.

Theorem 3.74. Let H be a nonempty functor let Y be an object in C0, and letu ∈ H(Y ). Then, there is a classifying space Y ′ for H containing Y such that

44

(Y ′, Y ) is a relative CW-complex and a universal element u′ ∈ H(Y ′) such thatu′|Y = u.

Proof. Using Lemma 3.72 and Lemma 3.73, we construct, by induction over n, asequence of objects Ynn≥0 in C0 and elements un ∈ H(Y n) such that:

(1) Y0 = Y and u0 = u;

(2) Yn+1 is obtained from Yn by attaching (n+ 1)-cells, where n ≥ 0;

(3) un+1|Yn = un.

(4) un is n-universal for n ≥ 1.

It follows from (2) above that Y ′ = ∪Yn, topologized with the topology coherentwith Yn, is a path-connected pointed space containing Y such that (Y ′, Y ) isa relative CW-complex. By Lemma 3.69, the homotopy class [jn] :

∨Yn →

Y ′ is an equalizer of the homotopy classes [∨in] :

∨Yn →

∨Yn and [

∨1n] :∨

Yn →∨Yn. By condition (2) of Definition 3.70, there is an element u ∈ H(

∨Yn)

such that u|Yn = un. It follows from (3) above that H(∨in)(u) = H(

∨1n)(u),

and by condition (1) of Definition 3.70 there is an element u′ ∈ H(Y ′) such thatH(jn)(u′) = u (that is, u′|Yn = un for n ≥ 0). Then u′|Y = u, and it remains toshow that u′ is universal.

By definition of Y ′ and u′, there is a commutative diagram for q ≥ 1

lim→πq(Yn) πq(Y′)

H(Sq)

≈

TunTb′

Since un is universal, Tun is an isomorphism for n > q, and so the left-hand map isan isomorphism. Therefore Tu′ is also an isomorphism, and u′ is universal.

This is what we need on classifying spaces and homotopy functors. In the nextsection we will use classifying spaces and Theorem 3.74 in particular to prove ourfirst result regarding extensions of functions in Theorem 4.11.

45

4. Obstruction Theory

Obstruction theory is a method for extending maps, exactly what we are lookingfor. To use it we will first look at Eilenberg-MacLane spaces in Section 4.1 and thiswill also give our first real extension result. This will rely on the important topicof homology and cohomology, which we address in the same section. Followingthis, we will then use a Moore-Postnikov factorization (defined in Section 4.2) togeneralize the results of Section 4.1. Finally, in Section 4.3 the generalization isfinalized and we prove the extension problem that motivated the use of algebraictopology in the first place.

4.1. Eilenberg-Maclane Spaces and Cohomology. In this section we will fi-nally be able to use algebraic topology to state when it is possible to extend maps,for certain spaces. Our big result of the section, Theorem 4.11, does this on acertain kind of space known as a Eilenberg-MacLane space, which we will discussshortly. This is the fundamental part of the extension we do by use of obstructiontheory in the next section. We will give two constructions of cohomology, one byStrom [25] and one by Spanier [24]. The argument for introducing two constructionsis that the former is explanatory, short and fits nicely with the flow of the essay.While the latter is necessary for much of the remainder of the essay. Unfortunaly,the construction due to [24] is quite involved and beyond the scope of this essay.Therefore, please see the first constrction as an explanation and perhaps as contextfor the Eilenberg-MacLane spaces, which we are going to introduce.

As said before: the higher homotopy groups of nearly all but the most trivial ofspaces are hard to compute. There is, however, one family of spaces, or perhapsmore accurately CW-complexes, that are particularly nice: the Eilenberg-MacLanespaces.

Definition 4.1. Let π be a group, and let n ≥ 1 be an integer. A space of type(π, n) is a path-connected pointed space Y such that πq(Y, y0) = 0 for q 6= n andπn(Y, y0) is isomorphic to π. We will often say that Y is a K(π, n) space.

Being a K(π, n) space is quite a restrictive condition on a space, which not allspaces satisfy. For instance, S1 is a K(Z, 1) space while no other sphere is anEilenberg MacLane space. However, the limitations on K(π, n) spaces are well out-weighed by their usefulness when it comes to mapping other spaces into EilenbergMaclane spaces. The tool we will use is cohomology, which is the homotopy classesof maps from a space X into a K(π, n) space. Our presentation of this will mostlyfollow [25].

Definition 4.2. Let X be a topological space, then the n-th cohomology groupwith coefficients in π, Hn(X;π), is defined as

Hn(X;π) = [X,K(π, n)].

For this definition to make sense we would need to know that there actuallyis a K(π, n) space for an arbitrary group π. This is not something we will proveexplicitly but instead we state restate a part of Theorem 17.49 of [25].

Proposition 4.3. For every abelian group π and every n ≥ 1, there is an Eilenberg-MacLane space K(π, n), which is a topological manifold.

46

Naturally, we are just interested in the existence part of Proposition 4.3. Thefact that the K(π, n) space is a manifold is just included as a curiosity for thereader.

Proof. For a proof please see Theorem 17.49 of [25].

Another property, which we will not prove, is that Hn(·;G) is a contravarianthomotopy functor. Instead we restate (a) of Theorem 21.2 in [25]. Note that ABGis the category of abelian groups.

Proposition 4.4. For any X in T∗ and any group π the n-th cohomology groupHn(X;π) is a contravariant homotopy functor T∗ → ABG .

Proposition 4.3 and Proposition 4.4 assures that there actually is a K(π, n) spacefor any π and that there is a homotopy functorHn to the category of abelian groups.Since Hn is contravariant, an element v of Hn(X;π) is a map v : K(π, n) → X,unique up to homotopy.

We will now give a minimalized version of Spaniers construction of cohomology.It is not intended to be formally correct or particularly explanatory, either of whichwould require a treatment beyond the scope of this essay. For a formal treamentplease see Chapter 5.1 of [24].

Now, let ∆n be a simplex with n dimensions and let X be a topological space.Then we say that the continuous mapping σn : ∆n → X is the singular n-simplex.The boundary of σn(∆n), often written as ∂nσn(∆n), is the formal sum of the(n− 1)-simplexes that form the boundary of ∆n. For a topological space, X, thereis a functor ∆, which sends X to its singular complex ∆(X), see Theorem 4.1.5 of[24]. By looking at different dimensions of the singular complex of a space, X, wecan form a chain of singular simplices where the boundary map ∂n : ∆n → ∆n−1

moves downwards trough the dimensions. We call this chain, C, together with theboundary map a singular chain complex and write it as C, ∂.

Linked to these chain complexes are the cochain complexes. These are chaincomplexes with a coboundary map, which is the boundary map reversed. This mapassigns to a (n − 1)-complex a n-complex. We denote the coboundary map by δ.Adopting the common notation, Cq for the q-th complex in a chain complex we candefine

δq = Cq → Cq+1.

This can then be used to give an alternative definition of Hn(X) as ker δn−1/ Im δn

where δ is defined on the cochain complex of X. To get the cohomology withcoefficients in π we define Hq(C;π), where C is a chain complex of some space X,by

Hq(C;π) = Hq(hom(C, π)).

Finally, the cohomology group Hn(X,A;π) is then given by the cohmology groupHn(hom (∆(X)/∆(A), π)), since ∆(X)/∆(A) forms a chain complex. Importantly,the map δn is then a morphism from Hn(X,A;π) to Hn+1(X,A;π).

It is difficult to convey the full gravity of cohomology while keeping the presenta-tion to a working minimum. The reader who is interested to learn more could lookat Chapter 3 of [16], Chapter 21 of [25] or the latter half of Chapter 5 of [24]. Weshall instead briefly work trough another important aspect of algebraic topology:homology.

47

In short, homology is the dual to cohomology and closely related to πn. Weshall need two things from homology: the n-characteristic classes of a cohomologygroup and the Hurewicz isomorphism theorem. The latter connects homology andhomotopy on the first nontrivial coefficient groups on a space. We will again follow[25] in the definition of homology.

Definition 4.5. A homology h∗ is a covariant homotopy functor, h∗ : T∗ → ABG ∗,such that:

(1) there is a natural isomorphism h∗ S∼=−→ S h∗;

(2) if A→ B → C is a cofiber sequence, then the sequence

h∗(A)→ h∗(B)→ h∗(C)

is exact.We will use Hn(X,A) to denote h∗(Sn(I), Sn(I);X,A).

An important aspect of homology is that the first nontrivial groups of Hn(X,A)and πn(X,A) are isomorphic. This is a famous result known as the Hurewiczisomorphism theorem. We will not go into the details of the proof. That wouldrequire a far deeper understanding of homology that what is given here. Instead werestate Theorem 7.5.4 of [24], which is the relative Hurewicz Isomorphism Theorem.

Theorem 4.6. Let x0 ∈ A → X and assume that A and X are path connected.If there is an n ≥ 2 such that πq(X,A, x0) = 0 for q < n, then Hq(X,A) = 0 forq < n and ϕ′ is an isomorphism

ϕ′ : π′n(X,A;x0) ≈ Hn(X,A).

Conversely, if A and X are simply conneced and there is an n ≥ 2 such thatHq(X,A) = 0 for q < n, then πq(X,A;x0) = 0 for q < n and ϕ is an isomorphism

ϕ : πn(X,A, x0) ≈ Hn(X,A).

Proof. For a proof, please see Theorem 7.5.4 of [24].

We will need another important concept, involving homotopy, homology andcohomology: the n-characteristic cohomology classes. This is yet again somethingthat will not elaborated upon. In particular, we will use a map h : Hq(C;π) →hom(Hq(C), π) defined by

(hf)∑

ci =∑

f(ci)

for f ∈ Hq) and∑ci ∈ Hq(C). Note that this is a simplification in order to

avoid the introduction of more homology and cohomology. A more fleshed outtheory together with the surrounding theory can be found on page 242 of [24].

Definition 4.7. Let (X,A) be a pointed pair, which is path connected. We saythat a cohomology class v ∈ Hn(X,A;π) is n-characteristic for (X,A) if it satisfieseither of the following:

(1) n = 1 and i#(π1(A)) is a normal subgroup of π1(X) whose quotient groupis mapped isomorphically onto π by the composite

π1(X)/i#(π1(A))ψ−→ H1(X)/i∗(H1(A))

j∗−→ H1(X,A)h(v)−−−→ π;

48

(2) n > 1 and the composite

πn(X,A)φ−→ Hn(X,A)

h(v)−−−→ π

is an isomorphism.

Lemma 4.8. Let i : A → X be a simple inclusion map between path-connectedpointed spaces such that the pair (X,A) is (n − 1)-connected, where n ≥ 1. Then,there exist cohomolgy classes v ∈ Hn(X,A;π) which are n-characteristic for (X,A),where π = π1(X)/i#(π1(A)) for n = 1 and π = πn(X,A) for n > 1.

Proof. If n = 1, it follows from the absolute Hurewicz isomorphism theorem appliedto A and to X that there are isomorphisms

π1(X)/i#(π(A))ϕ≈ H1(X)/i∗(H1(A))

j∗≈ H1(X,A).

By the universal-coefficient formula for cohomology, there is also an isomorphism

h : H1(X,A;π) ≈ Hom (H1(X,A), π).

Hence, if π = π1(X)/i#(π1(A)), then there exist 1-characteristic elements v ∈H1(X,A;π).

If n > 1, it follows, from the relative Hurewicz isomorphism theorem andthe universal-coefficient formula for cohomology, that there are isomorphisms ϕ :πn(X,A) ≈ Hn(X,A) and h : Hn(X,A;π) ≈ Hom (Hn(X,A), π). Therefore, ifπ = πn(X,A), there are n-characteristic elements v ∈ Hn(X,A;π).

This is all the homology and cohomology necessary for our purposes. We willnow continue on the track set by the Hurewicz isomorphism theorem and work withisomorphic coefficient groups of homotopy functors.

Lemma 4.9. Let F : H → H ′ be a natural transformation between homotopyfunctors which induces an isomorphism of their qth coefficient groups for q < nand a surjection of their nth coefficient groups (where 1 ≤ n ≤ ∞). For anypath-connected pointed CW-complex, W , the map

F (W ) : H(W )→ H ′(W )

is a bijection if dim W ≤ n− 1 and a surjection if dim W ≤ n.

Proof. For a proof please see Theorem 8.1.7 of [24].

Now, consider the following specialization of Lemma 4.9 to Hn and πn.

Theorem 4.10. Let π be an abelian group, Y a space of type (π, n), and ι ∈Hn(Y, y0;π) an n-characteristic element for Y . Let ψ : πY → Hn(·;π) be thenatural transformation, defined by ψ[f ] = f∗ι, for [f ] ∈ [X;Y ]. Then, ψ is anatural equivalence on the category of path-connected pointed CW-complexes.

Proof. By Lemma 4.9, it suffices to verify that ψ induces an isomorphism on allcoefficient groups of the two homotopy functors πY andHn(·;π). The only non-zerocoefficient groups are πn(Y, y0) and Hn(Sn, p0;π), and we need only verify that

ψ(Sn) : πn(Y, y0)→ Hn(Sn, p0;π)

is an isomorphism. If ν : Hn(Sn, p0;π) ≈ π is defined by ν(υ) = h(υ)(φ[1Sn ]),see the proof of lemma 8.1.4 of [24], then ν ψ(Sn) = h(ι) ϕ. Because ι isn-characteristic for Y , ν ψ(Sn) is an isomorphism, and thus, so is ψ(Sn).

49

We can now give our first result regarding obstructions. In essence this is justTheorem 3.67 with some additional detail but the reader should not dismiss it asuninteresting. Theorem 4.11 will incorporate most of the ideas we have developed inSection 4. Furthermore, it is this theorem that we will generalize in Theorem 4.35,the final result of the section.

Theorem 4.11. Let Y be a space of type (π, n), with n ≥ 1 and π abelian, and letι ∈ Hn(Y, y0;π) be n-characteristic for Y . If (X,A) is a relative CW-complex, amap, f : A→ Y , can be extended to X if, and only if, δf∗(ι) = 0 in Hn+1(X,A;π).

Proof. Assume that f = g i, where i : A → X and g : X → Y . Then, δf∗(ι) =δi∗g∗(ι) = 0, because δi∗ = 0. Hence, if f can be extended over X, then δf∗(ι) = 0.

Conversely, assume δf∗(ι) = 0. To extend f over X we need only extend fover each path component of X, and therefore there is no loss of generality inassuming X to be path connected and A to be non-empty. Let Y ′ be the spaceobtained from the disjoint union X ∪ Y by identifying a ∈ A with f(a) ∈ Y forall a ∈ A. Then Y is imbedded in Y ′, the pair (Y ′, Y ) is a relative CW-complex,and there is a cellular map, j : (X,A) → (Y ′, Y ), which induces an isomorphism,j∗ : H∗(Y ′, Y ) ≈ H∗(X,A), such that the following diagram commutes. Note thatthe map δ is the coboundary map from the constrction of the cohmology groups.

Hn(Y, y0) Hn+1(Y ′, Y )

Hn(A) Hn+1(X,A)

δ

f∗ j∗

δ

Since δf∗(ι) = 0, it follows that δ(ι) = 0, and there is υ ∈ Hn(Y ′, y0;π) such thatυ|(Y, y0) = ι. Since X and Y are path connected and A is nonempty, Y ′ is pathconnected.

Let Y = Y ′∨I (that is, y0 ∈ Y ′ is identified with 0 ∈ I) and let y0 =

1 ∈ Y . Then Y is a path-connected space with nondegenerate base point y0.Let r : (Y , I) → (Y ′, y0) be the retraction which collapses I to y0 and let υ =r∗(υ)|(Y , y0) ∈ Hn(Y , y0;π). By Theorem 3.74, there is an imbedding of Y in aspace Y ′′ which is a classifying space for the nth cohomology functor with coeffi-cients π and which has a universal element u ∈ Hn(Y ′′, y0;π) such that u|Y = υ.Then, Y ′′ is a space of type (π, n), and there is a unique n-characteristic element,u ∈ Hn(Y ′′, y0;π), such that u|Y ′′ = u|Y ′′. Then u|(Y, y0) = ι, and it follows fromTheorem 4.10 and the commutativity of the next diagram.

[Sq, p0;Y, y0] [Sq, p0;Y ′′, y0]

Hn(Sq, p0;π)

ψιψu

that Y ⊂ Y ′′ is a weak homotopy equivalence. Since the compositeXj|X−−→ Y ′ ⊂ Y ′′

is an extension of the composite X f−→ Y ⊂ Y ′′, it follows from Theorem 3.67 thatf can be extended to a map X → Y .

While Theorem 4.11 is interesting we would like it to apply to a wider classof spaces. In the next theorem section we will begin this work by introducingfactorizations of spaces.

50

4.2. The Tower of Postnikov. In Theorem 4.11 we gave a condition for extendinga map f : A → Y to X, given that Y was a K(n, π)-space. We will now work toweaken this condition. Ultimately we will require only that Y is simply connected.The idea is that we will factorize Y into a sequence of spaces. To do this we makethe following definition.

Definition 4.12. Let Enn∈N be a sequence of topological spaces. If there is asequence of functions fnn∈N such that fn : En+1 → En, then we say that

E1 ← E2 ← · · · ← En ← En+1 ← · · ·

is a tower. We will use E∞ to denote limn→∞En.

To see how this is applied, we consider the CW-complex (X,A). To extend amap A→ Y to a map X → Y we need to find a map g : X → Y such that f = g i.Let us now assume that Y can be switched for a tower Enn∈N. We will lookat the procedure for doing this later. Then, extending the map f ′ : A → E∞ isequivalent to the existence of g : X → E∞ of the following diagram.

A E∞

...

X E0

f ′

i

p∞

p1

f ′′

g

We will set aside the issue of what f ′ and f ′′ are for the moment. Instead we notethat the extension of f ′ now is equivalent to the existence of the map g, of thediagram. However, we can also see that the extension of f ′ would also solve thelifting of f ′′ and vice versa. Therefore, it is possible to solve the extension problemby inductively solving the problem of lifting f ′′ to a map X → En.

Let us now return to the issue of how to define the maps f ′ and f ′′. First, letan : E∞ → En be given by pn pn+1 · · · . This is just a convenience since, inparticular, a0 : E∞ → E0 will require only one map between E∞ and E0. With iand a0 as the two vertical maps in a square diagram we will introduce the mappingpair.

Definition 4.13. Let P ′, P ′′, Q′ and Q′′ be topological spaces with maps α : P ′′ →P ′ and β : Q′′ → Q′. We say that a pair of maps f ′′ : P ′′ → Q′′ and f ′ : P ′ → Q′

is a map pair, if the following diagram commutes.

P ′′ Q′′

P ′ Q′

f ′′

α β

f ′

To shorten the notation we will denote the map pair (f ′, f ′′) by f : α→ β.

If we are going to exchange Y for Enn∈N, we will need a map pair f : i→ a0.We will look at this in Corollary 4.27. Right now we shall instead look closer atthe properties of the Enn∈N side and, in particular, the maps pnn∈N.

51

Definition 4.14. Let E, B, and X be spaces. Let f : X → E, p : E → B, andF : X × I → B be mappings such that the following diagram commutes.

X × 0 E

X × I B

f

i p

F

If a homotopy F ′ : X × I → E exists such that the following diagram commutes aswell,

X × 0 E

X × I B

f

i p

F

F ′

then we say that p has the homotopy lifting property with respect to X.

Definition 4.15. Let E, B, and p be as given in Definition 4.14. If p has thehomotopy lifting property with respect to all spaces X, then we say that p is afibration.

We are interested in using fibrations to simplify the homotopy structure of Y .The way this work is by the use of a special kind of fibration: the principal fibration.Before we can define it we must introduce some auxiliary facts about fibrations ingeneral. First, for two maps f : B′ → B and p : E → B we define the fiberedproduct, E′, as

E′ = (b′, e) ∈ B′ × E|f(b′) = p(e).From the definition of E′ it follows that we have a map p′(b′, e) = b′ from E′ to B′and a map f ′(b′, e) = e from E′ to E.

Now, by using the fibered product we can take a fibration p : E → B and inducea fibration p′ : E′ → B′, via a map f : B′ → B. For details on this see [24, p.98].We say that fibration p′ : E′ → B′ is the fibration induced from p by f .

There is a natural way to induce fibrations from any base point preserving mapbwetween two spaces. To do this we shall first show that every space has certainkind of fibration. Formally, let B′ be a space with a basis point b0, then we say thatthe space PB′ = γ : I → B′|γ(0) = b0 is the path space of B′. For any space B′we can construct a fibration p : PB′ → B′ by setting p as the endpoint evaluationmap p(γ) = γ(1).

Definition 4.16. Let θ be a basis point preserving map θ : B → B′. For anyfibration, p : PB′ → B′, there is a fibered product Eθ and a fibration pθ : Eθ → Binduced by θ. We say that this is the principal fibration induced by θ.

We shall now look at how principal fibrations can help us to extend maps. To dothis we first need to make an observation about the nature of liftings to principalfibrations. Let pθ : Eθ → B be the principal fibration induced by θ : B → B′. Forany spaceW and map f : W → Eθ we can decompose f into two maps. Specificallywe can decompose f into f1 : W → B and f2 : W → PB′, since Eθ is the fiberedproduct of B and PB′.

As PB′ technically is B′I , then we can use the following theorem, which is statedwithout proof as Theorem 1.2.8 in [24].

52

Theorem 4.17. Let X be a locally compact Hausdorff space and Y and Z aretopological spaces, a map g : Z → Y X is continuous if and only if E (g × idX) :Z ×X → Y , where E is the evaluation map, is continuous.

Now, we can write PB′ as B′I . Therefore, by Theorem 4.17, there is a corre-spondence between the map f2 : W → PB′ and the homotopy F : W × I → B′

from the constant map to θ f1, given by

F : W × I f2×idI−−−−→ B′I × I E−→ B′

We now assume that there is a CW-complex (X,A) and a map pair f : i→ pθ.Furthermore, let

θ(f) : (A× I ∪X × I , X × 0)→ (B′, b′0)

be a map such that θ(f)(x, 0) = b′0, θ(f)(x, 1) = θ(f ′)(x), and θ(f)|(A×I) is thehomotopy from the constant map A → b′0 to the map θ f ′ i. By the previousparagraph, this map corresponds to the lifting f ′′ of f ′ i. Therefore, there is aone-to-one correspondence between liftings of f and extensions of θ(f) over X × I.

When we apply this to simplify the structure of Y we will need to assume thatB′ is a K(π, n) space, with n ≥ 1 and π abelian. Extending a map is particularlysimple. To see this assume that ι ∈ Hn(B′, b′0, π) is n-characteristic for B′ and letθ : B → B′ be a map. Furthermore, let pθ : Eθ → B be the principal fibrationinduced by θ. For any CW-complex (X,A) then (X,A) × (I, I) is also a CW-complex. Given a map g : A× I ∪X × I → B′, it follows from Theorem 4.11 thatg can be extended over X × I, if and only if, δg∗(ι) = 0 in Hn+1((X,A) × (I, I).In particular, given a map pair f : i → pθ, there is a lifting of f if and only ifδθ(f)∗(ι) = 0.

Let τ be an isomorphism between Hn(X,A;π) and Hn+1((X,A) × (I, I)). Fora definition please see Section 8.1 of [24]. We say that δθ(f)∗(ι) = (−1)nτ(c(f)) isthe obstuction to lifting f and denote it by c(f). In Theorem 4.18 we will computec(f) for a special case, which we are going to apply in the proof of Theorem 4.35.

Theorem 4.18. Let ι ∈ Hn(B′, b′0;π) and ι′ ∈ Hn−1(ΩB′, ω′0;π) be related char-acteristic elements. Let (X,A) be a relative CW-complex, with inclusion map i :

A → X. Given a map pair f : i→ p′, where p′ : ΩB′ → b′0, then c(f) = −δf ′′∗(ι′),where f ′′ : A→ ΩB′ is part of f .

Proof. Let f : (A × I, A × I) → (PB′,ΩB′) be the map defined by f(a, t)(t′) =f ′′(a)(tt′). Then

θ(f) : (A× I ∪ x× I , X × 0)→ (B′, b′0)

is the map such that θ(f)|A × I = p f and θ(f)(X × I) = b′0. Let ¯f : (A × I ∪X × I , X × I)→ (B′, b′0) be the map defined by θ(f) and let f ′ : (A× I , A× 0)→(ΩB′, ω′0) be the map defined by f . There is then a commutative diagram, in whichj and j′ are appropriate inclusion maps and h1 : A→ (X × I , A× 0) is defined byh1(a) = (a, 1).

53

Hn(A× I ∪X × I , X × 0)

Hn(A× I ∪X × I , X × I)Hn(B′, b′0) Hn+1((X,A)× (I, I))

Hn(A× I, A× I)Hn(PB′,ΩB′) Hn(X,A)

Hn−1(ΩB′, ω′0) Hn−1(A× I , A× 0) Hn−1(A)

θ(f)∗

j∗δ

¯f∗

δ

p∗ j′∗ ≈ (−1)n−1τ

f∗

f′∗ h∗1

≈

δ δ

(−1)n−1τ≈ δ

Furthermore, δ τ−1 j′∗ = τ−1 δ : Hn(A× I ∪X × I , X × I)→ Hn(X,A). Sincef ′′ = f ′ h1, then f

′′∗ = h∗1 f′∗, and we have

(−1)n−1τ−1(θ(f))∗(ι) = δf′′∗(ι′).

By definition, the left-hand side equals −c(f).

With a clear condition for both lifting and extending maps to principal fibrationswe will now look at how to replace Y with a sequence of such fibrations.

Definition 4.19. If, for any n < ∞, there is an Nn such that for pq from thesequence of fibrations E0

p1←− E1p2←− · · · is an n-equivalence for q < N , then we say

that the sequence is convergent.

To replace Y we are not going to factorize Y directly. Instead, it is the mapa0 : E∞ → E0, that we are going to factorize. However, we do not yet have anymeaningful way to talk about factorizations of maps. This is rectified with thefollowing definition.

Definition 4.20. Let f : Y ′ → Y be a map. Then, the sequence pq, Eq, fqq≥1 issaid to be a convergent factorization of f if it satisfies the following conditions:

(1) for q > 1, pq : Eq → Eq−1 is a fibration, and for q = 1, p1 : E1 → Y is afibration;

(2) for q ≥ 1, fq : Y ′ → Eq is a map fq = pq+1 fq+1 for q ≥ 1, and f = p1 f1;

(3) for any n <∞ there is Nn such that fq is an n-equivalence for q > Nn.

We now show that a0 : E∞ → E0 has a convergent factorization,

E0p1←− E1

p2←− · · · .

Theorem 4.21. If E0p1←− E1

p2←− · · · is a convergent sequence of fibrations thenpq, Eq, aqq≥1 is a convergent factorization of the map a0 : E∞ → E0.

Proof. Conditions (1) and (2) of Definition 4.20 are clearly satisfied. To prove thatcondition (3) is also satisfied, given 1 ≤ n <∞, choose N so that pq is an (n+ 1)-equivalence if q ≥ N . We prove that aq is an n-equivalence for q ≥ N , it suffices toprove that aN is an n-equivalence.

54

Let (P,Q) be a polyhedral pair such that dim P ≤ n, and let α : Q → E∞ andβ′N : P → EN be maps such that β′N |Q = aN α. We now prove that there is anextension β : P → E∞ with the desired properties, we must obtain a sequence ofmaps βq : P → Eq such that

βq|Q = αq, βq = pq+1 βq+1, βN = β′N .

Such a sequence of maps βq is defined, for q ≤ N , by βq = pq+1 · · · pN β′N ,and for q ≥ N , we use Theorem 3.67 to find a map β′N : P → Eq+1 such that:

(1) β′q+1|Q = αq+1,(2) βq ' pq+1 β′q+1 rel Q.

We use the fact that pq+1 is a fibration (and Theorem 7.2.6 of [24]) to alter β′q+1 by ahomotopy relative toQ to obtain a map, βq+1 : P → Eq+1, such that βq+1|Q = αq+1

and such that βq = pq+1 βq+1. Thus, the sequence, βq, can be found, and hencea map β : p→ E∞ with the requisite properties exist.

Taking P to be a single point and Q to be empty, we see that aN is surjective.Therefore, aN maps π0(E∞) injectively to π0(EN ). If instead, (P,Q) = (I, I), wesee that aN maps π0(E∞) injectively to π0(EN ). Then, aN induces a one-to-onecorrespondence between the set of path components of E∞ and the set of pathcomponents of EN

Let e∗ = (eq) ∈ E∞ be arbitrary and let 1 ≤ k ≤ n. Taking (P,Q) = (Sk, z0) itfollows that aN# maps πk(E∞, e∗) epimorphically to πk(EN , eN ). For 1 ≤ k < n,taking (P,Q) = (Ek+1, Sk), it follows that αN# maps πk(E∞, e∗) monomorphicallyto πk(EN , eN ). Hence, aN is an n-equivalence.

We have now shown that there is a convergent factorization of a0. However, weare not interested in just any factorization. We need the fibers to be principal fibra-tions of type (π, n). Such a sequence of fibrations is known as a Moore-Postnikovsequence of fibrations.

Definition 4.22. Let E0p1←− E1

p2←− · · · be a convergent sequence of fibrations.If pq : Eq → Eq−1 is a principal fibration of type (πq, nq) for q ≥ 1, then E0

p1←−E1

p2←− · · · is a Moore-Postnikov sequence of fibrations.

Similarly, given a map f : Y ′ → Y a Moore-Postnikov factorization is a conver-gent factorization pq, Eq, fq such that E0

p1←− E1p2←− · · · is a Moore-Postnikov

sequence.We are interested in a particular Moore-Postnikov factorization. To define it, let

f : Y ′ → Y be a map. A Postnikov factorization of Y ′ is then a Moore-Postnikovfactorization of the constant map Y ′ → y0. Another factorization, which is not aMoore-Postnikov factorization but very useful, is the n-factorization.

Definition 4.23. Let f : Y ′ → Y be a map between path-connected pointedspaces. For n ≥ 1 a n-factorization of f is a factorization of f as a composite

Y ′b′−→ E′

p′−→ Y such that(1) E′ is a path-connected pointed space, p′ is a fibration, and b′ is a lifting of

f ;

(2) b′# : πq(Y′)→ πq(E

′) is an isomorphism for 1 ≤ q < n and an epimorphismfor q = n;

55

(3) p′# : πq(E′) → πq(Y ) is an isomorphism for q > n and a monomorphism

for q = n.

Condition (2) actually implies that b′ is n-characteristic. It may be helpful tovisualize the n-factorization in a diagram and we do this below.

E′

Y Y ′

p′b′

f

We will use n-factorzations to show that we can replace the target space ofthe map we wish to extend with a Postnikov fibration. This will, however, geta bit technical. Should the reader feel tired she, or he, could jump directly toCorollary 4.27.

Lemma 4.24. Let (X,A) be a pointed set pair of path-connected spaces (n − 1)-connected for some n ≥ 1 and such that the inclusion map i : A → X is simple.

Then there is an n-factorization Ab′−→ E′

p′−→ X of i such that p′ is a principalfibration of type (π, n), where π = π1(X)/i#(π1(A)), if n = 1, and π = πn(X,A),if n > 1.

Proof. By Lemma 4.8, there is a class ∈ Hn(X,A;π) which is n-characteristic for(X,A). Let CA be the nonreduced cone over A and observe that X,CA is anexcisive couple in X ∪ CA. Therefore, there is an element v′ ∈ Hn(X ∪ CA;π)corresponding to v under the isomorphisms

Hn(X ∪ CA;π)≈←− Hn(X ∪ CA,CA;π)

≈−→ Hn(X,A;π).

It is possible to imbed X ∪ CA into a space X ′ of type (π, n) having an n-characteristic element ι′ such that ι′|X ∪ CA = v′. Let p′ : E′ → X be theprincipal fibration induced by the inclusion X → X ′ and let p′A : E′A → A bethe restriction of this fibration to A. There is a section s : A → E′A such thats(a) = (a, ωa) for a ∈ A, where ωa is the path from x0 to the vertex of CA followedby the path from the vertex of CA to a. That is, ωa(t) = [x0, 1− 2t] for 0 ≤ t ≤ 1

2

and ωa(t) = [a, 2t − 1] for 12 ≤ t ≤ 1. We define b′ : A → E′ to be the composite

As−→ E′A

iA−→ E′ and we shall prove that A b′−→ E′p′−→ X is an n-factorization of i.

The fiber of p′, hence also of p′A, is ΩX ′, and we define g : E′A → ΩX ′ byg(a, ω) = ω ∗ (s(a))−1. Then g|ΩX ′ : ΩX ′ → ΩX ′ is homotopic to the identitymap. If i′′ : ΩX ′ → E′A is the inclusion map, it follows from the exactness ofthe homotopy sequence of the fibration, p′A : E′ → A, that there is a direct-sumdecomposition

πq(E′A) ≈ i′′#πq(ΩX ′)⊕ S#πq(A) q ≥ 1,

For q = 1, this is a direct-product decomposition, but we shall still write it addi-tively. We define the homomorphism, λ : πq(X,A) → πq−1(ΩX ′), where q ≥ 1, tobe the composite

πq(X,A)p′−1#−−−→≈

πq(E′, E′A)

∂−→ πq−1(E′A)g#−−→ πq−1(ΩX ′).

We show that the following diagram commutes up to a sign:

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πq(A) πq(X) πq(X,A) πq−1(A)

πq(E) πq(X) πq(ΩX′) πq−1(E′)

i#

b′#

j#

=

∂

λ b′#

p′# ∂ i′#

In fact, the left-hand and middle squares are easily seen to be commutative. Weshall show that b# ∂ = −i′# λ.

For q = 1 this is because π0(A) = 0 implies that b′# ∂ is the trivial map andthe fact that j# is surjective and i′# λ j# = i′# ∂ = 0 implies that i′# λ is alsothe trivial map. For q > 1, we have

α = i′′#g#α+ s#p′A#α α ∈ πq−1(E′A).

Since the composite πq(E′, E′A)∂−→ πq−1(E′A)

i′A#−−→ πq−1(E′) is trivial, it followsthat for β ∈ πq(E′, E′A)

0 = i′A#∂β = i′A#i′′#g#∂β + i′A#s#p

′A#∂β = i′#g#∂β + b′#∂p#β.

By definition of λ, we see that λp#β = g#∂β. Therefore,

i′#λp#β + b′#∂p#β = 0.

Since p# : πq(E′, E′A) ≈ πq(X,A), this proves b′# ∂ = −i′# λ.

A straightforward verification shows that λ is also the composite

πn(X,A)→ πn(X ∪ CA,CA)←−≈πn(X ∪ CA)→ πn(X ′)

∂−→≈πn−1(ΩX ′).

The construction of X ′ and ι′ ∈ Hn(X ′, π) shows that there is a commutativediagram

πq(X,A) πn(X ∪ CA,CA) πn(X ∪ CA) πn(X ′)

Hn(X,A) Hn(X ∪ CA,CA) Hn(X ∪ CA) Hn(X ′)

π

≈ϕ ≈ ϕ ϕ ϕ≈

≈

≈h(v)

h(v′ )

h(ι′ )

≈

Therefore, λ : πn(X,A) ≈ πn−1(ΩX ′).In case n = 1, ∂ : π1(X) → π0(ΩX ′) is surjective [because π0(A) = 0], and so

E′ is path-connected. If n > 1, E′ is path connected because π : 0(ΩX ′) = 0.Therefore E′ is a path-connected pointed space. Since πq(ΩX ′) = 0 for q ≥ n itfollows from the exactness of the homotopy sequence of the fibration, p′ : E′ → X,that p′# : πq(E

′) → πq(X) is an isomorphism for q > n and a monomorphism forq = n.

Because λ : πq(X,A) → πq−1(ΩX ′) is a bijection for q ≤ n, with the only non-trivial case when q = n, it follows from Lemma 4.5.11 of [24] and the commutativity,up to a sign, of the first diagram of the proof that b′# : πq(A) → πq(E

′) is anisomorphism for 1 ≤ q < n and an epimorphism for q = n. Therefore b′ and p′ havethe properties required of an n-factorization of i.

57

Lemma 4.24 then readily extends to the following corollary.

Corollary 4.25. Let g : X ′ → X be a simple map between path-connected pointedspaces such that for some n ≥ 1 the map g# : πq(X

′)→ πq(X) is an isomorphismfor 1 ≤ q < n−1 and an epimorphism for q = n−1. Then there is an n-factorization

X ′b′−→ E′

p′−→ X of g such that p′ is a principal fibration of type (π, n) for someabelian group π.

Proof. Let Z be the reduced mapping cylinder of g. That is, the mapping cylinderof g|x′0 : x′0 → x0 has been collapsed to a point. Then (Z,X ′) is a pointed pair ofpath-connected spaces (n−1)-connected and with simple inclusion map i : X ′ → Z.

By Lemma 4.24, there is an n-factorization X ′ b′′−→ E′′p′′−→ Z of i such that p′′ is

a principal fibration of type (π, n). Let p′ : E′ → X be the restriction of p′′ to X.Then E′ → E′′ is a homotopy equivalence, so there is a map b′′ : X ′ → E′ suchthat b′′ is homotopic to the composite X ′ b

′′

−→ E′ → E′′. The composition p′ b′′ iseasily seen to be homotopic to g. By the homotopy lifting property of p′, there isa map b′ : X ′ → E′ homotopic to b′′ such that p′ b′ = g. It is easy to verify that,

X ′b′−→ E′

p′−→ X has the required properties.

The last corollary gave us a way to find a principal fibration between two pathconnected spaces. While the principal fibration is one of the conditions for a Moore-Postnikov fibration, it remains to check whether the remaining two are also satisfied.

Theorem 4.26. Let f : Y ′ → Y be a simple map between path connected pointedspaces. There is a Moore-Postnikov factorization pq, Eq, fqq≥1 of f such that forn ≥ 1 the sequence,

Y ′fn−→ En

p1···pn−−−−−−→ Y,

is an n-factorization of f .

Proof. By induction on q, we prove that the existence of a sequence, p1, Eq, fqq≥1,such that:

(1) for n = 1 the sequence Y ′ f1−→ E1p1−→ Y is a 1-factorization of f ;

(2) for n > 1 the sequence Y ′ fn−→ Enpn−→ En−1 is an n-factorization of fn−1;

(3) for n ≥ 1, p1 is a principal fibration of type (πn, n) for some πn.Once such a sequence pq, Eq, fqq≥1 has been found, it is easy to verify that

it is a Moore-Postnikov factorization of f with the desired property. Therefore welimit ourself to proving the existence of such a sequence.

By Corollary 4.25, with n = 1, there is a 1-factorization Y ′f1−→ E1

p1−→ Y off with p1 a principal fibration of type (π1, 1) for some π1. This defines p1, E1

and f1. Assume that pq, Eq, fq is defined for 1 ≤ q < n, where n > 1, tosatisfy (1), (2) and (3) above. Again by Corollary 4.25, there is an n-factorizationY ′

fn−→ Enpn−→ En−1 of fn−1 such that pn is a principal fibration of type (πn, n) for

some πn. Then Pn, En and fn have the desired properties.

This leaves us in a position to actually prove what we wanted with regards toreplacing a space Y ′ with a Postnikov fibration.

58

Corollary 4.27. Let Y ′ be a simple path-connected pointed space. Then Y ′ hasa Postnikov factorization pq, Eq, fqq≥1 in which πq(En) = 0 for q ≥ n and fn :Y ′ → En is an n-equivalence.

Proof. If Y ′ is a simple space, then the constant map Y ′ → y0 is a simple map.The result follows then from Theorem 4.26.

4.3. The Final Obstruction. Whether the reader has comfortably absorbed thematerial or feels left at the deep end of the pool, we are nearly done. This section iswhere we make use of Postnikov factorizations to extend maps. In this section wewill often use map pair and the reader might want to refer back to Definition 4.13

Using maps and map pairs one can define a related concept: homotopy pairs.These will be used in a sequence of three theorems that ultimately, in Theorem 4.34,proves the existence of a bijection between the inclusion map and a fibration.

Definition 4.28. A homotopy pair H : f0 ' f1 is a commutative square,

P ′′ × I Q′′

P ′ × I Q′

H′′

α×idI β

H′′

Before we can use homotopy pairs to prove Theorem 4.32 we need to talk aboutweak homotopy equivalences.

Definition 4.29. Let f be a continuous map X → Y . Then, if f is an n-characteristic for all n ≥ 1, we say that f is a weak homotopy equivalence.

Let us now prove a two things regarding these weak homotopy equivalences. Firstoff, a homotopy equivalence in the usual sense is also a weak homotopy equivalence.

Proposition 4.30. A homotopy equivalence is a weak homotopy equivalence.

Proof. This follows from Corollary 7.3.15 of [24].

We now show that the convergent factorization of Definition 4.20 can be used tocreate a weak homotopy equivalence.

Corollary 4.31. Let pq, Eq, fqq≥1 be a convergent factorization of a map f :Y ′ → Y and let f ′ : Y ′ → E∞ be the map such that aq f ′ = fq for q ≥ 1 anda0 f ′ = f . Then f ′ is a weak homotopy equivalence.

Proof. For any 1 ≤ n ≤ ∞ there is a q such that aq and fq are both n-equivalences,by Theorem 4.21. Because aq f ′ = fq, then f ′ is also an n-equivalence. Since thisholds for all n, it means that f ′ is a weak homotopy equivalence.

To understand Theorem 4.32 please note that a weak fibration is a fibration thathas the homotopy lifting property with respect to only cubes, Inn≥0, rather thanall spaces.

Theorem 4.32. Let (X,A) be a relative CW-complex, with inclusion map i : A →X, and let p : E → B be a weak fibration. Given a map f : X → E and a homotopypair H : i× 1I → p consisting of a homotopy H ′ : X × I → B starting at p f anda homotopy H ′′ : A× I → E starting at f i, there is a homotopy H : X × I → Estarting at f such that H ′ = p H and H ′′ = H (i× 1I).

59

Proof. Let g : X × 0 ∪ A× I → E be the map defined by g(x, 0) = f(x) for x ∈ Cand g(a, t) = H ′′(a, t) for a ∈ A and t ∈ I. Then H ′ is an extension of p g, andby the standard stepwise-extension procedure over the successive skeleta of (X,A)there is a map H : X × I → E such that p H = H ′ and H|X × 0 ∪ A × I = g.Then H has the desired properties.

Theorem 4.33. Let (X,A) be a relative CW-complex, with inclusion map i : A →X, and let g : p1 → p2 be a weak homotopy equivalence between weak fibrations.Given a map pair f : i→ p1 and a lifting h : X → E2 of the map pair g f , thereis a lifting f : X → E1 of f such that g′′ f and h are homotopic relative to g f

Proof. The proof involves two applications of Theorem 3.67 and then two applica-tions of Theorem 4.32. We shall not make specific reference to these when they areinvoked.

We have a commutative diagram

A E1 E2

X B1 B2

f ′′

i

g′′

p1 p2

f ′ g′

in which g′′ and g′ are weak homotopy equivalences, and we are given a maph : X → E2, such that h i = g′′ f ′′ and p2 h = g′ f ′. Then there is amap f : X → E1 such that f i = f ′′ and a homotopy G′′ : g′′ ¯f ' h rel A.The map p1 ¯f and f ′ agree on A and p2 G′′ is a homotopy relative to A fromg′p1 ¯f = p2g′′ ¯f to g′f ′ = p2h. Therefore, there is a homotopy F ′ : p1 ¯f ' f ′rel A and a homotopy H ′ : g′ F ′ ' p2 G′′ rel A× I ∪X × I

Let F ′′ : X × I → E1 be a lifting of F ′ such that F ′′(x, 0) = ¯f(x) for x ∈ X andF ′′(a, t) = f ′′(a) for a ∈ A and t ∈ I. Define f : X → E1 by f(x) = F ′′(x, 1). Weshow that f has the desired properties. It is clearly a lifting of f .

The maps g′′ F ′′ and G′′ are homotopies relative to A from g′′ ¯f to g′′ fand to h, respectively, and H ′ is a homotopy from p2 g′′ F ′′ to p2 G′′ relA× I ∪X × I. Since there is a homeomorphism of (X × I × I, A× I × I) onto itselftaking X × (I × I ∪ 0 × I) onto X × I × 0, there is a lifting H ′′ of H ′, which is ahomotopy from g′′ F ′′ to G′′ rel X × 0 ∪ A × I. Then the map H : X × I → E2

defined by H(x, t) = H ′′(x, 1, t) is a homotopy from g′′ f to h relative to g f .

Theorem 4.34. Let (X,A) be a relative CW-complex, with inclusion map i : A →X, and let g : p1 → p2 be a weak homotopy equivalence between weak fibrations.Given a map pair f : i→ p1, the map pair g induces a bijection

g′′# : [X;E1]f ≈ [X;E2]gf .

Proof. The fact that g′′# is surjective follows immediately from Lemma 4.33. Thefact that g′′# is injective follows from application of Lemma 4.33 to the relativeCW-complex (X,A)× (I, I).

Let us now look at how to extend a map f : A → Y to a map from X to Y .Our setting is that we have a (n− 1) connected space Y and a CW-complex (X,A)with a map f : A → Y . First, we replace Y with a Postnikov factorization, whichrespects the homotopy structure of Y . Within the factorization, we can inductivelylift the map, one level at a time. These lifts are particularly simple, since the

60

fibration by definition is a Principal fibration. Finally, as we are working witha Postnikov factorization, it follows that the lifting problem is equivalent to theextension problem. Now, let us prove the theorem.

Theorem 4.35. Let ι ∈ Hn(Y, y0;π) be n-characteristic for a simple (n − 1)-connected pointed space Y where n ≥ 1, and let (X,A) be a relative CW-complexsuch that Hq+1(X,A;πq(Y, y0)) = 0 for q > n. A map f : A→ Y can be extendedover X if, and only if, δf∗(ι) = 0 in Hn+1(X,A;π).

Proof. First, let i : A → X. For Y we can apply Corollary 4.27 to get a Postnikovfactorization pq, Eq, fqq≥1 in which πq(En) = 0 for q ≥ n. Since all Moore-Postikov factorizations are convergent, then Corollary 4.31 gives a weak homotopyequivalence f ′ : Y → E∞. Let aq : E∞ → Eq and in particular let a0 : E∞ → B.Then, Theorem 4.34 tells us that the lifting problem for a map pair from i to pis the equivalent to the one for a map pair from i to a0. Furthermore, since y0 isa point then the lifting problem is equivalent to the extension problem for a mapf ′′ : A→ E∞. Therefore we can solve the extension problem by solving the liftingone.

To solve the lifting problem we have to find a sequence of maps, fq : X → Eq,such that

(1) each fq is a lifting of fq,

(2) pq+1 fq+1 = fq.Let g(fq) be the map pair that forms the commutative square

A Eq+1

X Eq

f ′′q+1

i pq+1

fq

Then, a lifting of g(fq) has to be such that:(1) f1 is a lifting of f1;

(2) for q ≥ 1, fq+1 is a lifting of g(fq).In particular, these has to be such that f1 : X → E1 is an extension of α1 f ′′ andfq+1:X→Eq+1

for q ≥ 1 is a lifting of the map pair g(fq) : i→ pq+1 consisting of

A Eq+1

X Eq

aq+1f ′′

i pq+1

fq

Because pq, Eq, fqq≥1 is a Postnikov factorization, then pq is a principal fibrationof type (πq(Y, y0), q + 1). Hence, the obstruction to lifting g(fq) is an element ofHq+1(X,A;πq(Y, y0)) ≈ Hn+1(X,A;π). Therefore, the obstruction to extendingf ′′ is defined, and the (q + 1)th obstruction is a subset of Hn+1(X,A;π) Since Yis (n− 1)-connected, the lowest non-trivial obstruction lies in Hn+1(X,A;π) .

Since ι is n-characteristic for Y , it follows from Theorem 4.18 that the obstructionin Hn+1(X,A;π) is ±δf ′′∗(ι). Therefore, the obstruction vanishes if and only ifδf∗(ι) = 0.

61

63

5. Proof of Theorem 2.16

In Section 5 we looked at order isomorphisms and asked whether it was possibleto combine different isomorphisms into a single one. With the algebraic topol-ogy developed, it is finally possible answer this with the theorem by Chichilniskyand Heal. We restate this theorem here for easier reference. After proving Theo-rem 2.16 we will take look at an interesting connection between Arrow’s theoremand Theorem 2.16.

Theorem 2.16 Let P be the space of preference profiles. A continuous functionφ :∏i∈I P → P , which satisfies anonymity and unanimity, exists if and only if P

is contractible.

The idea of the proof of sufficiency is to first define φ as the vector average ona convex set K, such that P ⊂ K. Technically K could be defined the convex hullof P . We then extend the identity map of P to K, by use of Theorem 4.35. Thisextended map then gives a retract from K to

∏P .

Proof of sufficiency. To prove sufficiency we will show the following. Let K be aconvex set such that P ⊂ K. Then the map, φ :

∏K → K, defined by

φ(p1, . . . , pn) =1

n

n∑i=1

pi for (p1, . . . , pn) ∈ K,

is a continuous social choice rule, respecting anonymity and unanimity.We show that φ satisfies each of the properties in turn. First let (p′1, . . . p

′n) be a

permutation of (p1, . . . , pn). Since addition is commutative on K then φ(pini=1) =φ(pjnj=1). Hence, φ is invariant under permutations of (p1, . . . , pn) and there-fore satisfies anonymity. Since φ is the usual vector average we have for any(pk, . . . , pk) ∈ K,

φ(pk, . . . , pk) = pk,

and therefore φ is unanimous. Finally,∏ni=1K inherits its product topology from

the topology on K, see [21, p. 86]. It follows that 1n

∑ni=1 pi is continuous

The idea now is to retract the image of φ to P . A retraction is a space A ⊂ Xsuch that there is a continuous map r with the property r i = idA. This iswhere the obstruction theory comes in. Since P is contractible then the identitymap is trivial in every cohomology group Hn(P ;π). Therefore, δ id∗P (ι) = 0 andidP : P → P can be extended to a map r : K → P .

Since r|P = idP we can use r to retract K to P and form the function

(6) φ(p1, · · · , pn) = r

(1

n

n∑i=1

pi

).

By the previous paragraphs, we know that averaging is a continuous social choicerule that respects anonymity and unanimity. Therefore the φ defined in Equation 6is also a continuous, anonymous and unanimous social choice rule, completing theproof of sufficiency.

As the question of sufficiency has been settled we now turn to proving necessity.We must show that the existence of a social choice rule implies that πn(P ) = 0 forall n ≥ 1, which implies that P is contractible. To do this we will rely on someadditional algebraic structure. Specifically, we will use free and torsion subgroups.

64

Definition 5.1. Let G be an abelian group. The elements in G of finite orderforms a subgroup, known as the torsion subgroup.

Definition 5.2. Let G be an abelian group. We say that G is torsion free or simplyfree, if there is no nontrivial torsion subgroup.

Proposition 5.3. Each group can be divided into free and torsion groups.

Proof. This follows from the definitions.

Proof of necessity. To prove necessity we will show the following. If the preferencespace P admits a continuous social choice rule, satisfying anonymity and unanimity,then P must be contractible.

By assumption there is a continuous map φ : P k → P , which satisfies unanimityand anonymity. This map has a corresponding induced map φ∗ : πi(

∏P )→ πi(P )

for all i ≥ 1. Because of the Huruwicz Isomorphism Theorem, Theorem 4.6, weknow that the first nontrivial group πi(P ) is isomorphic to the abelian group Hi(P ),for i ≥ 2, with the case i = 1 left for later. The idea now is to show that, for eachi ≥ 2, πi(P ) is the trivial group. Then, for all q ≥ 2, we can split πq(P ) into atorsion and a torsion free part. If both of these are trivial then so is πq(P ), byProposition 5.3 .

To show that the free part of πi(P ) is trivial we will rely on three observations.First, by our construction, the group πi(

∏P ) is isomorphic to the product of k

copies of πi(P ). Second, for any x ∈ πi(P ) we have, for (x, x, . . . , x) ∈ πi(∏P ),

φ∗(x, x, . . . , x) = x,

by unanimity. Finally, since both πi(P ) and πi(∏P ) are groups, then they each

have binary operators and identity elements. Specifically let • be the binary opera-tion on πi(

∏P ) and let e be the identity element of πi(P ), implying that (e, . . . , e)

is the identity element of πq(∏P ). By the group structure on πi(

∏P ) we have

(x, x, . . . , x) = (x, e, . . . , e) • (e, x, e, . . . , e) • · · · • (e, e, . . . , e, x).

Now, let ∗ be the binary operation on πi(P ), then since φ∗ is a homomorphism wehave the following equation,

(7) φ∗(x, x, . . . , x) = φ∗(x, e, . . . , e) • (e, x, e, . . . , e) • · · · • (e, e, . . . , e, x)= φ∗(x, e, . . . , e) ∗ φ∗(e, x, e, . . . , e) ∗ · · · ∗ φ∗(e, . . . , e, x).

As we require the social choice rule to be anonymous, then all permutations of(x, e, . . . , e) must be equivalent. Consequentially, the right hand side of Equation 7can be rewritten as n(x, e, . . . , e), since πi(P ) is abelian. At the same time, una-nimity assures that the left hand side of Equation 7 must be equal to x. Therefore,we have

x = nφ∗(x, e, . . . , e) = ny,

for some y and n.Assume now that πi(P ) has a free part. Then there must be a generator x ∈

πi(P ) of that free part. However, by anonymity and unanimity xmust be equivalentto n ∗ y for some n and y. Since φ is unanimous, then x would have to be divisibleby every integer, which is a contradiction if x 6= e. As the only possible generatoris e then the free part of πq(P ) must be trivial, for all q ≥ 2.

Given that the torsion free part of πq(P ) is trivial, then the torsion part mustbe nontrivial, unless πq(P ) = 0. Now consider x a generator in the torsion part of

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πi(P ). Then x ∈ Zk, for some k. However, since x = nφ∗(x, e, . . . , e) for any k ∈ Z.It follows that Zpi is zero for all pi, which implies that the torsion part of πi(P )is trivial. Since πi(P ) has no torsion and no free part, it is the zero group, for alli ≥ 2.

In case i = 1, and πi(P ) is abelian, then the group π1(P ) is isomorphic to H1(P ).Thus, if we can show that π1(P ) abelian, the rest of the proof above applies wheni = 1. Therefore, we show next that the existence of the map φ implies immediatelythat π1(P ) is abelian. Because φ∗ is a group homomorphism, we have for any twoelements, x and y in π1(P ), and for any k ≥ 1

x ∗ y = kφ∗(x, e, . . . , e) ∗ kφ∗(e, y, e, . . . , e) = φ∗(kx, ky, e, . . . , e),

. But by the symmetry condition on φ we have

φ∗(kx, ky, e, . . . , e) = φ∗(ky, kx, e, . . . , e) = y ∗ x.

Therefore, x ∗ y = y ∗ x, and π1(P ) is abelian. This completes the proof thatπi(P ) = 0 for all i ≥ 1.

Since P is a connected CW complex and πi(P ) = 0 for all i ≥ 1, then P mustbe contractible.

Economically this result is handy. If we, for instance, relate back to Section 2where we defined the preference space P , then we can see the following. By con-struction, P is the sphere S1 but S1 is not contractible. Specifically, π1(S1) = Z.Therefore, we can see that there is no well behaved social choice rule on P , given aquite standard way of constructing the economical framework.

Furthermore, each social choice rule respecting the conditions of Theorem 2.16must be homotopically equivalent to Equation 6. This is a direct consequence ofTheorem 2.16 since it states that a space of preference profiles must be contractibleif there is a social choice function on it. For a general overview of the continuoussocial choice the reader is recommended to take a look at [15]. We will instead lookat more mathematical aspects of continuous social choice.

6. Intriguing Equivalences

Obstruction theory excluded, the proof of Theorem 2.16 is not that interesting,mathematically, and quite simple. It would be easy to take this simplicity as anindication of triviality. However, there is a very interesting connection betweencontinuous social choice rules and Brouwer’s Fixed Point Theorem, which is statedin Theorem 6.1. Curiously, there is also a connection between Brouwer’s FixedPoint Theorem and the original social choice theorem by Arrow. We will restatetwo theorems by Tanaka [26] as Theorem 6.3 and Theorem 6.5, which shows this.

First of all a fixed point, x0 ∈ X is such that for a function f : X → X we have

f(x0) = x0.

We now state the Brouwer Fixed point Theorem for reference.

Theorem 6.1 (Brouwer’s Fixed Point Theorem). Every continuous map f : En →En has at least one point x0 ∈ En such that f(x0) = x0.

Proof. For a proof see Corollary 2.15 of [16]; Theorem 4.7.5 of [24] or Theorem6.6.1 of [14]. The last contains a number of equivalent results.

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As recently said, the existence of a Chichilnisky social choice rule on a spaceconnected to Brouwer’s Fixed point theorem on that space. Strangely, the twotheorems actually imply each other. That is: the existence of a continuous, anony-mous and unanimous social rule is equivalent to the existence of a fixed point. Werestate the proof of [9], which is for two individuals. Do note that the proof isslightly edited for readability within the context of this essay.

Theorem 6.2. A social choice rule φ : P×P → P which is continuous, anonymousand respects unanimity exists if, and only if, there exists a continuous map f fromthe closed unit disk, D, of R2 into itself without fixed points.

Proof. First we shall prove that existence of a map φ : P × P → P is equivalent tothe existence of an extension of continuous map g : ∂D → ∂D to a map f : D → ∂D.Then, we note that such an extension exists if and only if g is homotopic to someconstant map on ∂D, and that this latter result, by 6.6.1 of Dieck [14], is equivalentto Brouwer’s Fixed Point Theorem on the disk. We shall therefore have proven thatthe non-existence of a continuous anonymous rule φ : P × P → P that respectsunanimity is equivalent to the nonexistence of maps from the disk into itself withoutfixed points .

To begin the proof of the first part let φ : P × P → P be a social choice rule.As argued in Section 2.2, P = S1. This implies that P × P = S1 × S1, whichis the torus. Let p0 be the distinguished point of this torus. Furthermore, letA(p) = (p0, p), B(p) = (p, p0), and let ∆ be the diagonal map, these are depictedin the left picture of Figure 6.

Note that we can always define on Im ∆ ∪ ImA ∪ ImB, or ∆ ∪A ∪B for short,a social choice rule, φ. Specifically, if this social choice rule is defined by

φ/∆ = id∆, φ(p0, p) = p0 and φ(p, p0) = p0, ∀p ∈ S1,

then φ satisfies all the conditions for a well behaved social choice rule. Therefore,on the set ∆ ∪ A ∪ B an adequate rule always exists. If we could extend φ from∆∪A∪B to the interior, which is the triangle T of Figure 6. Then, we would haveconstructed a social choice rule φ : S1 × S1 → S1 satisfying the desired conditions,because S1×S1 is the union of T and the symmetric set of T , i.e., S1×S1 = T ∪Tswhere Ts = (x, y) : (y, x) ∈ T . Therefore, if φ can be continuously defined on T asan extension of the map on ∂T then since S1 × S1 = T ∪ Ts, φ : S1 × S1 → S1

could be defined satisfying all conditions. Note that T and D are homeomorphic,i.e., there exists a continuous one to one map from T onto D. Therefore, the socialchoice problem is equivalent to that of extending a map g : ∂D → ∂D to a mapf : D → ∂D. We shall now move on to the second part of the argument.

A continuous map g : ∂D → ∂D can be extended to another continuous mapf : D → ∂D if and only if the map g can be deformed to a constant functionmapping all of ∂D into a given x0 in ∂D. Furthermore, this result is equivalent toBrouwer’s Fixed Point Theorem. For a proof of this equivalence see Theorem 6.6.1of Dieck [14].

Finally, we now show that if a continuous map h is defined on the boundary of T ,h : ∂T → S1 respecting unanimity and anonymity, then h cannot be deformed intoa constant map from ∂T into a fixed point x0 in S1. Unanimity requires that therestriction of h to ∆, h/∆, covers S1 exactly once; in fact since ∆ is homeomorphicto S1, h/∆ can be thought of as the identity map from S1 to S1. Since h/∆wraps around S1 in the opposite direction once, in order that h could be deformed

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P P

P P

T

B

A

B

A∆ PB ∆

A

Figure 6. The left picture depicts the standard torus, which is recre-ated by joining the A and B edges together with their opposite side.The point P is the distinguished point (p0, p0). The left picture showsthe same construction with all, equivalent, corners shown as one point.

to a constant map, h/A ∪ B must wrap around S1 in the opposite direction once.However, by anonymity h/A = h/B. This implies that if h/A wraps around S1 once(n-times) then h/A ∪B does so twice (2n times). If h/A does not wrap around S1

at least once, neither will h/A∪B, so that in all cases h cannot be deformed into aconstant map from ∂T into S1 when it is both anonymous and respects unanimity.Therefore such an h cannot be extended to all of T .

Since we have already shown that the social choice paradox is equivalent tothe impossibility of extending a map from ∂ D to D, and that the problem ofextendability of maps from ∂D to D is equivalent to a fixed point theorem, ittherefore follows that the social choice problem is equivalent to the problem ofexistence of fixed points of continuous maps from the disk to itself. This completesthe proof.

This shows that the social choice paradox as stated by Chichilnisky is equivalentto Brouwer’s Fixed Point Theorem in two dimensions. Supposedly, one can furtherextend this result to higher dimensions, as stated in Chichilnisky [9]. However,because of what we will look at next, two dimensions suffices.

While it might not be an surprise that Chichilnisky’s formulation of the socialchoice rule is equivalent to Brouwer’s Fixed Point Theorem. The following theo-rems prove a far more surprising equivalence: one between Brouwer’s Fixed PointTheorem and Arrow’s Theorem. Since Arrow’s model is discrete this is perhaps amore interesting connection.

This essay will use proofs from Tanaka [26], which establishes an equivalencebetween the theorems for n = 2. With this we will have proven that both themodel by Chichilnisky and Heal, and by Arrow are equivalent to Brouwer’s theoremfor n = 2. Thereby, we will also have proven that Arrow’s and Chichilnisky’sformulations are the same in the two dimensional case. There are reasons to believethat n = 2 is not a special case. Instead, the continuous and discrete social choiceproblems are probably equivalent for all n, see Baryshnikov [5], but this is beyondthe scope of the essay.

The following theorems, Theorem 6.3 and Theorem 6.5, with Lemma 6.4 in be-tween constitute the proof by Tanaka. The first theorem establishes an intermediateequivalence with Brouwer’s Fixed Point Theorem. It is then shown that Arrow’stheorem is equivalent to this intermediate theorem. From this the equivalence ofArrow and Brouwer follows. We state this in Theorem 6.5.

Theorem 6.3. The following two statements are equivalent.

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(1) If there exists a continuous function from an n-dimensional ball Dn to an(n − 1)-dimensional sphere Sn−1, (n = 2), F : Dn → Sn−1, then thefollowing function, which is obtained by restricting F to the boundary Sn−1

of Dn, F |Sn−1×Sn−1 → Sn−1 is homotopic to a constant mapping. Sincethe degree of mapping of a constant mapping is zero, the degree of mappingof F |Sn−1 is zero.

(2) (The Brouwer fixed point theorem) Any continuous function from Dn toDn (n = 2), G : Dn → Dn, has a fixed point.

Proof. (1)→ (2) Assume that G has no fixed point. Since we always have v 6= G(v)at any point v in Dn, there is a half line starting G(v) across v. Let F (v) be theintersection point of this half line and the boundary of Dn, which is Sn−1. Then,we obtain the following continuous function from Dn to Sn−1

F : Dn → Sn−1.

In particular, we have F (v) = v for v = Sn−1. Therefore, F |Sn−1 is an identitymapping. But, because an identity mapping on Sn−1 is not homotopic to anyconstant mapping, it is a contradiction.

(2)→ (1) We show that if there exists a continuous function F from Dn to Sn−1,(1) of this theorem is correct whether a continuous function G from Dn to Dn hasa fixed point or not. Define ft(v) = F [(1− t)v], (0 ≤ t ≤ 1) for any point v of Sn−1.Then, we get a continuous function ft : Sn−1 → Sn−1. (1 − t)v is a point whichdivide t : 1− t a line segment between v and the center of Dn, and it is transferredby F to a point on Sn−1. We have f0 = F |Sn−1, and f1 = F (0) is a constantmapping whose image is a point F (0). Since ft is continuous with respect to t, itis a homotopy from F |Sn−1 to a constant mapping, and the degree of mapping ofF |Sn−1 is zero.

Lemma 6.4. Suppose that there exists a social welfare function F : R × R → S1

which satisfies transitivity, Pareto principle and independence of irrelevant alter-natives. If F has no dictator, then the degree of mapping of F |∆∪A∪B is not zero,and hence it is not homotopic to a constant mapping.

Proof. The proof is quite technical and the reader is referred to Lemma 1 of [26].

Thus, Tanaka has shown that the conditions by Arrow creates a situation satis-fying the conditions of Theorem 6.3. And therefore the following result holds.

Theorem 6.5. The non-existence of social welfare function which satisfies transi-tivity, Pareto principle, independence of irrelevant alternatives and has no dictator(the Arrow impossibility theorem) is equivalent to the Brouwer fixed point theorem.

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The author can be reached at anton.vernersson@outlook.com