Homotopy Groups of the Spheres - Glannabijou/HomotopyProject.pdf · 2018-09-19 · 1 HIGHER...

Homotopy Groups of the Spheres

Fourth Year Project

Author: Navid Nabijou

Supervisor: Professor Richard Thomas

CID: 00600476

Abstract: The computation of the higher homotopy groups of the spheresis a fundamental problem in algebraic topology and an active area of currentresearch. In this article we use this problem to motivate the development ofsome tools and concepts in homotopy theory.

This is my own unaided work unless otherwise stated

Signed: Date:

Preface

The main idea of this article is to use the computation of the homotopy groupsof the spheres as motivation for the development of some concepts and tools inhomotopy theory.

We assume that the reader has taken a first course in algebraic topology,including at the very least the fundamental group, covering spaces, basic singularhomology and CW complexes (§§1-2 of [Hat01] should be more than sufficient).Occasionally we will make use of more advanced topics (e.g. universal coefficienttheorem); when we do, we always include references for the benefit of readersunfamiliar with the material.

Homotopy theory has deep connections to many other areas of study, andwe often make passing references to these in the text. Our hope is that readersfamiliar with the topics mentioned will be able to gain a sense of the full richnessof the theory we are developing, while on the other hand those who have not beenexposed to these topics might be intrigued enough to investigate the referencesprovided.

We tend to spell out arguments in a little more detail and spend more timeon motivation than one usually finds in the literature, in the hope of makingthe presentation accesible to a somewhat broader audience.

Notational conventions By “space” we mean topological space (or often,based topological space) and by “map” we mean continuous map (or often,basepoint-preserving continuous map). The n-skeleton of a CW complex X isdenoted by Xn. The unit interval [0, 1] is denoted by I.

Acknowledgements I am indebted to my supervisor, Richard Thomas, formany stimulating discussions and encouraging words, and for tolerating withgood humour my frequent changes of interest.

The cover image illustrating the Hopf fibration is due to Niles Johnson.

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CONTENTS

Contents

1 Higher Homotopy Groups 61.1 Definition of πn(X) . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2 Functoriality and homotopy invariance . . . . . . . . . . . . . . . 71.3 Basepoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.4 Higher homotopy groups are abelian . . . . . . . . . . . . . . . . 9

2 Spheres 10

3 Homotopy Groups of S1 12

4 Cellular Approximation 12

5 Hurewicz Theorem and Degree 145.1 The Hurewicz homomorphism . . . . . . . . . . . . . . . . . . . . 145.2 n-connected spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 155.3 Cubical homology . . . . . . . . . . . . . . . . . . . . . . . . . . . 165.4 The Hurewicz theorem . . . . . . . . . . . . . . . . . . . . . . . . 175.5 Degree theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

6 Interlude: the Road Ahead 20

7 Fibrations 217.1 The homotopy lifting property . . . . . . . . . . . . . . . . . . . 217.2 Replacing arbitrary maps by fibrations . . . . . . . . . . . . . . . 23

8 Suspensions and Loopspaces 248.1 Suspension-Loopspace adjunction . . . . . . . . . . . . . . . . . . 248.2 Freudenthal suspension and stable homotopy groups . . . . . . . 28

9 Long Exact Sequences 309.1 Puppe sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

9.1.1 Pointed sets . . . . . . . . . . . . . . . . . . . . . . . . . . 309.1.2 Statement of the Puppe sequence . . . . . . . . . . . . . . 319.1.3 Proving the Puppe sequence . . . . . . . . . . . . . . . . . 32

9.2 The long exact sequence of a fibration . . . . . . . . . . . . . . . 36

10 The Hopf Fibration 3810.1 Definition of the Hopf fibration . . . . . . . . . . . . . . . . . . . 3810.2 Picturing the Hopf fibration . . . . . . . . . . . . . . . . . . . . . 4010.3 Applications of the Hopf fibration . . . . . . . . . . . . . . . . . . 41

11 Postnikov Analysis 4111.1 Serre’s method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4111.2 Killing off homotopy groups . . . . . . . . . . . . . . . . . . . . . 4211.3 Recomputing π3(S2) . . . . . . . . . . . . . . . . . . . . . . . . . 44

4

CONTENTS

12 The Leray-Serre Spectral Sequence 4512.1 The spectral sequence associated to a filtration . . . . . . . . . . 4612.2 The Leray-Serre spectral sequence . . . . . . . . . . . . . . . . . 53

13 Applying Spectral Sequences to Spheres 5513.1 π3(S2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5513.2 π4(S2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

A CW Complexes 60

B Eilenberg-MacLane Spaces 61

References 64

Index 65

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1 HIGHER HOMOTOPY GROUPS

1 Higher Homotopy Groups

A first course in algebraic topology typically begins with the fundamental group.However, following a treatment of the basic definitions and examples (and, usu-ally, the connection to covering spaces) the tendency is to quickly move on to thestudy of homology and cohomology theories. The natural higher-dimensionalanalogues of the fundamental group, obtained simply by replacing S1 by Sn

in the definition, are rarely considered and often not mentioned at all. This isin spite of the fact that the study of such groups has motivated (and indeedcontinues to motivate) a considerable chunk of modern algebraic topology.

The aim of this article is to introduce the reader to the study of these higherhomotopy groups. We structure our development of the theory around themotivating example of the homotopy groups of the n-spheres.

We will see that, unlike the situation for homology, the structure of thesegroups is extremely complicated, and their computation is a highly nontrivialmatter (indeed, the general problem is still very much unsolved!).

Though this might seem disappointing at first, it should really be seen asa cause for celebration: for in our attempts to calculate these groups, we willuncover a rich theory which sheds a great deal of light on the intrinsic topologyof the spheres; furthermore, many of the tools which we are forced to developin order to carry out our computations end up being extremely useful moregenerally in other areas of algebraic topology.

We begin with the definition and basic properties of the higher homotopygroups. Our discussion is based primarily on §4.1 of [Hat01].

1.1 Definition of πn(X)

Recall that if X is a space with a chosen basepoint, then the fundamentalgroup of X is denoted π1(X) and is defined to be the set of homotopy classesof basepoint-preserving maps S1 → X (which we call loops in X).

Aside. In the above we require our homotopies to be basepoint-preserving. Wemust also, of course, choose a basepoint for S1, but this choice does not reallymake any difference. In fact, as long as X is path-connected, the choice ofbasepoint for X makes no difference either, so we are justified in suppressingthis information in the notation π1(X). In general, we tend thoughout thisarticle to tacitly assume our spaces have basepoints and our maps are basepoint-preserving, only being explicit with the notation when there is possibility forconfusion.

There is a natural group structure on π1(X) given by concatenating loops:if f and g are loops in X (beginning and ending at the chosen basepoint), wedefine f · g to be the loop given by first traversing f and then traversing g(note the order). This operation respects homotopy classes, so it descends to aproduct on π1(X) which makes it into a group.

There is an alternative description of this concatenation operation whichgeneralises immediately to higher dimensions. Suppose that f, g : S1 → X are

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loops in X. Then f · g is equal to the composition

S1 → S1 ∨ S1 → X

where the first map is the “pinch” obtained by collapsing two antipodal pointsof S1 together, and the second map is given by taking f on the first factor andg on the second factor (it is well-defined because the point in common of thetwo S1 factors is the basepoint, and f and g have the same value there).

We are now ready for our main definition. For any space X, the nth ho-motopy group of X is denoted πn(X) and is defined to be the set of homo-topy classes of maps Sn → X. The group operation is defined as follows: iff, g : Sn → X then the sum f + g is defined to be the composition

Sn → Sn ∨ Sn → X

where the first map is the “pinch” defined (in analogy to the pinch map for S1)by collapsing an equatorial Sn−1 ⊆ Sn to a point, and the second map is givenby taking f on the first factor and g on the second. Again, this is well-definedbecause the point in common of the two Sn factors is the basepoint, which fand g agree on (recall we are assuming all our maps are basepoint-preserving).

Using similar arguments to those used in the case of the fundamental group,we can show that this operation respects homotopy classes, so descends to asum on πn(X), and we can also show that this sum makes πn(X) into a group.We leave the verifications as worthwhile exercises for the reader. Notice thatwe use additive notation: the reason for this is given in Proposition 1.2.

Aside. Recall that S0 consists of two discrete points (see §2). Thus π0(X)makes sense as a set, and is in bijective correspondence with the set of pathcomponents of X. It also has a distinguished element, namely the path compo-nent containing the basepoint. However, it has no natural group structure, soone must be careful when dealing with it. For an example of where this becomesimportant, see Corollary 9.3

1.2 Functoriality and homotopy invariance

One of the most important facts about this construction is that it is functorial:given a map f : X → Y there is an induced group homomorphism f∗ : πn(X)→πn(Y ), defined by f∗[g] = [f g], and this clearly satisfies the usual functorialityproperties:

(fg)∗ = f∗g∗ id∗ = id

If f : X → Y is a homeomorphism then it follows purely formally from functo-riality that f∗ is an isomorphism, so we see that πn is a topological invariant.

In fact more is true: we only need to assume that f : X → Y is a homotopyequivalence to be able to conclude that f∗ is an isomorphism. This is more orless clear from the definition of f∗: if g : Y → X is a homotopy inverse for f ,then g∗ is a two-sided inverse for f∗. Therefore πn is a homotopy invariant, andas such we will implicitly be working in the homotopy category; that is, we willonly care about topological spaces up to homotopy equivalence.

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1.3 Basepoints

Implicit in all of the discussion above is a chosen basepoint for X. We will nowshow that (assuming X is path-connected) this choice is irrelevant, so that weare justified in ignoring it in our notation. The proof is a simple generalisationof the proof for the case of π1.

Suppose then that we have chosen basepoints x0 and x1 in X and denotethe corresponding homotopy groups by πn(X,x0) and πn(X,x1); our goal is toshow that these groups are isomorphic.

It will be helpful to this end to have an alternative description of πn. It isclear that Sn = In/∂In where In denotes the n-dimensional unit cube. There-fore a basepoint-preserving map Sn → X is the same thing as a map In → Xwhich sends ∂In to the basepoint of X (here we choose the basepoint of Sn tobe the image of ∂In under the quotient In → In/∂In). (Of course when n = 1this is just the description of a loop as a map I → X which takes the samevalue at the endpoints of I.) This gives us an alternative definition of πn(X) interms of maps In → X.

Now let [f ] ∈ πn(X,x0), so we may view f : In → X with f(∂In) = x0.The corresponding element [g] of πn(X,x1) is defined as follows. We shrink In

to some concentric sub-cube Jn; since Jn is homeomorphic to In we may viewf as a map Jn → X. We define g : In → X first on Jn by setting g|Jn = f . Onthe complement of Jn we take radial line segments from ∂Jn to ∂In and defineg to equal γ on these segments, where γ is any path in X from x0 to x1 (seethe figure below).

Since f(∂Jn) = x0 the map g is well-defined and continuous. It is clearfrom the definition that g(∂In) = x1, so g defines an element of πn(X,x1).Furthermore it is easy to see that this correspondence is a group homomorphism,because the sum operation on πn has a straightforward expression in the contextof maps In → X: viewing In from a “bird’s eye” perspective, with all but thefirst two co-ordinates pointing into the page, f1 + f2 is given by:

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Finally the fact that this is an isomorphism follows by considering the mapπn(X,x1) → πn(X,x0) induced by the path γ (which is γ traversed in theopposite direction).

We have thus shown that for path-connected spaces (which are the onlyspaces we consider) we do not need to worry about where we choose the base-point. Consequently we will often simply assume without comment that ourspaces have basepoints and that our maps are basepoint-preserving; we canget away with not talking about this explicitly because the precise choice ofbasepoint makes no difference.

There is one more subtlety regarding basepoints which we should address.Recall that in the definition of πn we require our homotopies to be basepoint-preserving. However on several occasions it will happen that we find ourselveswith a map Sn → X which is nullhomotopic, but via a homotopy which doesnot necessarily preserve basepoints. The following result ensures that we do nothave to be too careful here.

Proposition 1.1. A map Sn → X is nullhomotopic if and only if it is nullho-motopic with respect to basepoint-preserving homotopies.

Thus when we talk about nullhomotopy of maps Sn → X we do not haveto worry about whether or not the homotopy preserves basepoints. The proofis not too difficult, but since this really is a technical point we will not presentit here. The interested reader should see Theorem 1.6 of [Rot88].

1.4 Higher homotopy groups are abelian

Though in many respects the higher homotopy groups are considerably more dif-ficult to work with than the fundamental group, they do have one nice propertywhich the fundamental group does not.

Proposition 1.2. If n ≥ 2 then πn(X) is abelian.

Proof. Again we use the description of πn(X) in terms of maps In → X asexplained in the previous section. Let f and g be maps In → X with f(∂In) =g(∂In) = x0, so that f and g define elements of πn(X). We will construct ahomotopy between f + g and g + f . Recall from the previous section that themap f + g is given by:

Here the boundary of In as well as the hyperplane in the middle get sent tothe basepoint x0. Therefore we may pass to a homotopy equivalent map of theform

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2 SPHERES

with everything outside of the boxes labelled f and g getting sent to thebasepoint. We can now slide the domains of definition of f and g around eachother without affecting the homotopy class; futhermore we can do this withoutthe domains ever touching each other, since n ≥ 2. Thus we obtain

and by the same argument which we used to get from the first figure to thesecond figure, this is homotopy equivalent to the map

which is just g + f .

2 Spheres

We now turn to the spaces which we will be focusing on for rest of this article.Recall that the n-dimensional sphere is the subspace:

Sn = (x0, . . . , xn) ∈ Rn+1 | x20 + . . .+ x2

n = 1

Sometimes such a space is called an “n-sphere” but we will simply use the term“sphere” to refer to it, regardless of its dimension. Of course S1 is just a circle,and S2 is a sphere in the everyday sense of the word.

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2 SPHERES

Aside. We will not really be interested in the case n = 0, where S0 consistsof two discrete points, and will often assume without comment that n ≥ 1.Nonetheless we will occasionally have use for this space (see for instance Corol-lary 9.3).

The sphere can also be realised as the quotient

Sn = Dn/∂Dn

where Dn is the closed n-dimensional unit disc (in §1.3 we used the equivalentresult Sn = In/∂In). Because of this, Sn has the structure of a CW complex

Sn = en ∪ e0 (2.1)

with one zero-dimensional cell and one n-dimensional cell, the attaching mapbeing the only thing it can be, namely the constant map onto e0 = pt. Thisobservation will turn out to be extremely useful in what follows.

Our primary goal in this article is to compute some of the homotopy groupsπk(Sn). By definition these groups consist of homotopy classes of maps Sk → Sn

with domain and codomain both spheres.

The spheres are among the most perfectly symmetrical shapes imaginable,and as such one might expect that their homotopy groups are very simple.Indeed, this is certainly true in the case of homology.

Proposition 2.1. For n ≥ 1 we have:

Hk(Sn) =

Z for r = 0, n

0 otherwise

Proof sketch. For the reader familiar with cellular homology, this is immediatefrom the CW complex structure (2.1). Avoiding an appeal to cellular homology,we can proceed by induction on n, using Mayer-Vietoris. For the base case, writeS1 as the union of two contractible open segments U and V , with U∩V consistingof two points, and apply Mayer-Vietoris. The induction step is similar: wecan write Sn as the union of two contractible open sets U and V (given byslightly enlarged versions of the upper and lower hemispheres) such that U∩V 'Sn−1. Then a straightforward application of Mayer-Vietoris gives the desiredresult.

It turns out that the homotopy groups πk(Sn) are very regular as long ask ≤ n. What is perhaps surprising is that for k > n these groups becomeextremely complicated, with a pattern which is very difficult to predict. As wewill see, even the computation of very low-dimensional examples such as π3(S2)or π4(S2) will require some quite substantial machinery.

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4 CELLULAR APPROXIMATION

3 Homotopy Groups of S1

We begin our investigation with the first nontrivial example: that of the circle.This is a very easy example to get to grips with, because it is of low enoughdimension to be visualised explicitly.

Unfortunately, the theory of S1 is not at all representative of the generalcase. We will see that all the higher homotopy groups of S1 are trivial. By wayof constrast, to this day there is no n ≥ 2 for which all the homotopy groupsof Sn are known. Thus the case considered in this chapter is fairly degenerate;nevertheless it is still illuminating in its own way.

Recall that the universal cover of S1 is given by a map p : R→ S1 which canbe visualised as the projection of an infinite helix down onto the circle. It is byexamination of the deck transformations of this cover that we prove π1(S1) = Z(this is the main motivating example for the theory of covering spaces, and weassume the reader is familiar with it).

The universal cover is also extremely useful for studying the higher homotopygroups of S1: using it we can prove the following result with almost no effort.

Proposition 3.1. πk(S1) = 0 for k ≥ 2.

Proof. Consider a map f : Sk → S1. Since k ≥ 2 we have π1(Sk) = 0 (see[Hat01] Proposition 1.14, or §4 below). Therefore f∗(π1(Sk)) = 0 ⊆ π1(S1).Hence by the well-known Lifting Criterion ([Hat01] Proposition 1.33) there is alift f of f , i.e. a map f : Sk → R such that the following diagram commutes:

Rp

Skf//

f>>||||||||S1

Since R is contractible, we can find a nullhomotopy of f . Composing this withp gives a nullhomotopy of f . Since f was arbitrary we obtain πk(S1) = 0.

Aside. The argument in the previous proof actually shows something moregeneral: if X → X is a covering space with πk(X) = 0 (for some k ≥ 2) thenπk(X) = 0.

We have thus computed all the homotopy groups of S1. Of course thisapproach will not work on Sn for n ≥ 2, since in this case the universal cover isnot contractible (indeed since Sn is simply-connected it equals its own universalcover, so there is no extra information to be gained from studying this object).

4 Cellular Approximation

There is another large range within which πk(Sn) = 0, namely k < n. Inthis chapter we will show how to prove this result. In particular this gives analternative proof that Sn is simply-connected (for n ≥ 2), independent of the

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4 CELLULAR APPROXIMATION

usual argument using stereographic projection (although there is a similar ideaunderpinning both approaches).

The idea of the proof is to exploit the (very simple) CW complex structureon Sn (2.1). It makes use of a general result concerning maps between CWcomplexes, known as the cellular approximation theorem.

To motivate cellular approximation, we note that in mathematics it is com-mon to encounter maps which do not increase dimension. Linear maps providean obvious example (the dimension of the image is always less than or equal tothe dimension of the domain). Somewhat closer to algebraic topology, Sard’stheorem implies that a smooth map between manifolds cannot increase dimen-sion (see [Lee12] §6).

A related notion for CW complexes is that of a cellular map; a map f : X →Y between CW complexes is said to be cellular if and only if f(Xn) ⊆ Y n forall n. This is a strong form of not increasing dimension, since it requires thatf(Xn) doesn’t even touch, let alone cover, anything higher-dimensional.

Although it is easy to come up with examples of maps which are non-cellular(for instance, think of curves in S2) the cellular approximation theorem ensuresthat we are never too far from one which is.

Theorem 4.1 (Cellular Approximation Theorem). Any map between CW com-plexes is homotopy equivalent to a cellular map.

The proof is quite technical and we will not present it here. The interestedreader can find the details in §4.1 of [Hat01] or §10 of [May99]. Assuming thisresult, we then immediately have:

Proposition 4.2. πk(Sn) = 0 for k < n.

Proof. Consider any map f : Sk → Sn. By the previous theorem f is homotopyequivalent to a cellular map g. Then g(Sk) is a subspace of the k-skeleton ofSn. However since k < n this is just a single point, as is clear from (2.1). Henceg is constant, and so f is nullhomotopic.

We conclude this chapter by noting another immediate consequence of cel-lular approximation, which will be crucial for our arguments in §11.

Proposition 4.3. Let X be a CW complex. Then πn(X) = πn(Xn+1).

Proof. By cellular approximation any map Sn → X is homotopic to a mapwhich takes values in Xn ⊆ Xn+1; it follows that πn(X) is equal to the set ofmaps Sn → Xn+1 modulo homotopies through maps Sn → X.

Now consider a homotopy F : Sn × I → X in X. This is homotopic, bya second application of cellular approximation, to a map which takes values inXn+1 (since Sn×I is naturally an (n+1)-dimensional CW complex). Thereforeπn(X) is equal to the set of maps Sn → Xn+1 modulo homotopies through mapsSn → Xn+1, which is just πn(Xn+1).

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5 HUREWICZ THEOREM AND DEGREE

5 Hurewicz Theorem and Degree

When it comes to computing homology groups, we have an arsenal of powerfultools at our disposal: the Mayer-Vietoris sequence and the excision theoremare perhaps the most celebrated examples. Unfortunately the majority of theseresults simply do not hold for homotopy groups (this is the primary reason whyhomotopy groups are so much harder to compute than homology groups).

It would be extremely useful, therefore, if we could somehow reduce thecomputation of the homotopy groups of a space to that of its homology groups.Alas, in general such a thing is impossible. However there is a special case inwhich it can be done: as we will see, there exists a strong relation between thefirst nontrivial homotopy and homology groups of a space. This is expressed inthe so-called Hurewicz theorem (Theorem 5.4), which forms the heart of thischapter.

5.1 The Hurewicz homomorphism

For each k there is a natural map

h : πk(X)→ Hk(X)

called the Hurewicz homomorphism. It is defined by first choosing a gener-ator σ ∈ Hk(Sk) ∼= Z, and then defining

h([f ]) = f∗(σ)

for all f : Sk → X. This is well-defined (that is, it respects homotopy classesin πk(X)) by homotopy invariance of singular homology. To see that it is ahomomorphism, recall that for [f ], [g] ∈ πk(X) their sum is represented by themap f + g given by the composition:

Sk → Sk ∨ Sk → X

By functoriality, (f + g)∗ is equal to the composition

Hk(Sk)→ Hk(Sk)⊕Hk(Sk)→ Hk(X)

where we have used the fact that Hk(Sk∨Sk) = Hk(Sk)⊕Hk(Sk) (see Corollary2.25 of §[Hat01]). The first map sends α 7→ (α, α), and the second map sends(α, β) 7→ f∗(α) + g∗(β). Therefore

h([f + g]) = (f + g)∗(σ) = f∗(σ) + g∗(σ) = h([f ]) + h([g])

so indeed the map h is a homomorphism.Of course the definition of h depends on the choice of generator for Hk(Sk),

but only up to sign, so we will not worry too much about this and simply assumethat a generator has been fixed.

There is an intuitive description of the Hurewicz homomorphism which wenow discuss. The idea is simple: we subdivide Sk into k-dimensional simplices,

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and then view every map Sk → X as a collection of singular simplices in X byrestricting the map to each of these subdivisons. We give a sketch of this below,referring the reader to [Mun84] for full details and justifications.

We can choose a finite triangulation of Sk by k-dimensional simplices ∆1, . . . ,∆r

and since Sk is orientable we can orient these simplices in a consistent manner,meaning that the sum

∆1 + . . .+ ∆r

is a cycle in the kth chain group Ck(Sk) for singular homology. Here of coursewe view each ∆i as the image of a homeomorphism ϕi : ∆k → ∆i where ∆k isthe standard k-simplex, so that ∆i ∈ Ck(Sk).

In this case the homology class [∆1 + . . . + ∆r] is in fact a generator forHk(Sk). Therefore we have:

h([f ]) = f∗([∆1 + . . .+ ∆r]) = [f∗(∆1) + . . .+ f∗(∆r)]

Each of the singular simplices f∗(∆i) is given by f ϕi : ∆k → X, and shouldbe thought of as the restriction of f to ∆i. Therefore h([f ]) is obtained bysumming up the restrictions of f to each of the ∆i, as claimed.

5.2 n-connected spaces

The Hurewicz theorem gives a sufficient condition for the Hurewicz homomor-phism h to be an isomorphism. In order to state it properly, we must firstintroduce the concept of an n-connected space.

Definition 5.1. A space X is n-connected if and only if πk(X) = 0 for k ≤ n.

Thus a space is 0-connected if and only if it is path-connected and 1-connected if and only if it is simply-connected. Note of course that if X isn-connected then it is also m-connected for m ≤ n.

Roughly speaking, an n-connected space is one which has no holes of dimen-sion k ≤ n, where we think of a k-dimensional hole as being an obstruction to“filling in” a sphere of dimension k.

More precisely, suppose we have a map f : Sk → X, which we think of asdefining a “singular sphere” inside X. The question is whether or not we canextend f to a mapDk+1 → X, where we view f as being defined on ∂Dk+1 = Sk.If we can, then there is no hole in the inside of f , because we were able to fill itin by a solid disc.

Proposition 5.2. πk(X) = 0 if and only if every map Sk → X can be extendedto a map Dk+1 → X.

Proof. We will show that f : Sk → X is nullhomotopic if and only if it extendsto a map defined on Dk+1. Suppose first that f is nullhomotopic, so there is ahomotopy F : Sk × I → X such that F (y, 1) = f(y) and F (y, 0) = x0 where x0

is the basepoint of X. If we now take polar co-ordinates (θ1, . . . , θk, r) on Dk+1

we have a well-defined map f : Dk+1 → X given by

(θ1, . . . , θk, r) 7→ F ((θ1, . . . , θk), r)

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which is well-defined because F (y, 0) = x0 and is an extension of f becauseF (y, 1) = f(y).

Now suppose conversely that there is a map f : Dk+1 → X extending f . Lety0 denote the chosen basepoint for Sk and define a homotopy F : Sk × I → Xby:

(y, t) 7→ f(ty + (1− t)y0)

This is well-defined because Dk+1 is convex, so that ty + (1 − t)y0 ∈ Dk+1 forall y ∈ Sk ⊆ Dk+1. We have F (y, 0) = f(y0) = x0 and F (y, 1) = f(y) = f(y)since f extends f . Furthermore F (y0, t) = f(y0) = f(y0) = x0 for all t, so thatF is a basepoint-preserving homotopy.

As well as providing intuition for the notion of n-connectedness, the previousresult will be useful in our proof of the Hurewicz theorem 5.4.

There is another, extremely concrete, characterisation of n-connectednesswhich applies to CW complexes. Let X be a CW complex, and suppose for themoment that the n-skeleton of X consists of a single point (this means that Xhas only a single 0-cell and that all the other cells of X have dimension greaterthan n).

We claim that X must be n-connected. This follows by cellular approxima-tion since any map Sk → X with k ≤ n must be nullhomotopic.

Somewhat remarkably, every n-connected CW complex is of this form.

Proposition 5.3. Let X be a CW complex. Then X is n-connected if and onlyif it is homotopy equivalent to a CW complex whose n-skeleton is a point.

We proved the “if” direction in the paragraph above (since πk is homotopyinvariant). The “only if” direction follows from the theory of CW approxi-mation: the process by which one can form various “approximations” of anarbitrary space by a CW complex. For the details of the construction see §4.1of [Hat01].

5.3 Cubical homology

We saw earlier that the homotopy groups πn(X) can be described in terms ofmaps In → X whose domains are cubes. Since the Hurewicz theorem is aboutrelating homotopy and homology, it would be extremely useful if we had someway of thinking about homology in terms of cubes as well; this is given bycubical homology. We give a sketch of this now; some of the facts will needto be taken on trust. The skeptical reader can find a full account in §VII of[Mas91].

The idea is straightforward: whereas in the definition of ordinary singularhomology we use simplices ∆k as our basic objects, in cubical homology we usecubes Ik.

Thus the kth chain group is by definition the free abelian group with gener-ators the singular k-cubes in X, i.e. maps Ik → X.

16


Recall that the boundary of a k-simplex consists of a collection of (k − 1)-simplices; similarly, the boundary of a k-cube consists of a collection of (k− 1)-cubes, which we refer to as the facets of the k-cube. This allows us to define aboundary homomorphism

Ck(X)∂−→ Ck−1(X)

in much the same way as for ordinary singular homology: that is, by sendinga singular k-cube σ : Ik → X to a (signed) sum of the singular (k − 1)-cubesobtained by restricting σ to each facet of Ik. The boundary homomorphismsatisfies ∂2 = 0, so we obtain a chain complex whose homology groups are bydefinition the cubical homology groups of X.

Aside. As the reader who investigates the source [Mas91] will quickly discover,there is a subtlety involving “degenerate cubes” which we have ignored here, inview of keeping the exposition as readable as possible.

It is a fundamental fact that cubical homology is isomorphic to ordinarysingular homology; as such, this construction really just provides a differentway of thinking about singular homology, which turns out to be better suitedfor certain arguments.

5.4 The Hurewicz theorem

We are now ready to state and prove the main result of this chapter.

Theorem 5.4 (Hurewicz Theorem). Let X be an (n− 1)-connected space forsome n ≥ 2. Then Hk(X) = 0 for 1 ≤ k ≤ n− 1 and h : πn(X)→ Hn(X) is anisomorphism.

This says that the first nontrivial homotopy and (reduced) homology groupsoccur in the same dimension and are isomorphic, as long as that dimension isat least 2.

Aside. There does exist a statement dealing with the case n = 1, but it iscomplicated by the fact that π1(X) can be nonabelian whereas H1(X) cannot.We will not treat this here, but the interested reader can find a full account in§15 of [May99] or §4.2 of [Hat01].

Recall from §1.3 that we can view πk(X) as consisting of homotopy classesof maps Ik → X which send ∂Ik to the basepoint, i.e. as homotopy classes ofcertain singular k-cubes.

It now starts to become clear why cubical homology is so useful for ourpurposes. From this perspective the Hurewicz homomorphism takes the verysimple form

h : πk(X)→ Hk(X)

h([f ]) = [f ]

17


where the first [f ] indicates a homotopy class and the second [f ] indicates ahomology class.

We have ∂f = 0 because f sends ∂Ik to a single point: thus the homologyclass [f ] makes sense. Furthermore it is not too hard to see that homotopick-cubes must be homologous: a homotopy F between two singular k-cubes fand g defines a (k + 1)-cube whose boundary is f − g (the “sides” of F do notappear because they are degenerate cubes; see the remark at the end of theprevious section). Therefore h is well-defined on homotopy classes.

Proof of Hurewicz. We will prove the theorem in the case that X is a CWcomplex. This is sufficient for all of our applications, so that there is no realharm in this reduced generality. In any case, the general result follows from thisone by a quick application of CW approximation (Theorem A.2).

Suppose then that X is an (n− 1)-connected CW complex. By Proposition5.3 we may assume that the (n− 1)-skeleton of X is a point (which we denotex0 and choose to be the basepoint of X).

Using cellular homology it then follows immediately that Hk(X) = 0 fork = 1, . . . , n− 1 because all of the cellular chain groups in this range are trivial.

It remains to show that h : πn(X) → Hn(X) is an isomorphism. The ideais to define an inverse r : Hn(X)→ πn(X) to the Hurewicz homomorphism bytaking a singular n-cube, deforming it so that ∂In maps to the basepoint andthen considering the resulting homotopy class in πn(X).

We begin by defining r on the level of chains. So let σ : In → X be asingular n-cube in X. Note that In has a CW complex structure with a singlen-cell and whose (n − 1)-skeleton is just ∂In. Since the (n − 1)-skeleton of Xis equal to x0, we have by cellular approximation that σ is homotopic to ann-cube σ : In → X such that σ(∂In) = x0. We then define:

r(σ) = [σ] ∈ πn(X)

Clearly this definition is independent of the choice of σ because homotopy equiv-alence is an equivalence relation. Having defined r on the generators of Cn(X),we can naturally extend it to a homomorphism:

r : Cn(X)→ πn(X)

Now let τ : In+1 → X be a singular (n + 1)-cube and consider ∂τ ∈ Cn(X).We claim that r(∂τ) = 0. Once this is proven, it immediately follows that rdescends to a map Hn(X) → πn(X), because im ∂ ⊆ Cn(X) is generated bychains of the form ∂τ for τ a singular (n+ 1)-cube.

Now, the (n − 1)-skeleton of In+1 consists of the boundaries of each of thefacets of In+1. Again by cellular approximation there is a homotopy equivalentcube τ : In+1 → X which sends the (n − 1)-skeleton of In+1 to x0. We canwrite

∂τ = σ1 + . . .+ σ2n+2

18


where σi is obtained by restricting τ to the ith facet of In+1. By assumptionon τ we have σi(∂I

n) = x0 for all i, so that:

r(∂τ) = r(∂τ) = r(σ1) + . . .+ r(σ2n+2) = [σ1] + . . .+ [σ2n+2]

By the definition of the sum operation in πn(X), this homotopy class is repre-sented by the map f given by:

Since the boundary of the cube in this figure maps to x0, we can think of fas a map Sn → X. Furthermore, f has an extension to Dn+1, namely the mapτ (after composing with the appropriate identification Dn+1 ∼= In+1). Henceby Proposition 5.2 f is nullhomotopic, i.e:

[σ1] + . . .+ [σ2n+2] = 0 ∈ πn(X)

We have thus shown that r(∂τ) = 0 as required.To summarise, we get an induced homomorphism:

r : Hn(X)→ πn(X)

Finally we claim that r is inverse to h. It is more or less clear from the definitionsthat rh([f ]) = [f ] for [f ] ∈ πn(X).

On the other hand consider [σ] ∈ Hn(X); we claim that hr([σ]) = [σ].Again this is obvious if σ consists of a single n-cube (since homotopic cubes arehomologous). The result then follows by linearity.

The above proof was inspired by the treatment in [Hut11]. There are slicker(though less geometrical) proofs, but they tend to require prior knowledge ofπn(Sn), whereas we have opted to first prove Hurewicz and to then use it tocompute these groups (see below).

5.5 Degree theory

An immediate corollary of the Hurewicz theorem is the following importantresult.

Proposition 5.5. πn(Sn) ∼= Z

Proof. The case n = 1 follows from covering space theory. For n ≥ 2, we saw inProposition 4.2 that Sn is (n − 1)-connected. Therefore by Hurewicz we haveπn(Sn) ∼= Hn(Sn) ∼= Z.

19

6 INTERLUDE: THE ROAD AHEAD

If we let σ ∈ Hn(Sn) denote the generator which we chose when definingthe Hurewicz homomorphism (see §5.1), then σ determines an isomorphismHn(Sn) ∼= Z. It is then clear that the composition of the Hurewicz map withthis isomorphism is independent of the choice of σ, since replacing σ by −σreverses the signs of both maps. Thus we have a natural isomorphism

πn(Sn) = Z

which is called the degree map. This associates to every map f : Sn → Sn aninteger deg f called the degree of f . This is a homotopy invariant, and sincedeg is an isomorphism it is in fact a complete homotopy invariant (meaningthat two maps are homotopic if and only if they have the same degree).

There is a very nice geometric interpretation of degree, generalising the ideaof the winding number. We give a statement of this here; the interested readercan find proofs in [BT82] §4 (in the smooth setting) or [Hat01] §2.2 (in thetopological setting).

Suppose there is a point y ∈ Sn such that f−1(y) consists of finitely manypoints x1, . . . , xm (if f is smooth then almost every point y has this property;in general we can replace f by a homotopy equivalent smooth map). Then thedegree of f is given by

deg f =

m∑i=1

εi

where εi is equal to +1 if f is orientation-preserving near xi and −1 if f isorientation-reversing near xi (it follows that this value is independent of thechoice of y).

Thus the degree can be thought of as a signed measure of the size of an arbi-trary preimage of f ; that is, a measure of how “many times” (taking orientationsinto account) f wraps Sn around itself.

We can extend this idea to the other homotopy groups, thinking of πk(Sn)as measuring the different ways in which Sk can be wrapped around Sn. Thesecannot always be thought of as integers, since (as we will see) some of thehigher homotopy groups have torsion. Nonetheless it is a useful picture to havein mind, so long as it is taken with a pinch of salt.

6 Interlude: the Road Ahead

So far we have found πk(Sn) for k ≤ n. It remains then to consider the casek > n. Here we are entering into dark territory. To this day, the vast majorityof these groups are still unknown. Even for the very small values of k andn to which we restrict ourselves in this article, significant machinery must bedeveloped if we are to stand a chance.

In the next few chapters we will develop the concepts and tools necessaryfor further exploration. Our primary goal will be Theorem 9.10, the long exacthomotopy sequence associated to a fibration. Getting there will be quite ajourney, taking us on a whistlestop tour through several distinct topics.

20

7 FIBRATIONS

Since this theory is most elegantly developed in a general context, thesechapters may seem somewhat more abstract than those which have come before,and the connection to spheres might not be as apparent. On the plus side,working in greater generality, in addition to clarifying the exposition, meansthat the tools we develop can be applied in a wide variety of situations, andindeed will be invaluable for any later study of algebraic topology.

7 Fibrations

7.1 The homotopy lifting property

The essential ingredient in the study of the fundamental group is the theoryof covering spaces. Unfortunately, however, covering spaces are insufficient forstudying higher homotopy groups, because any covering projection E → Binduces isomorphisms on πn for n ≥ 2 ([Hat01] Proposition 4.1).

The search for a suitable analogue of covering spaces with which to studyhigher homotopy groups leads to the notion of a fibration. This is obtainedby abstracting one of the most useful facts about covering spaces, called thehomotopy lifting property.

We will see that this condition is loose enough to include a large class ofmaps (in fact, in a certain sense - see Proposition 7.2 - every map is fibration),but strong enough to impose stringent topological restrictions on the associatedspaces. For the remainder of this article, fibrations will be the preeminent toolwe use to explore higher homotopy groups.

We begin with a definition. A map p : E → B is said to satisfy the homo-topy lifting property with respect to the space Y if and only if for everyhomotopy G : Y × I → B and every lift G0 : Y → E of G0 = G(−, 0), there is alift G : Y × I → E of G to E which extends G0 in the sense that G(−, 0) = G0.In pictures

YG0 //

_

E

p

Y × I G //

G

<<yy

yy

yB

where the inclusion Y → Y × I is given by y 7→ (y, 0).

Aside. Despite the formal appearance of the diagram, there is no way in whichthis can be interpreted as a universal property, because the lift G is not assumedto be unique (and indeed in many cases is not so).

Probably the reader has already encountered special cases of the homotopylifting property. For instance, if Y = pt then we recover the path liftingproperty for covering spaces (with G0 : pt → E specifying the start point ofthe lift), and if Y = I then we recover the lifting of homotopies between paths.

21

7 FIBRATIONS

Definition 7.1. A map p : E → B is called a Hurewicz fibration (or just afibration) if and only if it has the homotopy lifting property with respect toall spaces Y .

The map is called a Serre fibration (or a weak fibration) if and only ifit has the homotopy lifting property with respect to cubes Ik for k = 0, 1, . . .

As is customary, we call E the total space and B the base space of thefibration.

Whenever we say “fibration” without qualification we mean Hurewicz fibra-tion. Most of the important facts about Hurewicz fibrations also hold for thelarger class of Serre fibrations. On the other hand, all of the Serre fibrationswhich we encounter in this article are also Hurewicz fibrations, so we have noreal need for this added generality. This being the case, we will not place toomuch emphasis on the difference between Hurewicz and Serre, and while thereader should bear in mind the distinction, the assumption that a map is aSerre fibration is usually sufficient for the purposes of algebraic topology.

It is a basic fact that every covering space is a Serre fibration. In fact wecan state a more general result which will come in useful later. Recall that acovering space of B consists of a map E → B with the fibers discrete spaces,and satisfying a local triviality condition.

If we remove the requirement that the fibers be discrete, we obtain the defi-nition of a fiber bundle over B. We assume the reader has had some experiencewith these already; if not then §4.2 of [Hat01] provides a good introduction.

Fiber bundles can be thought of as “twisted products” of the base B withthe fiber F , in the sense that every fiber bundle is locally of the form B × F ;the failure of this to hold globally arises from the “twisting” of the bundle.Such objects crop up all over mathematics: for instance, vector bundles (fiberbundles where each fiber is a vector space) are central to the field of differentialgeometry (see [Lee12]). The following proposition tells us that fiber bundles arealso of great importance in algebraic topology.

Proposition 7.2. Let p : E → B be a fiber bundle. Then p is a Serre fibration.

For the proof of this see Proposition 4.48 of [Hat01] or Theorem 11.49 of[Rot88]. There is a similar result for Hurewicz fibrations, the proof of which issomewhat technical and can be found in §2.7 of [Spa66]:

Proposition 7.3. Let p : E → B be a fiber bundle and assume that B isHausdorff and paracompact. Then p is a Hurewicz fibration.

The assumption on the base is quite mild, and is satisfied for instance byany manifold or CW complex (see [Lee10] Theorem 5.22). Therefore any fiberbundle we are likely to encounter will be a Hurewicz fibration, and we can applythe theory developed in this and subsequent chapters to it.

Although they are certainly more general, fibrations still have some proper-ties which are reminiscent of fiber bundles. For instance, we will see later that,though the fibers of a fibration may not be homeomorphic, they must at leastbe homotopy equivalent (Proposition 9.9).

22

7 FIBRATIONS

7.2 Replacing arbitrary maps by fibrations

In §9 we will prove a remarkable result about fibrations which will have pro-found consequences for the homotopy groups of the spheres. First, however,we will show that, despite their power, fibrations can be found (quite literally)everywhere, in the sense that every map is - up to homotopy equivalence - afibration, a statment which we make precise in the following.

Definition 7.4. Consider any maps f : X → Y and f ′ : X ′ → Y ′. Then f andf ′ are homotopy equivalent if and only if there exist homotopy equivalencesX ' X ′ and Y ' Y ′ making the following diagram commute:

X' //

f

X ′

f ′

Y' // Y ′

In much the same way as we can replace any space by a homotopy equivalentspace and continue on without further comment, we can replace any map wecome across by a homotopy equivalent map and not have to worry since, fromthe perspective of homotopy theory, the two maps are indistinguishable.

Proposition 7.5. Every map is homotopy equivalent to a Hurewicz fibration.

Proof. The proof is remarkably straightforward, relying on a clever pathspaceconstruction. Let f : X → Y be the map we are given, and let Ef be thecollection of pairs (x, γ) with x ∈ X and γ a path in Y with γ(1) = f(x). Wetopologise Ef as a subspace of X × Y I , the function space Y I being equippedwith the compact-open topology (see the appendix of [Hat01]).

Define p : Ef → Y by p(x, γ) = γ(0). We claim this is a fibration. Suppose

then that we have a homotopy G : Z×I → Y and a lift G0 : Z → Ef of G(−, 0).

We can write G0(z) = (h(z), γz) for h : Z → X and γz a curve in Y for all z.We then define:

G : Z × I → Ef

(z, t) 7→ (h(z), θ(z,t) · γz)

where θ(z,t) is the path in Y traced out by G(s, z) as s goes from t to 0. Then

θ(z,t) · γz(1) = γz(1) = fh(z)

so that G really does take values in Ef . Clearly G extends G0 since θ(z,0) is

trivial. Finally we must check that G is a lift of G:

pG(z, t) = θ(z,t) · γz(0) = θ(z,t)(0) = G(z, t)

Thus, the homotopy lifting property is verified.

23

8 SUSPENSIONS AND LOOPSPACES

To see how the fibration p relates to the original map f , note that thereis an inclusion i : X → Ef given by sending x to the pair (x, f(x)) (wheref(x) denotes the constant path in Y ). Furthermore, this map is a homotopyequivalence, since given a pair (x, γ) ∈ Ef we can restrict γ to shorter andshorter segments of the form [t, 1], and this defines a deformation retractionfrom Ef to i(X).

Notice also that pi(x) = p(x, f(x)) = f(x)(0) = f(x), so we have a commu-tative diagram

Xi //

f

Ef

p

Yid

Y

with the horizontal arrows homotopy equivalences.

Thus we are entitled to replace any map by a fibration, if we so desire. Thiswill come in useful in a number of arguments to follow, because every fibrationhas an associated long exact sequence of homotopy groups. The proof of thisresult is the subject of §9. To get there, though, we must first take a brief detourto discuss the suspension-loopspace adjunction.

8 Suspensions and Loopspaces

If the reader has taken a first course in algebraic topology, she will likely haveencountered the cone and the suspension (respectively CX and SX) of atopological space X. These are defined as follows:

CX = (X × I)/(X × 0)SX = CX/(X × 1)

One of the most basic properties of the suspension is that S(Sn) ' Sn+1, asillusrated in Figure 1.

In this section we introduce the loopspace construction, and explain its con-nection to the above via the so-called suspension-loopspace adjunction (Propo-sition 8.2). An effective consequence of this is to reduce the computation ofπn(X) to that of πn−1(ΩX), where ΩX is the loopspace of X whose definitionwill be given shortly. This has obvious utility for the study of higher homotopygroups, and is a vital step in the proof of the Puppe sequence, which we willrely on heavily.

8.1 Suspension-Loopspace adjunction

Begin by recalling the definition of the homotopy groups. As a set, πn(X) con-sists of homotopy classes of basepoint-preserving maps Sn → X. (Throughoutthis and the following sections we always assume implicitly that we have chosen

24


Figure 1: S(Sn) ' Sn+1

basepoints for our spaces and that the maps and homotopies are basepoint-preserving.)

More generally, let us write [Y,X] for the set of homotopy classes of mapsY → X. Then [Y,X] makes sense as a set for any space Y , but it is unclear ifthere is any natural way to put a group structure on it. Recall that the groupstructure on πn(X) = [Sn, X] is given by:

f + g : Sn → Sn ∨ Sn → X (8.1)

Where the second map is given by taking f on the first factor and g on thesecond, and the first map is the pinching map which collapses an equatorialSn−1 ⊆ Sn to a point. Of course, for an arbitrary space Y there is no analogousmap; however, if Y = SZ is a suspension, then there is a well-defined pinchSZ → SZ ∨ SZ given by collapsing Z × 1/2 ⊆ SZ.

We might hope that we can use this to define a group structure on [SZ,X]using a formula similar to (8.1). However there is a slight snag: the maps in[SZ,X] must preserve basepoints, and no matter where we choose the basepointof SZ, it is clear that for basepoint-preserving f and g, the map f+g will not ingeneral be basepoint-preserving (to see this, consider separately the cases wherethe basepoint lies on the equator Z ⊆ SZ and where it lies off of it).

One way we can get around this is to consider maps SZ → X which send anentire longitudinal line z0× I (for some z0 ∈ Z) to the basepoint of X. Sincethis longitudinal line is sent to two copies of itself by pinching (see Figure 3), itis clear then that the sum operation is well-defined for maps with this property.

By the universal property for quotients, a map SZ → X sending z0× I tothe basepoint is the same as a basepoint-preserving map ΣZ → X, where ΣZ

25


Figure 2: A general pinch map.

is the so-called reduced suspension :

ΣZ = SZ/(z0 × I)

And where we choose the basepoint for ΣZ to be the image of z0×I underthe quotient. To summarise, we have produced a natural group structure onthe set of (homotopy classes of) basepoint-preserving maps [ΣZ,X].

Although at first it might seem disappointing that we have to work withΣZ instead of SZ, in many examples these spaces are not very different. Forinstance, if Z is a CW complex and we choose z0 ∈ Z to be a 0-cell, than ΣZis homotopy equivalent to SZ. In particular we have:

πn(X) = [S(Sn−1), X] = [ΣSn−1, X]

And so the group structure on [ΣZ,X] really is just a generalisation of thegroup structure on πn(X).

We have managed to put a group structure on [Y,X] by assuming that Yholds a special property, namely that it is a suspension. Next we will assumeinstead that X has a special property, namely that it is a loopspace.

Definition 8.1. Let Z be a topological space with basepoint z0. Then theloopspace of Z is denoted and defined:

ΩZ = γ : I → Z | γ(0) = γ(1) = z0

In other words, ΩZ is the collection of loops based at z0. It is topologised as asubspace of ZI equipped with the compact-open topology (see the appendix of[Hat01]).

26


Figure 3: Effect of pinching on a longitudinal line.

Of course, π1(Z) is just ΩZ modulo homotopy equivalence. In exactly thesame way as we define the group operation on π1(Z), we can define a sum onΩZ by concatenating loops. This does not make ΩZ into a group (because wedo not identify homotopy-equivalent paths), but it induces a sum on [Y,ΩZ],given by

(f + g)(y) = f(y) · g(y)

which does make [Y,ΩZ] into a group: the identity is given by the map Y →ΩZ sending every element of y to the constant loop at z0, and the inverse off ∈ [Y,ΩZ] is given by the map sending y ∈ Y to the loop f(y) traversedin the opposite direction. We leave it to the reader to construct the relevanthomotopies.

The reason for introducing the loopspace construction is the following fun-damental result.

Proposition 8.2 (Suspension-Loopspace Adjunction). Let X and Y be topo-logical spaces. Then there is a natural isomorphism of groups:

[ΣY,X] = [Y,ΩX]

Proof. The definition of the isomorphism is clear (in fact, before reading on, tryto work out for yourself what it should be). If f ∈ [ΣY,X] then the correspond-ing element of [Y,ΩX] is defined as follows: each y ∈ Y maps to the loop in Xobtained by restricting f to y × I ⊆ ΣY . This gives a loop because in thereduced suspension the top and bottom vertices of the ordinary suspension areidentified, so that under the quotient y × I ' S1.

27


The reverse construction is equally straightforward. If g ∈ [Y,ΩX] then thecorresponding element of [ΣY,X] is given by sending (y, t) to g(y)(t). This iswell-defined under the quotient because g(y) is a loop.

After a moment’s thought it is clear that these constructions are mutuallyinverse, so it remains to show that they are group homomorphisms. But againthis is fairly clear, because the pinch operation on ΣY sends a longitudinal arcy×I to two copies of the same arc, and the sum f+g is given by restricting f tothe first copy and g to the second copy, which is by definition the concatenationf(y) · g(y), which is the sum operation in [Y,ΩX].

In the following sections we will see just how useful this result is. For themoment note that we have:

[Σ2Y,X] = [ΣY,ΩX] = [Y,Ω2X]

And by induction the more general property:

[ΣnY,X] = [Y,ΩnX] (8.2)

In particular

Corollary 8.3. πn(X) = π1(Ωn−1X) = π0(ΩnX)

where the second equality is one of pointed sets.

8.2 Freudenthal suspension and stable homotopy groups

The constructions Σ and Ω given above both produce covariant functors. Tosee how this works, suppose that we are given a map f : X → Y .

We define Σf : ΣX → ΣY by first composing f × idI with the quotientprojection Y × I → ΣY and then noting that by the universal property thisdescends to a map Σf : ΣX → ΣY . This has the usual functoriality propertiesΣ(f g) = Σf Σg and Σ(id) = id.

In a similar spirit we define Ωf : ΩX → ΩY by Ωf(γ) = f γ, whereγ : I → X is a loop starting at the basepoint x0. Again this satisfies thefunctoriality properties Ω(f g) = Ωf Ωg and Ω(id) = id. (We reiterate herethe importance of working with basepoint-preserving maps: these definitions donot work otherwise.)

Aside. The reader familiar with category theory may wish to note that thenaturality of the isomorphism in Proposition 8.2 means that it is an adjunctionbetween the functors Σ and Ω; see [Mac98] §IV.

Let X be any space and consider the homotopy group πn(X). If f : Sn → Xis a (representative of an) element of πn(X), we can take the suspension of f toobtain a map

Σf : ΣSn = Sn+1 → ΣX

28


which defines a homotopy class in πn+1(ΣX). Suppose that f is nullhomotopic:we claim the same holds for Σf . We saw in the previous section that there is anatural isomorphism:

[ΣSn,ΣX] = [Sn,ΩΣX]

Now, there is a natural map i : X → ΩΣX given by sending every x ∈ X tothe longitudinal loop in ΣX passing through (x, 1/2), i.e. the composition withthe quotient of the path t 7→ (x, t) in X × I. (If one prefers, this is the map in[X,ΩΣX] corresponding to the identity map in [ΣX,ΣX].)

Looking back at the proof of Proposition 8.2, we see that the map in [Sn,ΩΣX]corresponding to Σf ∈ [ΣSn,ΣX] is just the composition if . If f is nullhomo-topic it follows that if is also, and hence the same holds for the correspondingmap Σf (because the adjunction isomorphism is a group homomorphism, sopreserves the identity).

Therefore we have a well-defined map:

πn(X)→ πn+1(ΣX)

which we call the suspension homomorphism (we leave it to the reader toconvince herself that it really is a group homomorphism).

Theorem 8.4 (Freudenthal Suspension Theorem). The suspension homomor-phism πk(Sn)→ πk+1(Sn+1) is an isomorphism if 2n ≥ k + 2.

An immediate consequence of this is that for fixed k, the homotopy groupsπn+k(Sn) eventually stabilise: that is, they are independent of n for large enoughvalues of n. We call this limiting value the kth stable homotopy group(sometimes also the stable k-stem) of the spheres and denote it by πsk. Thecomputation of these stable homotopy groups is one of the most profound andstudied problems in all of algebraic topology.

We have already seen (Proposition 5.5) that πs0 = Z. The computation ofthe higher stable groups is a lot more difficult.

Notice that we know when each sequence πn+k(Sn) will stabilise. Indeed byFreudenthal, this happens when 2n ≥ (n+ k) + 2 or equivalently n ≥ k + 2, sothat we have:

πsk = π2k+2(Sk+2)

In particular πs1 = π4(S3), the computation of which we will be able to achieveby the end of this article.

Unfortunately proving Freudenthal is beyond the scope of our presentation.Nevertheless the proof is very nice (it relies on a weak version of excision forhomotopy groups), and the interested reader can find the deatils in [Hat01] §4.2or [May99] §11.

29

9 LONG EXACT SEQUENCES

9 Long Exact Sequences

9.1 Puppe sequence

When dealing with (co)homology we are inundated with long exact sequences:Mayer-Vietoris for singular (co)homology, long exact sequences for relative (co)homology,covariant Mayer-Vietoris for compactly supported cohomology ([BT82]), andmany more besides. These sequences are remarkably versatile tools, and with-out them most of the theory would be completely inaccessible.

Unfortunately such sequences are not as forthcoming in homotopy theory.The reason for this is clear: all of the common long exact sequences in (co)homologyare obtained by finding a suitable short exact sequence of chain complexes andthen applying the Snake Lemma. By the nature of the basic definitions of(co)homology theory, chain complexes are abundant there, but they rarely makean appearance when studying homotopy groups.

The Puppe sequence is an example of a long exact sequence in homotopy.In this chapter we will state and prove it in its general form, before focusingon the special case we are most interested in (Theorem 9.10). Unlike the longexact sequences described above, it is not obtained by considering any chaincomplexes: as we will see, its derivation is far more involved.

Our primary reference for this chapter is [Rot88] §11. An alternative proofof the main result, requiring less theory but a lot more technical, is given in[Hat01] §4.2. See also [May99] §8 for a condensed argument.

9.1.1 Pointed sets

Because the suspension-loopspace adjunction applies to all spaces and not justto spheres, we can work in a more general setting and consider, instead of justπn(X) = [Sn, X], the set [Y,X] for Y any space. This is not just generalityfor the sake of generality: such an approach will end up being very useful inclarifying our arguments.

In order to adopt this point of view, however, we will need to generalise ournotion of exactness. As we remarked earlier, [Y,X] is not a group, but it doeshave a distinguished element, namely the constant map from Y to the basepointof X.

A set with a distinguished element is called a pointed set, and when con-sidering maps ϕ : (A, a0) → (B, b0) between pointed sets we require thatϕ(a0) = b0. The kernel of a map between pointed sets is defined:

kerϕ = a ∈ A | ϕ(a) = b0

Notice that if A and B are groups (with a0 and b0 the corresponding identityelements) and ϕ is a group homomorphism, then ϕ is a map between the pointedsets (A, a0) and (B, b0), and its kernel is the same as its kernel in the sense ofgroup theory.

We say that a sequence of pointed sets

· · · → Aϕ−→ B

ψ−→ C → · · ·

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is exact at B if and only if im ϕ = kerψ. Again, if these happen to be groups(viewed as pointed sets with distinguished element the identity), with the mapsgroup homomorphisms, then this reduces to the ordinary definition of exactness.Predictably, a sequence of pointed sets is said to be exact if and only if it isexact at each point in the sequence.

This allows us to make sense of exactness for any sequence involving sets ofthe form [Y,X]. Note that even if we had restricted ourselves to consideringπn(X), we would still need this definition because π0(X) has no group structure.

9.1.2 Statement of the Puppe sequence

To every map between (based) topological spaces, there is an associated Puppesequence. We begin with some definitions.

Definition 9.1. Let f : X → Y be a map. The mapping fiber of f is thespace

Mf = (x, γ) ∈ X × Y I | γ(0) = y0, γ(1) = f(x)

topologised by taking the compact-open topology on Y I . The basepoint for Mf

is taken to be (x0, γ0) where γ0 is the constant path at y0.

Note that Mf is a subspace of the pathspace Ef which we constructed whenproving that every map can be replaced by a fibration (Proposition 7.2).

There is a natural inclusion j : ΩY → Mf given by γ 7→ (x0, γ), and anobvious projection q : Mf → X. Thus we get a sequence of spaces:

ΩXΩf−−→ ΩY

j−→Mfq−→ X

f−→ Y (9.1)

Applying the functor Ω to this sequence we get a new sequence ending in

ΩXΩf−−−→ ΩY , and as such we can join it up with (9.1) to obtain a longer

sequence. By iterating this construction we end up with a sequence:

· · · Ω2j−−−→ Ω2MfΩ2q−−−→ Ω2X

Ω2f−−−−→ Ω2YΩj−−−→ ΩMf

Ωq−−−→

ΩXΩf−−−→ ΩY

j−−−→Mfq−−−→ X

f−−−→ Y(9.2)

Which we call the Puppe sequence associated to the map f . The statementof the main theorem is then:

Theorem 9.2 (Puppe sequence). If Z is any space then applying the functor[Z,−] to (9.2) results in a long exact sequence of pointed spaces.

Corollary 9.3. For any map f : X → Y there is a long exact sequence:

· · · → π2(Mf )→ π2(X)→ π2(Y )→ π1(Mf )→π1(X)→ π1(Y )→ π0(Mf )→ π0(X)→ π0(Y )

Proof. Take Z = S0 in the previous theorem and use Corollary 8.3.

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Aside. Actually, there is a subtlety in the previous proof which we have skatedover: the maps in the sequence of π0’s are a priori only maps of pointed sets,even when we replace the π0’s by the appropriate πn’s. We have to check that,after this replacement, the maps are in fact group homomorphisms (wheneverit makes sense for them to be so). But this holds because we can take Z = S1

and use the fact that πn(X) = π1(Ωn−1X) as a group, as well as the fact that[S1,−] = π1(−) is a functor into the category of groups, hence sends continuousmaps to group homomorphisms.

A sequence · · · → Xk+1 → Xk → Xk−1 → · · · of topological spaces withthe property that, for any space Z, applying the funtor [Z,−] to the sequenceresults in a long exact sequence of pointed spaces, is called an exact sequenceof topological spaces. Therefore Theorem 9.2 can be rephrased as saying thatthe Puppe sequence is exact. We now turn to the proof of this result.

9.1.3 Proving the Puppe sequence

The proof consists of showing (by induction on n) that each of the sequences

Ωn+1XΩn+1f−−−−→ Ωn+1Y

Ωnj−−−→ ΩnMfΩnq−−−→ ΩnX

Ωnf−−−→ ΩnY

is exact (the result then follows since pasting together exact sequences yieldsan exact sequence). Though the strategy for the proof is clever and worthunderstanding, the details do get quite technical; as such, the reader may wish,on her first time through, to take some of the more involved calculations ontrust.

We begin with the base case n = 0, so we have:

ΩXΩf−−→ ΩY

j−→Mfq−→ X

f−→ Y (9.3)

Lemma 9.4. The sequence (9.3) is exact at X.

To prove this we must first establish the following result:

Lemma 9.5. Let r : Mf → Y be defined by r : (x, γ) 7→ γ(1). Then f isnullhomotopic (relative basepoints) if and only if there exists a map ϕ : X →Mf

such that the following diagram commutes:

Xϕ

//_______

f

????

????

Mf

r~~

Y

Proof. Suppose first that f is nullhomotopic (relative basepoints) and let F :X×I → Y be a nullhomotopy of f , so that F (x, 0) = y0 for all x and F (−, 1) =f . Define:

ϕ : X −→Mf

x 7→ (x, F (x,−))

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Where of course F (x,−) : I → Y is given by F (x,−)(t) = F (x, t). ThenF (x,−)(0) = y0 and F (x,−)(1) = f(x), so certainly (x, F (x,−)) ∈ Mf . Fur-thermore rϕ(x) = r(x, F (x,−)) = F (x, 1) = f(x) so the diagram commutes asrequired.

Now suppose conversely that such a ϕ exists, and write ϕ(x) = (g(x), γx)for g : X → X and γx ∈ Y I for each x. By commutativity of the diagram wethen have f(x) = rϕ(x) = r(g(x), γx) = γx(1) = g(x). Let us define:

F : X × I → Y

(x, t) 7→ γx(t)

This is continuous by the continuity of ϕ. Since (g(x), γx) ∈ Mf for all x, wehave F (x, 0) = γx(0) = y0 and F (x, 1) = γx(1) = g(x) = f(x), so we have anullhomotopy of f .

Finally this homotopy is relative basepoints because F (x0, t) = γx0(t) = y0

since ϕ : x0 7→ (x0, y0) (where y0 indicates the constant path) because (x0, y0)is the basepoint of Mf (recall that all our maps are basepoint-preserving byassumption).

Proof of Lemma 9.4. Let Z be any space. We must show that the sequence ofpointed sets

[Z,Mf ]q∗−→ [Z,X]

f∗−→ [Z, Y ]

is exact at [Z,X], i.e. that im q∗ = ker f∗.First let g ∈ [Z,Mf ] and consider q∗(g) = qg ∈ im q∗. We must show that

f∗(qg) = fqg is zero in [Z, Y ], i.e. that fqg is nullhomotopic. In fact, we willshow that fq is nullhomotopic (which certainly implies that fqg is).

Consider then fq : Mf → Y . We can consider the mapping fiber of fqitself, which by definition consists of triples (x, γ, η) ∈ X × Y I × Y I such that(x, γ) ∈Mf , η(0) = y0 and η(1) = fq(x, γ) = f(x). Define:

ϕ : Mf →Mfq

(x, γ) 7→ (x, γ, γ)

By the discussion in the previous paragraph this is well-defined (i.e. it reallydoes take values in Mfq). Then it is clear that the diagram

Mfϕ

//_______

fq A

AAAA

AAA

Mfq

r

Y

is commutative, and hence by Lemma 9.5 fq is nullhomotopic. We have thusshown that im q∗ ⊆ ker f∗.

Now let g ∈ [Z,X] lie in the kernel of f∗, so that fg : Z → Y is nullhomo-topic. Then again by Lemma 9.5, there is a map ϕ : Z → Mfg such that the

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following diagram commutes:

Zϕ

//_______

fg

????

????

Mfg

r

Y

And in fact ϕ takes the form ϕ(z) = (z, θz). Now, define ψ : Mfg →Mf by(z, γ) 7→ (g(z), γ), where z ∈ Z. Then ψϕ : Z →Mf , and:

qψϕ(z) = qψ(z, θz) = q(g(z), θz) = g(z)

So g = qψϕ = q∗(ψϕ) ∈ im q∗, and so ker f∗ ⊆ im q∗ and this completes theproof.

We have thus shown that (9.3) is exact at X. As for the rest of the sequence,however, such a direct approach will not work. Instead, we must investigate adifferent sequence of spaces (obtained by iterating the mapping fiber construc-tion), show exactness of that sequence, and then use this fact to prove exactnessof (9.3).

We can apply the mapping fiber construction to the map q : Mf → X toobtain a space Mq, and there is similarly a natural projection q′ : Mq → Mf .Carrying out this construction yet again for q′ we obtain a mapping fiber q′′ :Mq′ →Mq. Thus we have a sequence of spaces:

Mq′q′′−→Mq

q′−→Mfq−→ X

f−→ Y (9.4)

And this sequence is exact at X by 9.4. But the same result result alsoimplies that the sequence is exact at Mf and Mq, and hence the entire sequenceis exact.

Now we must find some way of relating this sequence to the sequence (9.3).Consider again Mq. By definition this consists of triples (x, γ, η) with (x, γ) ∈Mf and η a path in X with η(0) = x0 and η(1) = q(x, γ) = x. With thisinformation one can verify that the map:

i : ΩY →Mq

γ 7→ (x0, γ, x0)

is well-defined (where the second x0 really signifies the constant path at x0). Infact, i is a homotopy equivalence (for the proof see [Rot88] §11).

Lemma 9.6. There is a map l : ΩX →Mq′ such that the following diagram iscommutative (up to homotopy equivalence), with all the vertical arrows homo-topy equivalences:

ΩXΩf//

l

ΩYj//

i

Mfq//

id

Xf//

id

Y

id

Mq′q′′// Mq

q′// Mf

q// X

f// Y

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Proof. Commutativity of the rightmost two squares is tautological. To see thatthe second square commutes, let γ ∈ ΩY . Then by definition j(γ) = (x0, γ).On the other hand:

q′i(γ) = q′(x0, γ, x0) = (x0, γ)

So indeed the square commutes. The definition of l and the proof that with thisdefinition the leftmost square commutes is somewhat technical, so we defer itto [Rot88].

The previous lemma establishes a strong relationship between the sequences(9.3) and (9.4), an immediate consequence of which is the following.

Proposition 9.7. The sequence (9.3) is exact.

Proof. For any space Z, apply the functor [Z,−] to the diagram in the previouslemma. The resulting diagram commutes since the original diagram commutes,the vertical maps are isomorphisms in the new diagram because they are homo-topy equivalences in the old diagram, and the bottom row in the new diagramis an exact sequence of pointed sets since (9.4) is an exact sequence of spaces.Putting all these facts together, one can easily prove by an elementary diagramchase that the top row is exact, as required.

This completes the proof of the base case. Using the theory we have de-veloped thus far, the proof of the induction step is almost immediate, being adirect consequence of the following general fact:

Proposition 9.8. If · · · → Xk+1 → Xk → Xk−1 → · · · is an exact sequenceof topological spaces then so is the induced sequence · · · → ΩXk+1 → ΩXk →ΩXk−1 → · · · .

Proof. This is where we make use of the suspension-loopspace adjunction. Forany space Z and for any k we have a commutative diagram:

[Z,ΩXk+1] //

∼=

[Z,ΩXk] //

∼=

[Z,ΩXk−1]

∼=

[ΣZ,Xk+1] // [ΣZ,Xk] // [ΣZ,Xk−1]

Here the vertical arrows are the natural isomorphisms of the suspension-loopspaceadjunction (Proposition 8.2), and the diagram commutes by naturality. By as-sumption the bottom row is exact at [ΣZ,Xk], and hence by a diagram chase(similar to the one in the proof of Lemma 9.6) we have that the top row is exactat [Z,ΩXk], as required.

Using this result we see, by repeated application of Ω, that each of thesequences:

Ωn+1XΩn+1f−−−−→ Ωn+1Y

Ωnj−−−→ ΩnMfΩnq−−−→ ΩnX

Ωnf−−−→ ΩnY

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is exact. This completes the proof of the Puppe sequence (Theorem 9.2).

The Puppe sequence is an incredibly powerful tool for analysing homotopygroups, but it is limited by the fact that in general we do not know much aboutthe mapping fiber Mf . In the next section we will see that when f is a fibrationthere is a very nice interpretation for Mf . It is in its application to fibrations,then, that the Puppe sequences assumes its full strength.

9.2 The long exact sequence of a fibration

A basic fact about fiber bundles, pretty much built in to the definition, is thatthe fibers are all homeomorphic. For fibrations this is no longer the case; howeverthere is a weaker homogeneity property, reflecting the fact that, philosophicallyspeaking, we are working in the homotopy category.

Proposition 9.9. If p : E → B is a Hurewicz fibration with the base spacepath-connected, then the fibers of p are all homotopy equivalent.

Proof. Let b0 ∈ B be arbitrary. First note that p−1(b0) is nonempty, for we canchoose any point c in the image of p and then take a path in B from c to b0.Then c has a lift to E by assumption, and hence we can lift the whole path toE. The endpoint of the lifted path is then an element of p−1(b0). (We have justshown that every fibration over a path-connected base must be surjective.)

Hence we may choose x0 ∈ p−1(b0). We claim that p−1(b0) is homotopyequivalent to the mapping fiber Mp (see Definition 9.1), viewing p as a mapp : (E, x0)→ (B, b0).

This clearly implies the claim, because if b1 is any other point in B withp : (E, x1) → (B, b1) and we denote the mapping fibers of these pointed mapsby Mp,0 and Mp,1 respectively, then there is a natural homeomorphism

Mp,0

∼=−→Mp,1

(x, γ) 7→ (x, η · γ)

where η is a path in B from b1 to b0.

Thus it remains to show that p−1(b0) 'Mp. Write F = p−1(b0), and define:

λ : F →Mp

x 7→ (x, b0)

Where of course b0 indicates the constant path at b0. Since b0(0) = b0 andb0(1) = b0 = p(x), this is a well-defined map into Mp. In order to show that λis a homotopy equivalence, we will construct a homotopy inverse. We begin bydefining:

G : Mp × I → B

(x, γ, t) 7→ γ(1− t)

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Note that the natural projection q : Mp → E is a lift of G(−, 0) becausepq(x, γ) = p(x) = γ(1) = G(x, γ, 0). Hence, because p is a fibration, thereis a map G : Mp × I → E such that the following diagram commutes:

Mpq//

_

E

p

Mp × I G //

G

<<xx

xx

xB

From this diagram we can see that if (x, γ) ∈Mp, then pG(x, γ, 1) = G(x, γ, 1) =

γ(0) = b0, so G(x, γ, 1) ∈ F . Hence we can define:

µ : Mp → F

(x, γ) 7→ G(x, γ, 1)

We claim this is a homotopy inverse for λ. First consider the map:

H : F × I → F

(x, t) 7→ G(λ(x), t)

This is well-defined (i.e. it really does take values in F ) because:

pH(x, t) = pG(λ(x), t) = G(λ(x), t) = G(x, b0, t) = b0(1− t) = b0

And clearly:

H(x, 0) = G(λ(x), 0) = G(x, b0, 0) = q(x, b0) = x

H(x, 1) = G(λ(x), 1) = G(x, b0, 1) = µ(x, b0) = µλ(x)

So that H is a homotopy between idF and µλ. For the other composition,consider the map

K : Mp × I →Mp

(x, γ, t) 7→ (G(x, γ, t), θ(x,γ,t))

where θ(x,γ,t) : s 7→ γ(s(1− t)) is a path in B. To see that this really does takevalues in Mp, note that θ(x,γ,t)(0) = γ(0) = b0 and that:

θ(x,γ,t)(1) = γ(1− t) = G(x, γ, t) = pG(x, γ, t)

So indeed K is well-defined. Finally we have:

K(x, γ, 0) = (G(x, γ, 0), γ) = (q(x, γ, 0), γ) = (x, γ)

K(x, γ, 1) = (G(x, γ, 1), b0) = (µ(x, γ), b0) = λµ(x, γ)

So K is a homotopy between idMpand λµ, which completes the proof.

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10 THE HOPF FIBRATION

Because of this result, we are entitled to talk about the fiber of a fibration.We will often write F → E → B to indicate that E → B is a fibration withfiber F .

Now, in the proof of the previous proposition, we saw that the fiber F is ho-motopy equivalent to the mapping fiber Mp of the fibration. Having previouslynoted that the main practical obstruction to applying the Puppe sequence was alack of understanding of the mapping fiber, we see that this problem disappearsin the case of fibrations. Therefore we obtain the following result, which wehave been building towards for three chapters:

Theorem 9.10. Let F → E → B be a Hurewicz fibration. Then there is along exact sequence:

· · · → π2(F )→ π2(E)→ π2(B)→ π1(F )→π1(E)→ π1(B)→ π0(F )→ π0(E)→ π0(B)

Proof. This is just Corollary 9.3 applied to the map p : E → B, noting thatMp ' F by the proof of the previous result.

We call this the long exact homotopy sequence associated to the fibra-tion. This is the only form of the Puppe sequence which we will use in practice,and in some ways it is more fundamental (indeed, in many treatments it isproved independently). It will be our primary tool from here on out, and is themain reason why fibrations are so useful to us.

Aside. The same result also holds for Serre fibrations ([Hat01] Theorem 4.41).Although the proof is not particularly difficult, it would take us quite far offtrack to present it here. Anyway, since all the fibrations we will encounter areHurewicz (see the discussion after Proposition 7.3), we will not have any needfor this more general claim.

10 The Hopf Fibration

Following on from our general discussion of fibrations, we explore what is ar-guably the most important example: the Hopf fibration. This is a fibration (infact a fiber bundle) of the form S1 → S3 → S2 (so that the base, total space andfiber are all spheres). It is a fundamental object in algebraic topology; beautifulin its own right and incredibly useful for computations.

10.1 Definition of the Hopf fibration

Since we will have use for it later, we begin with a somewhat more generalconstruction: that of the complex projective space. For any positive integer n,there is a natural inclusion S2n+1 ⊆ Cn+1 as the set of complex (n + 1)-tupleswith unit norm:

S2n+1 = (z0, . . . , zn) ∈ Cn+1 : |z0|2 + . . .+ |zn|2 = 1

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Now, the scalar multiplication of C on Cn+1 restricts to an action of S1 ⊆ Con Cn+1. It is easy to check (we leave it to the reader) that S2n+1 is invariantunder this action, so that we have a well-defined action of S1 on S2n+1. Thuswe can form the quotient space S2n+1/S1; this is by definition the complexprojective space of dimension n, usually denoted CPn.

We assume the reader has had some exposure to projective spaces before(though she may have seen the definition in a slightly different form). Thesespaces are absolutely central to the field of algebraic geometry, and in suchcontexts they support a great deal of additional structure (see [Har77] §I.2, orfor the analytic point of view see [Huy05] §2.4]).

For our purposes, however, we will view CPn from the point of view ofalgebraic topology, meaning that we will only care about those properties whichremain unchanged under homotopy equivalence.

A basic fact we will need is the recursive construction of the projective spaces.Though we imagine the reader will already have seen this, we include an outlinefor the sake of completeness; the details (including the well-definedness of theconstructions) are left as exercises.

If (z0, . . . , zn) ∈ S2n+1 we denote the corresponding point of CPn by [z0, . . . , zn].Let:

U0 = [z0, . . . , zn] ∈ CPn : z0 6= 0

Then the map U0 → Cn given by [z0, . . . , zn] 7→ (z1/z0, . . . , zn/z0) is ahomeomorphism. Furthermore if we consider:

P0 = CPn \ U0 = [z0, . . . , zn] ∈ CPn : z0 = 0

Then the map P0 → CPn−1 given by [z0, . . . , zn] 7→ [z1, . . . , zn] is a homeo-morphism. Hence we may write CPn = Cn ∪ CPn−1, where we think of CPn−1

as being a “hyperplane at infinity”. In this way we may view CPn as beingobtained from CPn−1 by attaching a single 2n-cell. Since CP0 = pt, we haveby induction a cell complex structure:

CPn = e0 ∪ e2 ∪ . . . ∪ e2n (10.1)

Now consider the quotient map h : S2n+1 → CPn. The fiber over anypoint in the base is simply its orbit in S2n+1 under the action of S1, which ishomeomorphic to S1 since the action is faithful. Using affine co-ordinates onCPn it is not too hard to show that h is locally trivial, so we have a fiber bundle:

S1 → S2n+1 → CPn

Specialising to the case n = 1, we have:

S1 → S3 → CP1

But by (10.1), CP1 = e0 ∪ e2 = S2, since there is only one way to attacha 2-cell to a 0-cell. From a more conceptual point of view, we can consider S2

39


as the Riemann sphere C∪∞, the one-point compactification of the complexplane. Then our homeomorphism is given by:

CP1 ∼= C ∪ ∞[z0, z1] 7→ z0/z1

Therefore, after making the appropriate identifications, our fiber bundle hasthe following form:

S1 → S3 → S2

We call this the Hopf fibration. Since it is a fiber bundle, Proposition 7.3implies that it is a (Hurewicz) fibration, so that its title is justified.

Shortly we will derive a number of powerful consequences of the existence ofsuch a fibration; but first, we will take a brief detour to gain some intuition forthe map we are studying.

10.2 Picturing the Hopf fibration

From here on we will take for granted the identification S2 = C∪∞ discussedearlier. Then by definition of the quotient map and the identification CP1 ∼= S2

we have:h : (z0, z1) 7→ [z0, z1] 7→ z0/z1

Where z0, z1 ∈ C with |z0|2 + |z1|2 = 1. If we work in polar coordinates andwrite zj = rje

iθj for j = 1, 2 then:

h(z0, z1) = (r0/r1)ei(θ0−θ1)

Consider the circles CR of constant radius R in C ∪ ∞ for 0 < R < ∞(these correspond to lines of latitude on S2). Since r2

0 + r21 = 1 there is exactly

one solution to r0/r1 = R, namely:

r0 =R√

1 +R2, r1 =

1√1 +R2

On the other hand θ0 and θ1 are free to run over all of S1. Therefore,h−1(CR) ∼= S1×S1 = T 2, the 2-torus. Since the CR partition the image S2, thetori h−1(CR) partition the domain S3, as long as we include the limiting casesR = 0, R =∞. In these limiting cases we have h−1(C0) ∼= h−1(C∞) ∼= S1 sincethe fiber over any point is a circle. We think of these circles as being degeneratetori, obtained from the degenerate circles (that is to say, the points) C0 andC∞.

To picture this more concretely, we remove a point of h−1(C∞) from S3 andconsider the resulting stereographic projection S3 \ pt ∼= R3. Then the toridescribed above form a concentric partition of R3. The circle correspondingto C0 is just an ordinary circle in R3, whereas the circle corresponding to C∞becomes a line because one of its points (the point we are projecting from) hasbeen removed.

40

11 POSTNIKOV ANALYSIS

Once the reader has carried out the calculations above and convinced herselfof their veracity, the natural next step is to look at some pictures. Though thereare many praiseworthy illustrations in the literature, the author would like tosingle out for special mention the video [Joh11], available online.

10.3 Applications of the Hopf fibration

Having developed the theory of fibrations in a general context in the previouschapter, we can now apply it to the Hopf fibration. By 9.10 there is a long exactsequence of homotopy groups:

· · · → πk(S1)→ πk(S3)h∗−→ πk(S2)→ πk−1(S1)→ · · ·

As long as k ≥ 3 then we have by Proposition 3.1 that πk(S1) = πk−1(S1) = 0.Therefore the map h∗ : πk(S3) → πk(S2) in the sequence above is an isomor-phism. We have thus proven the following:

Theorem 10.1. πk(S2) ∼= πk(S3) for k ≥ 3

This wonderful result shows just how powerful the Hopf fibration is: it setsup a symmetry between S2 and S3 which, a priori, needn’t exist. In fact, thereare only two higher-dimensional analogues of Theorem 10.1, and neither is asstrong as this result.

As an immediate corollary, we can compute our first homotopy group of theform πk(Sn) for k > n.

Theorem 10.2. π3(S2) = Z

In fact, we can say a bit more by actually finding a generator for π3(S2).Recall from §5.5 that the degree map gives an isomorphism:

deg : π3(S3) ∼= Z

It is also clear from the discussion in that section that deg id = 1; hence we seethat π3(S3) is generated by the (homotopy class of the) identity map i : S3 →S3. But we saw above that h∗ : π3(S3) ∼= π3(S2). Hence a generator for π3(S2)is given by h∗(i) = h i = h. Thus, a generator for π3(S2) is given by the Hopffibration h : S3 → S2 itself.

11 Postnikov Analysis

11.1 Serre’s method

The utility of the Hurewicz theorem comes from the fact that homology groupsare in general much easier to compute than homotopy groups. For instance,we have the Mayer-Vietoris sequence for homology, an extremely powerful com-putational tool for which no analogue exists in homotopy theory. By reducing

41


the computation of homotopy to that of homology, we thus greatly increase ourchances of success.

We have exploited this fact already, when we calculated πn(Sn) (Proposition5.5). Notice that the real meat of the computation is contained in the Hurewicztheorem; once that is granted, the determination of πn(Sn) ∼= Hn(Sn) followseasily by Mayer-Vietoris (Proposition 2.1).

It would be nice if we could generalise the Hurewicz theorem to somehowrelate πn+k(Sn) to Hn+k(Sn) for k ≥ 1. On reflection, however, we can see thatsuch a hope is naıve: we know that the higher homology groups of Sn are alltrivial, whereas this is far from true for the homotopy groups.

Luckily, we don’t actually need a generalisation; the original Hurewicz the-orem will work just fine, provided we modify the space under consideration.Indeed, suppose we could find a space Y with πn+k(Y ) = πn+k(Sn) and withthe additional property that Y is (n + k − 1)-connected. Then we could applythe Hurewicz theorem to Y , obtaining πn+k(Sn) = πn+k(Y ) = Hn+k(Y ).

As we shall see shortly, not only does such a Y always exist, but there is anatural fibration relating Y , Sn and a suitable Eilenberg-MacLane space, whichwe can exploit to compute the homology of Y . This whole process is known asSerre’s method.

Our primary reference in this chapter, as well as in §13, is the (unpublished)[Hun93]. An approach along similar lines is presented in §4 of [Hat01] (see inparticular Example 4.17).

11.2 Killing off homotopy groups

Our first task will be to construct the space Y in the case k = 1. This space willhave the same homotopy groups as Sn in every dimension except for n, whereπn(Y ) = 0. Thus our task is to “kill off” πn.

We go about this in a somewhat roundabout way, first constructing a spacewhich kills off all homotopy groups in dimensions greater than n, and thenperforming a clever trick to produce the space we desire.

Although we will only be applying this construction to spheres, the heartof the argument is more apparent if we work in a more general context. So letX be an (n− 1)-connected CW complex with πn(X) nontrivial (of course, theexample we have in mind is X = Sn). First we will kill off πn+1(X); choose acollection of generators for πn+1(X), and choose a representative map for eachgenerator, so that we have a collection of maps gα : Sn+1 → X. We can thenuse these maps to attach (n+2)-cells en+2

α to X, resulting in a new CW complexwhich we call Yn+1.

Since we have only added cells in dimension n + 2, the (n + 1)-skeleton ofYn+1 is the same as that of X; therefore by cellular approxmation we haveπk(Yn+1) = πk(X) for k ≤ n.

We claim that πn+1(Yn+1) = 0. So consider a map g : Sn+1 → Yn+1. Bycellular approximation g is homotopic to a map Sn+1 → X, which we also denoteby g. Then g defines a class [g] ∈ πn+1(X), which can be written as a finitesum

∑α nα[gα] for nα ∈ Z. But each gα is nullhomotopic in Yn+1, since the

42


characteristic map for en+2α is an extension of gα to Dn+2 (see Proposition 5.2).

Hence [gα] = 0 for all α, and so [g] = 0. We have thus shown πn+1(Yn+1) = 0,and so we have killed off πn+1.

In order to kill of πn+2, we repeat the above construction with X replaced byYn+1. Thus we obtain a space Yn+2, with πk(Yn+2) = πk(Yn+1) for k ≤ n + 1and πn+2(Yn+2) = 0. Notice that we have πk(Yn+2) = πk(X) for k ≤ n andπn+1(Yn+2) = πn+2(Yn+2) = 0, so we have killed off πn+1 and πn+2 withoutaffecting the lower homotopy groups.

We continue in this way, attaching cells in dimensions n+ 2, n+ 3, n+ 4, . . .to kill off πn+1, πn+2, πn+3, . . .. Doing so, we end up with a (possibly infinite-dimensional) CW complex Y .

By construction there is a natural inclusion Yn+k → Y for all k ≥ 1, andunder this inclusion the (n+k+1)-skeleton of Y is equal to the (n+k+1)-skeletonof Yn+k. Therefore πn+k(Y ) = πn+k(Yn+k) = 0 for all k ≥ 1. Furthermore the(n + 1)-skeleton of Y is just the (n + 1)-skeleton of X, so that πr(Y ) = πr(X)for r ≤ n. In summary we have:

πr(Y ) =

πr(X) for r ≤ n0 for r ≥ n+ 1

In fact, since X is (n− 1)-connected, we see that Y has only one nontrivialhomotopy group, namely πn(Y ) = πn(X). Spaces with this property are calledEilenberg-MacLane spaces, and are absolutely central to algebraic topology. In§B we prove some basic facts about them which we will rely on later.

Again by construction there is a natural incluson X → Y , and by Proposi-tion 7.2 we may replace this map by a fibration, which we write as:

X(n+ 1)→ X → Y

(The fiber X(n+ 1) is uniquely defined up to homotopy equivalence becausethe base Y is connected.) We claim that X(n + 1) is the space we are lookingfor. Using the long exact homotopy sequence of the fibration, the homotopygroups of X(n+ 1) are easy to compute. We have:

· · · → πr+1(Y )→ πr(X(n+ 1))→ πr(X)→ πr(Y )→ · · ·If r is not equal to n−1 or n, we have that πr+1(Y ) = πr(Y ) = 0, and hence

πr(X(n+ 1)) ∼= πr(X). In the critical dimensions we have:

0→ πn(X(n+ 1))→ πn(X)→ πn(Y )→ πn−1(X(n+ 1))→ 0

The middle map i∗ : πn(X)→ πn(Y ) induced by the inclusion is clearly anisomorphism. Hence we have πn(X(n+ 1)) = Ker i∗ = 0 and πn−1(X(n+ 1)) =πn(Y )/ Im i∗ = 0. In summary we have:

πr(X(n+ 1)) =

0 for r ≤ nπr(X) for r ≥ n+ 1

43


(The reader should compare this to the homotopy of the space Y constructedabove.) In particular, we have πn+1(X(n+ 1)) = πn+1(X), but since X(n+ 1)is n-connected we also have πn+1(X(n+ 1)) = Hn+1(X(n+ 1)).

We have finally arrived at our destination: we have reduced the computationof πn+1(X) to that of Hn+1(X(n + 1)). It turns out that the computation ofthis group is often a highly nontrivial matter, but it is certainly less hopelessthan attempting to compute πn+1(X) by hand.

Aside. We can view this construction as a generalisation of the universal cover:we remarked earlier that a covering projection induces isomorphisms on πn forn ≥ 2 ([Hat01] Proposition 4.1), and consequently the universal cover X → Xis a space with π1(X) = 0 and πn(X) ∼= πn(X) for n ≥ 2. That is, X is a spacekilling off the first homotopy group of X.

We can repeat the previous constructions with the space X(n+ 1) in placeof X to kill off πn+2(X(n + 1)). This gives us a new space X(n + 2), whichsatisfies:

πr(X(n+ 2)) =

0 for r ≤ n+ 1

πr(X) for r ≥ n+ 2

And hence πn+2(X) = πn+2(X(n + 2)) = Hn+2(X(n + 2)). Clearly wecan continue in this way to construct spaces X(n + k) satisfying πn+k(X) =Hn+k(X(n+ k)).

11.3 Recomputing π3(S2)

As a first application of Postnikov analysis, we will rederive the result π3(S2) =Z which we proved earlier. Of course this is redundant: the primary aim hereis to acquaint the reader, via a relatively accessible example, with the details ofSerre’s method; thus paving the way for our later (much more difficult) compu-tation of π4(S2).

We apply the constructions in the previous section to the CW complex X =S2, killing off the second homotopy group to obtain a space S2(3). There is anatural map S2(3)→ S2, which induces isomorphisms on πr for r ≥ 3.

Let us replace this map with a homotopy equivalent fibration (as in Propo-sition 7.2), which for the moment we write as F → S2(3)→ S2. Again we mayuse the long exact homotopy sequence to determine the homotopy groups of F :

· · · → πr+1(S2(3))→ πr+1(S2)→ πr(F )→ πr(S2(3))→ πr(S

2)→ · · ·

As noted above, for r ≥ 3 we have πr(S2(3)) → πr(S

2) an isomorphism:hence πr(F ) = 0 for r ≥ 3. We also have

· · · → π3(S2(3))→ π3(S2)→ π2(F )→ 0

and since the leftmost map is an isomorphism we have π2(F ) = 0. Finally wehave

0→ π2(S2)→ π1(F )→ 0

44

12 THE LERAY-SERRE SPECTRAL SEQUENCE

so π1(F ) = π2(S2) = Z. To summarise, we have

πr(F ) =

Z for r = 1

0 for r ≥ 2

so that F is an Eilenberg-MacLane space K(Z, 1). Since the base and totalspace of the fibration are CW complexes, it is known that F has the homotopytype of a CW complex (see the discussion following Proposition 4.64 in [Hat01]).Hence by uniqueness (Proposition B.1) we have that F is homotopy equivalentto S1 (Proposition B.2). Therefore our fibration is:

S1 → S2(3)→ S2

Since we already know the homology of S1 and S2, it would be nice if wehad some way of relating the homology groups of spaces fitting into a fibration(analogous to the long exact sequence which we have for homotopy groups).In fact there does exist machinery for doing this, though it is considerablymore complicated than an exact sequence: it is called the Leray-Serre spectralsequence.

12 The Leray-Serre Spectral Sequence

The theory of spectral sequences is a rich one, and there exist many types ofspectral sequence, each adapted for a particular purpose. Here we will makeuse of the Leray-Serre spectral sequence (often just called the Serre spectralsequence): it is essentially a big machine which allows us to relate the homologiesof the base, total space and fiber of a fibration satisfying certain (fairly mild)properties.

When most people begin learning spectral sequences, the extremely algebraicnature of the arguments, along with the legions of subscripts and superscriptsinvolved, can make the task quite daunting. For this reason, we have decidedto give our presentation in the least general context possible. Our hope is thatthis increased concreteness will help give the first-time reader confidence indealing with spectral sequences, which she can build on later to gain a morecomprehensive view of the subject.

Having said this, the essentially algebraic nature of spectral sequences can-not be avoided, and consequently the present chapter will be by far the mostalgebraic in this article.

Our primary reference for this section is the excellent [Hat04]. An alternativeaccount, requiring more patience and dedication but providing a much broaderperspective, can be found in [McC01].

45


12.1 The spectral sequence associated to a filtration

Let F → Xp−→ B be a fibration, with B a finite-dimensional CW complex. We

can write B as an ascending chain of its skeleta

∅ = B(−1) ⊆ B0 ⊆ B1 ⊆ · · · ⊆ BN = B

(where by convention we think of the empty set as being the (−1)-skeleton ofthe space). A sequence like this, consisting of an ascending chain of subspacesof some topological space, is called a filtration. We can use the fibration p toobtain an induced filtration on the total space, namely:

∅ = p−1(B(−1)) ⊆ p−1(B0) ⊆ p−1(B1) ⊆ · · · ⊆ p−1(BN ) = p−1(B) = X

The idea of the Leray-Serre spectral sequence is to use this filtration of X toobtain information about its homology. For the moment it is helpful to considera somewhat more general situation: that of an arbitray (finite) filtration of X,which we can write as:

∅ = X−1 ⊆ X0 ⊆ X1 ⊆ · · · ⊆ XN = X (12.1)

The first thing to note is that this filtration induces a filtration on the homologygroups ofX. To see this, fix k and n and consider the image inHn(X) of the mapHn(Xk)→ Hn(X) induced by the inclusion. We claim that this lives inside theimage of Hn(Xk+1)→ Hn(k). The reason for this is that the inclusion Xk → Xcan be factored into a composition of inclusions

Xk → Xk+1 → X

and therefore by functoriality the induced map Hn(Xk) → Hn(X) is in factequal to the composition of the induced maps

Hn(Xk)→ Hn(Xk+1)→ Hn(X)

so in particular the image of Hn(Xk) → Hn(X) is contained inside that ofHn(Xk+1) → Hn(X). Writing this image as F kn we then have an ascendingchain of subgroups of Hn(X) given by

0 = F−1n ⊆ F 0

n ⊆ F 1n ⊆ · · · ⊆ FNn = Hn(X)

(where F−1n = Hn(∅) = 0). As such, we might hope to understand the structure

of Hn(X) by gaining information about the F kn .We will soon see that the machinery of spectral sequences allows us to com-

pute the successive quotients F kn/Fk−1n . It is important to note that this is

not always enough to recover Hn(X): for instance, consider Z with the simplefiltration

pZ ⊆ Zwhere p is any prime number. The quotient is then Z/pZ = Zp. On the otherhand we can consider Zp with the even simpler filtration:

0 ⊆ Zp

46


Clearly here the quotient is also Zp. So the successive quotients in the filtrationsof Z and Zp are the same, even though the groups are obviously not (thisphenomenon arises because there are short exact sequences of abelian groupswhich do not split).

Having said this, in many nice situations (for instance whenever all thegroups are free abelian) the succesive quotients F kn/F

k−1n do actually give enough

information to completely determine Hn(X). We will see several examples ofthis later on.

Our aim then is to find some practical way of computing F kn/Fk−1n . This

leads to the construction of the spectral sequence. Coming back to our filtration(12.1) of X, recall ([Hat01] §2.1) that for each pair (Xk, Xk−1) there is a longexact homology sequence of the form:

→ Hn(Xk−1)→ Hn(Xk)→ Hn(Xk, Xk−1)→ Hn−1(Xk−1)→

From here on, for reasons which will soon become apparent, we will write A1n,k =

Hn(Xk) and E1n,k = Hn(Xk, Xk−1) (think A for “absolute”). In this notation

the long exact homology sequence for the pair (Xk, Xk−1) is given by:

→ A1n,k−1 → A1

n,k → E1n,k → A1

n−1,k−1 → (12.2)

For varying k and n, we can fit these sequences together to form a diagram

// A1n+1,k

//

E1n+1,k

// A1n,k−1

//

E1n,k−1

// A1n−1,k−2

//

// A1n+1,k+1

//

E1n+1,k+1

// A1n,k

//

E1n,k

// A1n−1,k−1

//

// A1n+1,k+2

//

E1n+1,k+2

// A1n,k+1

//

E1n,k+1

// A1n−1,k

//

(12.3)which for the purposes of this article we will call the first A&E diagram.Since this diagram forms the starting point for the rest of the theory, we willtake some time to elaborate on it. The long exact homology sequences (12.2)are given, starting at any A1 term, by one vertical map down followed by twohorizontal maps to the right, with the pattern repeating itself from there. Thusthe relative homology sequences form “staircases” (albeit not very steep ones!)in the first A&E diagram.

The vertical maps, then, are those induced by the inclusions Xk ⊆ Xk+1.Each absolute homology group A1

n,k is contained in precisely two staircase se-quences: the one obtained by following the vertical map downwards, and the one

47


obtained by following the horizontal map to the right. This is what we shouldexpect, since each Xk is contained in two long exact sequences, namely thosecorresponding to the pairs (Xk+1, Xk) and (Xk, Xk−1). As a test of the reader’sfamiliarity with this diagram, we suggest that she tries to identify which of thestaircase sequences corresponds to which pair.

At the top of the A1 columns we have A1n,−1 = Hn(X−1) = Hn(∅) = 0,

whilst at the bottom we have A1n,N = Hn(XN ) = Hn(X).

Although the groupsA1n,k and E1

n,k have only been defined for k ∈ −1, . . . , N,it will be useful for our purposes to extend this definition. We therefore extendthe filtration of X by setting Xk = X−1 = ∅ for k < −1 and Xk = XN = Xfor k > N . Then the groups A1

n,k and E1n,k are defined for all n and k. Because

of this we now think of the first A&E diagram as being an infinite diagram(although outside the rows −1, . . . , N it is of course rather boring). This factwill help to simplify our arguments down the road.

The maps in the first A&E diagram fall into three distinct classes: those ofthe form A1 → A1, those of the form A1 → E1 and those of the form E1 → A1.We denote these by i1, j1 and k1, respectively. Thus we have:

i1 : A1n,k → A1

n,k+1

j1 : A1n,k → E1

n,k

k1 : E1n,k → A1

n−1,k−1

By exactness of the long exact homology sequence we have k1j1 = 0. We nowconsider the reverse composition j1k1. Suppose for the moment that X is a CWcomplex and that we have chosen our filtration to be the standard filtrationof X by its skeleta. Then E1

k,k = Hk(Xk, Xk−1) is the chain group in thechain complex for cellular homology, and by definition j1k1 is just the cellularboundary map:

j1k1 : E1k,k

// E1k−1,k−1

Hk(Xk, Xk−1) // Hk−1(Xk−1, Xk−2)

(The reader unfamiliar with cellular homology can find all the relevant detailsin [Hat01] §2.2.)

Thus we know a lot about j1k1 in this case, and so it is reasonable to hopethat investigating j1k1 will be worthwhile in the general case. As it turns out,such a line of enquiry will lead us quite quickly to the construction of the spectralsequence associated to the filtration on X. We begin this undertaking now.

At some points over the next few pages it might not be clear precisely whywe have chosen to introduce a certain concept or prove a certain fact; however,once we have constructed the spectral sequence, its usefulness will be apparentalmost immediately.

48


We return then to the case of an arbitrary filtration. Let d1 = j1k1. Notethat we have

d21 = j1k1j1k1 = 0

because k1j1 = 0. Since d1 is of the form E1 → E1, we can remove the A1

terms from the first A&E diagram (12.3), resulting in a diagram consisting ofd1 maps:

// E1n+2,k+1

// E1n+1,k

// E1n,k−1

// E1n−1,k−2

//

// E1n+2,k+2

// E1n+1,k+1

// E1n,k

// E1n−1,k−1

//

// E1n+2,k+3

// E1n+1,k+2

// E1n,k+1

// E1n−1,k

//

(12.4)

This diagram is by definition the first page of the spectral sequence associ-ated to the filtration on X, and we denote it by (E1, d1). Since d2

1 = 0, the rowsare chain complexes, so we can pass to their homology. We therefore define

E2n,k = ker d1/im d1

where of course the d1’s are those entering and leaving E1n,k. We would like to

relate these groups to the A1n,k in a similar manner to the first A&E diagram.

In order to make this work, we must restrict ourselves to certain subgroups ofthe A1. To this end, we define:

A2n,k = i1(A1

n,k−1) ⊆ A1n,k

The restriction of i1 to this group is clearly a well-defined map A2n,k → A2

n,k+1

and we denote it by i2. Building on this idea, we define:

j2 : A2n,k → E2

n,k−1

i1(a) 7→ [j1(a)]

To see that this is well-defined, first note that j1(a) ∈ ker d1 since d1(j1(a)) =j1k1j1(a) = 0. Furthermore if i1(a) = i1(b), then a − b ∈ ker i1. But ker i1 =im k1 by exactness of the staircase sequence, and therefore j1(a) − j1(b) ∈im j1k1 = im d1 so [j1(a)] = [j2(a)] in E2

n,k−1.Similarly we define:

k2 : E2n,k → A2

n−1,k−1

[e] 7→ k1(e)

Again we must check this is well-defined. Note first that since e ∈ ker d1 we havej1k1(e) = 0, so k1(e) ∈ ker j1 = im i1 by exactness of the staircase sequence,

49


so indeed k1(e) ∈ A2n−1,k−1. Furthermore if [e] = 0 in E2

n,k then e ∈ im d1 =im j1k1 ⊆ im j1. But im j1 = ker k1 by exactness, so k1(e) = 0 as required.

Putting all of this together, we have a new diagram

// A2n+1,k

77pppppppppppp

i2

E2n+1,k

k2 // A2n,k−1

99rrrrrrrrrr

i2

E2n,k−1

k2 // A2n−1,k−2

;;vvvvvvvvv

i2

// A2n+1,k+1

j299rrrrrrrrrr

i2

E2n+1,k+1

k2 // A2n,k

j2::vvvvvvvvv

i2

E2n,k

k2 // A2n−1,k−1

j2

==zzzzzzzzzz

i2

// A2n+1,k+2

j299rrrrrrrrrr

E2n+1,k+2

k2 // A2n,k+1

j2::vvvvvvvvv

E2n,k+1

k2 // A2n−1,k

j2

==zzzzzzzzzz

77pppppppppppp

99rrrrrrrrrr

(12.5)which we call the second A&E diagram. It is formally similar to the firstA&E diagram (12.3), except that the j maps now go one unit up in addition toone unit to the right.

Now, recall that in the first A&E diagram we had the long exact homologysequences forming staircases. These had the form:

· · · i1−→ · j1−→ · k1−→ · i1−→ · j1−→ · k1−→ · · ·

Consider then the sequences in the second A&E diagram of the form:

· · · i2−→ · j2−→ · k2−→ · i2−→ · j2−→ · k2−→ · · · (12.6)

Each of these traces a zig-zag path through the diagram much like the blade ofa saw. We claim that these sequences are exact. The proof is a diagram chase(using exactness of the staircase sequences in the first A&E diagram), and weleave it as a (valuable) exercise for the reader.

In particular, if we let d2 = j2k2 then d22 = 0 and we can remove the A2

terms from the second A&E diagram to obtain the second page of the spectralsequence associated to the filtration of X.

In fact, exactness of the sequences (12.6) means that all of the importantfacts about the first A&E diagram also hold true for the second A&E diagram.Therefore all the constructions which we just carried out for the first diagramcan also be applied to the second one: that is, we can construct a third A&Ediagram consisting of groups A3

n,k, E3n,k and maps i3, j3, k3.

We can repeat this process indefinitely. At the rth step, we replace Er−1n,k by

Ern,k = ker dr−1/im dr−1

and we replace Ar−1n,k by the subspace:

Arn,k = ir−1(Ar−1n,k−1)

50


We define the maps ir, jr, kr from the maps ir−1, jr−1, kr−1 in the same way asbefore, and in so doing we find that:

ir : Arn,k → Arn,k+1

jr : Arn,k → Ern,k−r+1

kr : Ern,k → Arn−1,k−1

(12.7)

(notice how the codomain of jr depends on r). These fit together to form therth A&E diagram, and taking dr = jrkr and removing all the A terms we obtainthe rth page of the spectral sequence, which we denote (Er, dr).

The spectral sequence associated to the filtration of X is then just thecollection of pages (Er, dr) for r ∈ N≥1 (the word “sequence” in the title alludesto the sequential nature of the pages, rather than to a sequence of groups ormodules as in the case of, say, an exact sequence).

The data (Er, dr) is enough to determine the groups Er+1 (just take ho-mology), but it is not in general sufficient to determine the maps dr+1. Forthis one needs to look at the rth A&E diagram, of which the page (Er, dr) isjust a sub-diagram. Generally, however, the full A&E diagram is difficult todetermine; for this reason, we usually rely on formal tricks which allow us tocompute (Er+1, dr+1) based only on the rth page (Er, dr). (In this light, theterminology “A&E diagram” takes on a double meaning: if we end up there itmust mean we’re in a bad way.)

Aside. In the literature, our so-called “A&E diagrams” are often presented in amuch more compact form, in the guise of objects called exact couples. Althoughthe formalism of exact couples is very elegant, we have decided not to developit here, in an effort to keep the presentation as concrete as possible.

After braving that relentless onslaught of algebra, it would be understand-able for the reader to have lost track of our goal; we therefore take a momentto recap.

We are trying to find information about the subgroups F kn ⊆ Hn(X). Moreprecisely, we are looking to see if the succesive quotient groups F kn/F

k−1n turn

up anywhere in our spectral sequence.We have good cause to be hopeful. Recall that by definition:

F kn = i∗(Hn(Xk))

where i : Xk → X is the inclusion. But we can factor this map as a sequenceof inclusions:

Xk → Xk+1 → · · · → XN = X

Then by functoriality, the map i∗ is equal to the composition of all the inducedmaps Hn(Xk+l)→ Hn(Xk+l+1) for l ∈ 0, . . . , N −k−1. But after going backand looking at the definitions, one can see that this is just the composition

irir−1 · · · i1

51


where r = N −k−1 and each ij is the map appearing in the jth A&E diagram.Thus we have:

F kn = irir−1 · · · i1(Hn(Xk))

= irir−1 · · · i1(A1n,k)

= irir−1 · · · i2(A2n,k+1)

= · · ·= Ar+1

n,(r+1)+(k−1)

In fact, the map ir is the identity map for sufficiently large r (since the filtrationof X eventually becomes stationary), so immediately we have the more generalresult that

F kn = Arn,r+k−1 (12.8)

for all sufficiently large r. Thus the groups we are looking for make a quiteexplicit appearance in the A&E diagram. However, as we already mentioned,the A&E diagram is often hard to compute: what we really want is for thesegroups to appear in the spectral sequence (Er, dr). We now turn ourselves tothis task.

Consider the rth page of the spectral sequence. By definition dr = jrkr, andso by (12.7) we can see that:

dr : Ern,k → Ern−1,k−r

Let us fix n and k. Since the filtration of X stabilises for indices M ≥ N orM ≤ −1, any relative group E1

n,M is zero for very large or very small M , andhence the same is true for Ern,M (since the latter group is obtained from theformer by taking subgroups and quotienting). Consequently if we take r largeenough, the maps dr entering and leaving Ern,k are zero, and therefore:

Er+1n,k = ker dr/im dr = Ern,k

and similarly for Er+2n,k and so on. In other words, the sequence of groups

(Ern,k)r∈N≥1becomes constant once r is large enough.

We denote this stable group by E∞n,k. The collection of such groups can bethought of as the “limit” of the spectral sequence. As we pass through the pagesof the sequence, taking homologies at each step, more and more of the groupson each page become stable. Although in general it can take a long time forcertain groups to stabilise (which is why we must assume in these argumentsthat r is very large), in the examples we will encounter the sequences actuallystabilise very quickly (always by the fourth page), so that it is within the realmof practicality to compute these limiting groups.

The claim we have been building towards this entire chapter can now bestated very simply:

E∞n,k = F kn/Fk−1n (12.9)

52


To see this, fix n and k and let r be large enough so that Ern,k = E∞n,k. Considerthe long exact sequence in the rth A&E diagram passing through Ern,k. Withreference to (12.7), we see that near Ern,k this looks like:

Ern+1,r+k−1kr−→ Arn,r+k−2

ir−→ Arn,r+k−1jr−→ Ern,k

kr−→ Arn−1,k−1

If we take r large enough then by the discussion above the first term is zero.On the other hand the last term Arn−1,k−1 is the image (under a composition of

ij maps) of A1n−1,k−r−1. For r very large, k − r − 1 ≤ −1 so Xk−r−1 = ∅ and

hence A1n−1,k−r−1 = Hn−1(∅) = 0. Thus for r large enough we have a short

exact sequence0→ Arn,r+k−2 → Arn,r+k−1 → E∞n,k → 0

and so we have

E∞n,k = Arn,r+k−1/ir(Arn,r+k−2) = Arn,r+k−1/A

r+1n,(r+1)+k−2 = F kn/F

k−1n

for r large enough by (12.8). Thus we have proved (12.9). In a sentence: thespectral sequence associated to the filtration of X converges to the successivequotients of the induced filtration on the homology of X.

12.2 The Leray-Serre spectral sequence

Everything in the previous section holds for an arbitrary filtration of X. Thesetup in this section is somewhat more specific. We suppose that we are given

a fibration F → Xp−→ B, with the base a finite-dimensional CW complex. We

then take the (finite) filtration on X induced by the filtration of B by its skeleta:

∅ = p−1(B(−1)) ⊆ p−1(B0) ⊆ p−1(B1) ⊆ · · · ⊆ p−1(BN ) = p−1(B) = X

In keeping with our notation in the previous section, we write Xk = p−1(Bk).We know that there is a spectral sequence associated to this filtration whichconverges to the homology of X (in the sense described above). The maintheorem of this section, due to Serre, gives an extremely nice formula for theterms of the second page of this sequence.

Before we can get to the statement of Serre’s result, there is a small issue ofindexing that needs to be taken care of. We have by definition

E1n,k = Hn(Xk, Xk−1)

and as it so happens this is trivial if n < k. We will not prove this fact, but(very) roughly speaking it holds because Bk/Bk−1 is a wedge of k-spheres andhence is (k − 1)-connected (so that its homology groups Hn vanish for n < kby the Hurewicz theorem), and then by the homotopy lifting property of p thesame holds for Xk/Xk−1.

As such E1n,k is nonzero only if k ≥ 0 and n ≥ k. The range of such values

in Z2 is not a particularly nice set to work with, so we re-index the spectral

53


sequence in such a way that this range of values gets sent to the first quadrant.To be explicit, we define new parameters p = k and q = n−k, and then we takeE1p,q to correspond to E1

n,k. This means that:

E1p,q = Hp+q(Xp, Xp−1)

This is nonzero only if p ≥ 0 and q ≥ 0, and hence with respect to theseparameters the first page of our spectral sequence (and therefore all the laterpages as well) is zero outside of the first quadrant.

Of course, this change of parameters also affects how we write down thedifferentials dr. Recall that in the old notation we had dr : Ern,k → Ern−1,k−r.Therefore with the new indexing, we have dr : Erp,q → Erp−r,q+r−1 since p− r =k− r and (p− r)+(q+ r−1) = n−1. Visually, this means that the differentialson the rth page go r units to the left and r − 1 units up.

Finally, note that once we translate (12.9) into the new parameters, it saysthat E∞p,n−p = F pn/F

p−1n .

Although the details of this adjustment might take a moment to percolate,we must stress that this a purely cosmetic change. The inner workings of thespectral sequence are exactly the same as in the previous section: for instance,we still take homologies in the rth page to compute the groups in the (r + 1)stpage. We have only made this change because it makes the spectral sequenceeasier to calculate in practice.

We are now in a position to state the main theorem of this section.

Theorem 12.1. Let F → X → B be a fibration with B a simply-connectedfinite-dimensional CW complex. Then there is a spectral sequence (Er, dr) withdr : Erp,q → Erp−r,q+r−1 which converges to the homology of X in the sense thatE∞p,n−p = F pn/F

p−1n and which satisfies:

E2p,q = Hp(B,Hq(F )) (12.10)

The only part of this theorem which we haven’t yet seen is (12.10). Theupshot of this result is that if we know the homology of B and the homologyof F then we know the terms E2

p,q (in the general case we can use the universalcoefficient theorem, though often this is not even necessary), and then we havea very good shot at finding the homology of X. We will see this in practice fora couple of important examples in the next chapter.

Aside. The theorem as stated is not as general as it could be. As it turnsout, the assumption that B is a finite-dimensional CW complex is not necessary(though the proof is much more illuminating in this case), and the assumptionthat B is simply-connected can be replaced by a weaker condition on π1(B).We will not have any need for this, but the reader interested in the full detailsshould consult §1 of [Hat04].

54

13 APPLYING SPECTRAL SEQUENCES TO SPHERES

13 Applying Spectral Sequences to Spheres

13.1 π3(S2)

Now we can return to the discussion of §11.3. Recall that, using Postnikovanalysis, we constructed a fibration of the form

S1 → S2(3)→ S2

where S2(3) is a 2-connected space such that π3(S2(3)) ∼= π3(S2). By Hurewiczit then suffices to find H3(S2(3)). This is the problem which motivated ourconstruction of the Leray-Serre spectral sequence.

All the conditions are satisfied to apply Theorem 12.1. Thus the second pageof the Leray-Serre spectral sequence consists of the groups:

E2p,q = Hp(S

2, Hq(S1))

For q 6= 0, 1 we have Hq(S1) = 0, so that E2

p,q = 0 also. On the other hand forq = 0, 1 we have Hq(S

1) = Z, so E2p,q = Hp(S

2,Z), which equals Z for p = 0, 2and 0 otherwise. Therefore the second page of the spectral sequence has theform

0 1 2 30123

Z ZZ Z

where the empty cells represent zero groups. Because d2 maps E2p,q to E2

p−2,q+1

the only differential whose domain and codomain are both nonzero is d2 : E22,0 →

E20,1 as shown above. As it happens, it is not too hard to show (using the fact

that S2(3) is 2-connected) that this map is an isomorphism, so that when wepass to the third page we have E3

2,0 = E30,1 = 0. However we have no real need

for this fact, so we leave its verification as an exercise for the interested reader.By the results of the previous chapter we have E∞p,n−p = F pn/F

p−1n . This

means that, for fixed n, the successive quotients arising from the filtration ofHn(S2(3)) form the nth stable diagonal of the spectral sequence, namely thegroups E∞p,q with p+ q = n:

E∞0,n, E∞1,n−1, . . . , E

∞n,0

We want to compute H3, so we look at the third diagonal of the sequence. Onthe second page, the only nonzero group is E2

2,1 = Z, and clearly the differentialsentering and leaving this cell will be zero on every subsequent page, so we haveE∞2,1 = Z.

On the other hand, all the other stable terms in the third diagonal are zero.Hence there is only one nontrivial quotient F p3 /F

p−13 and it is equal to Z. From

55


this it immediately follows that

H3(S2(3)) = Z

and hence we arrive at our result:

π3(S2) = π3(S2(3)) = H3(S2(3)) = Z

It is now no great effort to compute all the homology groups of S2(3) (thiswill come in useful in the next section). We already know that H1(S2(3)) =H2(S2(3)) = 0 because S2(3) is 2-connected, and we have just shownH3(S2(3)) =Z. For n > 3 clearly the nth stable diagonal of the spectral sequence is zero.Therefore we obtain:

Hn(S2(3)) =

Z if n = 0, 30 otherwise

13.2 π4(S2)

The reader has known π3(S2) = Z ever since the heady days of §10, and somay be feeling a little underwhelmed at this point. By the end of this section,however, any doubters will surely be converted, for here we will prove a resultthat was hopelessly out of reach beforehand: the computation of π4(S2).

As we saw in §11.2, we can apply the process of killing off homotopy groupsonce again to the space S2(3), obtaining a 3-connected space S2(4) along witha map S2(4)→ S2(3) inducing isomorphisms on homotopy groups πk for k ≥ 4.Note of course that since the map S2(3)→ S2 induces isomorphisms on πk fork ≥ 3, we have π4(S2(4)) ∼= π4(S2).

We apply our usual trick and replace the map S2(4)→ S2(3) by a homotopyequivalent fibration, whose fiber we denote for the moment by F . Passing to thelong exact sequence of homotopy groups and arguing in exactly the same wayas in §11.3, we see that F is an Eilenberg-MacLane space K(Z, 2). By the resultcited earlier, F is homotopy equivalent to a CW complex, and so we concludeby uniqueness that F ' CP∞ (see §B). Thus our fibration takes the form:

CP∞ → S2(4)→ S2(3)

This is the fibration whose spectral sequence we wish to compute. Howeverhere we run into a problem. If we use Theorem 12.1 and write out the spectralsequence (do it!) we see that once we reach the third page there are manynontrivial differentials, whose definitions are not at all clear. Trying to calculatethese maps directly would require us to consider the associated A&E diagram,and to do this properly we would need a thorough understanding of the CWcomplex structure of S2(3) as well as of the fibration S2(4) → S2(3). Thoughtechnically speaking it is possible to do this, it is well beyond the realms ofpracticality.

To get around this difficulty, we will make use of the Leray-Serre spectralsequence for cohomology. We will have to be a bit vague here, relying mostly on

56


the reader’s intuition for the homology spectral sequence. The two sequencesare formally similar in many ways, except that in the cohomology sequencethe differentials now go in the reverse direction (as one might expect, sincecohomology is obtained by dualising) and the groups on the second page arenow given by

Ep,q2 = Hp(B,Hq(F ))

(note the change in indexing convention). Furthermore, the stable terms Ep,n−p∞now compute the successive quotients in a filtration of the cohomology Hn(X),rather than the homology.

One of the major advantages of cohomology over homology is that there isa natural ring structure given by the cup product. Such a structure can also bedefined on each page of the cohomology spectral sequence. That is, for each rthere is a product (which we will denote by juxtaposition) of the form

Ep,qr × Es,tr → Ep+s,q+tr

which is related very closely to the cup product. We will not go into any moredetail than this, except to note that with respect to this product the differentialsdr are derivations, meaning that for x ∈ Ep,qr we have:

dr(xy) = dr(x)y + (−1)p+qxdr(y)

Now we will compute the cohomology spectral sequence for the fibrationCP∞ → S2(4) → S2(3) constructed earlier. The first thing to do is to writedown the cohomology of the base space and the fiber. In the previous section wefound the homology of S2(3), and since there is no torsion we have by universalcoefficients that:

Hn(S2(3)) =

Z if n = 0, 30 otherwise

(The reader unfamiliar with the universal coefficient theorem should consult[Hat01] §3.1 or [Mun84] §53.) On the other hand, the cohomology groups ofCP∞ are given (see §B) by

Hn(CP∞) =

Z if n even0 otherwise

and furthermore there is a very nice description of the cohomology ring: H∗(CP∞) =Z[x] where x is a generator ofH2(CP∞). This means that xk generatesH2k(CP∞)for all k ∈ N (see [Hat01] Theorem 3.12).

We are now ready to consider the spectral sequence. The second page has

57


the form:

0 1 2 3 40123456

Z Z

Z Z

Z Z

Z Z

The differentials go two units to the right and one unit down, and hence are alltrivial. As such, passing to the third page does not affect the groups, and so wehave:

0 1 2 3 40123456

Z Z

Z Z

Z Z

Z Z

Notice that on this page there is exactly one group in each diagonal. Further-more once we pass to the fourth page the sequence must stabilise because all thedifferentials will be zero. Because H3(S2(4)) = H2(S2(4)) = 0 we must there-fore have E3,0

4 = E0,24 = 0, and this means that the differential d3 : E0,2

3 → E3,03

at the bottom of the above diagram must be an isomorphism. Note that thedomain of this map is by definition

E0,23 = E0,2

2 = H0(S2(3), H2(CP∞)) = H2(CP∞)

and in keeping with the notation above we denote a generator of this group byx. Similarly we have

E3,03 = E3,0

2 = H3(S2(3), H0(CP∞)) = H3(S2(3))

and we let y = d3(x) ∈ E3,03 which is a generator since d3 is an isomorphism.

Now, by the ring structure of H∗(CP∞), a generator for E0,43 = H4(CP∞)

is given by x2. Then because d3 is a derivation we have:

d3(x2) = d3(x)x+ (−1)4xd3(x) = d3(x)x+ xd3(x) = yx+ xy = 2xy

More generally we see by induction that:

d3(xq+1) = qxqy

58


Moreover we note that xqy is a generator for:

E3,2q3 = H3(S2(3), H2q(CP∞)) = H3(S2(3), 〈xq〉)

Therefore the kernel of d3 : E0,2(q+1)3 → E3,2q

3 is zero, and the image is an indexq subgroup of E3,2q

3 . Therefore when we pass to the fourth page of the spectralsequence we obtain:

0 1 2 3 40123456

Z

Z2

Z3

Z4

This gives us the cohomology of S2(4):

Hn(S2(4)) =

Z if n = 0Zr if n = 2r + 10 otherwise

The universal coefficient theorem tells us that torsion shifts up a dimensionwhen we pass from homology to cohomology. Therefore we immediately obtain:

Hn(S2(4)) =

Z if n = 0Zr if n = 2r0 otherwise

In particular, H4(S2(4)) = Z2 and so we arrive, finally, at our goal.

Theorem 13.1. π4(S2) = Z2

Interestingly, this is the first homotopy group of a sphere we have seen whichexhibits torsion. As it turns out, the vast majority of nontrivial homotopygroups of spheres have no free part at all, so that torsion is the rule rather thanthe exception.

Due to the Hopf fibration (Proposition 10.1) we have π4(S3) = π4(S2) =Z2. Moreover we saw in the discussion following the Fruedenthal SuspensionTheorem 8.4 that the stable homotopy group πs1 = πn+1(Sn) stabilises for n ≥ 3,so we have as a corollary:

Theorem 13.2. πs1 = Z2

That is, we have found the first stable homotopy group. Thus we know allthe groups π5(S4), π6(S5), etc.

59

A CW COMPLEXES

A CW Complexes

Arbitrary topological spaces can be extremely pathological ([SS78]). For thisreason, algebraic topologists restrict themselves to the study of suitably “nice”spaces: typically manifolds or CW complexes. The main body of this arti-cle focuses on CW complexes, and in this appendix we collect together a fewinteresting and useful facts about these spaces.

There are two fundamental results in algebraic topology which, taken to-gether, place CW complexes at the heart of the subject. We say that a mapf : X → Y is a weak homotopy equivalence if and only if it induces isomor-phisms on all homotopy groups.

Theorem A.1 (Whitehead’s Theorem). A weak homotopy equivalence betweenCW complexes is a homotopy equivalence.

Theorem A.2 (CW Approximation). Every topological space is weakly homo-topy equivalent to a CW complex.

Aside. It is important to note that Whitehead’s Theorem does not say thatthe homotopy groups constitute a complete invariant for CW complexes: thereexist CW complexes with identical homotopy groups which are not homotopyequivalent, because there does not exist a map between them which inducesisomorphisms on all the groups simultaneously.

Unfortunately we do not have the space to discuss these results further, sowe refer to §4 of [Hat01] for more details. However the reader who knows someof the terminology of category theory should note the following corollary (whichis basically just a restatement of Theorems A.1 and A.2 taken together), whichsuggests that the category of CW complexes is the “correct” category in whichto study homotopy theory.

Proposition A.3. The homotopy category of CW complexes is equivalent toHoTop localised at the weak homotopy equivalences.

(Here HoTop denotes the homotopy category of topological spaces.) Thelocalisation of a category at a collection of morphisms S is defined in ananalagous manner to the localisation of a ring at a multiplicative set: that is,it is a new category equipped with a functor from the original category whichmakes all the morphisms in S into isomorphisms, and is universal with thisproperty. Of course, as with any universal construction, one has to show thatsuch a thing exists: ignoring set-theoretic difficulties, this can be done by justthrowing in formal inverses. For the details of this construction, as well as a moreextensive discussion of localisations (albeit in a somewhat different context) see§III.2 of [GM02].

As such, we think of the localisation of HoTop at the weak homotopy equiv-alences as being what happens when we declare all the weak homotopy equiva-lences to be isomorphisms. Whitehead’s Theorem says that the full subcategoryof HoTop consisting of the CW complexes is left unchanged by this procedure,

60

B EILENBERG-MACLANE SPACES

and CW approximation tells us that every topological space is isomorphic inthe localised category to some CW complex.

B Eilenberg-MacLane Spaces

In the discussion in §11.2 we noted that if X is (n−1)-connected then the spaceY obtained by killing off the homotopy groups πk(X) for k ≥ n+ 1 has exactlyone nontrivial homotopy group, namely πn(Y ) = πn(X). We now undertake asystematic investigation of such spaces.

A space X with πn(X) = G nontrivial and πk(X) = 0 for k 6= n is calledan Eilenberg-MacLane space and denoted K(G,n). Of course, to write X =K(G,n) involves an abuse of notation which, a priori, seems quite egregious.However we will soon see that - at least if we restrict ourselves to CW complexes- the space K(G,n) is actually uniquely defined (up to homotopy equivalence).

First we should show that a K(G,n) exists for any G and n (of course ifn ≥ 2 then we require that G is abelian due to Proposition 1.2). If we letX = Sn in the previous section, then the constructions there produce a K(Z, n)(which is also a CW complex). In fact, by the same argument we see that toconstruct a K(G,n) it is sufficient to construct an (n − 1)-connected space Xwith πn(X) = G. This can be done without too much difficulty by choosinga presentation of G in terms of generators and relations (and the constructionproduces a CW complex X, ensuring that the resulting K(G,n) is also a CWcomplex). Since we will only make use of the spaces K(Z, n) anyway, we willomit the general construction here. The interested reader can find the detailsin §4.2 of [Hat01].

Now we will show that, in the homotopy category of CW complexes, the spaceK(G,n) is unique. Again we will content ourselves to prove this in the case ofK(Z, n). The general case is similar but a bit more involved.

Suppose then that X is a CW complex with πn(X) = Z and πk(X) = 0otherwise. We will show that X is weakly homotopy equivalent to the K(Z, n)constructed above (which for the moment we denote by Y ). The result thenfollows by Whitehead’s Theorem (see §A for the definition of weak homotopyequivalence and the statement of Whitehead’s Theorem).

We need to define a weak homotopy equivalence f : Y → X. To start with,we will define f on the (n+ 1)-skeleton of Y , which by construction is just Sn.As such we need a map Sn → X. The obvious choice is to take (a representativeof) a generator for πn(X) = Z.

Suppose for the moment that we have extended f to all of Y (we will showthat this is always possible shortly). Since X and Y have trivial homotopy indimensions other than n, certainly f∗ : πi(Y ) → πi(X) is an isomorphism fori 6= n. It remains to check f∗ : πn(Y ) ∼= πn(X). Recall that the (n+ 1)-skeletonof Y is Sn, so that πn(Y ) = πn(Sn) = Z and is generated by the inclusion ofskeleta i : Sn → Y . Then f i = f |Sn : Sn → X, and by the choice of f thisrepresents a generator of πn(X). So f∗ sends the generator [i] of πn(Y ) to the

61


generator [f i] of πn(X), hence is an isomorphism. We have thus shown thatf is a weak homotopy equivalence, as required.

It remains to show that the map f : Sn → X can be extended to a mapY → X. We shall show how to extend f to the (n+ 2)-skeleton of Y ; the resultthen follows by induction on skeleta.

Choose an (n+2)-cell en+2; our goal is to extend f to en+2. Let ϕ : Sn+1 →Y be the attaching map for en+2. Since ϕ takes values in the (n+1)-skeleton ofY , the composition fϕ : Sn+1 → X makes sense. Since πn+1(X) = 0 this mapis nullhomotopic, and so by a well-known characterisation of nullhomotopicmaps on spheres (see Proposition 5.2), fϕ can be extend to the entire discDn+2 = en+2∪∂en+2. Since the characterstic map for en+2 is a homeomorphismon the interior of the cell, this allows us to extend f to en+2. Doing this for all(n+ 2)-cells, we can extend f to the (n+ 2)-skeleton. This completes the proofof:

Proposition B.1. In the homotopy category of CW complexes, the spaceK(G,n) (where G is any group, abelian if n ≥ 2) exists and is unique.

Thus whenever we encounter a CW complex which is a K(Z, n), we mayreplace it by any other CW complex which is a K(Z, n), safe in the knowledgethat the two spaces are homotopy equivalent.

This observation will come in handy because the K(Z, n) for very small n arewell-understood spaces. In particular we will have use for both of the followingresults:

Proposition B.2. K(Z, 1) = S1

Proof. This follows immediately from the results in §3.

Proposition B.3. K(Z, 2) = CP∞

Proof. Here we regard CP∞ as the limit of the CW complexes CPn as n→∞.This can be made rigorous with the language of inductive limits, but for ourpurposes it will be sufficient to simply note that CP∞ is a CW complex whose2n-skeleton is CPn. It therefore follows from Proposition 4.3 that

π1(CP∞) = π1(CP1) = π1(S2) = 0

π2(CP∞) = π2(CP1) = π2(S2) = Z

and that for k ≥ 2 we have

π2k−1(CP∞) = π2k−1(CPk) = 0

π2k(CP∞) = π2k(CPk) = 0

where π2k−1(CPk) = π2k(CPk) = 0 can be seen by considering the long exacthomotopy sequence of the fibration S1 → S2k+1 → CPk. Therefore CP∞ =K(Z, 2) as claimed.

62


For the purposes of our computations we will also need to know the coho-mology of CP∞; it is immediate from cellular homology that

Hn(CP∞) =


and therefore by universal coefficients the cohomology has the same pattern:

Hn(CP∞) =


There is also a simple description of the ring structure on H∗(CP∞), which weexplain in §13.2.

Though the indirectness of their definition makes Eilenberg-MacLane spacesseem mysterious at first, they arise naturally in many important contexts, mostof them sadly beyond the scope of this article. They play a fundamental roleas “representing” objects for (reduced) singular cohomology functors, meaningthat Hn(−, G) = [−,K(G,n)]. (Compare this to way in which spheres act asrepresenting objects for the functors πn(−) = [Sn,−].) For the details of thiscorrespondence (as well as an introduction to the “homotopical” approach toalgebraic topology), we refer the reader to [May99].

63

REFERENCES

References

[BT82] Raoul Bott and Loring Tu. Differential Forms in Algebraic Topology.Springer-Verlag, 1982.

[GM02] Sergei I. Gelfand and Yuri I. Manin. Methods of Homological Algebra.Springer-Verlag, second edition, 2002.

[Har77] Robin Hartshorne. Algebraic Geometry. Springer-Verlag, 1977.

[Hat01] Allen Hatcher. Algebraic Topology. Cambridge University Press, 2001.

[Hat04] Allen Hatcher. Spectral Sequences in Algebraic Topology. http://

www.math.cornell.edu/~hatcher/SSAT/SSATpage.html, 2004.

[Hun93] John Hunton. Algebraic Topology. Notes from a lecture course, takenby Ian Dowker, 1993.

[Hut11] Michael Hutchins. An Introduction to Higher Homotopy Groupsand Obstruction Theory. http://math.berkeley.edu/~hutching/

teach/215b-2011/homotopy.pdf, 2011.

[Huy05] Daniel Huybrechts. Complex Geometry. Springer-Verlag, 2005.

[Joh11] Niles Johnson. Hopf Fibration: Fibers and Base. https://www.

youtube.com/watch?v=AKotMPGFJYk, 2011.

[Lee10] John M. Lee. Introduction to Topological Manifolds. Springer-Verlag,second edition, 2010.

[Lee12] John M. Lee. Introduction to Smooth Manifolds. Springer-Verlag,second edition, 2012.

[Mac98] Saunders MacLane. Categories for the Working Mathematician.Springer-Verlag, second edition, 1998.

[Mas91] William S. Massey. A Basic Course in Algebraic Topology. Springer-Verlag, 1991.

[May99] J. P. May. A Concise Course in Algebraic Topology. University ofChicago Press, 1999.

[McC01] John McCleary. A User’s Guide to Spectral Sequences. CambridgeUniversity Press, second edition, 2001.

[Mun84] James R. Munkres. Elements of Algebraic Topology. Perseus Publish-ing, 1984.

[Rot88] Joseph J. Rotman. An Introduction to Algebraic Topology. Springer-Verlag, 1988.

[Spa66] Edwin H. Spanier. Algebraic Topology. McGraw-Hill, 1966.

[SS78] Lynn Arthur Steen and J. Arthur Seebach, Jr. Counterexamples inTopology. Springer-Verlag, second edition, 1978.

64

http://www.math.cornell.edu/~hatcher/SSAT/SSATpage.html

http://www.math.cornell.edu/~hatcher/SSAT/SSATpage.html

http://math.berkeley.edu/~hutching/teach/215b-2011/homotopy.pdf

http://math.berkeley.edu/~hutching/teach/215b-2011/homotopy.pdf

https://www.youtube.com/watch?v=AKotMPGFJYk

https://www.youtube.com/watch?v=AKotMPGFJYk

Index

n-connected space, 15CW complexes, 16

Adjunction, 28Algebraic geometry, 39

Cellular approximation, 12Cellular map, 13Compact-open topology, 23, 26, 31Concatenation of loops, 6Covering spaces, 12, 21Cube, 8, 16Cubical homology, 16CW approximation, 60CW complex, 60

Degree, 20

Eilenberg-MacLane space, 45, 61Exact couple, 51Exact sequence

of pointed sets, 30of topological spaces, 32

Fiber bundle, 22is a Hurewicz fibration, 22is a Serre fibration, 22

Fibration, 21definition, 21every map is a, 23long exact homotopy sequence, 38Serre vs. Hurewicz, 22

Filtrationof a group, 46of a space, 46

Freudenthal Suspension Theorem, 29Fundamental group, 6

Higher homotopy groupdefinition, 7is abelian, 9is functorial, 7is homotopy invariant, 7

Homotopy equivalence

of maps, 23Homotopy lifting property, 21Hopf fibration, 38Hurewicz homomorphism, 14Hurewicz Theorem, 17, 41

proof of, 18

Killing off homotopy groups, 41

Long exact sequence, 30Loopspace, 26

is functorial, 28

Mapping fiber, 31

Pinching map, 7, 25Pointed set, 30Projective space

complex, 38Puppe sequence, 30

Spectral sequence, 45A&E diagram, 47, 50associated to a filtration, 51convergence of, 52differentials, 48, 50for cohomology, 56Leray-Serre, 53limiting groups, 52

Sphere, 10CW complex structure of, 11homology of, 11

Stable homotopy group, 29Suspension, 24

is functorial, 28reduced, 25

Suspension homomorphism, 29Suspension-Loopspace adjunction, 27

Weak homotopy equivalence, 60, 61Whitehead’s Theorem, 60

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Homotopy Groups of the Spheres - Glannabijou/HomotopyProject.pdf · 2018-09-19 · 1 HIGHER...

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