A Concise Course in Algebraic Topology

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Chapter 23. Characteristic classes of vector bundles 187 1. The classification of vector bundles 187 2. Characteristic classes for vector bundles 189 3. Stiefel-Whitney classes of manifolds 191 4. Characteristic numbers of manifolds 193 5. Thom spaces and the Thom isomorphism theorem 194 6. The construction of the Stiefel-Whitney classes 196 7. Chern, Pontryagin, and Euler classes 197

8. A glimpse at the general theory 200

Chapter 24. An introduction to K-theory 203

1. The definition ofK-theory 203

2. The Bott periodicity theorem 206

3. The splitting principle and the Thom isomorphism 208 4. The Chern character; almost complex structures on spheres 211

5. The Adams operations 213

6. The Hopf invariant one problem and its applications 215

Chapter 25. An introduction to cobordism 219 1. The cobordism groups of smooth closed manifolds 219 2. Sketch proof thatN_∗ _{is isomorphic to}_π_∗(_{T O}₎ ₂₂₀

3. Prespectra and the algebraH∗(T O;Z2) 223 4. The Steenrod algebra and its coaction onH∗(T O) 226 5. The relationship to Stiefel-Whitney numbers 228 6. Spectra and the computation ofπ∗(T O) =π∗(M O) 230 7. An introduction to the stable category 232

Suggestions for further reading 235

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CONTENTS ix

3. Books on CW complexes 236

4. Differential forms and Morse theory 236

5. Equivariant algebraic topology 237

6. Category theory and homological algebra 237 7. Simplicial sets in algebraic topology 237 8. The Serre spectral sequence and Serre class theory 237 9. The Eilenberg-Moore spectral sequence 237

10. Cohomology operations 238

11. Vector bundles 238

12. Characteristic classes 238

13. K-theory 239

14. Hopf algebras; the Steenrod algebra, Adams spectral sequence 239

15. Cobordism 240

16. Generalized homology theory and stable homotopy theory 240

17. Quillen model categories 240

18. Localization and completion; rational homotopy theory 241

19. Infinite loop space theory 241

20. Complex cobordism and stable homotopy theory 242

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Introduction

The first year graduate program in mathematics at the University of Chicago consists of three three-quarter courses, in analysis, algebra, and topology. The first two quarters of the topology sequence focus on manifold theory and differential geometry, including differential forms and, usually, a glimpse of de Rham cohomol-ogy. The third quarter focuses on algebraic topolcohomol-ogy. I have been teaching the third quarter off and on since around 1970. Before that, the topologists, including me, thought that it would be impossible to squeeze a serious introduction to al-gebraic topology into a one quarter course, but we were overruled by the analysts and algebraists, who felt that it was unacceptable for graduate students to obtain their PhDs without having some contact with algebraic topology.

This raises a conundrum. A large number of students at Chicago go into topol-ogy, algebraic and geometric. The introductory course should lay the foundations for their later work, but it should also be viable as an introduction to the subject suitable for those going into other branches of mathematics. These notes reflect my efforts to organize the foundations of algebraic topology in a way that caters to both pedagogical goals. There are evident defects from both points of view. A treatment more closely attuned to the needs of algebraic geometers and analysts would include ˇCech cohomology on the one hand and de Rham cohomology and perhaps Morse homology on the other. A treatment more closely attuned to the needs of algebraic topologists would include spectral sequences and an array of calculations with them. In the end, the overriding pedagogical goal has been the introduction of basic ideas and methods of thought.

Our understanding of the foundations of algebraic topology has undergone sub-tle but serious changes since I began teaching this course. These changes reflect in part an enormous internal development of algebraic topology over this period, one which is largely unknown to most other mathematicians, even those working in such closely related fields as geometric topology and algebraic geometry. Moreover, this development is poorly reflected in the textbooks that have appeared over this period.

Let me give a small but technically important example. The study of gen-eralized homology and cohomology theories pervades modern algebraic topology. These theories satisfy the excision axiom. One constructs most such theories ho-motopically, by constructing representing objects called spectra, and one must then prove that excision holds. There is a way to do this in general that is no more dif-ficult than the standard verification for singular homology and cohomology. I find this proof far more conceptual and illuminating than the standard one even when specialized to singular homology and cohomology. (It is based on the approxima-tion of excisive triads by weakly equivalent CW triads.) This should by now be a

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2 INTRODUCTION

standard approach. However, to the best of my knowledge, there exists no rigorous exposition of this approach in the literature, at any level.

More centrally, there now exist axiomatic treatments of large swaths of homo-topy theory based on Quillen’s theory of closed model categories. While I do not think that a first course should introduce such abstractions, I do think that the ex-position should give emphasis to those features that the axiomatic approach shows to be fundamental. For example, this is one of the reasons, although by no means the only one, that I have dealt with cofibrations, fibrations, and weak equivalences much more thoroughly than is usual in an introductory course.

Some parts of the theory are dealt with quite classically. The theory of fun-damental groups and covering spaces is one of the few parts of algebraic topology that has probably reached definitive form, and it is well treated in many sources. Nevertheless, this material is far too important to all branches of mathematics to be omitted from a first course. For variety, I have made more use of the funda-mental groupoid than in standard treatments,1 and my use of it has some novel features. For conceptual interest, I have emphasized different categorical ways of modeling the topological situation algebraically, and I have taken the opportunity to introduce some ideas that are central to equivariant algebraic topology.

Poincar´e duality is also too fundamental to omit. There are more elegant ways to treat this topic than the classical one given here, but I have preferred to give the theory in a quick and standard fashion that reaches the desired conclusions in an economical way. Thus here I have not presented the truly modern approach that applies to generalized homology and cohomology theories.2

The reader is warned that this book is not designed as a textbook, although it could be used as one in exceptionally strong graduate programs. Even then, it would be impossible to cover all of the material in detail in a quarter, or even in a year. There are sections that should be omitted on a first reading and others that are intended to whet the student’s appetite for further developments. In practice, when teaching, my lectures are regularly interrupted by (purposeful) digressions, most often directly prompted by the questions of students. These introduce more advanced topics that are not part of the formal introductory course: cohomology operations, characteristic classes, K-theory, cobordism, etc., are often first intro-duced earlier in the lectures than a linear development of the subject would dictate. These digressions have been expanded and written up here as sketches without complete proofs, in a logically coherent order, in the last four chapters. These are topics that I feel must be introduced in some fashion in any serious graduate level introduction to algebraic topology. A defect of nearly all existing texts is that they do not go far enough into the subject to give a feel for really substantial applications: the reader sees spheres and projective spaces, maybe lens spaces, and applications accessible with knowledge of the homology and cohomology of such spaces. That is not enough to give a real feeling for the subject. I am aware that this treatment suffers the same defect, at least before its sketchy last chapters.

Most chapters end with a set of problems. Most of these ask for computa-tions and applicacomputa-tions based on the material in the text, some extend the theory and introduce further concepts, some ask the reader to furnish or complete proofs

1_{But see R. Brown’s book cited in}_§_{2 of the suggestions for further reading.}

2_{That approach derives Poincar´}_{e duality as a consequence of Spanier-Whitehead and Atiyah}

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INTRODUCTION 3

omitted in the text, and some are essay questions which implicitly ask the reader to seek answers in other sources. Problems marked ∗ are more difficult or more peripheral to the main ideas. Most of these problems are included in the weekly problem sets that are an integral part of the course at Chicago. In fact, doing the problems is the heart of the course. (There are no exams and no grades; students are strongly encouraged to work together, and more work is assigned than a student can reasonably be expected to complete working alone.) The reader is urged to try most of the problems: this is the way to learn the material. The lectures focus on the ideas; their assimilation requires more calculational examples and applications than are included in the text.

I have ended with a brief and idiosyncratic guide to the literature for the reader interested in going further in algebraic topology.

These notes have evolved over many years, and I claim no originality for most of the material. In particular, many of the problems, especially in the more classical chapters, are the same as, or are variants of, problems that appear in other texts. Perhaps this is unavoidable: interesting problems that are doable at an early stage of the development are few and far between. I am especially aware of my debts to earlier texts by Massey, Greenberg and Harper, Dold, and Gray.

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CHAPTER 1

The fundamental group and some of its

applications

We introduce algebraic topology with a quick treatment of standard mate-rial about the fundamental groups of spaces, embedded in a geodesic proof of the Brouwer fixed point theorem and the fundamental theorem of algebra.

1. What is algebraic topology?

A topological spaceX is a set in which there is a notion of nearness of points. Precisely, there is given a collection of “open” subsets ofX which is closed under finite intersections and arbitrary unions. It suffices to think of metric spaces. In that case, the open sets are the arbitrary unions of finite intersections of neighborhoods

Uε(x) ={y|d(x, y)< ε}.

A functionp:X −→Y is continuous if it takes nearby points to nearby points. Precisely, p−1₍_U_{) is open if} _U _{is open. If} _X _and _Y _{are metric spaces, this means} that, for anyx∈X andε >0, there existsδ >0 such thatp(Uδ(x))⊂Uε(p(x)).

Algebraic topology assigns discrete algebraic invariants to topological spaces and continuous maps. More narrowly, one wants the algebra to be invariant with respect to continuous deformations of the topology. Typically, one associates a groupA(X) to a spaceX and a homomorphism A(p) :A(X)−→A(Y) to a map

p:X −→Y; one usually writesA(p) =p∗.

A “homotopy” h : p ≃q between maps p, q : X −→ Y is a continuous map

h:X×I−→ Y such that h(x,0) =p(x) andh(x,1) =q(x), where I is the unit interval [0,1]. We usually want p∗=q∗ ifp≃q, or some invariance property close to this.

In oversimplified outline, the way homotopy theory works is roughly this.

(1) One defines some algebraic construction Aand proves that it is suitably homotopy invariant.

(2) One computes Aon suitable spaces and maps.

(3) One takes the problem to be solved and deforms it to the point that step 2 can be used to solve it.

The further one goes in the subject, the more elaborate become the construc-tions A and the more horrendous become the relevant calculational techniques. This chapter will give a totally self-contained paradigmatic illustration of the basic philosophy. Our constructionA will be the “fundamental group.” We will calcu-lateA on the circle S1 _{and on some maps from}_S1 _{to itself. We will then use the} computation to prove the “Brouwer fixed point theorem” and the “fundamental theorem of algebra.”

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6 THE FUNDAMENTAL GROUP AND SOME OF ITS APPLICATIONS

2. The fundamental group

LetX be a space. Two paths f, g:I−→X from xto yare equivalent if they are homotopic through paths fromx to y. That is, there must exist a homotopy

h:I×I−→X such that traversing firstf and theng, going twice as fast on each:

(g·f)(s) = be the constant loop atx: cx(s) =x. Composition of paths passes to equivalence classes via [g][f] = [g·f]. It is easy to check that this is well defined. Moreover, after passage to equivalence classes, this composition becomes associative and unital. It is easy enough to write down explicit formulas for the relevant homotopies. It is more illuminating to draw a picture of the domain squares and to indicate schematically how the homotopies are to behave on it. In the following, we assume given paths

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4. HOMOTOPY INVARIANCE 7

Moreover, [f−1_·_f_{] = [}_c_{x] and [}_f _·_f−1_{] = [}_c_{y]. For the first, we have the following} schematic picture and corresponding formula. In the schematic picture,

ft=f|[0, t] and ft−1=f−1|[1−t,1]. homotopy group of X. There are higher homotopy groups πn(X, x) defined in terms of mapsSn_−→_X_{. We will get to them later.} of the choice of the path class [a]. Thus, in this case, we have a canonical way to identifyπ1(X, x) withπ1(X, y).

4. Homotopy invariance

For a map p : X −→ Y, define p∗ : π1(X, x) −→ π1(Y, p(x)) by p∗[f] = [p◦f], where p◦f is the composite of p with the loop f : I −→ X. Clearly

p∗ is a homomorphism. The identity map id : X −→ X induces the identity homomorphism. For a mapq:Y −→Z,q∗◦p∗= (q◦p)∗.

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Proposition. _{The following diagram is commutative:}

π1(X, x)

Thus, going counterclockwise around the boundary starting at (0,0), we traverse

a−1_·₍_q_◦_f₎−1_·_a_·₍_p_◦_f_{). The map}_j _{induces a homotopy through loops between} this composite and cp(x). Explicitly, a homotopy k is given by k(s, t) = j(rt(s)), wherert:I−→I×I maps successive quarter intervals linearly onto the edges of the bottom left subsquare ofI×I with edges of lengtht, starting at (0,0):

Our first calculation is rather trivial. We take the origin 0 as a convenient basepoint for the real lineR.

Lemma. _π₁₍_R,_{0) = 0.}

Proof. _Define _k_: R×I −→R byk(s, t) = (1−t)s. Thenk is a homotopy from the identity to the constant map at 0. For a loop f : I −→ Rat 0, define

h(s, t) =k(f(s), t). The homotopyhshows that f is equivalent toc0.

Consider the circleS1 _{to be the set of complex numbers}_x₌_y₊_iz_{of norm 1,}

y2₊_z2_{= 1. Observe that}_S1 _{is a group under multiplication of complex numbers.} It is a topological group: multiplication is a continuous function. We take the identity element 1 as a convenient basepoint forS1_.

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5. CALCULATIONS:π1(R) = 0 AND π1(S1) =Z 9

Proof. _{For each integer}_n_{, define a loop} _f_n _in _S1 _by _f_n(_s_{) =}_e2πins_{. This is} the composite of the mapI−→S1_{that sends}_s_to_e2πis _{and the}_n_{th power map on}

S1_{; if we identify the boundary points 0 and 1 of}_I_{, then the first map induces the} evident identification of I/∂I withS1_{. It is easy to check that [}_f_m][_f_{n] = [}_f_m+n], and we define a homomorphismi:Z−→π1(S1,1) byi(n) = [fn]. We claim that

iis an isomorphism. The idea of the proof is to use the fact that, locally,S1 _looks just likeR.

Definep:R−→S1_by_p₍_s_{) =}_e2πis_{. Observe that}_p_{wraps each interval [}_{n, n}_+1] around the circle, starting at 1 and going counterclockwise. Since the exponential function converts addition to multiplication, we easily check thatfn=p◦f˜n, where

˜

fn is the path inRdefined by ˜fn(s) =sn.

This lifting of paths works generally. For any pathf :I−→S1 _with_f_{(0) = 1,} there is a unique path ˜f : I −→ R such that ˜f(0) = 0 and p◦f˜= f. To see this, observe that the inverse image in Rof any small connected neighborhood in

S1 _{is a disjoint union of a copy of that neighborhood contained in each interval} (r+n, r+n+ 1) for some r ∈ [0,1). Using the fact that I is compact, we see that we can subdivide I into finitely many closed subintervals such thatf carries each subinterval into one of these small connected neighborhoods. Now, proceeding subinterval by subinterval, we obtain the required unique lifting off by observing that the lifting on each subinterval is uniquely determined by the lifting of its initial point.

Define a functionj:π1(S1_,₁₎_−→_Z_by_j_[_f_{] = ˜}_f_{(1), the endpoint of the lifted} path. This is an integer since p( ˜f(1)) = 1. We must show that this integer is independent of the choice off in its path class [f]. In fact, if we have a homotopy

h : f ≃ g through loops at 1, then the homotopy lifts uniquely to a homotopy ˜

h:I×I−→Rsuch that ˜h(0,0) = 0 andp◦˜h=h. The argument is just the same as for ˜f: we use the fact that I×I is compact to subdivide it into finitely many subsquares such thathcarries each into a small connected neighborhood inS1_{. We} then construct the unique lift ˜hby proceeding subsquare by subsquare, starting at the lower left, say, and proceeding upward one row of squares at a time. By the uniqueness of lifts of paths, which works just as well for paths with any starting point,c(t) = ˜h(0, t) andd(t) = ˜h(1, t) specify constant paths sinceh(0, t) = 1 and

h(1, t) = 1 for allt. Clearlyc is constant at 0, so, again by the uniqueness of lifts of paths, we must have

˜

f(s) = ˜h(s,0) and g˜(s) = ˜h(s,1).

But then our second constant pathdstarts at ˜f(1) and ends at ˜g(1).

Sincej[fn] =nby our explicit formula for ˜fn, the compositej◦i:Z−→Zis the identity. It suffices to check that the functionjis one-to-one, since then bothi

andjwill be one-to-one and onto. Thus suppose thatj[f] =j[g]. This means that ˜

f(1) = ˜g(1). Therefore ˜g−1_·_f_˜_{is a loop at 0 in} _R_{. By the lemma, [˜}_g−1_·_f_˜_{] = [}_c_0]. It follows upon application ofp∗ that

[g−1][f] = [g−1·f] = [c1].

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6. The Brouwer fixed point theorem

LetD2 _{be the unit disk} _{_y₊_iz_|_y2₊_z2_≤_{1}. Its boundary is} _S1_{, and we let}

i:S1 _−→_D2 _{be the inclusion. Exactly as for} _R_{, we see that} _π₁₍_D2_{) = 0 for any} choice of basepoint.

Proposition. _{There is no continuous map}_r_:_D2_−→_S1 _{such that}_r_◦_i_{= id.}

Proof. _{If there were such a map}_r_{, then the composite homomorphism}

π1(S1_,₁₎ i∗

/

/π1(D2_,₁₎ r∗

/

/π1(S1_,₁₎

would be the identity. Since the identity homomorphism of Z does not factor through the zero group, this is impossible.

Theorem _{(Brouwer fixed point theorem)}. _{Any continuous map}

f :D2_−→_D2

has a fixed point.

Proof. _{Suppose that}_f₍_x₎₆₌_x_{for all}_x_{. Define}_r₍_x₎_∈_S1 _{to be the} intersec-tion withS1_{of the ray that starts at}_f₍_x_{) and passes through}_x_{. Certainly}_r₍_x_{) =}_x ifx∈S1_{. By writing an equation for}_r_{in terms of}_f_{, we see that}_r _{is continuous.}

This contradicts the proposition.

7. The fundamental theorem of algebra

Letι ∈π1(S1_,_{1) be a generator. For a map} _f _:_S1 _−→_S1_{, define an integer} deg(f) by letting the composite

π1(S1_,₁₎ f∗

/

/π1(S1_{, f}₍₁₎₎ γ[a] _/_/_π₁₍_S1_,₁₎

send ιto deg(f)ι. Herea is any path f(1)→1; γ[a] is independent of the choice of [a] since π1(S1_,_{1) is Abelian. If}_f _≃_g_{, then deg(}_f_{) = deg(}_g_{) by our homotopy} invariance diagram and this independence of the choice of path. Conversely, our calculation ofπ1(S1_,_{1) implies that if deg(}_f_{) = deg(}_g_{), then}_f _≃_g_{, but we will not} need that for the moment. It is clear that deg(f) = 0 iff is the constant map at some point. It is also clear that iffn(x) =xn, then deg(fn) =n: we built that fact into our proof thatπ1(S1_,_{1) =}_Z_.

Theorem _{(Fundamental theorem of algebra)}. _Let

f(x) =xn+c1xn−1+· · ·+cn−1x+cn

be a polynomial with complex coefficientsci, where n >0. Then there is a complex

numberxsuch thatf(x) = 0. Therefore there arensuch complex numbers (counted

with multiplicities).

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7. THE FUNDAMENTAL THEOREM OF ALGEBRA 11

thatf(x)6= 0 for allxsuch that|x| ≥1. This allows us to definej:S1_×_I_−→_S1 byj(x, t) =k(x, t)/|k(x, t)|, where

k(x, t) =tnf(x/t) =xn+t(c1xn−1+tc2xn−2+· · ·+tn−1cn).

Thenj is a homotopy fromfn to ˆf, and we conclude that deg( ˆf) =n. One of our

suppositions had better be false!

It is to be emphasized how technically simple this is, requiring nothing remotely as deep as complex analysis. Nevertheless, homotopical proofs like this are relatively recent. Adequate language, elementary as it is, was not developed until the 1930s.

PROBLEMS

(1) Letpbe a polynomial function onCwhich has no root onS1_{. Show that} the number of roots of p(z) = 0 with|z| < 1 is the degree of the map ˆ

p:S1−→S1 specified by ˆp(z) =p(z)/|p(z)|.

(2) Show that any mapf :S1_−→_S1_{such that deg(}_f₎_{6= 1 has a fixed point.} (3) Let Gbe a topological group and take its identity elemente as its base-point. Define the pointwise product of loops α and β by (αβ)(t) =

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CHAPTER 2

Categorical language and the van Kampen

theorem

We introduce categorical language and ideas and use them to prove the van Kampen theorem. This method of computing fundamental groups illustrates the general principle that calculations in algebraic topology usually work by piecing together a few pivotal examples by means of general constructions or procedures.

1. Categories

Algebraic topology concerns mappings from topology to algebra. Category theory gives us a language to express this. We just record the basic terminology, without being overly pedantic about it.

A categoryC _{consists of a collection of objects, a set} C₍_{A, B}_{) of morphisms}

(also called maps) between any two objects, an identity morphism idA ∈C₍_{A, A}₎

for each objectA(usually abbreviated id), and a composition law ◦:C₍_{B, C}₎_×C₍_{A, B}₎_−→C₍_{A, C}₎

for each triple of objects A,B,C. Composition must be associative, and identity morphisms must behave as their names dictate:

h◦(g◦f) = (h◦g)◦f, id◦f =f, and f◦id =f

whenever the specified composites are defined. A category is “small” if it has a set of objects.

We have the categoryS _{of sets and functions, the category}U _{of topological}

spaces and continuous functions, the category G _{of groups and homomorphisms,}

the categoryA_b _{of Abelian groups and homomorphisms, and so on.}

2. Functors

A functor F : C _−→ D _{is a map of categories. It assigns an object} _F₍_A_{) of} D _{to each object} _A _of C _{and a morphism} _F₍_f_{) :} _F₍_A₎ _−→ _F₍_B_{) of} D _{to each}

morphismf :A−→B ofC _{in such a way that}

F(idA) = idF(A) and F(g◦f) =F(g)◦F(f).

More precisely, this is a covariant functor. A contravariant functorF reverses the direction of arrows, so that F sends f :A −→ B to F(f) :F(B) −→F(A) and satisfies F(g◦f) = F(f)◦ F(g). A category C _{has an opposite category} Cop with the same objects and with Cop₍_{A, B}_{) =} _C₍_{B, A}_{). A contravariant functor}

F :C _−→D _{is just a covariant functor}Cop_−→_D_.

For example, we have forgetful functors from spaces to sets and from Abelian groups to sets, and we have the free Abelian group functor from sets to Abelian groups.

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14 CATEGORICAL LANGUAGE AND THE VAN KAMPEN THEOREM

3. Natural transformations

A natural transformationα:F −→Gbetween functorsC _−→D _{is a map of}

functors. It consists of a morphismαA : F(A) −→ G(A) for each object A of C such that the following diagram commutes for each morphismf :A−→B ofC_:

F(A) F(f) //

αA

F(B)

αB

G(A)

G(f)//G(B).

Intuitively, the mapsαA are defined in the same way for everyA.

For example, if F : S _−→ A_b _{is the functor that sends a set to the free}

Abelian group that it generates and U : A_b _−→ S _{is the forgetful functor that}

sends an Abelian group to its underlying set, then we have a natural inclusion of setsS−→U F(S). The functorsF andU are left adjoint and right adjoint to each other, in the sense that we have a natural isomorphism

A_b₍_F₍_S₎_{, A}₎∼₌S₍_{S, U}₍_A₎₎

for a set S and an Abelian group A. This just expresses the “universal property” of free objects: a map of setsS−→U(A) extends uniquely to a homomorphism of groupsF(S)−→A. Although we won’t bother with a formal definition, the notion of an adjoint pair of functors will play an important role later on.

Two categoriesC _and D _{are equivalent if there are functors}_F _:C _−→D _and G:D _−→C _{and natural isomorphisms}_{F G}_−→ _{Id and}_GF _−→_{Id, where the Id}

are the respective identity functors.

4. Homotopy categories and homotopy equivalences

LetT _{be the category of spaces} _X _{with a chosen basepoint}_x_∈ _X_{; its}

mor-phisms are continuous mapsX−→Y that carry the basepoint ofXto the basepoint ofY. The fundamental group specifies a functorT _−→G_{, where}G _{is the category}

of groups and homomorphisms.

When we have a (suitable) relation of homotopy between maps in a category

C_{, we define the homotopy category}_hC _{to be the category with the same objects}

as C _{but with morphisms the homotopy classes of maps. We have the homotopy}

categoryhU _{of unbased spaces. On}T_{, we require homotopies to map basepoint to}

basepoint at all timest, and we obtain the homotopy categoryhT _{of based spaces.}

The fundamental group is a homotopy invariant functor onT_{, in the sense that it}

factors through a functorhT _−→G_.

A homotopy equivalence inU _{is an isomorphism in}_hU_{. Less mysteriously, a}

mapf :X−→Y is a homotopy equivalence if there is a mapg:Y −→X such that both g◦f ≃id andf◦g ≃id. Working in T_{, we obtain the analogous notion of}

a based homotopy equivalence. Functors carry isomorphisms to isomorphisms, so we see that a based homotopy equivalence induces an isomorphism of fundamental groups. The same is true, less obviously, for unbased homotopy equivalences.

Proposition. _If_f _:_X_−→_Y _{is a homotopy equivalence, then}

f∗:π1(X, x)−→π1(Y, f(x))

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5. THE FUNDAMENTAL GROUPOID 15

Proof. _Let _g _: _Y _−→ _X _{be a homotopy inverse of} _f_{. By our homotopy} invariance diagram, we see that the composites

π1(X, x)_−→f∗

π1(Y, f(x))_−→g∗

π1(X,(g◦f)(x)) and

π1(Y, y)_−→g∗

π1(X, g(y))_−→f∗

π1(Y,(f◦g)(y))

are isomorphisms determined by paths between basepoints given by chosen homo-topies g◦f ≃id and f◦g ≃id. Therefore, in each displayed composite, the first map is a monomorphism and the second is an epimorphism. Taking y = f(x) in the second composite, we see that the second map in the first composite is an isomorphism. Therefore so is the first map.

A spaceX is said to be contractible if it is homotopy equivalent to a point.

Corollary. _{The fundamental group of a contractible space is zero.}

5. The fundamental groupoid

While algebraic topologists often concentrate on connected spaces with chosen basepoints, it is valuable to have a way of studying fundamental groups that does not require such choices. For this purpose, we define the “fundamental groupoid” Π(X) of a spaceX to be the category whose objects are the points ofXand whose morphismsx−→y are the equivalence classes of paths fromxto y. Thus the set of endomorphisms of the objectxis exactly the fundamental groupπ1(X, x).

The term “groupoid” is used for a category all morphisms of which are isomor-phisms. The idea is that a group may be viewed as a groupoid with a single object. Taking morphisms to be functors, we obtain the categoryG P _{of groupoids. Then}

we may view Π as a functorU _−→G P_.

There is a useful notion of a skeleton skC _{of a category} C_{. This is a “full”}

subcategory with one object from each isomorphism class of objects of C_{, “full”}

meaning that the morphisms between two objects ofskC _{are all of the morphisms}

between these objects inC_{. The inclusion functor}_J _:_skC _−→C _{is an equivalence}

of categories. An inverse functor F : C _−→ _skC _{is obtained by letting} _F₍_A₎

be the unique object in skC _{that is isomorphic to} _A_{, choosing an isomorphism} αA : A −→ F(A), and defining F(f) = αB ◦f ◦α−1A : F(A) −→ F(B) for a morphismf :A−→B in C_{. We choose}_α _{to be the identity morphism if}_A _{is in} skC_{, and then}_{F J} _{= Id; the}_α_A _{specify a natural isomorphism}_α_{: Id}_−→_JF_.

A categoryC _{is said to be connected if any two of its objects can be connected}

by a sequence of morphisms. For example, a sequence A ←− B −→C connects

A to C, although there need be no morphism A−→ C. However, a groupoidC

is connected if and only if any two of its objects are isomorphic. The group of endomorphisms of any objectC is then a skeleton of C_{. Therefore the previous}

paragraph specializes to give the following relationship between the fundamental group and the fundamental groupoid of a path connected spaceX.

Proposition. _Let _X _{be a path connected space. For each point} _x _∈ _X_{, the}

inclusion π1(X, x)−→Π(X)is an equivalence of categories.

Proof. _{We are regarding}_π₁₍_{X, x}_{) as a category with a single object}_x_{, and it}

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6. Limits and colimits

Let D _{be a small category and let}C _{be any category. A} D_{-shaped diagram}

in C _{is a functor}_F _:D _−→C_{. A morphism} _F _−→_F′ _of_D_{-shaped diagrams is a} natural transformation, and we have the category D_[C_{] of} D_{-shaped diagrams in} C_{. Any object} _C _ofC _{determines the constant diagram}_C _{that sends each object}

ofD _to_C _{and sends each morphism of}D_{to the identity morphism of} _C_.

The colimit, colimF, of aD_{-shaped diagram}_F _{is an object of}C _{together with}

a morphism of diagramsι:F −→colimF that is initial among all such morphisms. This means that ifη :F −→A is a morphism of diagrams, then there is a unique

The limit ofFis defined by reversing arrows: it is an object limF ofC _together

with a morphism of diagrams π : limF −→ F that is terminal among all such morphisms. This means that ifε:A−→F is a morphism of diagrams, then there is a unique map ˜ε:A−→limF in C _{such that}_π_◦_ε_˜₌_ε_{. Diagrammatically, this}

If D _{is a set regarded as a discrete category (only identity morphisms), then}

colimits and limits indexed on D _{are coproducts and products indexed on the set} D_{. Coproducts are disjoint unions in}S _orU_{, wedges (or one-point unions) in}T_,

free products inG_{, and direct sums in} A_b_{. Products are Cartesian products in all}

of these categories; more precisely, they are Cartesian products of underlying sets, with additional structure. IfD _{is the category displayed schematically as}

eoo d //f or d ////d′_,

where we have displayed all objects and all non-identity morphisms, then the co-limits indexed on D _{are called pushouts or coequalizers, respectively. Similarly, if} D _{is displayed schematically as}

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7. THE VAN KAMPEN THEOREM 17

then the limits indexed onD _{are called pullbacks or equalizers, respectively.}

A given category may or may not have all colimits, and it may have some but not others. A category is said to be cocomplete if it has all colimits, complete if it has all limits. The categoriesS_,U_,T_,G_{, and}A_b_{are complete and cocomplete.}

If a category has coproducts and coequalizers, then it is cocomplete, and similarly for completeness. The proof is a worthwhile exercise.

7. The van Kampen theorem

The following is a modern dress treatment of the van Kampen theorem. I should admit that, in lecture, it may make more sense not to introduce the fundamental groupoid and to go directly to the fundamental group statement. The direct proof is shorter, but not as conceptual. However, as far as I know, the deduction of the fundamental group version of the van Kampen theorem from the fundamental groupoid version does not appear in the literature in full generality. The proof well illustrates how to manipulate colimits formally. We have used the van Kampen theorem as an excuse to introduce some basic categorical language, and we shall use that language heavily in our treatment of covering spaces in the next chapter. Theorem _{(van Kampen)}. _Let O ₌ _{_U_} _{be a cover of a space} _X _{by path}

connected open subsets such that the intersection of finitely many subsets in O _is

again inO_{. Regard} O _{as a category whose morphisms are the inclusions of subsets}

and observe that the functor Π, restricted to the spaces and maps in O_{, gives a}

diagram

Π|O_:O_−→G P

of groupoids. The groupoid Π(X)is the colimit of this diagram. In symbols,

Π(X)∼= colimU∈OΠ(U).

Proof. _{We must verify the universal property. For a groupoid}C _{and a map} η : Π|O _−→ C _of O_{-shaped diagrams of groupoids, we must construct a map}

˜

η : Π(X) −→ C _{of groupoids that restricts to} _η_U _{on Π(}_U_{) for each}_U _∈ O_{. On}

objects, that is on points of X, we must define ˜η(x) = ηU(x) for x∈ U. This is independent of the choice ofU sinceO _{is closed under finite intersections. If a path} f :x→y lies entirely in a particularU, then we must define ˜η[f] =η([f]). Again, sinceO _{is closed under finite intersections, this specification is independent of the}

choice ofU iff lies entirely in more than one U. Any pathf is the composite of finitely many pathsfi, each of which does lie in a singleU, and we must define ˜η[f] to be the composite of the ˜η[fi]. Clearly this specification will give the required unique map ˜η, provided that ˜η so specified is in fact well defined. Thus suppose thatf is equivalent tog. The equivalence is given by a homotopyh:f ≃gthrough pathsx→y. We may subdivide the squareI×I into subsquares, each of which is mapped into one of the U. We may choose the subdivision so that the resulting subdivision ofI× {0}refines the subdivision used to decomposef as the composite of paths fi, and similarly forg and the resulting subdivision of I× {1}. We see that the relation [f] = [g] in Π(X) is a consequence of a finite number of relations, each of which holds in one of the Π(U). Therefore ˜η([f]) = ˜η([g]). This verifies the universal property and proves the theorem.

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Theorem _{(van Kampen)}. _Let _X _{be path connected and choose a basepoint}

x ∈ X. Let O _{be a cover of} _X _{by path connected open subsets such that the}

intersection of finitely many subsets in O _{is again in} O _and _x_{is in each} _U _∈O_.

RegardO _{as a category whose morphisms are the inclusions of subsets and observe}

that the functor π1(−, x), restricted to the spaces and maps inO_{, gives a diagram}

π1|O _:O_−→G

of groups. The groupπ1(X, x) is the colimit of this diagram. In symbols,

π1(X, x)∼= colimU∈Oπ1(U, x).

We proceed in two steps.

Lemma. _{The van Kampen theorem holds when the cover}O _{is finite.}

Proof. _{This step is based on the nonsense above about skeleta of categories.} We must verify the universal property, this time in the category of groups. For a groupGand a mapη :π1|O _−→_G_ofO_{-shaped diagrams of groups, we must show}

that there is a unique homomorphism ˜η : π1(X, x) −→ Gthat restricts to ηU on

π1(U, x). Remember that we think of a group as a groupoid with a single object and with the elements of the group as the morphisms. The inclusion of categories

J : π1(X, x) −→ Π(X) is an equivalence. An inverse equivalence F : Π(X) −→

π1(X, x) is determined by a choice of path classes x−→ y fory ∈ X; we choose

cx wheny =x and so ensure thatF◦J = Id. Because the coverO is finite and closed under finite intersections, we can choose our paths inductively so that the path x−→ y lies entirely in U whenevery is in U. This ensures that the chosen paths determine compatible inverse equivalences FU : Π(U) −→ π1(U, x) to the inclusionsJU :π1(U, x)−→Π(U). Thus the functors

Π(U) FU //π1(U, x) ηU //_G

specify an O_{-shaped diagram of groupoids Π|}O _−→ _G_. _{By the fundamental}

groupoid version of the van Kampen theorem, there is a unique map of groupoids

ξ: Π(X)−→G

that restricts toηU◦FU on Π(U) for eachU. The composite

π1(X, x) J //Π(X) ξ //_G

is the required homomorphism ˜η. It restricts toηU onπ1(U, x) by a little “diagram chase” and the fact that FU ◦JU = Id. It is unique becauseξ is unique. In fact, if we are given ˜η : π1(X, x) −→ G that restricts to ηU on each π1(U, x), then ˜

η◦F : Π(X)−→Grestricts toηU◦FU on each Π(U); thereforeξ= ˜η◦F and thus

ξ◦J = ˜η.

Proof of the van Kampen theorem. _{We deduce the general case from the} case just proved. LetF _{be the set of those finite subsets of the cover} O _{that are}

closed under finite intersection. ForS _∈F_{, let} _US _{be the union of the}_U _in S_.

ThenS _{is a cover of}_US _{to which the lemma applies. Thus}

colimU∈Sπ1(U, x)∼=π1(US, x).

RegardF _{as a category with a morphism}S _−→T _whenever_US ⊂UT. We claim

first that

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8. EXAMPLES OF THE VAN KAMPEN THEOREM 19

In fact, by the usual subdivision argument, any loopI−→X and any equivalence

h : I ×I −→ X between loops has image in some US. This implies directly that π1(X, x), together with the homomorphismsπ1(US, x)−→π1(X, x), has the

universal property that characterizes the claimed colimit. We claim next that

colimU∈Oπ1(U, x)∼= colimS∈Fπ1(US, x),

and this will complete the proof. Substituting in the colimit on the right, we have

colimS∈Fπ1(US, x)= colim∼ S∈FcolimU∈Sπ1(U, x).

By a comparison of universal properties, this iterated colimit is isomorphic to the single colimit

colim(U,S)∈(O,F)π1(U, x).

Here the indexing category (O_,F_{) has objects the pairs (}_U,S_{) with} _U _∈ S_;

there is a morphism (U,S₎ _−→ ₍_V,T_{) whenever both} _U _⊂ _V _and _US _⊂ _UT_.

A moment’s reflection on the relevant universal properties should convince the reader of the claimed identification of colimits: the system on the right differs from the system on the left only in that the homomorphismsπ1(U, x)−→π1(V, x) occur many times in the system on the right, each appearance making the same contribution to the colimit. If we assume known a priori that colimits of groups exist, we can formalize this as follows. We have a functorO _−→F _{that sends}_U _to

the singleton set{U}and thus a functor O _−→₍O_,F_{) that sends}_U _{to (}_U,_{_U_}).

The functorπ1(−, x) :O _−→G _{factors through (}O_,F_{), hence we have an induced}

map of colimits

colimU∈Oπ1(U, x)−→colim(U,S)∈(O,F)π1(U, x).

Projection to the first coordinate gives a functor (O_,F₎_−→O_{. Its composite with} π1(−, x) :O _−→G _{defines the colimit on the right, hence we have an induced map}

of colimits

colim(U,S)∈(O,F)π1(U, x)−→colimU∈Oπ1(U, x).

These maps are inverse isomorphisms.

8. Examples of the van Kampen theorem

So far, we have only computed the fundamental groups of the circle and of contractible spaces. The van Kampen theorem lets us extend these calculations. We now drop notation for the basepoint, writingπ1(X) instead ofπ1(X, x).

Proposition. _Let_X _{be the wedge of a set of path connected based spaces} _X_i,

each of which contains a contractible neighborhood Vi of its basepoint. Then π1(X)

is the coproduct (= free product) of the groups π1(Xi).

Proof. _Let _U_i _{be the union of} _X_i _{and the} _V_j _for _j ₆₌ _i_{. We apply the van} Kampen theorem with O _{taken to be the} _U_i _{and their finite intersections. Since}

any intersection of two or more of theUi is contractible, the intersections make no contribution to the colimit and the conclusion follows.

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Any compact surface is homeomorphic to a sphere, or to a connected sum of toriT2₌_S1_×_S1_{, or to a connected sum of projective planes}_R_P2₌_S2_/_Z

2(where we writeZ2=Z/2Z). We shall see shortly thatπ1(RP2) =Z2. We also have the following observation, which is immediate from the universal property of products. Using this information, it is an exercise to compute the fundamental group of any compact surface from the van Kampen theorem.

Lemma. _{For based spaces} _X _and_Y_,_π₁₍_X_×_Y₎∼₌_π₁₍_X₎_×_π₁₍_Y_).

We shall later use the following application of the van Kampen theorem to prove that any group is the fundamental group of some space. We need a definition.

Definition. _{A space}_X _{is said to be simply connected if it is path connected} and satisfiesπ1(X) = 0.

Proposition. _Let_X₌_U_∪_V_{, where}_U_,_V_{, and}_U_∩_V _{are path connected open}

neighborhoods of the basepoint of X and V is simply connected. Then π1(U)−→

π1(X) is an epimorphism whose kernel is the smallest normal subgroup of π1(U)

that contains the image of π1(U∩V).

Proof. _Let_N _{be the cited kernel and consider the diagram}

π1(U)

The universal property gives rise to the map ξ, andξis an isomorphism since, by an easy algebraic inspection,π1(U)/N is the pushout in the category of groups of the homomorphismsπ1(U∩V)−→π1(U) andπ1(U∩V)−→0.

PROBLEMS

(1) Compute the fundamental group of the two-holed torus (the compact sur-face of genus 2 obtained by sewing together two tori along the boundaries of an open disk removed from each).

(2) The Klein bottleKis the quotient space ofS1×Iobtained by identifying (z,0) with (z−1_,_{1) for}_z_∈_S1_{. Compute}_π₁₍_K_).

(3) ∗ Let X ={(p, q)|p6=−q} ⊂ Sn_×_Sn_{. Define a map}_f _: _Sn _−→ _X _by

f(p) = (p, p). Prove thatf is a homotopy equivalence.

(4) LetC _{be a category that has all coproducts and coequalizers. Prove that} C _{is cocomplete (has all colimits). Deduce formally, by use of opposite}

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CHAPTER 3

Covering spaces

We run through the theory of covering spaces and their relationship to fun-damental groups and funfun-damental groupoids. This is standard material, some of the oldest in algebraic topology. However, I know of no published source for the use that we shall make of the orbit category O₍_π₁₍_{B, b}_{)) in the classification of}

coverings of a space B. This point of view gives us the opportunity to introduce some ideas that are central to equivariant algebraic topology, the study of spaces with group actions. In any case, this material is far too important to all branches of mathematics to omit.

1. The definition of covering spaces

While the reader is free to think about locally contractible spaces, weaker con-ditions are appropriate for the full generality of the theory of covering spaces. A space X is said to be locally path connected if for any x∈X and any neighbor-hood U of x, there is a smaller neighborhood V of xeach of whose points can be connected toxby a path inU. This is equivalent to the seemingly more stringent requirement that the topology ofX have a basis consisting of path connected open sets. In fact, if X is locally path connected and U is an open neighborhood of a pointx, then the set

V ={y|y can be connected toxby a path inU}

is a path connected open neighborhood ofxthat is contained inU. Observe that ifX is connected and locally path connected, then it is path connected. Through-out this chapter, we assume that all given spaces are connected and locally path connected.

Definition. _{A map}_p_:_E_−→_B _{is a covering (or cover, or covering space) if} it is surjective and if each pointb∈Bhas an open neighborhoodV such that each component ofp−1₍_V_{) is open in}_E_{and is mapped homeomorphically onto} _V _by_p_. We say that a path connected open subset V with this property is a fundamental neighborhood ofB. We callE the total space,B the base space, andFb=p−1(b) a fiber of the coveringp.

Any homeomorphism is a cover. A product of covers is a cover. The projection

R−→S1 _{is a cover. Each} _f

n :S1−→S1 is a cover. The projection Sn −→RPn is a cover, where the real projective spaceRPn _{is obtained from}_Sn _{by identifying} antipodal points. Iff :A−→B is a map (whereA is connected and locally path connected) andD is a component of the pullback off alongp, thenp:D−→Ais a cover.

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22 COVERING SPACES

2. The unique path lifting property

The following result is abstracted from what we saw in the case of the particular cover R −→ S1_{. It describes the behavior of} _p _{with respect to path classes and} fundamental groups.

Theorem_{(Unique path lifting)}. _Let_p_:_E_−→_B _{be a covering, let}_b_∈_B_{, and}

lete, e′_∈_F_b.

(i) A path f :I−→B withf(0) =b lifts uniquely to a pathg:I−→E such

that g(0) =e andp◦g=f.

(ii) Equivalent paths f ≃f′ _:_I _−→ _B _{that start at} _b _{lift to equivalent paths}

g≃g′_:_I_−→_E _{that start at} _e_{, hence}_g_{(1) =}_g′_(1).

(iii) p∗:π1(E, e)−→π1(B, b)is a monomorphism.

(iv) p∗(π1(E, e′₎₎_{is conjugate to}_p_∗(_π₁₍_{E, e}_)).

(v) Ase′ _{runs through}_F_{b, the groups}_p_∗(_π₁₍_{E, e}′₎₎_{run through all conjugates}

of p∗(π1(E, e))in π1(B, b).

Proof. _{For (i), subdivide} _I _{into subintervals each of which maps to a} fun-damental neighborhood under f, and lift f to g inductively by use of the pre-scribed homeomorphism property of fundamental neighborhoods. For (ii), let

h:I×I−→Bbe a homotopyf ≃f′_{through paths}_b_−→_b′_{. Subdivide the square} into subsquares each of which maps to a fundamental neighborhood underf. Pro-ceeding inductively, we see thathlifts uniquely to a homotopyH :I×I−→Esuch that H(0,0) =eandp◦H =h. By uniqueness,H is a homotopy g≃g′ _through pathse−→e′_{, where}_g_{(1) =}_e′₌_g′_{(1). Parts (iii)–(v) are formal consequences of} (i) and (ii), as we shall see in the next section.

Definition. _{A covering} _p _: _E _−→ _B _{is regular if} _p_∗(_π₁₍_{E, e}_{)) is a normal} subgroup ofπ1(B, b). It is universal ifE is simply connected.

As we shall explain in §4, for a universal cover p: E −→ B, the elements of

Fb are in bijective correspondence with the elements ofπ1(B, b). We illustrate the force of this statement.

Example. _For_n_≥_2,_Sn _{is a universal cover of}_RPn_{. Therefore}_π₁₍_RPn_{) has} only two elements. There is a unique group with two elements, and this proves our earlier claim thatπ1(RPn_{) =}_Z_2.

3. Coverings of groupoids

Much of the theory of covering spaces can be recast conceptually in terms of fundamental groupoids. This point of view separates the essentials of the topol-ogy from the formalities and gives a convenient language in which to describe the algebraic classification of coverings.

Definition. _{(i) Let} C _{be a category and}_x_{be an object of}C_{. The category} x\C _{of objects under}_x_{has objects the maps}_f _:_x_−→_y_inC_{; for objects}_f _:_x_−→_y

andg :x−→z, the morphismsγ :f −→gin x\C _{are the morphisms}_γ_:_y _−→_z

in C _{such that} _γ_◦_f ₌_g _:_x_−→_z_{. Composition and identity maps are given by}

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3. COVERINGS OF GROUPOIDS 23

(ii) LetC _{be a small groupoid. Define the star of}_x_{, denoted}_St₍_x_{) or}_StC₍_x_),

to be the set of objects of x\C_{, that is, the set of morphisms of}C _{with source}_x_.

WriteC₍_{x, x}_{) =}_π₍C_{, x}_{) for the group of automorphisms of the object}_x_.

(iii) LetE _andB _{be small connected groupoids. A covering}_p_:E _−→B_{is a}

functor that is surjective on objects and restricts to a bijection

p:St(e)−→St(p(e))

for each object eofE_{. For an object}_b _ofB_{, let}_F_b _{denote the set of objects of}E

such thatp(e) =b. Thenp−1(St(b)) is the disjoint union overe∈Fb ofSt(e). Parts (i) and (ii) of the unique path lifting theorem can be restated as follows.

Proposition. _If _p_:_E_−→_B _{is a covering of spaces, then the induced functor} Π(p) : Π(E)−→Π(B)is a covering of groupoids.

Parts (iii), (iv), and (v) of the unique path lifting theorem are categorical consequences that apply to any covering of groupoids, where they read as follows.

Proposition. _Let _p_:E _−→B _{be a covering of groupoids, let} _b _{be an object}

of B_{, and let}_e_and_e′ _{be objects of} _F

b.

(i) p:π(E_{, e}₎_−→_π₍B_{, b}₎_{is a monomorphism.}

(ii) p(π(E_{, e}′₎₎_{is conjugate to} _p₍_π₍_E_{, e}_)).

(iii) As e′ _{runs through} _F_{b, the groups} _p₍_π₍_{E, e}′₎₎ _{run through all conjugates}

of p(π(E_{, e}₎₎_in _π₍B_{, b}_).

Proof. _{For (i), if}_{g, g}′_∈_π₍E_{, e}_{) and}_p₍_g_{) =}_p₍_g′_{), then}_g₌_g′_{by the injectivity} ofponSt(e). For (ii), there is a mapg:e−→e′_since_E _{is connected. Conjugation} byg gives a homomorphismπ(E_{, e}₎_−→_π₍E_{, e}′_{) that maps under}_p_{to conjugation} of π(B_{, b}_{) by its element}_p₍_g_{). For (iii), the surjectivity of} _p_on _St₍_e_{) gives that}

anyf ∈π(B_{, b}_{) is of the form}_p₍_g_{) for some}_g_∈_St₍_e_{). If}_e′ _{is the target of}_g_{, then}

p(π(E_{, e}′_{)) is the conjugate of}_p₍_π₍_E_{, e}_{)) by} _f_. The fibersFb of a covering of groupoids are related by translation functions.

Definition. _Let _p _: E _−→ B _{be a covering of groupoids. Define the fiber}

translation functorT =T(p) :B_−→S _{as follows. For an object}_b_ofB_,_T₍_b_{) =}_F_b.

For a morphism f :b −→b′ _of _B_,_T₍_f_{) :}_F

b −→Fb′ is specified by T(f)(e) = e′,

wheree′ _{is the target of the unique} _g_in _St₍_e_{) such that}_p₍_g_{) =}_f_.

It is an exercise from the definition of a covering of a groupoid to verify thatT

is a well defined functor. For a covering spacep:E−→B and a pathf :b−→b′_,

T(f) :Fb −→Fb′ is given byT(f)(e) =g(1) whereg is the path in E that starts

ateand coversf.

Proposition. _{Any two fibers} _F_b _and _F_b′ of a covering of groupoids have the

same cardinality. Therefore any two fibers of a covering of spaces have the same cardinality.

Proof. _For_f _:_b_−→_b′_, _T₍_f_{) :}_F

b −→Fb′ is a bijection with inverseT(f−1).

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24 COVERING SPACES

4. Group actions and orbit categories

The classification of coverings is best expressed in categorical language that involves actions of groups and groupoids on sets.

A (left) action of a group Gon a set S is a function G×S −→ S such that

es = s (where e is the identity element) and (g′_g₎_s ₌_g′₍_gs_{) for all} _s _∈ _S_{. The}

isotropy groupGsof a pointsis the subgroup{g|gs=s}ofG. An action isfreeif

gs=simpliesg=e, that is, ifGs=efor everys∈S.

The orbit generated by a point sis {gs|g ∈G}. An action is transitiveif for every pair s, s′ _{of elements of} _S_{, there is an element} _g _of _G _{such that} _gs ₌ _s′_. Equivalently, S consists of a single orbit. If H is a subgroup of G, the set G/H

of cosetsgH is a transitive G-set. When Gacts transitively on a setS, we obtain an isomorphism of G-sets between S and the G-set G/Gs for any fixed s∈S by sendinggsto the cosetgGs.

The following lemma describes the group of automorphisms of a transitive

G-setS. For a subgroupH ofG, letN Hdenote the normalizer ofH inGand define

W H=N H/H. Such quotient groupsW H are sometimes called Weyl groups.

Lemma. _Let _G _{act transitively on a set} _S_{, choose} _s _∈ _S_{, and let} _H ₌ _G_s.

Then W H is isomorphic to the group AutG(S)of automorphisms of theG-setS.

Proof. _For_n _∈ _{N H} _{with image ¯}_n _∈_{W H}_{, define an automorphism} _φ_(¯_n_{) of}

S byφ(¯n)(gs) =gns. For an automorphismφ of S, we haveφ(s) =ns for some

n ∈ G. For h ∈ H, hns = φ(hs) = φ(s) = ns, hence n−1_hn _∈ _G

s = H and

n∈N H. Clearlyφ=φ(¯n), and it is easy to check that this bijection betweenW H

and AutG(S) is an isomorphism of groups.

We shall also need to considerG-maps between differentG-setsG/H.

Lemma. _A _G_-map_α_:_G/H _−→_G/K _{has the form} _α₍_gH_{) =}_gγK_{, where the}

elementγ∈Gsatisfiesγ−1_hγ_∈_K _{for all}_h_∈_H_.

Proof. _If_α₍_eH_{) =}_γK_{, then the relation}

γK=α(eH) =α(hH) =hα(eH) =hγK

implies thatγ−1_hγ_∈_K _for_h_∈_H_.

Definition. _{The category} O₍_G_{) of canonical orbits has objects the} _G_-sets G/H and morphisms theG-maps ofG-sets.

The previous lemmas give some feeling for the structure of O₍_G_{) and lead to}

the following alternative description.

Lemma. _{The category}O₍_G₎_{is isomorphic to the category}G _{whose objects are}

the subgroups of G and whose morphisms are the distinct subconjugacy relations

γ−1_Hγ_⊂_K _for _γ_∈_G_.

If we regardGas a category with a single object, then a (left) action ofGon a setSis the same thing as a covariant functorG−→S_{. (A right action is the same}

thing as a contravariant functor.) If B_{is a small groupoid, it is therefore natural}

to think of a covariant functorT :B_−→S _{as a generalization of a group action.}

For each object b of B_, _T _{restricts to an action of} _π₍B_{, b}_{) on}_T₍_b_{). We say that}

the functorT is transitiveif this group action is transitive for each object b. If B

A Concise Course in Algebraic Topology - Area