Notes on Algebraic Topologymaraston/Topologia2/Topo2... · 2010-11-15 · Notes on Algebraic Topology deﬁned(30): this procedure is said to be an amalgamation of the free product

Notes on Algebraic Topology

1.5 Theorem of Van Kampen

The theorem of Van Kampen(28) allows the computation of the fundamental group of aspace in terms of the fundamental groups of the open subsets of a suitable cover.

Let X be an arcwise connected topological space, (Λ,≤) an ordered set, Uλ : λ ∈ Λ anopen cover of X such that:

(a) all Uλ are arcwise connected;

(b) λ ≤ µ if and only if Uλ ⊆ Uµ;

(c) the family Uλ : λ ∈ Λ is stable under finite intersections;

(d) there exists x0∈

λ∈Λ Uλ.

Let us denote by ιλ : Uλ −→ X and ιλ,µ : Uλ −→ Uµ the canonical inclusions (whereλ ≤ µ), and write for short π1(Uλ) instead of π1(Uλ, x0

). We then have morphismsιλ# : π1(Uλ) −→ π1(X) and ιλ,µ# : π1(Uλ) −→ π1(Uµ) (where λ ≤ µ), which simplyassociate to the class of a loop the class of the same loop viewed in the larger space. Inparticular, one has an inductive system π1(Uλ), ιλ,µ : λ, µ ∈ Λ (see Appendix A.1).

Let M ⊂ Λ such that X =

λ∈M Uλ.

Proposition 1.5.1. π1(X) is generated by ιλ#(π1(Uλ)) : λ ∈ M.

Proof. Let γ : I −→ X be a loop at x0 , δ > 0 the Lebesgue number relative to the cover γ−1(Uλ) : λ ∈ Mof I, 0 = t0 < t1 < · · · < tk−1 < tk = 1 with tj − tj−1 < δ and λj ∈ M (with j = 1, . . . , k) such thatγ([tj−1, tj ]) ⊂ Uλj . Since γ(tj) ∈ Uλj ∩ Uλj+1 (arcwise connected), let σj (for j = 1, . . . , k − 1) be an arcinto Uλj ∩ Uλj+1 from x0 to γ(tj). Setting γj = γ|

[tj−1,tj ](reparametrized by sending tj−1 to 0 and tj to

1) it clearly holds [γ] = [γ1] · · · · · [γk] = [γ1 · σ−1

1] · [σ1 · γ2 · σ−1

2] · · · · · [σk−2 · γk−1 · σ−1

k−1][σk−1 · γk].

An obvious consequence of Proposition 1.5.1 is:

Corollary 1.5.2. If the open subsets Uλ (λ ∈ M) are simply connected, such is also X.

Example. Sn is simply connected for n ≥ 2. Namely, let N = en+1 be the North pole, S = −N the

South pole, and set U = Sn \ N and V = S

n \ S: noting that U ∩ V is arcwise connected and that

both U and V are simply connected, just apply Corollary 1.5.2. Note that this argument does not apply

for S1 (in that case U ∩ V is not arcwise connected).

In general one has the following result (see Appendix A.1 for the notion of “inductivelimit” of an inductive system).

Theorem 1.5.3. (Van Kampen) In the category Groups it holds

π1(X) = lim−→λ∈Λ

π1(Uλ).

(28)The result has been proved independently also by Karl Seifert in the 30s of last century; in fact, it isoften referred to as “Seifert - Van Kampen theorem”.

Corrado Marastoni 22


Proof. Consider a group L and a family of morphisms ψλ : π1(Uλ) −→ L such that ψλ = ψµιλ,µ# for λ ≤ µ,and let us see if there exists a unique morphism ψ : π1(X) −→ L such that ψλ = ψιλ# for any λ ∈ Λ. So let[γ] ∈ π1(X): by Proposition 1.5.1 we may write [γ] = ιλ1#([γ1]) · · · · · ιλk#([γk]) with [γj ] ∈ π1(Uλj ). If ψexists, it must be necessarily unique because ψ([γ]) = ψ(ιλ1#[γ1]) · · ·ψ(ιλk#[γk]) = ψλ1([γ1]) · · ·ψλk ([γk])(the products are in L). We are left with showing that this is actually a well-posed definition for ψ, i.e. that if[cx0

] = ιλ1#([σ1])· · · · ·ιλk#([σk]) then also ψλ1([σ1]) · · ·ψλk ([σk]) = e (where e is the identity element of L).

Let σ = σ1 · · · · ·σk be in X (hence, if t ∈ [ j−1

k,j

k] it holds σ(t) = σj(kt−(j−1))), and let h : I×I −→ X be a

homotopy rel ∂I between σ and cx0. Let ε > 0 the Lebesgue number relative to the cover h−1(Uλ) : λ ∈ Λ

of I × I, and let r ∈ N be such that√2

kr < ε. Hence, setting Ri,j =i−1

kr ,i

kr

×

j−1

kr ,j

kr

⊂ I × I (for

i, j = 1, . . . , kr), there exists λi,j ∈ Λ such that h(Ri,j) ⊂ Uλi,j . Let vi,j =

i

kr ,j

kr

(hence Ri,j is the

square with side k−r and opposed vertices vi−1,j−1 and vi,j), Uµ(i,j) the intersection of the (one, two or

four) Uλl,m such that vi,j ∈ Rl,m, and γi,j a path in Uµ(i,j) from x0 to h(vi,j). Let αi,j(t) = h

t+(i−1)

kr ,j

kr

(path from h(vi−1,j) to h(vi,j)) and βi,j(t) = h

i

kr ,t+(j−1)

kr

(path from h(vi,j−1) to h(vi,j)): note that

α(m−1)kr−1+1,0 · · · · · αmkr−1,0

= [σm] (for m = 1, . . . , k) and that αi,kr (t) = β0,j(t) = βkr,j(t) ≡ x0 (for

t ∈ I and i, j = 1, . . . , kr). From the equality [αi,j−1 · βi,j ] = [βi−1,j · αi,j ] one gets (by inserting the pathsγi,j and their inverses to base at x0) the relations

(γi−1,j−1 · αi,j−1) · γ−1

i,j−1

·(γi,j−1 · βi,j) · γ−1

i,j

=

(γi−1,j−1 · βi−1,j) · γ−1

i−1,j

·(γi−1,j · αi,j) · γ−1

i,j

in the group π1(Uλi,j ). Applying ψλi,j and setting

ai,j = ψλi,j

(γi−1,j · αi,j) · γ−1

i,j

and bi,j = ψµ(i,j)

(γi,j−1 · βi,j) · γ−1

i,j

, one then has the equality

ai,j−1 bi,j = bi−1,j ai,j in L. Knowing that a1,kr · · · akr,kr = e and that b0,j = bkr,j = e (for any j =1, . . . , kr), one has e = a1,kr · · · akr,kr = (b0,kr a1,kr )a2,kr · · · akr,kr = a1,kr−1(b1,kr a2,kr ) · · · akr,kr =· · · = a1,kr−1 · · · akr,kr−1; by repeating the procedure one obtains a1,0 · · · akr,0 = e, as required.

Now the problem is to understand what lim−→λ∈Λ

π1(Uλ) seems like.

In general, the free product ∗λ∈ΛGλ of a family of groups Gλ : λ ∈ Λ is the groupformed by finite “words” a1 · · · ak constructed with “letters” aj ∈ Gλj

(j = 1, . . . , k; k ≥ 1),where any letter is different from the identity element in the respective group and wheretwo adjacent letters must belong to different groups (one often says “reduced letters”);also the “empty word” is considered to be an element. The operation is given by thenatural juxtaposition (a1 · · · ak) · (b1 · · · bh) = a1 · · · akb1 · · · bh where, in the case ak andb1 belong to a same group, the expression “akb1” should be replaced by their product inthat group (and possibly removed if akb1 is the identity, causing then the same procedurefor ak−1b2, and so on); the identity element is clearly the empty word.

Example. The free product of any number of copies of Z is called free group, in the sense that there

is one generator for each copy of Z and the elements of the group are words formed by powers of these

generators. For example, Z ∗Z is formed by the words r1s1 · · · rksk where all rj ’s and sj ’s are integer (the

rj ’s are meant to belong to the first copy of Z, and the sj ’s to the second); or also, in abstract notation,

by ak1b

h1 · · · akrbhr where a and b denote the two generators and the exponents are integers.

Note that for any µ ∈ Λ there is a natural monomorphism Gµ −→ ∗λ∈ΛGλ; in fact one seesthat the free product ∗λ∈ΛGλ is the inductive limit in Groups of the system Gλ : λ ∈ Λwith trivial preorder, i.e. without considering morphisms(29). More generally, if morphismsfλ,µ : Gλ −→ Gµ are given for some pairs (λ, µ) with fλ,µ fµ,ν = fλ,ν whenever defined,then to obtain the inductive limit of the system Gλ, fλ,µ : λ, µ ∈ Λ one must quotientout the previous free product ∗λ∈ΛGλ by its normal subgroup N generated by all theelements of type fλ,µ(a)fλ,ν(a)−1 for a ∈ Gλ whenever the morphisms fλ,µ and fλ,ν are

(29)Namely, given a family of morphisms ψλ : Gλ −→ L, the definition (necessary, hence unique)ψ(a1 · · · ak) = ψλ1(a1) · · ·ψλk (ak) is a morphism.



defined(30): this procedure is said to be an amalgamation of the free product with respectto the given morphisms fλ,µ.

When applying the above notions to the framework of Van Kampen, the groups are π1(Uλ)and the morphisms are the maps ιλ,µ# : π1(Uλ) −→ π1(Uµ), which send the class of a loopin the small open subset Uλ to the class of same loop viewed in the larger open subsetUµ: by Proposition 1.5.1 and the subsequent discussion it is then enough to consider thefree product of the groups π1(Uλ)’s of a selected family of open subsets Uλ with indicesλ ∈ M which cover X and are not contained in each other, and then to amalgamate onlywith respect to the double intersections Uλ ∩ Uµ for λ, µ ∈ M .(31)

Example. (Wedge sums) Consider a family of pointed spaces (Xλ, xλ) for λ ∈ M , and let X =

λ∈MXλ

be their wedge sum (see p. 9). For each λ ∈ M let Vλ be a open subset of xλ in Xλ which has xλ

as deformation retract, and set Uλ = Xλ ∨ (

µ =λVµ). It is clear that the Uλ’s cover X; moreover, any

intersection of two or more of them is always

λ∈MVλ, which is arcwise connected and, since it deformation-

retracts to the base point, has trivial fundamental group and hence causes no effective amalgamation.

Finally, since each Uλ deformation-retracts to the corresponding Xλ, by Van Kampen theorem it follows

that π1(X) ∗λ∈M π1(Xλ).

The Theorem of Van Kampen is used mainly in the case of two open subsets X = U ∪ V ,with U , V and U ∩ V arcwise connected: in that case π1(X) will be the inductive limitof the system π1(U ∩ V ),π1(U),π1(V ); , ιU∩V,U , , ιU∩V,V , and its universal property isexpressed by the following diagram:

(1.4) L

π1(U)ιU#

ψU

π1(X)

∃!ψ

π1(U ∩ V )ιU∩V,V #

ιU∩V,U#

ιU∩V #

ψU∩V

π1(V )

ιV #

ψV

One usually denotes the free product of two groups G and H by G ∗H and, given anothergroup K and morphisms f : K −→ G and g : K −→ H, the free product of G and Hamalgamated on K by G ∗K H.(32) By what has been said we get:

(30)Namely, if in the previous notation ψλ αλ = ψµ αµ, for any λ, µ ∈ Λ then N ⊂ ker(ψ) and hence ψ

factorizes uniquely through the quotient ∗λ∈Λ Gλ/N .(31)Namely, a typical element of ∗λ∈Λ π1(Uλ) is [γ1] · · · [γk] where γj is a loop in Uλj but the class[γj ] is taken as loop in X (in fact, we should have written more precisely iλj#([γj ]Uλj

)): hence, by the

compatibility of the various morphisms of type i#, the class [γj ] can be thought as coming from someUλ with λ ∈ M , and this shows that ∗λ∈Λ π1(Uλ) ∗λ∈M π1(Uλ). Similar considerations hold for theamalgamation: the objects iλ,µ#([γ]) · iλ,ν#([γ])−1 coming from intersections of three or more Uλ withλ ∈ M can be thought as having already come from some double intersection.(32)In the language of categories, the group G ∗K H is usually called the pushout of the morphismsf : K −→ G and g : K −→ H. Note that the notation G ∗K H does not show explicitly what are f and g,but of course it is important to take them into account.



Corollary 1.5.4. Let X be an arcwise connected topological space, X = U ∪ V an opencover with U , V and U ∩ V arcwise connected open subsets. Then

π1(X) π1(U) ∗π1(U∩V )

π1(V ).

In particular, let us emphasize the following two cases.

(i) If U ∩ V is simply connected, then π1(X) π1(U) ∗ π1(V );

(ii) If V is simply connected, and N is the normal subgroup generated by the image (byιU∩V,U#) of π1(U ∩ V ) in π1(U), then π1(X) π1(U)/N .

Examples. (1) (Plane with k holes) Let x1, . . . , xk be a family of k distinct points of the plane R2, andlet X = R

2 \ x1, . . . , xk. Then π1(X) is the free group with k generators. Namely for k = 1 we alreadyknow that π1(X) Z; given k > 1, let sj be closed half lines with origin xj and with empty intersection(33),set U = X \ s1, . . . , sk−1 and V = X \ sk. Now, X = U ∪ V , all the sets are arcwise connected andU ∩ V is simply connected (even contractible), and so π1(X) π1(U) ∗ π1(V ); but U is homotopicallyequivalent to S

1, while π1(V ) is free with k − 1 generators by inductive hypothesis. The same holds forY = S

2 \ y1, . . . , yk+1 (with y1, . . . , yk+1 distinct points of S2).(34) (2) (Bouquet of k circles) Thefundamental group of a “bouquet” of k circles (i.e. the wedge sum of k circles) is again a free group withk generators: this follows immediately from what has been said in general for wedge sums. Alternativelyone could also note that the bouquet is in fact a strong deformation retract of the plane with k holes;another proof is to use induction and Van Kampen, by choosing for any circle Cj a point xj differentfrom the center of the bouquet (j = 1, . . . , k), and then taking U = X \ x1 and V = (C1 ∪ C2) \ x2:then U ∩ V is contractible, V has the homotopy of a circle and U of a bouquet of (k − 1) circles. (3)(Removing an annulus from R

3) Let S1

(x,y)= (x, y, z) ∈ R

3 : z = 0, x2 + y2 = 1, and let us compute

π1(X) where X = R3 \ S

1

(x,y). Let Rz = (x, y, z) ∈ R

3 : x = y = 0 (the z-axis), and set U = X \ Rz

and V = (x, y, z) ∈ X : x2 + y2< 1. Obviously X = U ∪ V , all are arcwise connected and V is simply

connected (even contractible). On the other hand, U ∩ V is homotopically equivalent to S1 and hence

π1(U ∩V ) Z, while, setting T = (x, y, z) ∈ R3 : x > 0, y = 0 \ (1, 0, 0), U is homeomorphic to T ×S

1

(exercise) and hence π1(U) π1(T ) × π1(S1) Z × Z = Z

2; moreover, one may identify the morphismπ1(U ∩ V ) −→ π1(U) π1(T )× π1(S

1) with the morphism Z −→ Z2, 1 → (0, 1) (the generator of π1(U ∩ V )

goes into the generator of the second factor). One therefore has π1(X) Z2/Z Z. (4) (Torus) On the

surface of X = T2 = (S1)2 (the 2-dimensional torus viewed as a doughnut in R

3, see Example 1.4) makea small circular hole F , and let U = X \ F (open); let V be an open neighborhood of F in X (a “patch”above F ). We are in fact in the hypotheses of Van Kampen’s theorem; it is evident that V is contractibleand that U ∩ V is homotopically equivalent to a circle (hence π1(V ) is trivial and π1(U ∩ V ) Z). Onthe other hand U is homotopically equivalent to two tangent circles (this can be easily understood in theinterpretation of T2 as a square modulo identifications, as recalled in the cited Example 1.4: making ahole in the interior of the square, the latter deformation-retracts radially on its boundary; as an usefulexercise, we suggest to interpret this retraction on the doughnut), and then to a plane with two holes:hence π1(U) is free on two generators. Now, the normal subgroup of π1(U) Z∗Z generated by the imageof π1(U ∩ V ) is the subgroup of commutators(35) of Z ∗ Z (in the interpretation of the square, a generatorof π1(U ∩ V, x0) is a loop based at a vertex which surrounds the hole: such loop is clearly homotopic rel∂I to the boundary of the square run twice forth and back, which is exactly the commutator of the two

(33)For example, draw the lines rl,m = xl + t(xm − xl) : t ∈ R (for 1 ≤ l < m ≤ k), then choosey ∈ R

2 \

1≤l<m≤krl,m and set sj = y + t(xj − y) : t ≥ 1 (with j = 1, . . . , k).

(34)Y is homeomorphic to X by the stereographic projection from one of the yj (recall that, considering

in R3 the sphere S = x2 + y

2 + (z − 1)2 = 1 S2 and the plane Π = z = 0 R

2, the “stereographicprojection” from the North pole N = (0, 0, 2) ∈ S identifies diffeomorphically S\N with Π by associatingto x = (x, y, z) ∈ S \ N the intersection point between Π and the half line coming from N and passingthrough x).(35)If G is a group, the subgroup of commutators of G is denoted by [G,G] and it is the normal subgroupgenerated by the elements of the form xyx

−1y−1 for x, y ∈ G. Obviously, G is abelian if and only if [G,G]

is trivial. Moreover, if GA is the free group generated by a set A, then GA/[GA , GA ] Z(A) (for the

notation Z(A) see Appendix A.1).



generators γ1 and γ2 of π1(U, x0)), so one has π1(X) Z∗Z/[Z∗Z,Z∗Z] Z2, as we have already seen. (5)

(Real projective line) Let P1 be the real projective line, endowed with the quotient topology with respect tonatural map p : R2

× −→ P1. Let q = p|

S1(Hopf map): since q : S1 −→ P

1 is continuous, surjective and closed,then q is still quotient. The map ( · )2 : S1 −→ S

1 is also quotient for the same reason, and has exactlythe same fibers of q: it follows that S

1 and P1 are canonically homeomorphic. hence π1(P

1) Z. (6)(Real projective plane) Let P

2 be the projective plane, endowed with the quotient topology with respectto the map p : B

2 −→ P2 obtained by identifying in P

2 the pairs of antipodal points on S1 ∂B

2 (ifx = (x, y) ∈ B

2 and [x0, x1, x2] are homogeneous coordinates in P2 one can set p(x) = [1− |x|, x, y]): note

that p(S1) P1 (one can also identify p|

S1with the Hopf map). The quotient map p is closed but not

open; nevertheless, if B2

× = B2 \ 0 and B2 = B

2 \ S1, then U = p(B2

×) and V = p(B2) are open in

P2.(36) From now on let us choose 1

2∈ B

2

× = U ∩V ⊂ C as base point for the computation of fundamental

groups. The set V (homeomorphic image of B2) is clearly contractible, while U ∩V (homeomorphic image

of B2

×) has fundamental group Z generated by [γ] obtained by (the image by p of) t → e2πit

2. As for

B2

×, it strong deformation-retracts to S1 by the affine homotopy h(x, t) = (1− t)x+ t

x

|x| , homotopy which

descends via p to a strong deformation retraction h of U a p(S1) P1.(37) Let r = h( · , 1) : U −→ p(S1):

by r# : π1(U, 1

2)

∼−→ π1(p(S1), 1) Z, the canonical generator of the second member comes from the

generator [ψ] of π1(U, 1

2) obtained by t → p(eπit

/2): hence ιU∩V,U# sends the generator [γ] in [ψ]2, andhence π1(P

2) Z/2Z (analogously to P3 and, as we shall show, to any P

n with n ≥ 2). (7) (Removingone or two annuli from R

3) If A is an annulus in R3 and X = R

3 \ A, we already computed above thatπ1(X) Z: another method is to observe that X can be deformation-retracted first to a 2-sphere S

2 plusa diameter, then to the a wedge sum S

1 ∨ S2 (by slowly approaching the endpoints of the diameter along

an equator), hence π1(X) π1(S1) ∗ π1(S

2) Z.

X : X : X

:

Figure 5: Deforming R3minus one annulus; minus two unlinked annuli; minus two linked annuli.

Let us use the same approach for two other similar situations. • If B is another annulus of R3 unlinked

with A, then X = R

3 \ (A B) can be deformation-retracted to S1 ∨ S

1 ∨ S2 ∨ S

2 and hence π1(X) is

free on two generators. • If C is a third annulus of R3 linked with A, then X = R

3 \ (A C) can be

deformation-retracted to S2 ∨ T

2 and hence π1(X) is isomorphic to π1(T

2), i.e. a free abelian group of

rank two. (8) (Klein bottle) Let us compute the fundamental group of the Klein bottle K by using its

description in terms of fundamental polygon (i.e. a quotient of a polygon); the argument will be suitable to

compute again the fundamental group of the torus T2 (see Figure 6). In both fundamental polygons take

(36)The map p is closed since B2 is compact and P

2 is Hausdorff (the finite points of P2 have the sameneighborhoods of the points of B2, while a basis of neighborhoods of a point at infinity p(x) with x ∈ S

1

is given by A ∪ (−A) where A = B2 ∩ U with U ⊂ C a small open ball centered in x; hence it is still

possible to separate the points of P2), but p is not open (the above A is open in B2, but its p-saturated

p−1(p(A)) = A ∪ (−A ∩ S

1) is not open: hence p(A) is not open in the (quotient) topology of P2). On theother hand the open subsets B2

× and B2 are already p-saturated, hence their images by p are open in P2.

(37)Recall the factorization property of quotient functions (Proposition 1.1.14): given a quotient functionf : X −→ Y and a continuous function g : X −→ Z, there exists a unique continuous function h : Y −→ Z

such that f = h g if and only if g is constant on the fibers of f . Here we mean X = B2

× × I, Y = U × I,Z = U , f = p × idI and g = p h, and the factorization hypothesis are satisfied. The situation would bedifferent if we would instead consider the strong deformation retraction of B2

× to αS1 for a 0 < α < 1 (e.g.

α = 1

2), for example the affine one hα(x, t) = (1− t)x+ αt

x

|x| : namely, note that p hα is not constant on

the fibers of p× idI , since (p× idI)(x, t) = (p× idI)(−x, t) but p(hα(x, t)) = −p(hα(−x, t)) = p(hα(−x, t))for any x ∈ S

1 and t ∈ I.



γ1

γ1

T2 = γ2 γ2

x0

x0

x0

x0

γ1

γ1

K = γ2 γ2

x0

x0

x0

x0

Figure 6: The torus, the Klein bottle and a suitable open cover for both.

U to be a central open square (yellow) whose edges are at some small distance δ > 0 from the boundary;

and V to be the open square crown (grey) of the points of the polygon whose distance from the boundary is

< 2δ. It is clear that U is contractible, and that U ∩V (the overlapped yellow-grey zone) is homotopically

equivalent to a circle; on the other hand, V can be deformation-retracted to the boundary, which can

be identified to a “figure eight” (a bouquet of two circles) and hence has fundamental group free on two

generators, i.e. Z ∗ Z. So, by Corollary 1.5.4, the fundamental group is in both cases (Z ∗ Z)/N where

N is the normal subgroup generated by the image of π1(U ∩ V ) Z: hence, what makes the difference

between T2 and K will be the different images of π1(U ∩V ) into π1(V ). Namely, a generator of π1(U ∩V )

is a square-shaped loop, e.g. run counterclockwise: when deformed on the boundary, this loop becomes

γ1γ2γ−1

1γ−1

2in the case of T2, and γ1γ2γ

−1

1γ2 in the case of K. Hence for T2 the subgroup N is generated

by the commutator of [γ1] and [γ2], and hence π1(T2) is the abelianization of Z ∗ Z, i.e. Z

2 (as we saw

above); while π1(K) is the group with generators a = [γ1] and b = [γ2] with relation aba−1

b = id, i.e.

bab = a. (9) (g-fold torus) A g-fold torus is a orientable closed surface of genus g; its fundamental polygon

is a 4g-gon with pairwise identifications of edges allowing g junctions naturally generalizing the one of the

(1-)torus (the Figure 7 shows the case n = 2). To compute the fundamental group of the 2-fold torus from

Figure 7: The double torus.

its fundamental polygon we can proceed exactly as we did above for the Klein bottle: U is contractible,

U ∩V is homotopically equivalent to a circle, while V can be deformation-retracted to the boundary, which

in this case can be identified to a bouquet of four circles and hence has fundamental group free on four

generators. Since a counterclockwise loop generating π1(U ∩V ), when deformed on the boundary, becomes

bab−1

a−1

cdc−1

d−1, we get that the fundamental group of the 2-fold torus is the free group generated by

a, b, c, d modulo the normal subgroup generated by bab−1

a−1

cdc−1

d−1. More generally, the fundamental

group of the g-fold torus is the free group generated by a1, b1, · · · , ag, bg modulo the normal subgroup

generated by a1b1a−1

1b−1

1· · · agbga

−1

g b−1

g . (10) (Spaces with fundamental group Z/nZ) Given any n ∈ N,

using the above technique of fundamental polygons it is then easy to construct a space whose fundamental

group is Z/nZ: just consider a regular n-gon and identify all its edges with a chosen direction (e.g.

conterclockwise). Namely, here we have π1(V ) Z (say with generator a) and the image of a generator

of π1(U ∩ V ) Z into π1(V ) is an, hence the quotient π1(V )/N is isomorphic to Z/nZ. (11) (Graphs)

In a connected graph, a tree is a contractible subgraph; a tree is called maximal if it contains all vertices

of X. If T is a maximal tree in a connected graph X, let dλ : λ ∈ Λ be the family of edges of X − T :

then π1(X) is a free group with generators [γλ] corresponding to each edge dλ. This can be proved by



Figure 8: Graphs and maximal trees.

considering, for any λ ∈ Λ, an open neighborhood Uλ of T +dλ which deformation-retracts to T +dλ: then

each Uλ deformation-retracts into a circle, and the intersections of two or more Uλ’s is contractible since

it deformation-retracts to T . For example, the fundamental group of the graph X on the left of Figure 8

is free on four generators, each one corresponding to a loop containing only one of the edges not in any

chosen maximal tree (whose edges are represented in black). Similarly, the graph Y on the right of Figure

8 — which can also be interpreted as the suspension of the three red vertices Pj with j = 1, 2, 3 — has

fundamental group free on two generators (a maximal tree is depicted in black). As for this last example

note that, setting Uj = Y \ Pj (for j = 1, 2, 3), then the open cover U1, U2 is suitable for applying

Van Kampen and confirms that π1(Y ) Z ∗ Z, while the open cover U1, U2, U3 is not suitable since

U1∩U2∩U3 = Y \P1, P2, P3 is not arcwise connected (hence one cannot conclude that π1(Y ) Z∗Z∗Z,a statement that would be false).



1.6 Covering spaces

The prototype of a covering space is the exponential map : R −→ S1

given by (t) = e2πit : the key property is that any small open intervalof S1 is “well-covered” by this map, i.e. its inverse image is a family ofpairwise disjoint homeomorphic copies of itself. We have already usedthis map to prove (see Proposition 1.4.2) that the fundamental groupof S1 is free with one generator: in fact, we shall see that there is adeep relation between the classification of the covering spaces of some

topological space X and the structure of the fundamental group of X.

1.6.1 Fiber bundles and covering spaces

Definition 1.6.1. Let X be a topological space. A space on X is a pair (Y,π) where Y isa topological space and π : Y −→ X is a surjective continuous function. A morphism from(Y1,π1) to (Y2,π2) is a continuous function f : Y1 −→ Y2 such that π1 = π2 f .

Given x ∈ X and a space (Y,π) onX, we denote by Yx = π−1(x) the fiber on x. Note that amorphism of spaces on X respects the fibers, in the sense that f(Y1,x) ⊂ Y2,x; in particular,if f is a isomorphism, for any x ∈ X it is induced a homeomorphism fx : Y1,x

∼−→ Y2,x.The simplest case of space on X is the one of type (X × F, p

X) where F is a topological

space and pX

the projection on X. More generally:

Definition 1.6.2. A space (Y,π) on X is called trivial if there exists a topological spaceF and an isomorphism f : (Y,π)

∼−→ (X × F, pX): in this case, such an isomorphism of

spaces on X is called a trivialization of (Y,π).

Anyway, the most important notion is the one of “locally trivial space”, or “fiber bundle”.

Definition 1.6.3. Given a space (Y,π) on X and an open subset U ⊂ X, the restrictionof (Y,π) to U (sometimes denoted by Y |

U) is the space on U given by (π−1(U), π|

π−1(U)).

The space (Y,π) on X is called locally trivial (or also fiber bundle, or bundle) on X if thereexists an open cover U = Uλ : λ ∈ Λ of X such that Y |

Uλis trivial for any λ ∈ Λ; i.e.,

for any x ∈ X there exists an open neighborhood U ⊂ X of x such that Y |Uis trivial. A

local trivialization of (Y,π) on Uλ is a trivialization of Y |Uλ

.

If the space (Y,π) on X is trivial, then obviously the map π is open and all fibers of (Y,π)on X are homeomorphic.(38) This is still true for any bundle on an arcwise connectedspace:

(38)If f : (Y,π)∼−→ (X ×F, pX ) is a trivialization, then all fibers of Y are homeomorphic to F (the fiber of

X×F ). As for the openness, since f is a homeomorphism we are left with proving that pX : X×F −→ X isopen. Let V be an open subset of X×F , and (x, f) ∈ V : then there exist open subsets U ⊂ X and W ⊂ F

such that (x, f) ∈ U ×W ⊂ V , and hence U = pX (U ×W ) ⊂ pX (V ). Therefore pX (V ) is a neighborhoodof pX (x, f) = x because it contains U (an open neighborhood of x), and this proves that pX (V ) is open.



Proposition 1.6.4. If (Y,π) is a bundle on X, then π is open. Moreover, if X is arcwiseconnected, then all fibers of (Y,π) on X are homeomorphic.

Proof. Let U = Uλ : λ ∈ Λ be an open cover of X such that (Y,π) is trivial on every U ∈ U . Let V ⊂ Y

be open, y ∈ V and U ∈ U such that x = π(y) ∈ U : then W = π−1(U) ∩ V is an open neighborhood of

y, and since the map π|π−1(U)

is open and W ⊂ π−1(U), one has that π(W ) ⊂ U ∩ π(V ) ⊂ π(V ) is an

open neighborhood of x. Hence π(V ) is open in X. Now let X be arcwise connected; given x0 , x1 ∈ X,let γ : I −→ X be a path between them and let λ0, . . . ,λk ∈ Λ be such that γ(I) ⊂

k

j=0Uλj , x0 ∈ Uλ0 ,

x1 ∈ Uλk and Uλj ∩Uλj+1 = ∅. We are left with proving that Y |Uλj

and Y |Uλj+1

are isomorphic if further

restricted to Uλj ∩Uλj+1 : it is enough to observe that, given local trivializations ψj : π−1(Uλj )∼−→ Uλj ×Fj ,

the isomorphism ψj+1 ψ−1

jof trivial spaces on Uλj ∩ Uλj+1 induces a homeomorphism Fj

∼−→ Fj+1.

If (Y,π) is a bundle on X, then Y is usually referred to as the “total space” and X as the“base” of the bundle; moreover, if X is arcwise connected, thanks to Proposition 1.6.4 onedirectly talks about “bundle with fiber F” or “F -bundle”, where F is a topological spacehomeomorphic to the fibers of π.

Remark 1.6.5. Since here we are interested only in topological matters, in the previousbrief exposition of the notion of bundle we have not paid so much attention to the structureof the fiber, which has been required to be nothing more than a topological space. In otherwords: we saw that, if (Y,π) is a bundle on the arcwise connected space X with fiber Fand U1, U2 are two open subsets of X with trivializations ψj : π−1(Uj)

∼−→ Uj × F , thenfor any x ∈ U1∩U2 there is an induced homeomorphism (ψ2 ψ−1

1)(x, · ) : F ∼−→ F , and we

have not requested this homeomorphism to respect also possible further structures of F(hence, for example, that it should be a linear map if F is a vector space, or a orthogonaltransformation if F is a sphere). In a more motivated exposition the structure of F has tobe respected by these homeomorphisms, which are commonly called transition functions.In fact, the proper notion of bundle requires also a group structure operating effectivelyon the fiber.(39) More precisely, a bundle of base X with total space Y , fiber F and groupstructure G (where X, Y and F are topological spaces and G is a topological group) isthe datum of:

(1) a space (Y,π) on X with fibers homeomorphic to F ;

(2) an effective action of G as a group of homeomorphisms on F ;

(3) an open cover U = Uλ : λ ∈ Λ of X with a family of local trivializations ψλ :π−1(Uλ)

∼−→ Uλ × F (the map ψλ is usually called a local chart of Y over Uλ);

(4) for any λ, µ ∈ Λ such that Uλ ∩ Uµ = ∅, a continuous transition function αλ,µ :Uλ ∩Uµ −→ G such that ψλψ−1

µ (x, t) = (x,αλ,µ(x) · t) for any x ∈ Uλ ∩Uµ and t ∈ F .

In particular, the bundle will be called: (a) vector bundle if F is a real or complex euclideanspace (for example, real vector bundle of rank n if the fiber is R

n) and G is the generallinear group (or a subgroup of it) of the same euclidean space; (b) sphere bundle if F isa sphere in an euclidean space and G is the orthogonal group (or a subgroup of it) of thesame euclidean space; (c) principal bundle if F is the same group G operating on itself

(39)Recall that a (left) action of a group G on a topological space F is a morphism from G to the groupof autohomeomorphisms of F : in other words, the identity element of G acts as the identity of F , andg1(g2(f)) = (g1g2)(f) for any g1, g2 ∈ G and f ∈ F . The action is called effective if the only element of Gwhich operates trivially on F is the identity.



by right translation. In the case of particular structures on Y , X, F and G (e.g. if theyare real or complex manifolds, or algebraic varieties) it is usual to require more regularitythan continuity to the trivializations and to the transition functions, and so one can alsotalk about continuous, differentiable, holomorphic, algebraic ... bundles.

Examples. (1) If H ⊂ G are closed subgroups of GL(n;C) and π : G −→ G/H is the canonical projection,

then (G,π) is a bundle with fiber H and structure group N (H)/H, where N (H) is the normalizer of H

in G (see for example Bredon [2, II.14, pp. 110-111]). (2) Given a real manifold M of class C1 and

dimension n, the tangent bundle TM = (x, v) : x ∈ M, v ∈ TxM and the cotangent bundle T∗M =

(x,α) : x ∈ M, α ∈ T∗xM are real vector bundles on M of rank n; if N ⊂ M is a submanifold

of dimension k, there are real vector bundles on N of rank n− k by considering the normal bundle

TNM = (x, v) : x ∈ N, v ∈ TxM/TxN and the conormal bundle T∗NM = (x,α) : x ∈ N, α ∈ (TxN)⊥ ⊂

T∗xM. Note that the tangent bundle TS

1 is trivial, being homeomorphic to S1 × R by S

1 × R∼−→ TS

1,

((x, y), t) → ((x, y), (−ty, tx)).

From now on we shall assume that the topological space X is arcwise connected.

Definition 1.6.6. A covering space of X is a bundle (Y,π) on X with discrete fiber. Inthis case one also says that π : Y −→ X is a covering map. The cardinality of the fibers(well-defined thanks to Proposition 1.6.4) is called multiplicity of the covering (if suchmultiplicity is finite, say n, one also talks about a “n-sheet covering”). A morphism ofcovering spaces on X is a morphism as spaces on X.

In other words, the fact that π : Y −→ X is a covering mapmeans that for any x ∈ X there exists a neighborhood U ⊂ X ofx such that π−1(U) is the disjoint union of homeomorphic copiesof U , i.e. π−1(U) =

λ∈Λ Vλ, with π|

Vλ: Vλ

∼−→ U : such an opensubset U is said to be evenly covered. It is then clear that, if(Y,π) is a covering space of X and U ⊂ X is open, then Y |

U(see

Definition 1.6.3) is a covering space of U .

Proposition 1.6.7. Any covering space is a local homeomorphism.(40) Conversely, a localhomeomorphism (Y,π) where Y is Hausdorff and whose fibers are finite sets with the samecardinality is a covering.

Proof. The first statement follows immediately from the definitions and Proposition 1.6.4. For the second,let k be the cardinal of the fibers; fixed x0 ∈ X, let π−1(x0) = y1, . . . , yk. Thanks to the hypotheses, wemay choose by recurrence some neighborhoods Vyj ⊂ Y of yj such that π|

Vyj: Vyj

∼−→ Uyj = π(Vyj ) and

Vyj ⊂ Y \

j−1

i=1Vyi . So set U =

k

j=1Uyj ⊂ X and Vj = π

−1(U)∩Vyj : it clearly holds π−1(U) =

k

j=1Vj ,

and π|Vj

: Vj −→ U is a homeomorphism.

Examples. (1) Let p(z) =

n

j=0ajz

j be any polynomial with complex coefficients and an = 0, and let

Γ = p(p−1(0)) (the set of critical values of p). Then, setting X = C\Γ and Y = p−1(X), the space (Y, p) is

a n-sheet covering space of X. For example, if p(z) = zn the open subset which are evenly covered are those

U ⊂ X = C× such that the inclusion map j : U → X is nullhomotopic, i.e. those which do not contain

(40)A space (Y,π) on X is called local homeomorphism if π is open and if for any y ∈ Y there exists anopen neighborhood V ⊂ Y of y such that π|

V: V −→ π(V ) is a homeomorphism.



loops of nonzero index in 0.(41) Among them we find all simply connected open subsets of X, for examplealso the “tickened spiral” z = re

iθ ∈ X : θ ∈ R>0, θ − φ(θ) < r < θ + φ(θ) where φ : R>0 −→]0, 1[ is anystrictly increasing continuous function with φ(θ) < θ

2, (limθ−→0+ φ(θ) = 0+ and) limθ−→+∞ φ(θ) = 1−, or

also the open subsets U of X contained in Uα = C \ reiα : r ≥ 0 ⊂ X for some α ∈ R. (2) The mapexp : C −→ C

× is a covering of X of countable multiplicity: for any α ∈ R, the already known open subsetsUα = C \ reiα : r ≥ 0 ⊂ X are evenly covered, being exp−1(Uα) =

k∈Z

z ∈ C : α + 2kπ < Im(z) <

α+ 2(k + 1)π. If one then considers the open subset U = C \ 0, 1 ⊂ X, one has exp−1(U) = C \ 2πiZ:hence exp : C \ 2πiZ −→ C \ 0, 1, as the restriction of a covering space, is itself a covering space. (3) Theabove covering spaces of C× induce covering spaces of S1, which are respectively the maps z

n : S1 −→ S1

(n-fold) and : R −→ S1, (t) = e

2πit (countable). In fact, we shall show that these examples exhaust(up to isomorphism) all connected covering spaces of S

1. Given z0 ∈ S1, the open subset S

1 \ z0 isevenly covered by these covering spaces: for example, if z0 = 1 then

−1(S1 \ 1) =

k∈Z]k, k + 1[, and

(zn)−1(S1 \ 1) =

n−1

k=0eiθ : 2kπ

n< θ <

2(k+1)π

n.

Figure 9: Connected covering spaces of the circle.

(4) Let H be a discrete topological group operating on the left on a topological space Y in a properly

discontinuous way (i.e., for any y ∈ Y there exists a open neighborhood V ⊂ Y of y such that g1V ∩g2V = ∅

if g1 = g2). Let X be the space of orbits of H in Y , endowed with the quotient topology: then the canonical

projection π : Y −→ X is a covering space. Namely, given y ∈ Y let V be an open neighborhood of y with

the properties of discontinuity just defined, and let U = π(V ) (an open neighborhood of π(y), because

π is open): one has π−1(U) =

g∈H

gV , and π|gV

: gV −→ U is a homeomorphism.(42) For example,

let Y = G be a topological group and be H a discrete subgroup operating by multiplication on the left:

such action is properly discontinuous, and the projection π : Y = G −→ X = G/H is a covering.(43) In

the case (2), we had Y = C and H = 2πiZ operating by translation; in the case (3), we had Y = R

and H = Z. (5) Setting Y =]0, 2[ and X = S1, the map π = |

]0,2[: Y −→ X, π(t) = e

2πit is a local

(41)Suppose that U ⊂ X is an open subset containing a loop γ (say based at a point z) of index nonzeroin 0. We shall show that, for w ∈ p

−1(z) (i.e. wn = z), there exists a unique “lifting” of γ based at w,i.e. a path δ completely contained in the inverse image V = p

−1(U) in Y = C such that δ(0) = w andp δ = γ (here the computation can be performed also explicitly: if δ(t) = r(t)eiθ(t) and γ(t) = ρ(t)eiϕ(t)

with γ(0) = γ(1) = z, from p δ = γ one gets r(t) = n

ρ(t) and θ(t) = ϕ(t)+2kπ

nfor some 0 ≤ k ≤ n− 1,

and the good k can be found by requiring that δ(0) = w), whose extremity is another w in the inverse

image of z certainly different from w (namely, if w = w then δ should be nullhomotopic because C issimply connected, and hence also γ = p δ would be nullhomotopic): but then V could not be a disjointunion of copies homeomorphic to U by p, hence U is not evenly covered. Conversely, if U is not evenlycovered there exist a point z in U , two distinct points w and w

in the inverse image p−1(z) of z and a

path α from w to w completely contained in V . Now, p α is surely a loop in U based at z; on the other

hand, if ψ is the shortest path from w to w along the circle containing both of them, it is clear that α

and ψ are paths homotopic rel ∂I (because C is simply connected) hence also p α and p ψ are loopshomotopic rel ∂I: but p ψ is the loop based at z which describes the circle one or more times, hence itsindex in 0 is nonzero, and hence also the index in 0 of p α is nonzero.(42)It is clearly continuous, open and surjective; it is also injective, because from π(gy1) = π(gy2) one getsgy2 = hgy1 for some h ∈ H, hence gU ∩ hgU = ∅, hence g = hg, i.e. h = e and gy1 = gy2.(43)This fact will explain in a more general framework the properties of the maps of canonical projection(see §1.4) of lifting uniquely paths and homotopies. In that case we were considering the right classes (i.e.G/H = gH : g ∈ G), hence H was acting on G on the right instead than on the left.



homeomorphism with discrete fibers, but it is not a covering (the cardinality of the fibers is not the same

for any point). The same conclusion holds with Y = R>0 (no neighborhood of 1 is evenly covered). (6)

Setting Y = (z, w) ∈ C2 : z = w

2 and V = (z, w) ∈ C2 : z = w

2, z = 0 (open subset of Y ), the first

projection p1 : Y −→ C ((z, w) → z) is not a covering space, while p1 : V −→ C× is: namely in the first case

the map is even not a local homeomorphism, while in the second there is a local homeomorphism with

fibers finite and of the same cardinality.(44)

1.6.2 Liftings and the Monodromy lemma

A crucial feature of covering maps π : Y −→ X is that they are able to lift maps from thebase X to the full space Y in a unique way.

Let X be an arcwise connected topological space.

Definition 1.6.8. Let (Y,π) be a space on X, f : Z −→ X a continuous function. Alifting of f by π is a continuous function f : Z −→ Y such that f = π f . In particular, ifZ = A ⊂ X and f = ιA is the canonical inclusion, a lifting of ιA is called a (continuous)section of π over A.

Y

π

Z

f

f

X

Example. If X is a differential manifold and U ⊂ X is open, a section of the tangent bundle on U is a

vector field in U .

Proposition 1.6.9. If π : Y −→ X is a local homeomorphism, two liftings of f : Z −→X which coincide in one point, coincide in a whole neighborhood of the point itself. Ifmoreover Z is connected and π is a covering space, or if Z is connected and Y is Hausdorff,then they are equal.

Proof. If f1(z0) = f2(z0) = y0 , and if V ⊂ Y is a neighborhood of y0 on which π : V∼−→ U = π(V ) is a

homeomorphism, then f1 and f2 must necessarily coincide on the neighborhood W = f−1(U) ∩ f

−1

1(V ) ∩

f−1

2(V ) of z0 . This says that Z

= z ∈ Z : f1(z) = f2(z) is an open subset of Z. If π is a covering space,or if Y is of Hausdorff, Z is also a closed subset of Z (in the first case, if z ∈ Z \ Z

let U ⊂ X be anevenly covered neighborhood of f(z): then π

−1(U) =

λ∈ΛVλ with f1(z) ∈ Vλ1 and f2(z) ∈ Vλ2 (where

λ1 = λ2), so that z ∈ f−1

1(Vλ1) ∩ f

−1

2(Vλ2) ⊂ Z \ Z

, i.e. Z \ Z is open; in the second see Lemma 1.2.2),

and this implies that f1 = f2 because Z is connected.

Definition 1.6.10. The space (Y,π) on X has the property of lifting paths (uniquely) iffor any path γ : I −→ X and any initial point y

0∈ π−1(γ(0)) there exists a (unique) path

(44)π1 is not a local homeomorphism in (0, 0); while it is in (z0, w0) ∈ V , by taking as neighborhood a

small open ball not containing (0, 0).



γy0: I −→ Y such that γ = π γy

0and γy

0(0) = y

0; if Z is a topological space, we say

that (Y,π) has the property of lifting homotopies (uniquely) with respect to Z if for anyhomotopy h : Z × I −→ X and any lifting α0 : Z −→ Y of the base h0 (i.e., π α0 = h

0)

there exists a (unique) homotopy hα0: Z × I −→ Y such that h = π hα0

and (hα0)0 = α0.

Proposition 1.6.11. Let (Y,π) be a space on X which lifts paths uniquely, and let γ,φ :I −→ X be two paths with x

0= γ(0) and x1 = γ(1) = φ(0). Then, given y

0∈ Yx

0and set

y1 = γy0(1) ∈ Yx1

, it holds (γ · φ)y0

= γy0· φy1

.

Proof. Obvious.

Lemma 1.6.12. The covering spaces have the property of lifting paths uniquely.

Proof. Uniqueness is given by Proposition 1.6.9; as for the existence, let π : Y −→ X be the covering space,γ : I −→ X a path in X with γ(0) = x0 , and let y0 ∈ π

−1(x0). Let us define the lifting γy0piecewise.

There exist 0 = t0 < t1 < · · · < tm−1 < tm = 1 and evenly covered open subsets Uj ⊂ X such thatγ([tj−1, tj ]) ⊂ Uj (where j = 1, . . . ,m). Let s1 : U1

∼−→ V1 be the section of π over U1 with y0 ∈ V1 (i.e.

s1 =π|

V1

−1

), and set γy0

[t0,t1]

= s1 γ|[t0,t1]

; then, constructed γy0

[tj−1,tj ]

, let sj+1 : Uj+1

∼−→ Vj+1

be the section of π over Uj+1 with γy0(tj) ∈ Vj+1 and set γy0

[tj ,tj+1]

= sj+1 γ|[tj ,tj+1]

. The path γy0

obtained joining the paths γy0

[tj−1,tj ]

for j = 1, . . . ,m will be continuous by the Gluing lemma.

Actually, a local homeomorphism lifting paths uniquely does much more:

Lemma 1.6.13. Any local homeomorphism which has the property of lifting paths uniquelyhas also the property of lifting homotopies uniquely.

Proof. Let π : Y −→ X be a local homeomorphism with the property of lifting paths uniquely, andlet h : Z × I −→ X be a homotopy and α0 : Z −→ Y with π α0 = h0 . For any z ∈ Z, the pathγz : I −→ X, γz(t) = h(z, t) lifts uniquely to γ

z : I −→ Y such that γz(0) = α0(z): this leads necessarily

to define h : Z × I −→ Y as h(z, t) = γz(t). We are left with showing the continuity of h. Now, since

π is a local homeomorphism, any continuous function with values in X admits locally liftings aroundany point: given (z0 , s) ∈ Z × I, let Vs ⊂ Y be an open neighborhood of h(z0 , s) = γ

z0 (s) such thatπ|

Vs: Vs

∼−→ Us = π(Vs), let Ws × Js be a neighborhood of (z0 , s) such that h(Ws × Js) ⊂ Us, and

define hs = (π|Vs)−1 h|

Ws×Js: Ws × Js −→ Vs. Observe that hs(z0 , s) = γ

z0 (s) = h(z0 , s): this

implies that hs(z0 , t) = h(z0 , t) for any t ∈ Js (the paths hs(z0 , · ) and h(z0 , · ) on Js are both liftings ofh(z0 , · ) and coincide for t = s). Since z0 × I is compact, there exist 0 < s1 < · · · < sr < 1 such thatz0× I ⊂

r

j=1(Wsj × Jsj ). As we saw above, the functions hsj coincide on z0× I: hence the sections

(π|Vsj

)−1 of π must coincide on the connected compact subset h(z0×I) ⊂ X. But, since by Proposition

1.6.9 two sections of a local homeomorphism which coincide on a connected compact subspace coincide ona whole open neighborhood of the compact itself, there exists an open neighborhood W0 ⊂

r

j=1Wsj of

z0 such that hsj and hsj+1 coincide on W0 × (Jsj ∩ Jsj+1) (where j = 1, . . . , r − 1) giving rise in this way

to a continuous function h : W0 × I −→ Y ; but it will hold also h

= h

W0×I

(again by the uniqueness of

lifting of paths defined by fixing repeatedly a z ∈ W0), and hence h is continuous in all of W0 × I.

Hence we get the fundamental property of covering spaces:

Proposition 1.6.14. The covering spaces have the property of lifting homotopies uniquely.

Proof. Follows from Lemmas 1.6.12 and 1.6.13.

As a consequence, homotopic paths are lifted to paths ending at the same point, and evenhomotopic (see Figure 10):



Corollary 1.6.15. (Monodromy lemma) Let π : Y −→ X be a covering space, α and β betwo paths in X with [α] = [β]. Then, if α and β are liftings of α and β with α(0) = β(0),it holds also α(1) = β(1) and, even, [α] = [β].

Proof. Let h : I×I −→ X be a homotopy rel ∂I between α and β: by Proposition 1.6.14 (with Z = I in thesecond factor), there exists a unique homotopy h : I×I −→ Y such that h = π h and h(0, τ) ≡ y0 = α(0) =β(0) for any τ ∈ I. By the property of lifting paths uniquely, one gets α(t) = h(t, 0) and β(t) = h(t, 1).Now, the map h(1, · ) : I −→ Y is continuous; since h = π h, that map must take values in the (discrete)fiber of π on α(1) = β(1): hence it is constant, and in particular α(1) = h(1, 0) = h(1, 1) = β(1).

Figure 10: The Monodromy lemma.

From the Monodromy lemma it follows that the only connected covering of a simplyconnected space is, up to homeomorphisms, the space itself:

Corollary 1.6.16. Let X be a simply connected topological space, π : Y −→ X a coveringwith Y arcwise connected. Then π is a homeomorphism.

Proof. We already know that π is a local homeomorphism: it is enough to prove that π is injective. Solet y1, y2 ∈ Yx0

, and let γ : I −→ Y be a path from y1 to y2. The path π γ is a loop based at x0 , hence

[π γ] = [cx0] by hypothesis. By the Monodromy lemma (with α = π γ, β = cx0

, α = γ and β = cy1) weget y2 = γ(1) = cy1(1) = y1.

1.6.3 Classification of covering spaces

LetX be an arcwise connected topological space and x0∈ X. Let us see how the subgroups

of π1(X,x0) are in corrispondence with the covering spaces of X.

Proposition 1.6.17. Let π : Y −→ X be a covering space, and y0∈ Yx

0. Then the

morphism π# : π1(Y, y0) −→ π1(X,x

0) is injective.

Proof. Let π#([γ]) = [π γ] = [cx0]: observing that (π γ)

y0= γ and (cx0

)y0

= cy0, by the Monodromy

lemma one has [γ] = [cy0 ].



Definition 1.6.18. We denote by G(Y, y0) (characteristic subgroup of the covering space)

the isomorphic image of π1(Y, y0) in π1(X,x

0) by π#:

G(Y, y0) = π#(π1(Y, y0

)) = [γ] ∈ π1(X,x0) : γy

0is a loop based at y

0.

Proposition 1.6.19. Let π : Y −→ X be a covering space, and y0∈ Yx

0. Then the

subgroups of π1(X,x0) conjugated to G(Y, y

0) are exactly the subgroups G(Y, y1) with y1 ∈

Yx0in the same arc-component of y

0in Y .

Proof. Exercise.

Remark 1.6.20. (Monodromy action) The group π1(X,x0) acts on the right on the fiber

Yx0: in other words, there is a monodromy morphism µ : π1(X,x

0) −→ SYx

0

, where SYx0

denotes the group of permutations of the fiber Yx0. This action is described as follows:

given y ∈ Yx0and γ a loop in X based at x

0, let γy be the lifting of γ with starting point

y, and define µ([γ])(y) = y · [γ] := γy(1). Hence the stabilizer of some y ∈ Y is preciselyG(Y, y). Moreover, if the covering space Y is arcwise connected then by Proposition1.6.19 the subgroup acting trivially on Yx

0is the heart(45) of G(Y, y

0) in π1(X,x

0), for any

y0∈ Yx

0.

Examples. (1) The fiber of the covering space exp : C −→ C× over z0 = re

iθ is the set of complexlogarithms wk = log r+ i(θ+2kπ) for k ∈ Z, and the generator re2πit of π1(C

×, z0) sends wk to wk+1. (2)

Let us consider the following 3-sheet covering spaces of S1:

p1−→ p2−→ p3−→

where p1 : S1 −→ S1, p1(z) = z

3; p2 : S1 S1 −→ S

1, p2(z) = z2 or p2(z) = z according to the fact that z

belongs to the first or to the second copy of S1; and p3 : S1 S1 S

1 −→ S1, p3(z) = z. Then, denoting the

fiber of pj always by y1, y2, y3 (where y1 is the external one, y2 the middle one and y3 the internal one),

the action of a generator of π1(S1) on the fiber is (in the standard notation of S3) the cyclic permutation

(1 2 3) for p1, the transposition (2 3) for p2, and the identity for p3.

Lemma 1.6.21. Let π : Y −→ X be a covering space, α and β two paths in X from x0

to x1, and let y0∈ Yx

0. Then αy

0and βy

0have the same ending point if and only if

[α · β−1] ∈ G(Y, y0).

Proof. Exercise (apply Proposition 1.6.11).

We saw that every covering space π : Y −→ X has the property of lifting paths uniquely:given a path (continuous function) f : I −→ X and a starting point in the covering space(i.e. a point y

0∈ Y in the fiber of x

0= f(0)), there exists a unique path (continuous

function) f : I −→ Y such that π f = f and f(0) = y0. If we aim to replace (I, 0)

(45)If G is a group and H is a subgroup of G, the heart of H is the largest normal subgroup of G containedin H: hence it is

g∈G

gHg−1. Dually, the smallest normal subgroup of G containing H is the normal

subgroup generated by the subset

g∈GgHg

−1. The latter shoul not be confused with the normalizer

NH = g ∈ G : gHg−1 = H, which is the largest subgroup of G containing H as a normal subgroup.

Hence, H is normal in G if and only if the heart of H and the normal subgroup generated by H coincidewith H, and NH = G.



by any pointed topological space (Z, z0) the solution of the same problem depends on

the properties of the function f to be lifted and also of the space Z, for which we mustintroduce a topological notion.

Definition 1.6.22. A topological space is said to be locally (arcwise) connected if anypoint has a basis of open and (arcwise) connected neighborhoods.

Example. The “comb space” (Figure 2(a)) is arcwise connected but not locally arcwise connected.

Proposition 1.6.23. (Lifting criterion) Let X, Y and Z be topological spaces with Zarcwise connected and locally arcwise connected, π : (Y, y

0) −→ (X,x

0) a covering space

and f : (Z, z0) −→ (X,x

0) a continuous function. Then f admits a unique lifting f :

(Z, z0) −→ (Y, y

0) if and only if f#(π1(Z, z0)) ⊂ G(Y, y

0).

Proof. Necessity is an immediate consequence of the functoriality of π1 (exercise); let us see now thesufficience. The uniqueness of f comes from Proposition 1.6.9. As for the existence, given any z ∈ Z letus choose a path α : I −→ Z from z0 to z: its image f α : I −→ X is a path from x0 to f(z), which lifts to

a unique path (f α)y0. The definition f(z) = (f α)

y0(1) is well-posed: if β is another path in Z from

z0 to z, then f α and f β are two paths in X from x0 to f(z), and their liftings from y0 have the sameendpoint if and only if (by Lemma 1.6.21) [(f α) · (f β)−1] = [f (α · β−1)] = f#([α · β−1]) ∈ G(Y, y0),a fact ensured by the hypotheses. We are left with showing the continuity of f . Let z ∈ Z and V ⊂ Y

be an open neighborhood of f(z): we may assume that the open U = π(V ) ⊂ X is evenly covered. LetW ⊂ f

−1(U) be open, arcwise connected and containing z (such a W exists since Z is locally arcwiseconnected): let us prove that f(W ) ⊂ V . Let ζ ∈ W , and let βζ be a path in W from z to ζ. We have

f(ζ) = ((f (α · βζ))y0(1) = (f α)

y0· (f βζ)f(z)(1): now, (f βζ)f(z) is a path from f(z) ∈ V which

lifts f βζ (path in U), and hence also the endpoint (f βζ)f(z)(1) is in V (recall that U = π(V ) is evenlycovered, and V is one of the disjoint sheets above U).

Figure 11: A non locally arcwise connected space for which the Lifting criterion does not work.

Remark 1.6.24. The hypothesis of locally arcwise connectedness cannot be dropped inthe proof of the Lifting criterion (Proposition 1.6.23). For example, let Z be the “quasi-circle” of Figure 11 (starting with a vertical straight segment which will be later approachedby a part of type “sin 1

x”), and let f : Z −→ S

1 be a quotient map which collapses all thepoints of Z contained in the dashed box into the point 1 of S1. If we consider the usualexponential covering : R −→ S

1 given by (t) = e2πit, since the fundamental group ofZ is trivial(46) the hypothesis on the characteristic subgroup is satisfied; however, if welift f by to f : Z −→ R with base point 0 in R, then the upper point of the straight

(46)Namely, given a path α : I −→ Z, the support α(I) ⊂ Z is compact and hence it cannot collapse on thevertical straight segment: then α is nullhomotopic, in other words Z is simply connected.



segment goes to 0 while the part of type “sin 1

x” is sent to 1, but this shows that f is not

continuous.

We now show that, in the hypothesis of local arcwise connectedness, a covering space ofX is covered by any other covering space of X having a smaller characteristic subgroup.

Proposition 1.6.25. Let π : (Y, y0) −→ (X,x

0) and p : (Z, z

0) −→ (X,x

0) be two covering

spaces of X with Z arcwise connected and locally arcwise connected. Then there existsa morphism of covering spaces ϕ : ((Z, z

0), p) −→ ((Y, y

0),π) if and only if G(Z, z

0) ⊂

G(Y, y0), and in such a case ϕ itself is a covering space.

Proof. The first statement follows from the Lifting criterion (Proposition 1.6.23); we are left with provingthat ϕ itself is a covering space. Given y ∈ Y , let β be a path in Y from y0 to y, and consider the path

π β in X from x0 to π(y), which lifts uniquely to α = (π β)z0

in Z: since π (ϕ α) = p α = π β,

using the Monodromy lemma one has ϕ(α(1)) = β(1) = y. Hence ϕ is surjective. Let Vy0⊂ Y be

an open neighborhood of y0 such that U = π(Vy0) is an arcwise connected open neighborhood of x0

evenly covered both for p and for π: one then has π−1(U) =

y∈π−1(x0 )

Vy (with π|Vy

: Vy

∼−→ U for

any y ∈ π−1(x0)) and p

−1(U) =

z∈p−1(x0 )Wz (with p|

Wz: Wz

∼−→ U for any z ∈ p−1(x0)). One

then shows that ϕ−1(Vy0

) =

z∈ϕ−1(y0 )Wz (note that ϕ

−1(y0) ⊂ p−1(x0)): namely, if z ∈ ϕ

−1(y0) then

ϕ|Wz

: Wz

∼−→ Vy0,(47) and hence ζ ∈ Wz for a certain z ∈ ϕ

−1(y0) if and only if ϕ(ζ) ∈ Vy0, i.e. if and

only if ζ ∈ ϕ−1(Vy0

).

Corollary 1.6.26. (Uniqueness theorem) Two arcwise connected and locally arcwise con-nected covering spaces of a (connected and locally arcwise connected) topological space areisomorphic if and only if they have the same characteristic subgroup.(48)

The statement of a Existence theorem for a covering space with prescribed characteristicsubgroup requires a slightly stronger topological hypothesis.

Definition 1.6.27. A topological space X is said locally simply connected if any x ∈ Xhas a basis of open simply connected neighborhoods; more generally, X is said semi-locallysimply connected if any x ∈ X admits a open neighborhood U ⊂ X such that any loopin U based at x is nullhomotopic in X (i.e., with homotopies not necessarily with valuesonly in U).

Examples. (1)Obviously, manifolds are locally (hence also semi-locally) simply connected. (2) (Shrinking

wedge of circles) Let C =

n∈NCn, where Cn is the circle of center (−1/n, 0) and radius 1/n: then C is

locally arcwise connected but neither locally nor semi-locally simply connected. The fundamental group

of C turns out to be very complicated. In fact, the topology of C is the one induced from R2, so that

a neighborhood of (0, 0) must contain all but a finite number of the Cn: hence this topology is much

weaker than the wedge sum topology of

NS1. In particular, since the Cn collapse to (0, 0), this allows

also infinite junctions of loops on different Cn to be continuous as loops in C based at 0, and hence to

contribute to the complication of π1(C). In particular, for each sequence (rn) of integers one can construct

(47)From U = p(Wz) = π(ϕ(Wz)) one has that the connected subset ϕ(Wz) is in p−1(U) =

y∈π−1(x0 )

Vy

and contains y0 , hence ϕ(Wz) ⊂ Vy0and it holds even ϕ(Wz) = Vy0

because otherwise p|Wz

= π|Vy0

ϕ|Wz

would not be an isomorphism.(48)It must be noted that we are always working in the framework of pointed topological spaces, so weare keeping track of a base point both in the base space (i.e. a x0), and in the covering (i.e. a point inthe fiber of x0). In the case we do not keep track of base points, the result should be stated as follows:Two arcwise connected and locally arcwise connected covering spaces of a (connected and locally arcwiseconnected) topological space are isomorphic if and only if they have conjugated characteristic subgroups.



Figure 12: The sets C (shrinking wedge of circles), T (a cone of C) and X (union of two copies of T ).

a loop γ(rn) in C winding rk times at each Ck, and these loops are mutually nonhomotopic: this fact

provides a a surjective morphisms π1(C) −→

NZ and so, the direct product

NZ being uncountable,

also π1(C) is uncountable and hence deeply different from π1(

NS1) ∗N Z (which has countably many

generators, and hence is countable). (3) Let T be a cone in R3 with base C (where all the Cn are meant

to be e.g. in the plane (x, y) with center (−1/n, 0, 0)): then T is locally arcwise connected and clearly

contractible, hence simply connected; in particular T is semi-locally simply connected, but not locally

simply connected. (4) Let X be the union of two copies of T at the base point, e.g. X = T ∪ (−T ) where

−T = (x, y, z) ∈ R3 : (−x,−y,−z) ∈ T is the opposite to T (hence T and −T have only the point (0, 0, 0)

in common): then X is connected and locally arcwise connected, but neither simply connected nor semi-

locally (hence, nor locally) simply connected. The argument is as follows: using the notation introduced

above for C, the loops γ(rn) with all but a finite number of the rn equal to zero are nullhomotopic in X

(namely, if N ∈ N is the largest number such that rN = 0, then all extremities (± 2

n, 0, 0) of the loops

which constitute γ(rn) keep being at “security distance” 1

N> 0 from (0, 0, 0), hence they can be sent to

the vertices of ±T for 0 ≤ t ≤ 1

2, and then down to (0, 0, 0) 1

2≤ t ≤ 1 without breaking the continuity of

the homotopy), while the γ(rn) with infinitely many rn different from zero are not.

Proposition 1.6.28. (Existence theorem) Given an arcwise connected, locally arcwiseconnected and semi-locally simply connected topological space (X,x

0) and a subgroup H ⊂

π1(X;x0), there exists a unique (up to a canonical isomorphism) covering space π : Y −→ X

such that G(Y, y0) = H.

Proof. (Sketch) The idea is to consider on the set Ωx0 ,x of paths from x0 to x ∈ X the equivalence relation

given by α ∼ β if [α · β−1] ∈ H, then to define Y =

x∈X

Ωx0 ,x/ ∼

, y0 as the class of cx0

in Ωx0 ,x0and

π : Y −→ X given by [γ] −→ γ(1), then finally to endow Y with a suitable topology using the hypotheses onX. For more details we refer for example to Janich [9, from p. 144].

Examples. (1) The subgroups of π1(S1) Z are Z itself, nZ (for n ∈ N) and 0: they correspond to the

coverings (S1, id), (S1

, zn) (for n ∈ N) and (R, ) (recall that (t) = exp(2πit)), which therefore represent

—up to isomorphism— all arcwise connected and locally acwise connected covering spaces of S1. (2) As

for the bouquet X = S1 ∨ S

1, the family of subgroups of π1(X) Z ∗ Z is much richer than the one of

Z, and hence the classification of arcwise connected and locally arcwise connected covering spaces of X is

much more interesting (see e.g. [8, §1.3], or the example at p. 41).

Remark 1.6.29. The hypothesis of semi-local simple connectedness is necessary for theproof of Proposition 1.6.28. For example, the above double cone of shrinking wedge ofcircles X = T ∪ (−T ) has been proved to have nontrivial fundamental group, but it is



possible to prove that any arcwise connected covering space of X is necessarily trivial(49):hence the proper subgroups of π1(X;x

0) do not correspond to any covering space of X.

1.6.4 Covering automorphisms

Let X and Y be topological spaces, π : Y −→ X a covering space, and consider the set ofendomorphisms of (Y,π), i.e. End(Y |X) = ϕ : Y −→ Y : π ϕ = π. The subset

Aut(Y |X) = ϕ : Y −→ Y : ϕ homeomorphism, π ϕ = π

(the “covering automorphisms”, or deck transformations) has a natural structure of group,given by the composition.

Examples. (1) The deck transformations of : R −→ S1 (where (t) = e

2πit) are the translations τk : R −→ R

given by τk(t) = t+ k for k ∈ Z, hence Aut(R|S1) Z. (2) The deck transformations of zn : S1 −→ S1 are

the rotations of multiples of 2π/n, hence Aut((S1, z

n)|S1) Z/nZ.

An immediate consequence of Corollary 1.6.26 is the following

Proposition 1.6.30. Let π : Y −→ X be a covering space with X and Y arcwise connectedand locally arcwise connected topological spaces, x

0∈ X, y

0, y1 ∈ Yx

0. Then there exists

ϕ ∈ Aut(Y |X) with ϕ(y0) = y1 if and only if G(Y, y

0) = G(Y, y1).

What does this condition mean? By Proposition 1.6.19 we know that G(Y, y1) is conju-gated to G(Y, y

0) in π1(X,x

0): if γ is a path in Y from y

0to y1, setting α = π γ it holds

G(Y, y1) = [α−1] ·G(Y, y0) · [α]. Hence, the condition G(Y, y

0) = G(Y, y1) is equivalent to

[α] ∈ NG(Y, y

0)(the normalizer of G(Y, y

0) in π1(X,x

0)).

Theorem 1.6.31. Let π : (Y, y0) −→ (X,x

0) be a covering space with X and Y arcwise

connected and locally arcwise connected topological spaces. Then for any [α] ∈ NG(Y, y

0)

there exists one and only one covering automorphism ϕ[α] such that ϕ[α](y0) = αy

0(1).

The application NG(Y, y

0)−→ Aut(Y |X) obtained in this way is a surjective morphism

of groups with kernel G(Y, y0), and provides an isomorphism of groups

NG(Y, y

0)

G(Y, y0)

∼−→ Aut(Y |X).

Proof. Let y1 = αy0(1): then the hypothesis [α] ∈ N

G(Y, y0)

is equivalent to the fact that G(Y, y0) =

G(Y, y1), and the result follows from the Lifting criterion (Proposition 1.6.23). In particular, one constructsexplicitly ϕ[α] ∈ Aut(Y |X) as follows: given y ∈ Y and a path βy from y0 to y, one sets ϕ[α](y) =

(π βy)y1(1) (note that ϕ[α](y0) = cy1(1) = y1, as required).

(49)The idea is that a arcwise connected covering space π : Y −→ X induces on T and −T (which are simplyconnected) trivial covering spaces (±T )×F ; if F would not be a point (i.e., if such induced covering spaceswould not be homeomorphisms), two points of the same fiber would stay in different arcwise connectedcomponents, and that would contradict the fact that π is arcwise connected (just think that X = T ∪(−T ),and that x0 = (0, 0, 0) is the only point in common between T and −T ...); hence F = pt, and sinceX = T ∪ (−T ) this implies that π is a homeomorphism.



Definition 1.6.32. A connected and locally arcwise connected covering space π : (Y, y0) −→

(X,x0) is called normal, or Galois, if G(Y, y

0) is a normal subgroup of π1(X,x

0).

The following proposition shows the properties of normal covering spaces. In particularthey act transitively on the fibers, and hence these covering spaces can be viewed as thosehaving a complete symmetry among different sheets.

Proposition 1.6.33. A connected and locally arcwise connected normal covering spaceπ : (Y, y

0) −→ (X,x

0) has the following properties.

(i) π1(X,x0)/G(Y, y

0)

∼−→ Aut(Y |X).

(ii) Aut(Y |X) operates on the left on Y in a properly discontinuous way (hence freely)(50),and the orbits are the fibers of π: in particular, the multiplicity of the covering spaceis equal to the index of G(Y, y

0) in π1(X,x

0).

(iii) Denoted by Y/Aut(Y |X) the space of orbits of Aut(Y |X) in Y (i.e., the space offibers of π) with the quotient topology, the natural bijection Y/Aut(Y |X) −→ X is ahomeomorphism.

(iv) Either all liftings of loops in X based at x0are loops in Y , or no one of them is.

(Such condition is also sufficient in order that a covering space be Galois.)

Proof. (i) follows from Theorem 1.6.31, as well as the fact that the action on the left of Aut(Y |X) has thefibers of π as orbits. Let U ⊂ X be a evenly covered open neighborhood of x0 : hence one has π

−1(U) =λ∈Λ

Vλ, and let y0 ∈ Vλ0. If ϕ1(Vλ0

) ∩ ϕ2(Vλ0) = ∅, let y1, y2 ∈ Vλ0

be such that ϕ1(y1) = ϕ2(y2) ∈ U ;in particular y1 and y2 belong to the same fiber of π, and also to the same sheet Vλ0

: hence y1 = y2 ≡ y,which implies ϕ1 = ϕ2 because they coincide in the point y (uniqueness of lifting). This shows (ii). Thebijection in (iii) is continuous by definition of quotient topology, and is open because such is also π. Finally,to be Galois is equivalent to the fact that G(Y, y1) = G(Y, y2) for any y1 and y2 in the same fiber of π,and this is equivalent to (iv).

Remark 1.6.34. From Remark 1.6.20 it follows immediately that, if the covering spaceπ : (Y, y

0) −→ (X,x

0) is Galois, the subgroup of π1(X,x

0) acting trivially on the fiber Yx

0

is G(Y, y0).

Example. If π1(X,x0) is commutative, then obviously all arcwise connected and locally arcwise connectedcovering spaces of X are normal. On the other hand, let us show an example of connected and locallyarcwise connected covering space which is not normal. Consider the function

X =

Y =

x0

y2y0y1

π

β α

(50)Recall that the action of a group G on a set Z si said properly discontinuous if any z ∈ Z has aneighborhood U ⊂ Z such that g1U ∩ g2U = ∅ if g1 = g2, and free (a weaker notion) if any point hastrivial stabilizer, i.e. Gz = 1 for any z ∈ Z.



(We mean that π(yj) = x0 (j = 0, 1, 2) and that the arcs of Y denoted by −→ (resp. by ) are sent intoα (resp. into β) in the specified direction. Note that π is a surjective local homeomorphism with fiber ofcardinality 3: by Proposition 1.6.7, π is a 3-sheet covering space. Consider the morphism of pointed spacesπ : (Y, y0) −→ (X,x0), and the injective morphism π# : π1(Y, y0) −→ π1(X,x0). The space X is the bouquetof two circles and hence π1(X,x0) is free on two generators [α] and [β]; on the other hand, π1(Y, y0) isgenerated by the classes of loops

y0y0y0y0

γ4 =γ3 =γ2 =γ1 =

and one has π#([γ1]) = [α · β · α], π#([γ2]) = [α2], π#([γ3]) = [β2], π#([γ4]) = [β · α · β]: hence the

characteristic subgroup G(Y, y0) is generated by [α · β · α], [α2], [β2] and [β · α · β]. Would there exist a

covering automorphism sending y0 for example into y1, the liftings from y0 and y1 of the same loop based at

x0 should be either both loops or no one of them: actually αy0and αy1 are not loops (the first one goes from

y0 to y1, the second from y1 to y0), but βy0is not a loop (goes from y0 to y2) while βy1 is. Therefore one

has Aut(Y |X) = idY , which implies NG(Y, y0)

= G(Y, y0) (hence the covering space is not normal).

Hence G(Y, y1) = [α−1] · G(Y, y0) · [α] = G(Y, y0) and G(Y, y2) = [β−1] · G(Y, y0) · [β] = G(Y, y0). The

heart

j=0,1,2G(Y, yj) is the subgroup of π1(X,x0) formed by the loops whose liftings from the yj ’s are

all loops: it is generated by [α2] and [β2]. The action of π1(X,x0) on π−1(x0) = y0 , y1, y2 is given by

[α] = (0 1) and [β] = (0 2): hence the morphism π1(X,x0) −→ S3 is surjective.

A particular case of Galois covering space is, if it exists, to one with G(Y, y0) = 1.

Definition 1.6.35. A connected and locally arcwise connected covering space π : Y −→ Xis called universal cover of X if Y is simply connected.

Theorem 1.6.36. If X is a connected, locally arcwise connected and semi-locally sim-ply connected topological space, there exists a unique —up to canonical isomorphisms—universal cover π : (X, x

0) −→ (X,x

0), with the following properties.

(i) π1(X,x0) Aut(X|X).

(ii) Aut(X|X) operates on the left on X in a properly discontinuous way (hence freely),and the orbits are the fibers of π.

(iii) if π : (Y, y0) −→ (X,x

0) is another connected and locally arcwise connected covering

space, there exists one and only one covering space πY : (X, x0) −→ (Y, y

0) such that

π = π πY . (Hence, the universal cover of X is also the universal cover of any otherconnected and locally arcwise connected covering space of X.)

(iv) The universal cover determines all other arcwise connected and locally arcwise con-nected covering spaces of X, in the following sense: if Γ ⊂ Aut(X|X) is a subgroup,denoted by X/Γ the space of orbits of Γ in X endowed with the quotient topology, thenatural map πΓ : (X/Γ, [x

0]) −→ (X,x

0) is a connected and locally arcwise connected

covering of X, and moreover all arcwise connected and locally arcwise connectedcovering spaces of X are obtained in this way, up to a canonical isomorphism.

Proof. Existence and uniqueness follow from Proposition 1.6.28 and Corollary 1.6.26. (i) and (ii) followfrom Proposition 1.6.33, (iii) from Proposition 1.6.25, (iv) follows from (iii) and Proposition 1.6.33 (whichsays that Y X/Aut(X|Y ), and Γ = Aut(X|Y ) ⊂ Aut(X|X)).



In particular, it is useful to remark that (i) provides a method —often the preferable one—for computing the fundamental group of X:

π1(X,x0) Aut(X|X).

Some examples of this method will be shown in the next Section.



1.7 Exercises and complements

(1) The universal covering of S1 is the exponential map : R −→ S1 ⊂ C, (t) = e2πit;

more generally, the universal cover of the torus Tn = (S1)n is the map : Rn −→ T

n,(t1, . . . , tn) = (e2πit1 , . . . , e2πitn), and it holds Aut(Rn|Tn) = τ : Rn −→ R

n, τ(t1, . . . , tn) =(t1 + k1, . . . , tn + kn) for some kj ∈ Z Z

n, which yields again π1(Tn) Zn.

(2) The real space projective Pn (with n ≥ 1) is defined as the space of orbits of the

multiplicative group R× in (Rn+1)× (i.e., the family of homogeneous lines of Rn+1), en-

dowed with the quotient topology. Consider the Hopf map q : Sn −→ Pn (the restriction

to Sn of the projection (Rn+1)× −→ P

n), which is a 2-sheet covering of Pn. If n ≥ 2

this map is the universal cover; since the covering automorphisms of q are ± idSn , weget π1(Pn) Aut(Sn|Pn) = Z/2Z. On the other hand, for n = 1 we saw that the map( · )2 : S1 −→ S

1 has the same fibers of q, hence there exists a homeomorphism γ : S1∼−→ P

1

such that q = γ ( · )2. Therefore π1(P1) Z.

An interesting application is a particular case of the Borsuk-Ulam theorem (if n ≥ 2, theredoes not exist any continuous functions of Sn into itself which is odd and nullhomotopic):

Corollary 1.7.1. (Borsuk-Ulam, particular case) If n ≥ 2, there does not exist odd con-tinuous functions of Sn with values in S

1.

Proof. By absurd let f : Sn −→ S1 be a odd continuous function, and denote by qj : Sj −→ P

j the Hopfmap. By Proposition 1.1.14, there exists g : Pn −→ P

1 continuous such that g qn = q1 f (note thatq1 f is constant on the fibers of qn). Now, fixed x0 ∈ S

n, the morphism g# : π1(Pn, qn(x0)) = Z/2Z −→

π1(P1, g(qn(x0))) = Z must be zero, hence by the Lifting criterion (Proposition 1.6.23) there exists a unique

lifting g : Pn −→ S1 such that g = q1 g and g(qn(x0)) = f(x0). But one has also f = g qn (namely f and

g qn are two liftings of g qn which coincide in x0): this is a contradiction because f(−x) = −f(x) whileg(qn(−x)) = g(qn(x)) = f(x).

A consequence of Corollary 1.7.1 is, for example, that at any particular time there are twoantipodal places on the Earth with the same temperature and the same pressure. Namelylet, at a certain fixed moment, t : S2 −→ R be the temperature and p : S2 −→ R be thepressure: if the statement would be false, the function ϕ : S2 −→ R

2 given by ϕ(x) =(t(x) − t(−x), p(x) − p(−x)) would never take the value (0, 0) and hence the functionf : S2 −→ S

1 given by f(x) = ϕ(x)/|ϕ(x)| would be continuous and odd.

(3) We now deal with the fundamental group of topological groups.

Proposition 1.7.2. Let G a topological group with identity element 1. Then:

(i) the fundamental group π1(G, 1) is commutative;

(ii) if G is a connected and locally arcwise connected topological group, and π : (E, e) −→(G, 1) is a connected and locally arcwise connected covering, then there exists oneand only one multiplication on E for which (a) E is a topological group with identityelement e, (b) π is a morphism.

Proof. (The student should verify by exercise the unproven statements.) (i) Let µ : G × G −→ G be themultiplication. This map induces a pointwise product between loops based at 1, by defining (α ∗ β)(t) =µ(α(t),β(t)) for any t ∈ I; and this product induces a product ∗ in π1(G, 1). We then have two operations



(∗ and ·) in π1(G, 1). Now, by a elementary algebraic fact, if a set S is endowed with two binary operations∗ and · with a same identity element and such that (a ∗ b) · (c ∗ d) = (a · c) ∗ (b · d) for any a, b, c, d ∈ S,then ∗ and · coincide and are associative and commutative: such condition is verified in our case. (ii) Themap µ (π×π) : (E×E, (e, e)) −→ (G, 1) (given n covering spaces πj : Yj −→ Xj , and set π = π1 × · · ·×πn,Y = Y1 × · · · × Yn and X = X1 × · · · × Xn, also π : Y −→ X is a covering) lifts uniquely to a mapµ : (E × E, (e, e)) −→ (E, e): this follows from Proposition 1.6.23, since (µ (π × π))#([α], [β]) = [µ(π α,π β)] = [π α] ∗ [π β] = [π α] · [π β] = π#([α · β]) ∈ G(E, e). The uniqueness of lifting shows allthe properties which are required to µ to be the desired operation in E: for example, to show that e is theidentity element of µ we define µe : (E, e) −→ (E, e) by µe(y) = µ(y, e): since π µe = π, we get µe = idE

(because idE and µe are two morphisms of the covering space π which coincide in e).

(4) Let us study the covering spaces of manifolds and, in particular, what happens to thefundamental group when we remove, from a given manifold, a closed submanifold whichis “small enough”.

Let M be a (arcwise) connected C0 manifold of dimension m, N ⊂ M a C0 submanifoldof dimension n ≤ m, and let ι : M \N −→ M be the open embedding.

Proposition 1.7.3. If π : P −→ M is a local homeomorphism, then on P is naturallyinduced a structure of C0 manifold of dimension m, and on π−1(N) a structure of sub-manifold of dimension n.

Proof. The local charts on P and π−1(N) are just the local pullbacks of local charts on M and N .

Proposition 1.7.4. The following statements hold.

(i) If m− n ≥ 2, then M \N is connected.

(ii) If m − n ≥ 3, any covering space of M \ N extends to one of M (i.e., given acovering space q : P −→ M \ N there exist a covering space q : P −→ M and ainjective morphism of manifolds ιP : P −→ P such that q ιP = ι q).

Proof. (i) It is clear that Rm \ Rn is connected if and only if m− n ≥ 2. In general, it is enough to showthat, given x, y ∈ M \N , there exists a path in M \N from x to y. Let α : I −→ M be a path from x toy, and (Uλ,ϕλ) : λ ∈ Λ be an atlas of M such that Uλ ∩N = ∅ or ϕλ(Uλ ∩N) = R

n. By compactness,there exist 0 = t0 < t1 < · · · < tr−1 < tr = 1 and λ1, . . . ,λr ∈ Λ such that α([tj−1, tj ]) ⊂ Uλj ,where j = 1, . . . , r. Since any Uλj \ (Uλj ∩ N) is connected, we can construct a sequence of paths γj

such that (1) γj(I) ⊂ Uλj \ (Uλj ∩ N); (2) γ1(0) = x, γj+1(0) = γj(1), γr(1) = y. (In particular, ifα([tj−1, tj ]) ⊂ Uλj \ (Uλj ∩N) we can choose γj = α|

[tj−1,tj ]with a reparametrization taking tj−1 into 0

and tj into 1.) The path γ obtained by joining the paths γj has the required properties: hence M \N isconnected. (ii) For the costruction of the covering space q, obtained through a gluing procedure of coveringspaces on local charts, we refer e.g. to Godbillon [5, X.2].

Theorem 1.7.5. For x0∈ M \N consider the morphism of fundamental groups

ι# : π1(M \N, x0) −→ π1(M,x

0).

Then:

(i) if m− n ≥ 2, then ι# is surjective;

(ii) if m− n ≥ 3, then ι# is a isomorphism.



Proof. (i) It is enough to show that, given x, y ∈ M \ N and a path α : I −→ M from x to y, thereexists a path γ : I −→ M \ N from x to y with [α] = [γ]. Following the proof of the first statement ofProposition 1.7.4, we may also require that (3) γj be homotopic (not necessarily rel ∂I) to α|

[tj ,tj+1]. The

path γ obtained by joining the paths γj has the required properties. (ii) Let H = ker(ι#), and let usprove that H = 1. Consider the connected covering space q : P −→ M \ N having H as characteristic

subgroup.(51) From Proposition 1.7.4 we know that there exist a covering q : P −→ M and an injectivemorphism of manifolds ιP : P −→ P such that qιP = ιq. By definition of H, one has (q# ιP#)(π1(P )) =

(ι# q#)(π1(P )) = ι#(H) = 1, hence (q# being injective and ιP# surjective) it holds π1( P ) = 1: i.e.,P is the universal cover of M . On the other hand, since P \ ιP (P ) is a submanifold of codimension

m− n ≥ 3 of P (namely, from ιP (P ) = q−1(M \N) we get P \ ιP (P ) = q

−1(N), and it is enough to recall

Proposition 1.7.3), the universal cover π : S −→ P extends to π : S −→ P , which ( P being simply connected)is a homeomorphism: hence also π is a homeomorphism, because the fiber has cardinality 1. Therefore P

is simply connected, i.e. H = 1.

(5) We now deal with the fundamental group of some classical real groups. Given n ∈ N,let M(n,K) be the vector space of square matrices of order n with coefficients in K = R,C.If A ∈ M(n,C), we denote by A∗ = tA the adjoint matrix, and:

GL(n,C) = A ∈ M(n,C) : det(A) = 0 (complex general linear group)

U(n,C) = A ∈ GL(n,C) : A−1 = A∗ (unitary group)

SU(n,C) = A ∈ U(n,C) : det(A) = 1 (special unitary group)

GL(n,R) = M(n,R) ∩GL(n,C) (real general linear group)

GL±(n,R) = A ∈ GL(n,R) : det(A) ≷ 0O(n,R) = M(n,R) ∩ U(n,C) (real orthogonal group)

SO(n,R) = M(n,R) ∩ SU(n,C) (real special orthogonal group)

and moreover

H(n,C) = A ∈ M(n,C) : A = A∗ (hermitian matrices)

H+(n,C) = A ∈ H(n,C) : txAx > 0 ∀x ∈ Cn \ 0 (positive definite h. m.)

S(n,R) = M(n,R) ∩H(n,C) (symmetric matrices)

S+(n,R) = M(n,R) ∩H+(n,C) (positive definite s. m.).

We briefly recall the following facts (for further details we refer e.g. to Godbillon [5, II.2]):

(1) M(n,C) (resp. M(n,R)) is a vector space on C (resp. on R) of dimension n2, andH(n,C) (resp. S(n,R)) is a real subspace of M(n,C) (resp. M(n,R)) of dimensionn2 (resp. n(n+1)/2). The application A → A =

tr(AA∗) is a norm on M(n,C),

which induces the norm A =tr(A tA) on M(n,R).

(2) GL(n,C) is an open subset of M(n,C) and a multiplicative topological group.

(3) The exponential exp : M(n,C) −→ GL(n,C) (where exp(A) =∞

n=0An/n!) satisfies

exp(A + B) = exp(A) exp(B) if AB = BA, and is also a diffeomorphism betweenan open neighborhood of 0 ∈ M(n,C) and an open neighborhood of the identity

(51)Such a q exists, because the manifolds —and M \ N is so— are locally simply connected, and inparticular locally arcwise connected and semi-locally simply connected.



1n ∈ GL(n,C): this makes GL(n,C) into a real Lie group(52) of dimension 2n2.Moreover, exp induces a homeomorphism of H(n,C) on H+(n,C) and of S(n,R) onS+(n,R).

(4) U(n,C), SU(n,C), GL(n,R), GL+(n,R), O(n,R) and SO(n,R) are closed Lie sub-groups of GL(n,C) of dimensions n2, n2 − 1, n2, n2, n(n − 1)/2 and n(n − 1)/2.Local charts around the identity 1n of these groups, seen as real submanifolds ofGL(n,C), are obtained by inverting the restriction of exp respectively to the realsubspaces u(n,C) = A ∈ M(n,C) : A + A∗ = 0, su(n,C) = A ∈ M(n,C) :A + A∗ = 0, tr(A) = 0, gl(n,R) = M(n,R) (for GL(n,R) and GL+(n,R)) andso(n,R) = A ∈ M(n,R) : A + tA = 0 (for O(n,R) and SO(n,R))(53). More-over, U(n,C), SU(n,C), O(n,R) and SO(n,R) are compact (if A ∈ U(n,C) thenA =

√n).

(5) There are isomorphisms of Lie groups U(n,C) S1×SU(n,C) andO(n,R) ±1×

SO(n,R), given by A → (det(A), A/det(A)). Moreover, SU(n,C) and SO(n,R) arearcwise connected, and hence U(n,C) is arcwise connected while O(n,R) has twoconnected components.

(6) GL(n,C) is homeomorphic to the product H+(n,C)×U(n,C) (polar decomposition)and in particular, by (3) and (5), GL(n,C) is arcwise connected. A homeomorphismis induced between GL(n,R) and S+(n,C) × O(n,R) and in particular betweenGL+(n,R) and S+(n,C)×SO(n,R), hence GL(n,R) has two connected componentsGL±(n,R).

From (3) one has π1(H+(n,C), 1n) 1 and π1(S+(n,R), 1n) 1: hence from (5) and(6) one has

π1(GL(n,C)) π1(U(n,C)) Z× π1(SU(n,C)),

π1(GL(n,R), 1n) π1(O(n,R), 1n) π1(SO(n,R)).

We are left with computing the fundamental groups of SO(n;R) and SU(n,C).

• It holds SO(1;R) = 1, and SO(2;R) =

cos θ sin θ− sin θ cos θ

: θ ∈ R

S

1. For n = 3,

note that S3 (intended as the group of quaternions of unitary norm, see Example

1.4) operates on R3 by S

3 × R3 (q, u) → quq−1: such trasformation is linear and

preserves the norms, i.e. it is in SO(3;R), and one obtains in this way a morphismS3 −→ SO(3;R), which is surjective and has ±1 as kernel. This shows that SO(3;R)

(52)A real Lie group of dimension m is a topological group with a structure of real C1 manifold of dimensionm which makes the multiplication and the inversion into differentiable maps. By the way, GL(n,C) wouldbe of course also a complex Lie group, but here we are interested only in its structure of real differentiablemanifold.(53)In the terminology of Lie theory, these vector subspaces are the Lie algebras associated to the Liesubgroups: in the ambient vector space M(n,C), they are the tangent space to the Lie subgroups at theidentity 1n (hence, one uses also the notation gl(n,C) = M(n,C)). In general, a Lie algebra on a field K

is a vector space V on K endowed with a internal multiplication [ · , · ] which is bilinear, antisymmetricand satisfying the Jacobi identity [x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0 for any x, y, z ∈ V : in our case, forA,B ∈ M(n,C), it is [A,B] = AB − BA (the commutator of A and B). Note that each one of the realsubspaces of M(n,C) considered here is stable with respect to such operation.



is homeomorphic to P3, and hence π1(SO(3;R)) Z/2Z. In the general case,

note that SO(n;R) is a submanifold of the open subset GL+(n;R) of M(n;R), ofdimension n(n − 1)/2. We embed N = SO(n − 1;R) into M = SO(n;R) as thoseorthogonal transformations that fix the North pole en: note that dimM − dimN =n − 1, hence for n ≥ 4 we have π1(SO(n;R) \ SO(n − 1;R)) π1(SO(n;R)). Nowwe aim to show that SO(n;R) \ SO(n − 1;R) is an open subset of SO(n;R) whichdeformation-retracts to a manifold homeomorphic to SO(n − 1;R), and this willimply that π1(SO(n;R)) Z/2Z for n ≥ 4. Let ρ : SO(n;R) −→ S

n−1 be themap ρ(A) = Aen: setting V = S

n−1 \ en, we have precisely SO(n;R) \ SO(n −1;R) = ρ−1(V ). Now define a function f : V × SO(n − 1;R) −→ ρ−1(V ). Lets : Sn−1\−en −→ SO(n) be the continuous application under which, given x = −en(the South pole), s(x) induces the identity on x, en⊥ and the rotation sending eninto x in the plane x, en. Let a : S

n−1 −→ Sn−1 be the antipodal map: since

a(V ) = Sn−1 \ −en, for x ∈ V the trasformation s(a(x)) is well defined. So let us

set f(x,α) = as(a(x))α (note that f is well defined because f(x,α)(en) = x = en).Well, such f is a homeomorphism: namely it is continuous, and its inverse is givenby g(β) = (β(en), s(a(β(en)))−1 a β). Since V is contractible, we have proven theclaim. Summarizing up, one has

π1(GL(n,R), 1n) π1(O(n,R), 1n) π1(SO(n;R))

1 (n = 1)Z (n = 2)Z/2Z (n ≥ 3)

• As for SU(n,C), let us begin by observing that SU(1,C) = 1 is simply connected.Since dimR(SU(n,C)) = n2 − 1, if n ≥ 2 one has dimR(SU(n,C)) − dimR(SU(n −1,C)) = 2n−1 ≥ 3, and hence π1(SU(n,C)\SU(n−1,C))

∼−→ π1(SU(n,C)). On theother hand, arguing as before one shows that for n ≥ 2 there exists a homeomorphismf : SU(n,C)\SU(n−1,C) −→ V ×SU(n−1,C), where V = S

2n−1\e2n ⊂ Cn R

2n,contractible. This implies that SU(n,C) is simply connected and therefore

π1(SU(n,C)) = 1, π1(GL(n,C)) π1(U(n,C)) Z.


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Notes on Algebraic Topologymaraston/Topologia2/Topo2... · 2010-11-15 · Notes on Algebraic Topology deﬁned(30): this procedure is said to be an amalgamation of the free product