Download Theorem of Van Kampen, covering spaces, examples and

Notes on Algebraic Topology
1.5
Theorem of Van Kampen
The theorem of Van Kampen(28) allows the computation of the fundamental group of a
space 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} an
open 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 (where
λ ≤ µ), and write for short π1 (Uλ ) instead of π1 (Uλ , x0 ). We then have morphisms
ιλ# : π1 (Uλ ) →
− π1 (X) and ιλ,µ# : π1 (Uλ ) →
− π1 (Uµ ) (where λ ≤ µ), which simply
associate to the class of a loop the class of the same loop viewed in the larger space. In
particular, 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 ∈ M }
of I, 0 = t0 < t1 < · · · < tk−1 < tk = 1 with tj − tj−1 < δ and λj ∈ M (with j = 1, . . . , k) such that
γ([tj−1 , tj ]) ⊂ Uλj . Since γ(tj ) ∈ Uλj ∩ Uλj+1 (arcwise connected), let σj (for j = 1, . . . , k − 1) be an arc
into 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
][σk−1 · γk ].
1) it clearly holds [γ] = [γ1 ] · · · · · [γk ] = [γ1 · σ1−1 ] · [σ1 · γ2 · σ2−1 ] · · · · · [σk−2 · γk−1 · σk−1
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 = Sn \ {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 “inductive
limit” of an inductive system).
Theorem 1.5.3. (Van Kampen) In the category Groups it holds
π1 (X) = lim π1 (Uλ ).
−→
λ∈Λ
(28)
The result has been proved independently also by Karl Seifert in the 30s of last century; in fact, it is
often referred to as “Seifert - Van Kampen theorem”.
Corrado Marastoni
22
Notes on Algebraic Topology
Proof. Consider a group L and a family of morphisms ψλ : π1 (Uλ ) −
→ L such that ψλ = ψµ ◦ιλ,µ# for λ ≤ µ,
and let us see if there exists a unique morphism ψ : π1 (X) −
→ L such that ψλ = ψ ◦ιλ# 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 ψ
exists, 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
, kj ] it holds σ(t) = σj (kt−(j −1))), and let h : I ×I −
→ X be a
k
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 kr2 < ε. Hence, setting Ri,j = i−1
, kir × j−1
, kjr ⊂ I × I (for
kr
kr
�
�
j
i, j = 1, . . . , kr ), there exists λi,j ∈ Λ such that h(Ri,j ) ⊂ Uλi,j . Let vi,j = kir , 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)
, kjr
kr
�
�
(path from h(vi,j−1 ) to h(vi,j )): note that
(path from h(vi−1,j ) to h(vi,j )) and βi,j (t) = h kir , t+(j−1)
kr
�
�
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−1paths
��
−1
=
· (γi,j−1 · βi,j ) · γi,j
γi,j and their inverses to base at x0 ) the relations (γi−1,j−1 · αi,j−1 ) · γi,j−1
��
��
�
�
−1
−1
in the group π1 (Uλi,j ). Applying ψλi,j and setting
(γi−1,j−1 · βi−1,j ) · γi−1,j
· (γi−1,j · αi,j ) · γi,j
��
��
��
��
−1
−1
, one then has the equality
and bi,j = ψµ(i,j) (γi,j−1 · βi,j ) · γi,j
ai,j = ψλi,j (γi−1,j · αi,j ) · γi,j
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 π1 (Uλ ) seems like.
−→
λ∈Λ
In general, the free product ∗λ∈Λ Gλ of a family of groups {Gλ : λ ∈ Λ} is the group
formed 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 where
two 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 the
natural juxtaposition (a1 · · · ak ) · (b1 · · · bh ) = a1 · · · ak b1 · · · bh where, in the case ak and
b1 belong to a same group, the expression “ak b1 ” should be replaced by their product in
that group (and possibly removed if ak b1 is the identity, causing then the same procedure
for ak−1 b2 , 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 r1 s1 · · · rk sk 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 ak1 bh1 · · · akr bhr 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∈Λ Gλ ; in fact one sees
that the free product ∗λ∈Λ Gλ is the inductive limit in Groups of the system {Gλ : λ ∈ Λ}
with trivial preorder, i.e. without considering morphisms(29) . More generally, if morphisms
fλ,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 quotient
out the previous free product ∗λ∈Λ Gλ by its normal subgroup N generated by all the
elements 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.
Corrado Marastoni
23
Notes on Algebraic Topology
defined(30) : this procedure is said to be an amalgamation of the free product with respect
to 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 loop
in the small open subset Uλ to the class of same loop viewed in the larger open subset
Uµ : by Proposition 1.5.1 and the subsequent discussion it is then enough to consider the
free product of the groups π1 (Uλ )’s of a selected family of open subsets Uλ with indices
λ ∈ M which cover X and are not contained in each other, and then to amalgamate only
with 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∈M
Xλ
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�=λ Vµ ). It is clear that the Uλ ’s cover X; moreover, any
�
intersection of two or more of them is always λ∈M Vλ , 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 limit
of the system {π1 (U ∩ V ), π1 (U ), π1 (V ); , ιU ∩V,U , , ιU ∩V,V }, and its universal property is
expressed by the following diagram:
(1.4)
������ L
������������������
�
�
�
�
�
�
����� ������ ������
����� ���∃!ψ
�
�
�
�
�
�
�
�
�
�
���
�� ���
π1 (U ) ιU # � π1 (X)
��
�
�
�� �
��
�
��
�
��
�
�
�
�
�� ���� ψU ∩V ��� ψV
ιU ∩V # ��
�
�
ιU ∩V,U #
�
� ���� ιV # ���
�
�
�
�
�
�����
��
����ιU ∩V,V #
��
� π1 (V )
π1 (U ∩ V )
ψU
One usually denotes the free product of two groups G and H by G ∗ H and, given another
group K and morphisms f : K →
− G and g : K →
− H, the free product of G and H
amalgamated 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 ]Uλ )): hence, by the
j
compatibility of the various morphisms of type i# , the class [γj ] can be thought as coming from some
Uλ with λ ∈ M , and this shows that ∗λ∈Λ π1 (Uλ ) � ∗λ∈M π1 (Uλ ). Similar considerations hold for the
amalgamation: the objects iλ,µ# ([γ]) · iλ,ν # ([γ])−1 coming from intersections of three or more Uλ with
λ ∈ 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 morphisms
f :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.
Corrado Marastoni
24
Notes on Algebraic Topology
Corollary 1.5.4. Let X be an arcwise connected topological space, X = U ∪ V an open
cover 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 (by
ιU ∩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 , and
let X = R2 \ {x1 , . . . , xk }. Then π1 (X) is the free group with k generators. Namely for k = 1 we already
know 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 and
U ∩ V is simply connected (even contractible), and so π1 (X) � π1 (U ) ∗ π1 (V ); but U is homotopically
equivalent to S1 , while π1 (V ) is free with k − 1 generators by inductive hypothesis. The same holds for
Y = S2 \ {y1 , . . . , yk+1 } (with {y1 , . . . , yk+1 } distinct points of S2 ).(34) (2) (Bouquet of k circles) The
fundamental group of a “bouquet” of k circles (i.e. the wedge sum of k circles) is again a free group with
k generators: this follows immediately from what has been said in general for wedge sums. Alternatively
one 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 different
from 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 R3 ) Let S1(x,y) = {(x, y, z) ∈ R3 : z = 0, x2 + y 2 = 1}, and let us compute
π1 (X) where X = R3 \ S1(x,y) . Let Rz = {(x, y, z) ∈ R3 : x = y = 0} (the z-axis), and set U = X \ Rz
and V = {(x, y, z) ∈ X : x2 + y 2 < 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 × S1
(exercise) and hence π1 (U ) � π1 (T ) × π1 (S1 ) � Z × Z = Z2 ; moreover, one may identify the morphism
π1 (U ∩ V ) −
→ π1 (U ) � π1 (T ) × π1 (S1 ) 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 R3 , see Example 1.4) make
a 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 contractible
and that U ∩ V is homotopically equivalent to a circle (hence π1 (V ) is trivial and π1 (U ∩ V ) � Z). On
the other hand U is homotopically equivalent to two tangent circles (this can be easily understood in the
interpretation of T2 as a square modulo identifications, as recalled in the cited Example 1.4: making a
hole in the interior of the square, the latter deformation-retracts radially on its boundary; as an useful
exercise, 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 image
of π1 (U ∩ V ) is the subgroup of commutators(35) of Z ∗ Z (in the interpretation of the square, a generator
of 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 choose
�
y ∈ R2 \ 1≤l<m≤k rl,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} � R2 , the “stereographic
projection” from the North pole N = (0, 0, 2) ∈ S identifies diffeomorphically S\{N } with Π by associating
to x = (x, y, z) ∈ S \ {N } the intersection point between Π and the half line coming from N and passing
through x).
(35)
If G is a group, the subgroup of commutators of G is denoted by [G, G] and it is the normal subgroup
generated by the elements of the form xyx−1 y −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).
Corrado Marastoni
25
Notes on Algebraic Topology
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 to
natural map p : R2× −
→ P1 . Let q = p|S1 (Hopf map): since q : S1 −
→ P1 is continuous, surjective and closed,
2
1
1
then q is still quotient. The map ( · ) : S −
→ S is also quotient for the same reason, and has exactly
the same fibers of q: it follows that S1 and P1 are canonically homeomorphic. hence π1 (P1 ) � Z. (6)
(Real projective plane) Let P2 be the projective plane, endowed with the quotient topology with respect
to the map p : B2 −
→ P2 obtained by identifying in P2 the pairs of antipodal points on S1 � ∂B2 (if
2
x = (x, y) ∈ B 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|S1 with the Hopf map). The quotient map p is closed but not
open; nevertheless, if B2× = B2 \ {0} and B˙2 = B2 \ S1 , then U = p(B2× ) and V = p(B˙2 ) are open in
P2 .(36) From now on let us choose 12 ∈ B˙2× = U ∩ V ⊂ C as base point for the computation of fundamental
groups. The set V (homeomorphic image of B˙2 ) is clearly contractible, while U ∩ V (homeomorphic image
2πit
of B˙2× ) has fundamental group � Z generated by [γ] obtained by (the image by p of) t �→ e 2 . As for
x
B2× , it strong deformation-retracts to S1 by the affine homotopy h(x, t) = (1 − t)x + t |x|
, homotopy which
descends via p to a strong deformation retraction h̃ of U a p(S1 ) � P1 .(37) Let r = h̃( · , 1) : U −
→ p(S1 ):
∼
1
1
by r# : π1 (U, 2 ) −
→ π1 (p(S ), 1) � Z, the canonical generator of the second member comes from the
generator [ψ] of π1 (U, 12 ) obtained by t �→ p(eπit /2): hence ιU ∩V,U # sends the generator [γ] in [ψ]2 , and
hence π1 (P2 ) � Z/2Z (analogously to P3 and, as we shall show, to any Pn with n ≥ 2). (7) (Removing
one or two annuli from R3 ) If A is an annulus in R3 and X = R3 \ A, we already computed above that
π1 (X) � Z: another method is to observe that X can be deformation-retracted first to a 2-sphere S2 plus
a diameter, then to the a wedge sum S1 ∨ S2 (by slowly approaching the endpoints of the diameter along
an equator), hence π1 (X) � π1 (S1 ) ∗ π1 (S2 ) � Z.
X :
X� :
X �� :
Figure 5: Deforming R3 minus 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 � = R3 \ (A � B) can be deformation-retracted to S1 ∨ S1 ∨ S2 ∨ S2 and hence π1 (X � ) is
free on two generators. • If C is a third annulus of R3 linked with A, then X �� = R3 \ (A � C) can be
deformation-retracted to S2 ∨ T2 and hence π1 (X �� ) is isomorphic to π1 (T2 ), 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 P2 is Hausdorff (the finite points of P2 have the same
neighborhoods of the points of B˙2 , while a basis of neighborhoods of a point at infinity p(x) with x ∈ S1
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 ∩ S1 ) is not open: hence p(A) is not open in the (quotient) topology of P2 ). On the
other hand the open subsets B2× and B˙2 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 function
f :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 be
different if we would instead consider the strong deformation retraction of B2× to αS1 for a 0 < α < 1 (e.g.
x
α = 21 ), for example the affine one hα (x, t) = (1 − t)x + αt |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 ∈ S1 and t ∈ I.
Corrado Marastoni
26
Notes on Algebraic Topology
x0
� ✲γ1 �x0
T2 = γ2 ✻
✻
x0
�✲
γ1
x0
γ2
✻
✻ �
�x
0
K=
γ2
x0
� ✲γ1 �x0
❄
❄
�✲
γ1
γ2 �
✻
✻
�x
0
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
γ1 γ2 γ1−1 γ2−1 in the case of T2 , and γ1 γ2 γ1−1 γ2 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. Z2 (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
−1
−1 −1
generated by a1 b1 a−1
1 b1 · · · ag bg ag bg . (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
Corrado Marastoni
27
Notes on Algebraic Topology
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).
Corrado Marastoni
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Notes on Algebraic Topology
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 interval
of S1 is “well-covered” by this map, i.e. its inverse image is a family of
pairwise disjoint homeomorphic copies of itself. We have already used
this map to prove (see Proposition 1.4.2) that the fundamental group
of S1 is free with one generator: in fact, we shall see that there is a
deep 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 is
a 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) on X, we denote by Yx = π −1 (x) the fiber on x. Note that a
morphism 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, pX ) 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 space
∼
F 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 restriction
of (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 there
exists an open cover U = {Uλ : λ ∈ Λ} of X such that Y |Uλ is trivial for any λ ∈ Λ; i.e.,
for any x ∈ X there exists an open neighborhood U ⊂ X of x such that Y |U is 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 connected
space:
(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 is
open. 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 neighborhood
of pX (x, f ) = x because it contains U (an open neighborhood of x), and this proves that pX (V ) is open.
Corrado Marastoni
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Notes on Algebraic Topology
Proposition 1.6.4. If (Y, π) is a bundle on X, then π is open. Moreover, if X is arcwise
connected, 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) ⊂ kj=0 Uλj , x0 ∈ Uλ0 ,
x1 ∈ Uλk and Uλj ∩ Uλj+1 �= ∅. We are left with proving that Y |Uλ and Y |Uλ
are isomorphic if further
j
j+1
∼
restricted to Uλj ∩Uλj+1 : it is enough to observe that, given local trivializations ψj : π −1 (Uλj ) −
→ Uλj ×Fj ,
∼
−1
→ Fj+1 .
the isomorphism ψj+1 ◦ ψj of trivial spaces on Uλj ∩ Uλj+1 induces a homeomorphism Fj −
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 one
directly talks about “bundle with fiber F ” or “F -bundle”, where F is a topological space
homeomorphic to the fibers of π.
Remark 1.6.5. Since here we are interested only in topological matters, in the previous
brief exposition of the notion of bundle we have not paid so much attention to the structure
of the fiber, which has been required to be nothing more than a topological space. In other
words: we saw that, if (Y, π) is a bundle on the arcwise connected space X with fiber F
∼
and U1 , U2 are two open subsets of X with trivializations ψj : π −1 (Uj ) −
→ Uj × F , then
∼
for any x ∈ U1 ∩ U2 there is an induced homeomorphism (ψ2 ◦ ψ1−1 )(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 orthogonal
transformation if F is a sphere). In a more motivated exposition the structure of F has to
be respected by these homeomorphisms, which are commonly called transition functions.
In fact, the proper notion of bundle requires also a group structure operating effectively
on the fiber.(39) More precisely, a bundle of base X with total space Y , fiber F and group
structure G (where X, Y and F are topological spaces and G is a topological group) is
the 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 ψλ ψµ−1 (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 euclidean
space (for example, real vector bundle of rank n if the fiber is Rn ) and G is the general
linear group (or a subgroup of it) of the same euclidean space; (b) sphere bundle if F is
a sphere in an euclidean space and G is the orthogonal group (or a subgroup of it) of the
same 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 group
of autohomeomorphisms of F : in other words, the identity element of G acts as the identity of F , and
g1 (g2 (f )) = (g1 g2 )(f ) for any g1 , g2 ∈ G and f ∈ F . The action is called effective if the only element of G
which operates trivially on F is the identity.
Corrado Marastoni
30
Notes on Algebraic Topology
by right translation. In the case of particular structures on Y , X, F and G (e.g. if they
are real or complex manifolds, or algebraic varieties) it is usual to require more regularity
than continuity to the trivializations and to the transition functions, and so one can also
talk 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 C 1 and
dimension n, the tangent bundle T M = {(x, v) : x ∈ M, v ∈ Tx M } and the cotangent bundle T ∗ M =
{(x, α) : x ∈ M, α ∈ Tx∗ M } 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
∗
TN M = {(x, v) : x ∈ N, v ∈ Tx M/Tx N } and the conormal bundle TN
M = {(x, α) : x ∈ N, α ∈ (Tx N )⊥ ⊂
∼
→ T S1 ,
Tx∗ M }. Note that the tangent bundle T S1 is trivial, being homeomorphic to S1 × R by S1 × R −
((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. In
this 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 such
multiplicity is finite, say n, one also talks about a “n-sheet covering”). A morphism of
covering spaces on X is a morphism as spaces on X.
In other words, the fact that π : Y →
− X is a covering map
means that for any x ∈ X there exists a neighborhood U ⊂ X of
x such that π −1 (U ) �
is the disjoint union of homeomorphic copies
∼
of U , i.e. π −1 (U ) = λ∈Λ Vλ , with π|Vλ : Vλ −
→ U : such an open
subset 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 local
homeomorphism (Y, π) where Y is Hausdorff and whose fibers are finite sets with the same
cardinality 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, we
∼
may choose by recurrence some neighborhoods Vyj ⊂ Y of yj such that π|Vy : Vyj −
→ Uyj = π(Vyj ) and
j
�j−1
�k
�
−1
Vyj ⊂ Y \ i=1 Vyi . So set U = j=1 Uyj ⊂ X and Vj = π (U ) ∩ Vyj : it clearly holds π −1 (U ) = kj=1 Vj ,
and π|Vj : Vj −
→ U is a homeomorphism.
�
j
Examples. (1) Let p(z) = n
j=0 aj z 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) = z n 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 an
open neighborhood V ⊂ Y of y such that π|V : V −
→ π(V ) is a homeomorphism.
Corrado Marastoni
31
Notes on Algebraic Topology
loops of nonzero index in 0.(41) Among them we find all simply connected open subsets of X, for example
also the “tickened spiral” {z = reiθ ∈ X : θ ∈ R>0 , θ − φ(θ) < r < θ + φ(θ)} where φ : R>0 −
→]0, 1[ is any
+
−
strictly increasing continuous function with φ(θ) < θ2 , (limθ−
→+∞ φ(θ) = 1 , or
→0+ φ(θ) = 0 and) limθ−
iα
also the open subsets U of X contained in Uα = C \ {re : r ≥ 0} ⊂ X for some α ∈ R. (2) The map
exp : C −
→ C× is a covering of X of countable multiplicity: for any α ∈ R,
� the already known open subsets
Uα = 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) The
above covering spaces of C× induce covering spaces of S1 , which are respectively the maps z n : S1 −
→ S1
1
2πit
(n-fold) and � : R −
→ S , �(t) = e
(countable). In fact, we shall show that these examples exhaust
(up to isomorphism) all connected covering spaces of S1 . Given z0 ∈ S1 , the open �
subset S1 \ {z0 } is
−1
1
evenly covered by these covering spaces: for example, if z0 = 1 then � (S \ {1}) = k∈Z ]k, k + 1[, and
�
iθ
< θ < 2(k+1)π
}.
(z n )−1 (S1 \ {1}) = n−1
: 2kπ
k=0 {e
n
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 g1 V ∩g2 V = ∅
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) = e2πit is a local
(41)
Suppose that U ⊂ X is an open subset containing a loop γ (say based at a point z) of index nonzero
in 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 and
p ◦ δ = γ (here the computation can be performed also�explicitly: if δ(t) = r(t)eiθ(t) and γ(t) = ρ(t)eiϕ(t)
for some 0 ≤ k ≤ n − 1,
with γ(0) = γ(1) = z, from p ◦ δ = γ one gets r(t) = n ρ(t) and θ(t) = ϕ(t)+2kπ
n
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 is
simply connected, and hence also γ = p ◦ δ would be nullhomotopic): but then V could not be a disjoint
union of copies homeomorphic to U by p, hence U is not evenly covered. Conversely, if U is not evenly
covered 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 loops
homotopic rel ∂I: but p ◦ ψ is the loop based at z which describes the circle one or more times, hence its
index 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 gets
gy2 = 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.
Corrado Marastoni
32
Notes on Algebraic Topology
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 = w2 } and V = {(z, w) ∈ C2 : z = w2 , 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 the
base 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. A
˜
lifting of f by π is a continuous function f : Z →
− Y such that f = π ◦ f˜. In particular, if
Z = A ⊂ X and f = ιA is the canonical inclusion, a lifting of ιA is called a (continuous)
section of π over A.
�Y
f˜
Z
π
�
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. If
moreover Z is connected and π is a covering space, or if Z is connected and Y is Hausdorff,
then they are equal.
∼
Proof. If f˜1 (z0 ) = f˜2 (z0 ) = y0 , and if V ⊂ Y is a neighborhood of y0 on which π : V −
→ U = π(V ) is a
−1
˜
˜
homeomorphism, then f1 and f2 must necessarily coincide on the neighborhood W = f (U ) ∩ f˜1−1 (V ) ∩
f˜2−1 (V ) of z0 . This says that Z � = {z ∈ Z : f˜1 (z) = f˜2 (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 an
evenly covered neighborhood of f (z): then π −1 (U ) = λ∈Λ Vλ with f˜1 (z) ∈ Vλ1 and f˜2 (z) ∈ Vλ2 (where
λ1 �= λ2 ), so that z ∈ f˜1−1 (Vλ1 ) ∩ f˜2−1 (Vλ2 ) ⊂ Z \ Z � , i.e. Z \ Z � is open; in the second see Lemma 1.2.2),
and this implies that f˜1 = f˜2 because Z is connected.
Definition 1.6.10. The space (Y, π) on X has the property of lifting paths (uniquely) if
for any path γ : I →
− X and any initial point y0 ∈ 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).
Corrado Marastoni
33
Notes on Algebraic Topology
γ̃y0 : I →
− Y such that γ = π ◦ γ̃y0 and γ̃y0 (0) = y0 ; if Z is a topological space, we say
that (Y, π) has the property of lifting homotopies (uniquely) with respect to Z if for any
homotopy h : Z × I →
− X and any lifting α0 : Z →
− Y of the base h0 (i.e., π ◦ α0 = h0 )
there exists a (unique) homotopy h̃α0 : Z × I →
− Y such that h = π ◦ h̃α0 and (h̃α0 )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 x0 = γ(0) and x1 = γ(1) = φ(0). Then, given y0 ∈ Yx0 and set
�
y1 = γ̃y (1) ∈ Yx , it holds (γ
· φ) = γ̃y · φ�y .
0
y0
1
0
1
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 γ̃y0 piecewise.
There exist 0 = t0 < t1 < · · · < tm−1 < tm = 1 and evenly covered open subsets Uj ⊂ X such that
∼
→ V1 be the section of π over U1 with y0 ∈ V1 (i.e.
γ([tj−1 , tj ]) ⊂ Uj (where j = 1, . . . , m). Let s1 : U1 −
�
�
�
�−1
∼
s1 = π|V1
), and set γ̃y0 �[t ,t ] = s1 ◦ γ|[t0 ,t1 ] ; then, constructed γ̃y0 �[t ,t ] , let sj+1 : Uj+1 −
→ Vj+1
0 1
j−1 j
�
be the section of π over Uj+1 with γ̃y0 (tj ) ∈ Vj+1 and set γ̃y0 �[t ,t ] = sj+1 ◦ γ|[tj ,tj+1 ] . The path γ̃y0
j j+1
�
obtained joining the paths γ̃y0 �[t ,t ] for j = 1, . . . , m will be continuous by the Gluing lemma.
j−1
j
Actually, a local homeomorphism lifting paths uniquely does much more:
Lemma 1.6.13. Any local homeomorphism which has the property of lifting paths uniquely
has also the property of lifting homotopies uniquely.
Proof. Let π : Y −
→ X be a local homeomorphism with the property of lifting paths uniquely, and
let h : Z × I −
→ X be a homotopy and α0 : Z −
→ Y with π ◦ α0 = h0 . For any z ∈ Z, the path
γz : I −
→ X, γ z (t) = h(z, t) lifts uniquely to γ̃ z : I −
→ Y such that γ̃ z (0) = α0 (z): this leads necessarily
to define h̃ : Z × I −
→ Y as h̃(z, t) = γ̃ z (t). We are left with showing the continuity of h̃. Now, since
π is a local homeomorphism, any continuous function with values in X admits locally liftings around
any point: given (z0 , s) ∈ Z × I, let Vs ⊂ Y be an open neighborhood of h̃(z0 , s) = γ̃ z0 (s) such that
∼
π|Vs : Vs −
→ Us = π(Vs ), let Ws × Js be a neighborhood of (z0 , s) such that h(Ws × Js ) ⊂ Us , and
define h̃s = ( π|Vs )−1 ◦ h|Ws ×Js : Ws × Js −
→ Vs . Observe that h̃s (z0 , s) = γ̃ z0 (s) = h̃(z0 , s): this
implies that h̃s (z0 , t) = h̃(z0 , t) for any t ∈ Js (the paths h̃s (z0 , · ) and h̃(z0 , · ) on Js are both liftings of
h(z0 , · ) and�coincide for t = s). Since {z0 } × I is compact, there exist 0 < s1 < · · · < sr < 1 such that
{z0 } × I ⊂ rj=1 (Wsj × Jsj ). As we saw above, the functions h̃sj coincide on {z0 } × I: hence the sections
( π|Vs )−1 of π must coincide on the connected compact subset h({z0 } × I) ⊂ X. But, since by Proposition
j
1.6.9 two sections of a local homeomorphism which coincide on a connected compact subspace�coincide on
a whole open neighborhood of the compact itself, there exists an open neighborhood W0 ⊂ rj=1 Wsj of
z0 such that h̃sj and h̃sj+1 coincide on W0 × (Jsj ∩ Jsj+1 ) (where j = 1,
� . . . , r − 1) giving rise in this way
�
�
�
to a continuous function h̃ : W0 × I −
→ Y ; but it will hold also h̃ = h̃�
(again by the uniqueness of
W0 ×I
lifting of paths defined by fixing repeatedly a z ∈ W0 ), and hence h̃ 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 even
homotopic (see Figure 10):
Corrado Marastoni
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Notes on Algebraic Topology
Corollary 1.6.15. (Monodromy lemma) Let π : Y →
− X be a covering space, α and β be
two paths in X with [α] = [β]. Then, if α̃ and β̃ are liftings of α and β with α̃(0) = β̃(0),
it holds also α̃(1) = β̃(1) and, even, [α̃] = [β̃].
Proof. Let h : I × I −
→ X be a homotopy rel ∂I between α and β: by Proposition 1.6.14 (with Z = I in the
second factor), there exists a unique homotopy h̃ : I × I −
→ Y such that h = π ◦ h̃ and h̃(0, τ ) ≡ y0 = α̃(0) =
β̃(0) for any τ ∈ I. By the property of lifting paths uniquely, one gets α̃(t) = h̃(t, 0) and β̃(t) = h̃(t, 1).
Now, the map h̃(1, · ) : I −
→ Y is continuous; since h = π ◦ h̃, that map must take values in the (discrete)
fiber of π on α(1) = β(1): hence it is constant, and in particular α̃(1) = h̃(1, 0) = h̃(1, 1) = β̃(1).
Figure 10: The Monodromy lemma.
From the Monodromy lemma it follows that the only connected covering of a simply
connected space is, up to homeomorphisms, the space itself:
Corollary 1.6.16. Let X be a simply connected topological space, π : Y →
− X a covering
with 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. So
let 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 , α̃ = γ and β̃ = cy1 ) we
get y2 = γ(1) = cy1 (1) = y1 .
1.6.3
Classification of covering spaces
Let X 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 ∈ Yx0 . Then the
morphism π# : π1 (Y, y0 ) →
− π1 (X, x0 ) is injective.
�
�
Proof. Let π# ([γ]) = [π ◦ γ] = [cx0 ]: observing that (π
◦ γ)y = γ and (c
= cy0 , by the Monodromy
x0 )
y
0
lemma one has [γ] = [cy0 ].
Corrado Marastoni
0
35
Notes on Algebraic Topology
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, x0 ) by π# :
G(Y, y0 ) = π# (π1 (Y, y0 )) = {[γ] ∈ π1 (X, x0 ) : γ̃y0 is a loop based at y0 }.
Proposition 1.6.19. Let π : Y →
− X be a covering space, and y0 ∈ Yx0 . Then the
subgroups of π1 (X, x0 ) conjugated to G(Y, y0 ) are exactly the subgroups G(Y, y1 ) with y1 ∈
Yx0 in the same arc-component of y0 in 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, x0 ) →
− SYx , where SYx
0
0
denotes the group of permutations of the fiber Yx0 . This action is described as follows:
given y ∈ Yx0 and γ a loop in X based at x0 , let γ̃y be the lifting of γ with starting point
y, and define µ([γ])(y) = y · [γ] := γ̃y (1). Hence the stabilizer of some y ∈ Y is precisely
G(Y, y). Moreover, if the covering space Y is arcwise connected then by Proposition
1.6.19 the subgroup acting trivially on Yx0 is the heart(45) of G(Y, y0 ) in π1 (X, x0 ), for any
y 0 ∈ Y x0 .
Examples. (1) The fiber of the covering space exp : C −
→ C× over z0 = reiθ is the set of complex
logarithms 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 :
p
p
1
−→
2
−→
p
3
−→
where p1 : S1 −
→ S1 , p1 (z) = z 3 ; p2 : S1 � S1 −
→ S1 , p2 (z) = z 2 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 � S1 −
→ 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 ∈ Yx0 . Then α̃y0 and β̃y0 have 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 y0 ∈ Y in the fiber of x0 = 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 contained
in 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∈G gHg −1 . The latter shoul not be confused with the normalizer
−1
N H = {g ∈ G : gHg
= 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 coincide
with H, and N H = G.
Corrado Marastoni
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Notes on Algebraic Topology
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 must
introduce a topological notion.
Definition 1.6.22. A topological space is said to be locally (arcwise) connected if any
point 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 Z
arcwise connected and locally arcwise connected, π : (Y, y0 ) →
− (X, x0 ) a covering space
and f : (Z, z0 ) →
− (X, x0 ) a continuous function. Then f admits a unique lifting f˜ :
(Z, z0 ) →
− (Y, y0 ) if and only if f# (π1 (Z, z0 )) ⊂ G(Y, y0 ).
Proof. Necessity is an immediate consequence of the functoriality of π1 (exercise); let us see now the
sufficience. The uniqueness of f˜ comes from Proposition 1.6.9. As for the existence, given any z ∈ Z let
us 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) . The definition f (z) = (f ◦ α) (1) is well-posed: if β is another path in Z from
y0
y0
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 same
endpoint 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. Let
W ⊂ f −1 (U ) be open, arcwise connected and containing z (such a W exists since Z is locally arcwise
connected): 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 )) (1) = (f
◦ α) · (f�
◦ βζ ) ˜ (1): now, (f�
◦ βζ ) ˜ is a path from f˜(z) ∈ V which
y0
y0
f (z)
f (z)
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 evenly
covered, 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 in
the proof of the Lifting criterion (Proposition 1.6.23). For example, let Z be the “quasicircle” of Figure 11 (starting with a vertical straight segment which will be later approached
by a part of type “sin x1 ”), and let f : Z →
− S1 be a quotient map which collapses all the
points of Z contained in the dashed box into the point 1 of S1 . If we consider the usual
exponential covering � : R →
− S1 given by �(t) = e2πit , since the fundamental group of
(46)
Z is trivial
the hypothesis on the characteristic subgroup is satisfied; however, if we
˜
lift 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 the
vertical straight segment: then α is nullhomotopic, in other words Z is simply connected.
Corrado Marastoni
37
Notes on Algebraic Topology
segment goes to 0 while the part of type “sin x1 ” 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 of
X is covered by any other covering space of X having a smaller characteristic subgroup.
Proposition 1.6.25. Let π : (Y, y0 ) →
− (X, x0 ) and p : (Z, z0 ) →
− (X, x0 ) be two covering
spaces of X with Z arcwise connected and locally arcwise connected. Then there exists
a morphism of covering spaces ϕ : ((Z, z0 ), p) →
− ((Y, y0 ), π) if and only if G(Z, z0 ) ⊂
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 proving
that 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 x to π(y), which lifts uniquely to α = (π
◦ β) in Z: since π ◦ (ϕ ◦ α) = p ◦ α = π ◦ β,
z0
0
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
�
∼
→ U for
evenly covered both for p and for π: one then has π −1 (U ) = y∈π−1 (x ) Vy (with π|Vy : Vy −
0
�
∼
any y ∈ π −1 (x0 )) and p−1 (U ) = z∈p−1 (x ) Wz (with p|Wz : Wz −
→ U for any z ∈ p−1 (x0 )). One
0
�
then shows that ϕ−1 (Vy0 ) = z∈ϕ−1 (y ) Wz (note that ϕ−1 (y0 ) ⊂ p−1 (x0 )): namely, if z ∈ ϕ−1 (y0 ) then
∼
0
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 connected covering spaces of a (connected and locally arcwise connected) topological space are
isomorphic if and only if they have the same characteristic subgroup.(48)
The statement of a Existence theorem for a covering space with prescribed characteristic
subgroup requires a slightly stronger topological hypothesis.
Definition 1.6.27. A topological space X is said locally simply connected if any x ∈ X
has a basis of open simply connected neighborhoods; more generally, X is said semi-locally
simply connected if any x ∈ X admits a open neighborhood U ⊂ X such that any loop
in U based at x is nullhomotopic in X (i.e., with homotopies not necessarily with values
only in U ).
Examples. (1) Obviously, manifolds are locally (hence also semi-locally) simply connected. (2) (Shrinking
�
wedge of circles) Let C = n∈N Cn , 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 N S1 . 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
�
From U = p(Wz ) = π(ϕ(Wz )) one has that the connected subset ϕ(Wz ) is in p−1 (U ) = y∈π−1 (x ) Vy
0
and contains y0 , hence ϕ(Wz ) ⊂ Vy0 and it holds even ϕ(Wz ) = Vy0 because otherwise p|Wz = π|Vy ◦ ϕ|Wz
(47)
0
would not be an isomorphism.
(48)
It must be noted that we are always working in the framework of pointed topological spaces, so we
are keeping track of a base point both in the base space (i.e. a x0 ), and in the covering (i.e. a point in
the 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 arcwise
connected) topological space are isomorphic if and only if they have conjugated characteristic subgroups.
Corrado Marastoni
38
Notes on Algebraic Topology
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) −
→ N Z and so, the direct product N Z being uncountable,
� 1
also π1 (C) is uncountable and hence deeply different from π1 ( N S ) � N Z (which has countably many
3
∗
generators, and hence is countable). (3) Let T be a cone in R 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 semilocally (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 (± n2 , 0, 0) of the loops
which constitute γ(rn ) keep being at “security distance”
the vertices of ±T for 0 ≤ t ≤
1
,
2
1
N
and then down to (0, 0, 0)
> 0 from (0, 0, 0), hence they can be sent to
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 arcwise
connected and semi-locally simply connected topological space (X, x0 ) 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 ,x0 and
π:Y −
→ X given by [γ] −
→ γ(1), then finally to endow Y with a suitable topology using the hypotheses on
X. For more details we refer for example to Jänich [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 , z n ) (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 ∨ S1 , 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 the
proof of Proposition 1.6.28. For example, the above double cone of shrinking wedge of
circles X = T ∪ (−T ) has been proved to have nontrivial fundamental group, but it is
Corrado Marastoni
39
Notes on Algebraic Topology
possible to prove that any arcwise connected covering space of X is necessarily trivial(49) :
hence the proper subgroups of π1 (X; x0 ) 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 of
endomorphisms 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) = e2πit ) 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 z n : 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 connected
and locally arcwise connected topological spaces, x0 ∈ X, y0 , y1 ∈ Yx0 . Then there exists
ϕ ∈ Aut(Y |X) with ϕ(y0 ) = y1 if and only if G(Y, y0 ) = G(Y, y1 ).
What does this condition mean? By Proposition 1.6.19 we know that G(Y, y1 ) is conjugated to G(Y, y0 ) in π1 (X, x0 ): if γ is a path in Y from y0 to y1 , setting α = π ◦ γ it holds
G(Y, y1 )�= [α−1 ] ·�G(Y, y0 ) · [α]. Hence, the condition G(Y, y0 ) = G(Y, y1 ) is equivalent to
[α] ∈ N G(Y, y0 ) (the normalizer of G(Y, y0 ) in π1 (X, x0 )).
Theorem 1.6.31. Let π : (Y, y0 ) →
− (X, x0 ) be a covering space with X and �Y arcwise�
connected and locally arcwise connected topological spaces. Then for any [α] ∈ N G(Y, y0 )
there exists one and only one covering automorphism ϕ[α] such that ϕ[α] (y0 ) = α̃y0 (1).
�
�
The application N G(Y, y0 ) →
− Aut(Y |X) obtained in this way is a surjective morphism
of groups with kernel G(Y, y0 ), and provides an isomorphism of groups
�
�
N G(Y, y0 )
∼
−
→ Aut(Y |X).
G(Y, y0 )
�
�
Proof. Let y1 = α̃y0 (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 constructs
explicitly 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 ) (1) (note that ϕ[α] (y ) = cy (1) = y1 , as required).
y1
0
1
(49)
The idea is that a arcwise connected covering space π : Y −
→ X induces on T and −T (which are simply
connected) trivial covering spaces (±T ) × F ; if F would not be a point (i.e., if such induced covering spaces
would not be homeomorphisms), two points of the same fiber would stay in different arcwise connected
components, 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 since
X = T ∪ (−T ) this implies that π is a homeomorphism.
Corrado Marastoni
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Notes on Algebraic Topology
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, y0 ) is a normal subgroup of π1 (X, x0 ).
The following proposition shows the properties of normal covering spaces. In particular
they act transitively on the fibers, and hence these covering spaces can be viewed as those
having a complete symmetry among different sheets.
Proposition 1.6.33. A connected and locally arcwise connected normal covering space
π : (Y, y0 ) →
− (X, x0 ) has the following properties.
∼
(i) π1 (X, x0 )/G(Y, y0 ) −
→ 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 space
is equal to the index of G(Y, y0 ) in π1 (X, x0 ).
(iii) Denoted by Y /Aut(Y |X) the space of orbits of Aut(Y |X) in Y (i.e., the space of
fibers of π) with the quotient topology, the natural bijection Y /Aut(Y |X) →
− X is a
homeomorphism.
(iv) Either all liftings of loops in X based at x0 are 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 the
fibers
of π as orbits. Let U ⊂ X be a evenly covered open neighborhood of x0 : hence one has π −1 (U ) =
�
V
λ∈Λ λ , 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). The
bijection 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 space
π : (Y, y0 ) →
− (X, x0 ) is Galois, the subgroup of π1 (X, x0 ) acting trivially on the fiber Yx0
is G(Y, y0 ).
Example. If π1 (X, x0 ) is commutative, then obviously all arcwise connected and locally arcwise connected
covering spaces of X are normal. On the other hand, let us show an example of connected and locally
arcwise connected covering space which is not normal. Consider the function
✛✛
✬
Y
=
π
❄
X
=
✫
✛
✬
✩
✛✛
✬
✩
✲
✫
y1✪
✲✲
✫
y✪
0
�
β✛
✛
✬
✫
�
α
✛
✬
✩
�
✫
x✪
0
✩
✬
✩
✲
✫
y2✪
✪
�
✩
✪
(50)
Recall that the action of a group G on a set Z si said properly discontinuous if any z ∈ Z has a
neighborhood U ⊂ Z such that g1 U ∩ g2 U = ∅ if g1 =
� g2 , and free (a weaker notion) if any point has
trivial stabilizer, i.e. Gz = {1} for any z ∈ Z.
Corrado Marastoni
41
Notes on Algebraic Topology
(We mean that π(yj ) = x0 (j = 0, 1, 2) and that the arcs of Y denoted by −
→ (resp. by �) are sent into
α (resp. into β) in the specified direction. Note that π is a surjective local homeomorphism with fiber of
cardinality 3: by Proposition 1.6.7, π is a 3-sheet covering space. Consider the morphism of pointed spaces
π : (Y, y0 ) −
→ (X, x0 ), and the injective morphism π# : π1 (Y, y0 ) −
→ π1 (X, x0 ). The space X is the bouquet
of two circles and hence π1 (X, x0 ) is free on two generators [α] and [β]; on the other hand, π1 (Y, y0 ) is
generated by the classes of loops
γ1 =
✛
✛ ✏
✛
✓
✓
✒
✲
✑
✒
✏
✑
γ2 =
y0
✛
✓
✲
✒
✏
✑
γ3 =
y0
✛
✛ ✏
✓
✲
✲
✒
y0
✑
γ4 =
✛
✛ ✓
✓
✏
✲
✲
✒
y0
✲
✑
✒
✏
✑
and one has π# ([γ1 ]) = [α · β · α], π# ([γ2 ]) = [α2 ], π# ([γ3 ]) = [β 2 ], π# ([γ4 ]) = [β · α · β]: hence the
characteristic subgroup G(Y, y0 ) is generated by [α · β · α], [α2 ], [β 2 ] 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 α̃y0 and α̃y1 are not loops (the first one goes from
y0 to y1 , the second from y1 to y0 ), but β̃y0 is not a loop (goes from y0 to y2 ) while β̃y1 is. Therefore one
�
�
has Aut(Y |X) = {idY }, which implies N G(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=0,1,2 G(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 [β 2 ]. 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 →
− X
is 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 simply connected topological space, there exists a unique —up to canonical isomorphisms—
universal cover π̃ : (X̃, x̃0 ) →
− (X, x0 ), with the following properties.
(i) π1 (X, x0 ) � Aut(X̃|X).
(ii) Aut(X̃|X) operates on the left on X̃ in a properly discontinuous way (hence freely),
and the orbits are the fibers of π̃.
(iii) if π : (Y, y0 ) →
− (X, x0 ) is another connected and locally arcwise connected covering
space, there exists one and only one covering space π̃Y : (X̃, x̃0 ) →
− (Y, y0 ) such that
π̃ = π ◦ π̃Y . (Hence, the universal cover of X is also the universal cover of any other
connected and locally arcwise connected covering space of X.)
(iv) The universal cover determines all other arcwise connected and locally arcwise connected covering spaces of X, in the following sense: if Γ ⊂ Aut(X̃|X) is a subgroup,
denoted by X̃/Γ the space of orbits of Γ in X̃ endowed with the quotient topology, the
natural map π̃Γ : (X̃/Γ, [x̃0 ]) →
− (X, x0 ) is a connected and locally arcwise connected
covering of X, and moreover all arcwise connected and locally arcwise connected
covering 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) follow
from Proposition 1.6.33, (iii) from Proposition 1.6.25, (iv) follows from (iii) and Proposition 1.6.33 (which
says that Y � X̃/Aut(X̃|Y ), and Γ = Aut(X̃|Y ) ⊂ Aut(X̃|X)).
Corrado Marastoni
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Notes on Algebraic Topology
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(X̃|X).
Some examples of this method will be shown in the next Section.
Corrado Marastoni
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Notes on Algebraic Topology
1.7
Exercises and complements
(1) The universal covering of S1 is the exponential map � : R →
− S1 ⊂ C, �(t) = e2πit ;
n
1
n
more generally, the universal cover of the torus T = (S ) is the map � : Rn →
− Tn ,
�(t1 , . . . , tn ) = (e2πit1 , . . . , e2πitn ), and it holds Aut(Rn |Tn ) = {τ : Rn →
− Rn , τ (t1 , . . . , tn ) =
n
(t1 + k1 , . . . , tn + kn ) for some kj ∈ Z} � Z , 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 ), endowed with the quotient topology. Consider the Hopf map q : Sn →
− Pn (the restriction
to Sn of the projection (Rn+1 )× →
− Pn ), 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 , we
get π1 (Pn ) � Aut(Sn |Pn ) = Z/2Z. On the other hand, for n = 1 we saw that the map
∼
( · ) 2 : S1 →
− S1 has the same fibers of q, hence there exists a homeomorphism γ : S1 −
→ P1
such that q = γ ◦ ( · )2 . Therefore π1 (P1 ) � Z.
An interesting application is a particular case of the Borsuk-Ulam theorem (if n ≥ 2, there
does 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 continuous functions of Sn with values in S1 .
Proof. By absurd let f : Sn −
→ S1 be a odd continuous function, and denote by qj : Sj −
→ Pj the Hopf
map. By Proposition 1.1.14, there exists g : Pn −
→ P1 continuous such that g ◦ qn = q1 ◦ f (note that
q1 ◦ f is constant on the fibers of qn ). Now, fixed x0 ∈ Sn , 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 g̃ : Pn −
→ S1 such that g = q1 ◦ g̃ and g̃(qn (x0 )) = f (x0 ). But one has also f = g̃ ◦ qn (namely f and
g̃ ◦ qn are two liftings of g ◦ qn which coincide in x0 ): this is a contradiction because f (−x) = −f (x) while
g̃(qn (−x)) = g̃(qn (x)) = f (x).
A consequence of Corollary 1.7.1 is, for example, that at any particular time there are two
antipodal places on the Earth with the same temperature and the same pressure. Namely
let, at a certain fixed moment, t : S2 →
− R be the temperature and p : S2 →
− R be the
2
2
pressure: if the statement would be false, the function ϕ : S →
− R given by ϕ(x) =
(t(x) − t(−x), p(x) − p(−x)) would never take the value (0, 0) and hence the function
f : S2 →
− S1 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 one
and only one multiplication on E for which (a) E is a topological group with identity
element e, (b) π is a morphism.
Proof. (The student should verify by exercise the unproven statements.) (i) Let µ : G × G −
→ G be the
multiplication. 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
Corrado Marastoni
44
Notes on Algebraic Topology
(∗ 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) The
map 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 map
µ̃ : (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 all
the properties which are required to µ̃ to be the desired operation in E: for example, to show that e is the
identity element of µ̃ we define µ̃e : (E, e) −
→ (E, e) by µ̃e (y) = µ̃(y, e): since π ◦ µ̃e = π, we get µ̃e = idE
(because idE and µ̃e 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 the
fundamental group when we remove, from a given manifold, a closed submanifold which
is “small enough”.
Let M be a (arcwise) connected C 0 manifold of dimension m, N ⊂ M a C 0 submanifold
of 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 naturally
0
induced a structure of C manifold of dimension m, and on π −1 (N ) a structure of submanifold 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 a
covering space q : P →
− M \ N there exist a covering space q̃ : P̃ →
− M and a
�
injective morphism of manifolds ιP : P →
− P such that q̃ ◦ 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 show
that, 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 to
y, and {(Uλ , ϕλ ) : λ ∈ Λ} be an atlas of M such that Uλ ∩ N = ∅ or ϕλ (Uλ ∩ N ) = Rn . 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, if
α([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 is
connected. (ii) For the costruction of the covering space q̃, obtained through a gluing procedure of covering
spaces 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, x0 ).
Then:
(i) if m − n ≥ 2, then ι# is surjective;
(ii) if m − n ≥ 3, then ι# is a isomorphism.
Corrado Marastoni
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Notes on Algebraic Topology
Proof. (i) It is enough to show that, given x, y ∈ M \ N and a path α : I −
→ M from x to y, there
exists a path γ : I −
→ M \ N from x to y with [α] = [γ]. Following the proof of the first statement of
Proposition 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 us
prove 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 q̃ : P� −
→ M and an injective
morphism of manifolds ιP : P −
→ P� such that q̃ ◦ ιP = ι ◦ q. By definition of H, one has (q̃# ◦ ιP # )(π1 (P )) =
(ι# ◦ q# )(π1 (P )) = ι# (H) = {1}, hence (q̃# 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 ) = q̃ −1 (M \ N ) we get P� \ ιP (P ) = q̃ −1 (N ), and it is enough to recall
Proposition 1.7.3), the universal cover π : S −
→ P extends to π̃ : 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.
t the adjoint matrix, and:
If A ∈ M (n, C), we denote by A∗ = A
GL(n, C) =
U (n, C) =
SU (n, C) =
GL(n, R) =
{A ∈ M (n, C) : det(A) �= 0}
(complex general linear group)
{A ∈ U (n, C) : det(A) = 1}
(special unitary group)
{A ∈ GL(n, C) :
A−1
=
A∗ }
M (n, R) ∩ GL(n, C)
(real general linear group)
GL (n, R) = {A ∈ GL(n, R) : det(A) ≷ 0}
±
O(n, R) =
SO(n, R) =
(unitary group)
M (n, R) ∩ U (n, C)
M (n, R) ∩ SU (n, C)
(real orthogonal group)
(real special orthogonal group)
and moreover
H(n, C) =
+
{A ∈ M (n, C) : A = A∗ }
H (n, C) = {A ∈ H(n, C) :
S(n, R) =
+
S (n, R) =
t xAx
> 0 ∀x ∈
Cn
M (n, R) ∩ H(n, C)
M (n, R) ∩
H + (n, C)
(hermitian matrices)
\ {0}} (positive definite h. m.)
(symmetric matrices)
(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 , and
H(n, C) (resp. S(n, R)) is a real subspace of M (n, �
C) (resp. M (n, R)) of dimension
2
n (resp. n(n + 1)/2). The application
A �→ �A� = tr(AA∗ ) is a norm on M (n, C),
�
t
which induces the norm �A� = tr(A A)
on M (n, R).
(2) GL(n, C) is an open subset of M (n, C) and a multiplicative topological group.
�
n
(3) The exponential exp : M (n, C) →
− GL(n, C) (where exp(A) = ∞
n=0 A /n!) satisfies
exp(A + B) = exp(A) exp(B) if AB = BA, and is also a diffeomorphism between
an 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 in
particular locally arcwise connected and semi-locally simply connected.
Corrado Marastoni
46
Notes on Algebraic Topology
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) on
S + (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 subgroups 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 of
GL(n, C), are obtained by inverting the restriction of exp respectively to the real
subspaces 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)) and
t = 0} (for O(n, R) and SO(n, R))(53) . Moreso(n, R) = {A ∈ M (n, R) : A + A
over, U (n, C), SU (n, C), O(n, R) and SO(n, R) are compact (if A ∈ U (n, C) then
√
�A� = n).
(5) There are isomorphisms of Lie groups U (n, C) � S1 ×SU (n, C) and O(n, R) � {±1}×
SO(n, R), given by A �→ (det(A), A/ det(A)). Moreover, SU (n, C) and SO(n, R) are
arcwise connected, and hence U (n, C) is arcwise connected while O(n, R) has two
connected 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 homeomorphism
is induced between GL(n, R) and S + (n, C) × O(n, R) and in particular between
GL+ (n, R) and S + (n, C)×SO(n, R), hence GL(n, R) has two connected components
GL± (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
�
� S1 . For n = 3,
note that S3 (intended as the group of quaternions of unitary norm, see Example
1.4) operates on R3 by S3 × 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 morphism
S3 →
− 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 C 1 manifold of dimension
m which makes the multiplication and the inversion into differentiable maps. By the way, GL(n, C) would
be of course also a complex Lie group, but here we are interested only in its structure of real differentiable
manifold.
(53)
In the terminology of Lie theory, these vector subspaces are the Lie algebras associated to the Lie
subgroups: in the ambient vector space M (n, C), they are the tangent space to the Lie subgroups at the
identity 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, antisymmetric
and satisfying the Jacobi identity [x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0 for any x, y, z ∈ V : in our case, for
A, B ∈ M (n, C), it is [A, B] = AB − BA (the commutator of A and B). Note that each one of the real
subspaces of M (n, C) considered here is stable with respect to such operation.
Corrado Marastoni
47
Notes on Algebraic Topology
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), of
dimension n(n − 1)/2. We embed N = SO(n − 1; R) into M = SO(n; R) as those
orthogonal transformations that fix the North pole en : note that dim M − dim N =
n − 1, hence for n ≥ 4 we have π1 (SO(n; R) \ SO(n − 1; R)) � π1 (SO(n; R)). Now
we aim to show that SO(n; R) \ SO(n − 1; R) is an open subset of SO(n; R) which
deformation-retracts to a manifold homeomorphic to SO(n − 1; R), and this will
imply that π1 (SO(n; R)) � Z/2Z for n ≥ 4. Let ρ : SO(n; R) →
− Sn−1 be the
map ρ(A) = Aen : setting V = Sn−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 ). Let
n−1
s:S
\{−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 en
into x in the plane �x, en �. Let a : Sn−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) = a◦s(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 given
by g(β) = (β(en ), s(a(β(en )))−1 ◦ a ◦ β). Since V is contractible, we have proven the
claim. Summarizing up, one has

(n = 1)
 {1}
Z
(n = 2)
π1 (GL(n, R), 1n ) � π1 (O(n, R), 1n ) � π1 (SO(n; R)) �

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 the
other hand, arguing as before one shows that for n ≥ 2 there exists a homeomorphism
f : SU (n, C)\SU (n−1, C) →
− V ×SU (n−1, C), where V = S2n−1 \{e2n } ⊂ Cn � R2n ,
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.
Corrado Marastoni
48