INTRODUCTION TO ALGEBRAIC TOPOLOGY
MARTIN ˇCADEK
11. Fundamental group
The fundamental group of a space is the first homotopy group. In this section we
describe two basic methods how to compute it.
11.1. Covering space. A covering space of a space X is a space X together with a
map p : X → X such that (X, X, p) is a fibre bundle with a discrete fibre.
In the previous section we have proved that every fibre bundle has homotopy lifting
property with respect to CW-complexes. In the case of covering spaces the lifts of
homotopies are unique:
Proposition. Let p : X → X be a covering space and let Y be a space. Given a
homotopy F : Y × I → X and a map f : Y × {0} → X such that F(−, 0) = pf, there
is a unique homotopy F : Y × I → X making the following diagram commutative:
Y × {0}

f
// X
p

Y × I F
//
F
;;wwwwwwwwww
X
Proof. Since the proof follows the same lines as the proof of the analogous proposition
in 10.5, we outline only the main steps.
(1) Using compactness of I we show that for each y ∈ Y there is a neighbourhood
U such that F can be defined on U × I.
(2) F is uniquely determined on {y} × I for each y ∈ Y .
(3) The lifts of F defined on U1 × I and U2 × I concide on (U1 ∩ U2) × I.
From the uniquiness of lifts of loops and their homotopies starting at a fixed point
we get immediately the following
Corollary. The group homomorphism p∗ : π1(X, x0) → π1(X, x0) induced by a covering
space (X, X, p) is injective. The image subgroup p∗(π1(X, x0)) in π1(X, x0) consists
of loops in X based at x0 whose lifts in X starting at x0 are loops.
11.2. Group actions. A left action of a discrete group G on a space Y is a map
G × Y → Y, (g, y) → g · y
1
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such that 1 · y = y and (g1g2) · y = g1 · (g2 · y). We will call this action properly
discontinuous if each point y ∈ Y has an open neighbourhood U such that g1U ∩g2U =
∅ implies g1 = g2.
An action of a group G on a space Y induces the equivalence x ∼ y if y = g · x for
some g ∈ G. The orbit space Y/G is the factor space Y/ ∼.
A space Y is called simply connected if it is path connected and π1(Y, y0) is trivial
for some (and hence all) base point y0.
The following theorem provides a useful method for computation of fundamental
groups.
Theorem. Let Y be a path connected space with a properly discontinuous action of a
group G. Then
(1) The natural projection p : Y → Y/G is a covering space.
(2) G ∼= π1(Y/G, p(y0))/p∗π1(Y, y0). Particularly, if Y is simply connected, then
π1(Y/G) ∼= G.
Proof. Let y ∈ Y and let U be a neighbourhood of y from the definition of properly
discontinuous action. Then p−1
(p(U)) is a disjoint union of gU, g ∈ G. Hence
(Y, Y/G, p) is a fibre bundle with the fibre G.
Applying the long exact sequence of homotopy groups of this fibration we obtain
0 = π1(G, 1) → π1(Y, y0)
p∗
−→ π1(Y/G; p(y0))
δ
−→ π0(G) = G → π0(Y ) = 0.
In general π0 of a fibre is only the set with distinguished point. However, here it has
the group structure given by G. Using the definition of δ from 10.3 one can check that
δ is a group homomorphism. Consequently, the exact sequence implies that
G ∼= π1(Y/G, p(y0))/p∗π1(Y, y0).
Example A. Z acts on real numbers R by addition. The orbit space is R/Z = S1
.
According to the previous theorem
π1(S1
, s) = Z.
The fundamental group of the sphere Sn
with n ≥ 2 is trivial. The reason is that any
loop γ : S1
→ Sn
is homotopic to a loop which is not a map onto Sn
and Sn
without
a point is contractible.
Next, the group Z2 = {1, −1} has an action on Sn
, n ≥ 2 given by (−1) · x = −x.
Hence
π1(RPn
) = Z2.
Example B. The abelian group Z ⊕ Z acts on R2
(m, n) · (x, y) = (x + m, y + n).
The factor R2
/(Z ⊕ Z) is two dimensional torus S1
× S1
. Its fundamental group is
Z ⊕ Z.
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Example C. The group G given by two generators α, β and the relation β−1
αβ = α−1
acts on R2
by
α · (x, y) = (x + 1, y), β · (x, y) = (1 − x, y + 1).
The factor R2
/G is the Klein bottle. Hence its fundamental group is G.
11.3. Free product of groups. As a set the free product ∗αGα of groups Gα, α ∈ I
is the set of finite sequences g1g2 . . . gm such that 1 = gi ∈ Gαi
, αi = αi+1, called
words. The elements gi are called letters. The group operation is given by
(g1g2 . . . gm) · (h1h2 . . . hn) = (g1g2 . . . gmh1h2 . . . hn)
where we take gmh1 as a single letter gm ·h1 if both elements belong to the same group
Gα. It is easy to show that ∗αGα is a group with the empty word as the identity
element. Moreover, for each β ∈ I there is the natural inclusion iβ : Gβ → ∗αGα.
Up to isomorhism the free product of groups is characterized by the following universal
property: Having a system of group homomorphism hα : Gα → G there is just
one group homomorphism h : ∗αGα → G such that hα = hiα.
Exercise. Describe Z2 ∗ Z2.
11.4. Van Kampen Theorem. Suppose that a space X is a union of path connected
open subsets Uα each of which contains a base point x0 ∈ X. The inclusions
Uα → X induce homomorphisms jα : π1(Uα) → π1(X) which determine a unique
homomorphism ϕ : ∗απ1(Uα) → π1(X).
Next, the inclusions Uα ∩ Uβ → Uα induce the homomorphisms iαβ : π1(Uα ∩ Uβ) →
π1(Uα). We have jαiαβ = jβiβα. Consequently, the kernel of ϕ contains elements of
the form iαβ(ω)iβα(ω−1
) for any ω ∈ π1(Uα ∩ Uβ).
Van Kampen Theorem provides the full description of the homomorphism ϕ which
enables us to compute π1(X) using groups π1(Uα) and π1(Uα ∩ Uβ).
Theorem (Van Kampen Theorem). If X is a union of path connected open sets Uα
each containing a base point x0 ∈ X and if each intersection Uα ∩Uβ is path connected,
then the homomorhism ϕ : ∗απ1(Uα) → π1(X) is surjective. If in addition each intersection
Uα ∩ Uβ ∩ Uγ is path connected, then the kernel of ϕ is the normal subgroup
N in ∗απ1(Uα) generated by elements iαβ(ω)iβα(ω−1
) for any ω ∈ π1(Uα ∩ Uβ). So ϕ
induces an isomorphism
π1(X) ∼= ∗απ1(Uα)/N.
Example. If Xα are path connected spaces, then
π1( Xα) = ∗απ1(Xα).
Outline of the proof of Van Kampen Theorem. For simplicity we suppose that X is a
union of only two open subsets U1 and U2.
Surjectivity of ϕ. Let f : I → X be a loop starting at x0 ∈ U1 ∪ U2. This loop
is up to homotopy a composition of several paths, for simplicity suppose there are
three such that f1 : I → U1, f2 : I → U2 and f3 : I → U1 with end points succesively
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x0, x1, x2, x0 ∈ U1 ∩U2. Since U1 ∩U2 is path connected there are paths g1 : I → U1 ∩U2
and g2 : I → U1 ∩ U2 from x0 to x1 and x2, respectively. Then the loop f is up to
homotopy the composition of loops f1 − g1 : I → U1, g1 + f2 − g2 : I → U2 and
g2 + f3 : I → U1. Consequently, [f] ∈ π1(X) lies in the image of ϕ.
f2
f1
f3
x0
x1
x2
g1
g2
Figure 11.1. [f] = [f1 + f2 + f3] = [f1 − g1] + [g1 + f2 − g2] + [g2 + f3]
Kernel of ϕ. Suppose that the image under ϕ of a word with m letters [f1][g1][f2] . . . ,
where [fi] ∈ π1(U1), [gi] ∈ π1(U2), is zero in π1(X). Then there is a homotopy
F : I × I → X such that
F(s, 0) = f1 + g1 + f2 + . . . , F(s, 1) = x0, F(0, t) = F(1, t) = x0
where we suppose that fi is defined on [2i−2
m
, 2i−1
m
] and gi is defined on [2i−1
m
, 2i
m
]. Since
I × I is compact, there is an integer n, a multiple of m, such that
F
i
n
,
i + 1
n
×
j
n
,
j + 1
n
is a subset in U1 or U2. Using homotopy extension property, we can construct a
homotopy from F to F rel J1
such that again
F
i
n
,
i + 1
n
×
j
n
,
j + 1
n
is a subset in U1 or U2, and moreover,
F
i
n
,
j
n
= x0.
Further, F(s, 0) = f 1 + g 1 + f 2 + . . . where f i ∼ fi, g i ∼ gi in U1 and U2, respectively,
rel the boundary of the domain of definition. We want to show that the word
[f 1]1[g 1]2[f 2]1 . . . belongs to N. Here [ ]i stands for an element in π1(Ui).
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We can decompose
I × I =
i
Mi
where Mi is a maximal subset with the properties:
(1) Mi is a union of several squares [ i
n
, i+1
n
] × [ j
n
, j+1
n
].
(2) int Mi is path connected.
(3) F(Mi) is a subset in U1 or U2.
For simplicity suppose that we have four sets Mi as indicated in the picture.
x0
x0 x0
M1
M2
M3
M4
p
k
l
f1 g1 f2
Figure 11.2. [f1]1[g1]2[f2]1 ∈ Kerϕ
In this situation there are three loops k, l and p starting at x0 and lying in U1 ∩ U2.
They are defined by F on common boundary of M1 and M2, M2 and M3, M3 and M4,
respectively. Now, we get
[f 1]1[g 1]2[f 2]1 = [k]1[−k + l]2[−l + p]1 = [k]1[−k]2[l]2[−l]1[p]1
= [k]1[−k]2[l]2[−l]1 ∈ N.
Corollary. Let X be a union of two open subsets U and V where V is simply connected
and U ∩ V is path connected. Then
π1(X) = π1(U)/N
where N is the normal subgroup in π1(U) generated by the image of π1(U ∩ V ).
Exercise. Use the previous statement to compute the fundamental group of the Klein
bottle and other 2-dimensional closed surfaces.
11.5. Fundamental group and homology. Here we compare the fundamental
group of a space with the first homology group. We obtain a special case of Hurewitz
theorem, see 13.6.
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Theorem. By regarding loops as 1-cycles, we obtain a homomorphism h : π1(X, x0) →
H1(X). If X is path connected, then h is surjective and its kernel is the commutator
subgroup of π1(X). So h induces isomorphism from the abelization of π1(X, x0) to
H1(X).
For the proof we refer to [Hatcher], Theorem 2A.1, pages 166–167.
CZ.1.07/2.2.00/28.0041
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