Download 1. Fundamental Group Let X be a topological space. A path γ on X is

FUNDAMENTAL GROUP
1. Fundamental Group
Let X be a topological space. A path γ on X is a continuous map γ : [0, 1] → X. The
points γ(0) and γ(1) are called initial point and terminal point respectively. Two points
p, q in X are said to be connected by a path if there is a curve γ whose initial point is p
and terminal point is q.
We say that X is path connected if every two points of X can be connected by a path;
it is locally path connected if every point x ∈ X has an open neighborhood U such that U
is path connected.
Example 1.1. Let Rn be the Euclidean n-space with the Euclidean topology. Since every
two points of Rn can be connected by a line segment, Rn is path connected. In fact, given
p, q ∈ Rn , set γ(t) = (1 − t)p + tq for t ∈ [0, 1]. Then γ is a line segment connecting p and q.
Suppose γ0 , γ1 are two paths connecting p, q in X. We say that γ0 is homotopic to γ1
with end points fixed, denoted by γ0 ' γ1 (rel {0, 1}) if there is a continuous map
F : [0, 1] × [0, 1] → X
such that
(1) F (t, 0) = γ0 (t) for all t ∈ [0, 1]
(2) F (t, 1) = γ1 (t) for all t ∈ [0, 1]
(3) F (0, s) = p and F (1, s) = q for all s ∈ [0, 1].
Proposition 1.1. ' (rel {0, 1}) is an equivalence relation.
Let α and β be two paths in X. Define the concatenation of α and β by
(
u(2t)
if 0 ≤ t ≤ 1/2
α?β =
u(2t − 1) if 1/2 ≤ t ≤ 1.
Define the inverse curve α− of α by
α− (t) = α(1 − t),
t ∈ [0, 1].
Proposition 1.2. Let us denote the relation ' (rel {0, 1}) by ∼ . Suppose α1 , α2 , β1 , β2
are paths on X. If α1 , α2 connect p, q with α1 ∼ α2 and β1 , β2 connect q, r with β1 ∼ β2 ,
then α1 ? β1 ∼ α2 ? β2 and α1− ∼ α2− .
A constant path γ on X is a constant map, i.e. γ(t) = p for all t ∈ [0, 1] for some p ∈ X.
Proposition 1.3. Lett α, β, γ be paths connecting p, q and q, r and r, s in X respectively.
Assume that γ0 is a constant path with γ0 = p. Then
(1) γ0 ? α ∼ α ? γ0
(2) α ? α− ∼ γ0
(3) (α ? β) ? γ ∼ α ? (β ? γ).
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FUNDAMENTAL GROUP
A loop in X is a path γ such that γ(0) = γ(1). The space of loops on X at x0 is the set
Ω(X, x0 ) = {γ : [0, 1] → X : γ(0) = γ(1) = x0 }.
The set Ω(X, x0 ) modulo the relation ' (rel {0, 1}) is denoted by π1 (X, x0 ). The set
π1 (X, x0 ) has a group structure defined as follows. For any [α], [β] in π1 (X, x0 ), set [α][β] =
[α ? β], and set [α]−1 = [α− ]. The identity of π1 (X, x0 ) is [ex0 ], where ex0 : [0, 1] → X is the
constant loop ex0 (t) = x0 for t ∈ [0, 1].
Definition 1.1. The group π1 (X, x0 ) is called the fundamental group of X with base point
x0 .
If x0 and x1 can be connected by a path α in X, for any given loop γ ∈ Ω(X, x0 ) at x0
then α ? γ ? α− ∈ Ω(X, x1 ). The path α induces a group homomorphism
α∗ : π1 (X, x0 ) → π1 (X, x1 ),
[γ] 7→ [α ? γ ? α− ].
Proposition 1.4. The group homomorphism α∗ : π1 (X, x0 ) → π1 (X, x1 ) is a group isomorphism.
If X is path connected, the any two points can be connected by a path. In this case, any
x0 , x1 ∈ X, one has π1 (X, x0 ) ∼
= π1 (X, x1 ). We may define
π1 (X) = π1 (X, x0 )
for any point x0 ∈ X. Then π1 (X) is unique up to noncanonical isomorphisms. In general,
if π1 (X, x0 ) ∼
= π1 (X, x1 ) for any two points x0 , x1 ∈ X, we may also define π1 (X) to be
π1 (X, x0 ) for any x0 .
Definition 1.2. A path connected space X is called simply connected if π1 (X) = 0.
Here are some examples of simply connected spaces: Rn is simply connected for n ≥ 1;
the n-dimensional closed unit sphere S n is simply connected for n ≥ 2; every convex set in
Rn is simply connected.
Proposition 1.5. Let X, Y be topological spaces with x0 ∈ X and y0 ∈ Y. Let f : X → Y
be a continuous map such thatf (x0 ) = y0 . Then f induces a group homomorphism
π1 (f ) : π1 (X, x0 ) → π1 (Y, y0 ),
[γ] 7→ [f ◦ γ].
Proposition 1.6. Let X, Y, Z be topological spaces with x0 ∈ X, y0 ∈ Y, and z0 ∈ Z.
Suppose f : X → Y and g : Y → Z are continuous maps such that f (x0 ) = y0 and
g(y0 ) = z0 . Then
π1 (g ◦ f ) = π1 (g) ◦ π1 (f )
A pair (X, x0 ) is called a pointed (topological) space if X is a topological space with a
point x0 ∈ X. A morphism f : (X, x0 ) → (Y, y0 ) is a continuous map f : X → Y with
f (x0 ) = y0 . All pointed spaces together with their morphisms defines a category. Then π1
defines a functor from the category of pointed spaces into the category of groups:
π1 : Category of pointed spaces → Category of groups
sending (X, x0 ) to π1 (X, x0 ) and f : (X, x0 ) → (Y, y0 ) to π1 (f ) : π1 (X, x0 ) → π1 (Y, y0 ).
Proposition 1.7. Let (X, x0 ) and (Y, y0 ) be pointed spaces. Then (X × Y, (x0 , y0 )) is also
a pointed space. Moreover, we have an isomorphism of groups:
∼ π1 (X, x0 ) ⊕ π1 (Y, y0 ).
π1 (X × Y, (x0 , y0 )) =
FUNDAMENTAL GROUP
3
Let f, g : (X, x0 ) → (Y, y0 ) be morphisms of pointed spaces. We say that f is homotopic
to g written as f ' g if there is a continuous map
F : X × [0, 1] → Y
such that
(1) F (x, 0) = f (x) for x ∈ X
(2) F (x, 1) = g(x) for x ∈ X
(3) F (x0 , t) = y0 for t ∈ [0, 1].
Proposition 1.8. Suppose f, g : (X, x0 ) → (Y, y0 ) are morphisms of pointed spaces such
that f ' g. Then
π1 (f ) = π1 (g).
We say that a morphism g : (Y, y0 ) → (X, x0 ) is a homotopy inverse of a morphism
f : (X, x0 ) → (Y, y0 ) if g ◦ f ' id(X,x0 ) and f ◦ g ' id(Y,y0 ) . In this case, we say f :
(X, x0 ) → (Y, y0 ) is a homotopy equivalence, (X, x0 ) and (Y, y0 ) are homotopic equivalent.
Corollary 1.1. The fundamental groups of homotopic equivalent pointed spaces are isomorphic.
Proof. Let f : (X, x0 ) → (Y, y0 ) be a homotopy equivalence with homotopy inverse g :
(Y, y0 ) → (X, x0 ). Then g ◦ f ' id(X,x0 ) and f ◦ g ' id(Y,y0 ) and Proposition 1.8 imply
π1 (g) ◦ π1 (f ) = idπ1 (X,x0 ) ,
π1 (f ) ◦ π1 (g) = idπ1 (Y,y0 ) .
This shows that π1 (g) : π1 (Y, y0 ) → π1 (X, x0 ) is the inverse of the group homomorphism
π1 (f ) : π1 (X, x0 ) → π1 (Y, y0 ).
A topological invariant is a property of a space that depends only on the topology of a
space. In other words, for any space X, we assign to X a property F (X) which can be a set,
a group, a ring, a vector space, a module over a ring, e.t.c such that F (X) can be identified
with (equal to, isomorphic to, e.t.c.) F (Y ) whenever X and Y are homeomorphic. The next
corollary shows that the fundamental group of a (path) connected space is a topological
invariant:
Corollary 1.2. Let X, Y be path connected spaces. If X is homeomorphic to Y, then
π1 (X) ∼
= π1 (Y ).
In other words, the functor π1 is a topological invariant.
Hence if two path connected spaces have non isomorphic fundamental groups, they are
not homeomorphic. Suppose that we know the fundamental group π1 (D) of the closed unit
disk D = {z ∈ C : |z| ≤ 1} is the trivial group (D is a convex set in C and hence is simply
connected) while the fundamental group π1 (S 1 ) of the unit circle S 1 is Z. Corollary 1.2
shows that D is not homeomorphic to S 1 . Of course, we will compute π1 (S 1 ) later.