Graphics Reference
In-Depth Information
Figure 7.21.
Proving that the fundamental group
is independent of the base point.
a(t)
g (t)
x
0
= g (1)
x
1
= g (0)
The fundamental group of
S
2
is trivial.
7.4.1.9. Theorem.
Proof.
Here is a sketch of the proof. Let a :(
I
,∂
I
) Æ (
S
2
,
e
1
). The Simplicial Approx-
imation Theorem implies that a is homotopic (relative to ∂
I
) to a map that misses a
point
p
π
e
1
in
S
2
. Since
S
2
-
p
can be contracted to
e
1
this proves that a is homo-
topic to the constant map and the theorem is proved.
Just in case the reader is beginning to think that the fundamental group is
always trivial, we give some examples of simple spaces for which the group is
nontrivial.
7.4.1.10. Theorem.
(1) p
1
(
S
1
) ª
Z
.
(2) p
1
(
P
2
) ª
Z
2
.
(3) The fundamental group of a wedge of two circles (figure eight) is a free group
on two generators.
Proof.
See [Mass67] or [Cair68]. One way to prove part (1) is to show that the iso-
morphism is defined by the degree of the map as sketched in Section 5.7 and defined
rigorously in Section 7.5.1. Later in Corollary 7.4.2.23 we shall see alternate proofs of
(1) and (2).
Next, we look at how the fundamental group behaves with respect to continuous
maps. Let (
X
,
x
0
) and (
Y
,
y
0
) be pointed spaces. Let f : (
X
,
x
0
) Æ (
Y
,
y
0
) be a continuous
map. Define
(
)
Æ
(
)
f
*
:
p
Xx
,
p
Yy
,
1
0
1
0
by
*
[
()
=
[
]
f
a
f
o
a
.
7.4.1.11. Lemma.
The map f
*
is a well-defined homomorphism of groups.
Proof.
This is Exercise 7.4.1.3.
