Graphics Reference
In-Depth Information
[c] is the identity for *. Lemma 7.4.1.4 shows that each element of p
1
(
X
,
x
0
) has an
inverse with respect to *.
Definition.
The group p
1
(
X
,
x
0
) is called the
fundamental group
or
first homotopy
group
of the pointed space (
X
,
x
0
). The point
x
0
is called the
base point
for this fun-
damental group.
So now we have another group associated to a space, but what information does
it give us? Before we compute the group for some spaces we look at some general
properties of it. First of all, the isomorphism type of the fundamental group is inde-
pendent of the base point if the space is path-connected.
7.4.1.6. Theorem.
If
X
is path-connected, then p
1
(
X
,
x
0
) is isomorphic to p
1
(
X
,
x
1
)
for all
x
0
,
x
1
Œ
X
.
Proof.
Let g :
I
Æ
X
be a path from
x
1
to
x
0
. Define a map
(
)
Æ
(
)
T:
p
Xx
,
p
Xx
,
1
0
1
1
by
[
()
=
[
,
T a
g
(
)
Æ
where
a∂
g
:
II
X
is defined by
1
3
Œ
È
˘
˙
()
=
()
a
g
t
g
3
t
,
for t
0
,
,
Í
1
3
2
3
Œ
È
Í
˘
˙
(
)
=
a
31
t
-
,
for t
,
,
2
3
Œ
È
˘
˙
(
)
=-
g
33
t
,
for t
,
1
.
Í
The map a
g
(t) is the path that walks along g(t), then a(t), and then backtracks along
g(t). See Figure 7.21. It is easy to check that T is a well-defined isomorphism (Exer-
cise 7.4.1.1).
Note.
Because of Theorem 7.4.1.6, the base point is often omitted for path-
connected spaces
X
and p
1
(
X
) is used to denote p
1
(
X
,
x
0
) for some
x
0
Œ
X
.
7.4.1.7. Theorem.
A contractible space has a trivial fundamental group.
Proof.
The homotopy that shows the space is contractible to a point easily provides
a homotopy between every closed path in the space with the constant path (Exercise
7.4.1.2).
A point,
R
n
, and
D
n
all have a trivial fundamental group.
7.4.1.8. Corollary.
