Graphics Reference
In-Depth Information
Definition.
The homomorphism f
*
is called the homomorphism
induced
by the con-
tinuous map f.
7.4.1.12. Theorem.
(1) If f, g : (
X
,
x
0
) Æ (
Y
,
y
0
) are homotopic continuous maps, then
(
)
Æ
(
)
f
=
g
:
p
Xx
,
p
Yy
,
.
*
*
1
0
1
0
(2) If f : (
X
,
x
0
) Æ (
Y
,
y
0
) and g : (
Y
,
y
0
) Æ (
Z
,
z
0
) are continuous maps, then
=
()
(
Æ
(
)
gf
o
gf
o
:
p
Xx
,
p
Zz
,
.
**
1
0
1
0
*
Proof.
This is Exercise 7.4.1.4.
We can now prove the homotopy invariance of the fundamental group.
7.4.1.13. Theorem.
Homotopy equivalent spaces have isomorphic fundamental
groups.
Proof.
This is an easy consequence of Theorem 7.4.1.12.
Note that Theorem 7.4.1.7 is actually an easy consequence of Theorem 7.4.1.13
since a contractible space has the same homotopy type as a point.
There is a nice relationship between the fundamental group of two spaces and
that of their product.
7.4.1.14. Theorem.
Let (
X
,
x
0
) and (
Y
,
y
0
) be pointed spaces and let
p
:
XY X
¥Æ
and
q
:
XY Y
¥Æ
be the natural projections defined by p(
x
,
y
) =
x
and q(
x
,
y
) =
y
. Then the map
(
)
Æ
(
)
¥
(
)
sp
:
XYx y
¥
,
¥
p
Xx
,
p
Yy
,
1
0
0
1
0
1
0
defined by
[
()
=
[
(
] [
]
)
sa
pq
o
a
,
o
a
,
is an isomorphism.
Proof.
See [Mass67].
Theorem 7.4.1.14 enables us to compute many more fundamental groups.
p
1
(
S
1
¥
S
1
) ª
Z
≈
Z
.
7.4.1.15. Corollary.
