Graphics Reference
In-Depth Information
Figure 7.30.
Proving the commutativity of
homotopy groups.
t
n
t
n
t
n
t
2
t
2
t
2
x
0
g
g
f
g
f
f
t
1
t
1
t
1
t
n
t
n
t
2
t
2
x
0
g
g
f
f
t
1
t
1
Continuous maps induce homomorphisms on the homotopy groups in a natural
way, similar to how it was done in the case of the fundamental group. Let (
X
,
x
0
) and
(
Y
,
y
0
) be pointed spaces. Given a continuous map
(
)
Æ
(
)
f:
Xx
,
Yy
,
0
0
define
(
)
Æ
(
)
f
:
p
Xx
,
p
Yy
,
,
n
≥
0
,
*
n
0
n
0
by
*
[
()
=
[
]
f
a
f
o
a
.
7.4.3.3. Lemma.
The map f
*
is well defined. It is a homomorphism of groups when
n ≥ 1.
Proof.
Exercise 7.4.3.2.
Definition.
The map f
*
is called the homomorphism
induced
by the continuous map
f.
The fact that the higher (n ≥ 2) homotopy groups are abelian sets them apart from
the fundamental group. In other ways, they satisfy similar properties however. For
example, one can show, just like in Theorem 7.4.1.14, that there are isomorphisms
(
)
Æ
(
)
¥
(
)
p
XYx y
¥
,
¥
p
Xx
,
p
Yy
,
.
n
0
0
n
0
n
0
(By the way, no such isomorphism exists for homology. A theorem, the Künneth
theorem, relates the homology of the product of two spaces to that of the spaces but
it is much more complicated.) There are also natural homomorphisms
(
)
Æ
()
mp
:
Xx
,
H
X
,
(7.10)
n
0
n
