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(3) The covering transformations of the universal covering space of a space have
no fixed points and are in one-to-one correspondence with the elements of the fun-
damental group of the space. They act transitively on the fibers, that is, any point in
the total space can be mapped into any other point belonging to the same fiber as the
first.
(4) Every conjugacy class of a subgroup of the fundamental group of a space
defines a covering space for that space.
See [Jäni84] for a variety of applications of covering space theory.
7.4.3
Higher Homotopy Groups
There is an important generalization of the fundamental group of a space that leads
to higher-dimensional homotopy groups.
Let n ≥ 2. Given maps a, b : (
I
n
,∂
I
n
) Æ (
X
,
x
0
), define a map
Definition.
(
)
Æ
(
nn
)
ab
*
:
II
,
∂
Xx
,
0
by
1
2
(
)(
)
=
(
)
ab
*
t
,
t
,...,
t
a
2
t
,
t
,...,
t
,
if
0
££
t
,
12
n
12
n
1
1
2
(
)
=
b
21
t
-
,
t
,...,
t
,
if
££
t
1
.
1
2
n
1
Definition.
Let
{
(
)
Æ
(
}
(
)
=
nn
)
p
Xx
,
a
a
:
I
,
∂
I
Xx
,
n
0
0
n
∂
I
be the set of equivalence classes of maps a with respect to the equivalence relation
∂
I
n
. Define a product * on p
n
(
X
,
x
0
) as follows: If [a], [b] Œp
n
(
X
,
x
0
), then
[]
*
[]
=*
[
]
.
ab ab
7.4.3.1. Theorem.
The operation * on p
n
(
X
,
x
0
) is well defined and makes p
n
(
X
,
x
0
)
into a group called the
nth homotopy group
of the pointed space (
X
,
x
0
).
Proof.
The proof is similar to the one for the fundamental group. Exercise 7.4.3.1.
There is a perhaps easier way to visualize the product in p
n
(
X
,
x
0
). First, we need
a definition.
Definition.
Let
X
,
Y
, and
Z
be pointed spaces with base points
x
0
,
y
0
, and
z
0
, respec-
tively. If f :
X
Æ
Z
and g :
Y
Æ
Z
are continuous maps with f(
x
0
) =
z
0
and g(
y
0
) =
z
0
,
then define a map
