Graphics Reference
In-Depth Information
Definition.
If f :
X
Æ
Y
is a continuous map, then the
homotopy class of f
, denoted
by [f], is the equivalence class of f with respect to
. The set of homotopy classes of
maps from
X
to
Y
will be denoted by [
X
,
Y
].
If
X
consists of a single point
p
, then a homotopy between two maps f, g : {
p
} Æ
Y
is just a path in
Y
from the point
y
0
= f(
p
) to the point
y
1
= g(
p
). See Figure 5.12(b).
In particular, it is easy to see that the set of homotopy classes [{
p
},
Y
] is in one-to-one
correspondence with the path-components of
Y
.
Definition.
A continuous map f :
X
Æ
Y
is called a
homotopy equivalence
if there is
a continuous map g :
Y
Æ
X
with g f
1
X
and f g
1
Y
. In this case we shall write
o
o
X
Y
and say that
X
and
Y
have the same
homotopy type
.
5.7.3. Theorem.
Homotopy equivalence is an equivalence relation on topological
spaces.
Proof.
This is straightforward.
Since the general homeomorphism problem is much too difficult except in certain
very special cases, a weaker classification is based on homotopy equivalence.
Definition.
A space is said to be
contractible
if it has the homotopy type of a single
point.
5.7.4. Example.
The unit disk
D
n
is contractible. To see this we show that it has the
same homotopy type as the point
0
. Define maps f :
D
n
Æ
0
and g :
0
Æ
D
n
by f(
p
) =
0
and g(
0
) =
0
. Clearly, f g = 1
0
. Define h :
D
n
¥ [0,1] Æ
D
n
by h(
p
,t) = t
p
. Then h is a
homotopy between g f and the identity map on
D
n
, and we are done. Another way to
state the result is to say that both f and g are homotopy equivalences.
o
o
Definition.
A subspace
A
of a space
X
is called a
retract
of
X
if there exists a con-
tinuous map r :
X
Æ
A
with r(
a
) =
a
for all
a
in
A
. The map r is called a
retraction of
X
onto
A
.
If
x
0
is any point in a space
X
, the constant map r(
x
) =
x
0
shows that any point
of a space is a retract of the space. A less trivial example is
5.7.5. Example.
The unit circle in the plane is a retract of the cylinder
{
}
(
)
2
2
[]
X
=
x y z
,,
X
+=
Y
1
and z
0 1
,
(5.10)
because we have the retraction r(x,y,z) = (x,y,0).
Definition.
Let
A
be a subspace of a space
X
. A
deformation retraction of
X
onto
A
is a continuous map h :
X
¥
I
Æ
X
satisfying
(
)
=
(
Œ
h
x
,
0
x
,
h
x
,
1
A
,
all
x
Œ
X
,
and
