Graphics Reference
In-Depth Information
()
=
h
aa
,
,
all
aA
Œ
.
In this case
A
is called a
deformation retract
of
X
.
The argument in Example 5.7.4 also shows that the map f :
D
n
Æ
0
defined by
f(
p
) =
0
is a deformation retraction of
D
n
onto
0
.
5.7.6. Example.
The unit circle is a deformation retract of the cylinder
X
defined
by equation (5.10). To see this simply define h :
X
¥
I
Æ
X
by h((x,y,z),t) = (x,y,(1-t)z).
5.7.7. Theorem.
Let
A
be a subspace of a space
X
. If
A
is a deformation retract of
X
, then the inclusion map i :
A
Æ
X
is a homotopy equivalence. In particular,
A
and
X
have the same homotopy type.
Proof.
Let h :
X
¥
I
Æ
X
be a deformation retraction of
X
onto
A
. Define f :
X
Æ
A
by f(
x
)
= h(
x
,1). Since f i=1
A
and h is homotopy between i f and 1
X
, we are done.
o
o
Intuitively speaking, a subset
A
is a deformation retract of a space
X
if we can
shrink
X
down to
A
without “cutting” anything. We shall see later (Corollary 7.2.3.3
and Theorem 7.2.3.4) that a circle does not have the same homotopy type as a point.
Therefore, no point of the circle is a deformation retract of the circle. The only way
to “shrink” the circle to a point would be to cut it first.
Often it is convenient to talk about “pointed” homotopies, or more generally
“relative homotopies.”
Definition.
The notation f : (
X
,
A
) Æ (
Y
,
B
) will mean that f is a map from
X
to
Y
and
f(
A
)
B
.
Definition.
Let f, g : (
X
,
A
) Æ (
Y
,
B
) be continuous maps. A
homotopy between f and
g relative
A
is a continuous map
¥
[
Æ
h:
X
0
,
Y
such that h(
x
,0) = f(
x
), h(
x
,1) = g(
x
), and h(
a
,t)
B
for all
x
X
,
a
A
, and
t
[0,1]. In that case, we shall also say that
f is homotopic to g relative
A
and write
f
A
g.
5.7.8. Theorem.
The homotopy relation
A
is an equivalence relation on the set of
continuous maps f : (
X
,
A
) Æ (
Y
,
B
).
Proof.
The proof is similar to the proof of Theorem 5.7.2. We just have to be careful
that the homotopies keep sending
A
to
B
.
Definition.
The set of homotopy classes of maps f : (
X
,
A
) Æ (
Y
,
B
) with respect to
the equivalence relation
A
will be denoted by [(
X
,
A
),(
Y
,
B
)].
A natural question to ask at this point is how many homotopy classes of maps
there are between spaces in general and what this number measures. We are not ready
