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Figure 7.17.
The homotopies in Lemma 7.4.1.2.
s
s
g
f
1
1
g
f
t
t
0
a
0
b
1
1
s
a¢*b¢
h
1
f
g
t
0
1
a*b
(
) Æ (
)
h:
II
¥¥»¥
,
0
I
1
I
Xx
,
0
defined by
1
2
() = (
)
h t s
,
f
2
t s
,
,
if
0
££
t
,
1
2
(
)
=-
gt
21
,,
s
f
££
t
1
,
is a homotopy between a*b and a¢*b¢. See Figure 7.17.
7.4.1.2. Lemma.
Given maps a, b, g :( I ,∂ I ) Æ ( X , x 0 ), then (a*b) *g
I a*(b*g).
Proof.
It is easy to check that
1
4
(
(
) *
)( ) =
()
ab g
*
t
a
4
t
,
for
0
££
t
,
1
4
1
2
(
)
=
b
41
t
-
,
for
££
t
,
1
2
(
)
=-
g
21
t
,
for
££
t
1
,
and
1
2
(
(
)
)( ) =
()
abg
**
t
a
2
t
,
for
0
££
t
,
1
2
3
4
(
)
=
b
42
t
-
,
for
££
t
,
3
4
(
)
=
g
43
t
-
,
for
££
t
1
.
The map h defined by
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