Graphics Reference
In-Depth Information
Figure 7.18.
Associativity of closed path
composition.
s
2
h
h is constant
along these
lines
a
b
g
1
t
a
g
0
b
1
-1
4
t
££
+
s
1
Ê
Ë
ˆ
¯
()
=
hts
,
a
,
for
0
t
,
s
+
1
4
s
+
1
££
+
s
2
(
)
=
b
4
t
-
s
-
1
,
for
t
,
4
4
4
ts
s
--
-
2
s
+
2
Ê
Ë
ˆ
¯
=
g
,
for
££
t
1
,
2
4
is a homotopy between (a*b) *g and a*(b*g). See Figure 7.18.
Define the constant path c :
I
Æ
X
by c(t) =
x
0
.
7.4.1.3. Lemma.
For any a :(
I
,∂
I
) Æ (
X
,
x
0
), a*c
∂
I
a
∂
I
c *a.
Proof.
Define a map h by
2
t
££
+
s
1
Ê
Ë
ˆ
¯
()
=
hts
,
a
,
for
0
t
,
s
+
1
2
s
+
1
=
x
,
for
££
t
1
.
0
2
Then h is a homotopy between a*c and a. See Figure 7.19. A similar homotopy can
be defined between c *a and a, proving the lemma.
7.4.1.4. Lemma.
Given a, define b by b(t) =a(1 - t). Then a*b
∂
I
c
∂
I
b*a.
Proof.
First observe that
1
2
(
)( )
=
()
ab
*
t
a
2
t
,
for
0
££
t
,
1
2
(
)
=-
a
22
t
,
for
££
t
1
.
