Graphics Reference
In-Depth Information
=
()
[]
=-
()
[]
=-
i
i
¢
n
¢
1
o
1
o
n
.
i
i
If s = i, then the new ordering of the vertices of s is
(
)
v
,...,
v
,
v
,
v
,...,
v
,
v
...,
v
0
i
-
1
t
i
+
1
t
-
1
t
+
1
q
and
=
()
[]
=
()()
t
t
t i
--
1
i
¢
[]
=-
()
[]
=-
n
¢
1
o
1
1
o
1
o
n
.
i
i
i
A similar equation holds if t = i. Therefore, interchanging two distinct vertices of o
always results in a change of sign of n. Since the representative o for m is well defined
up to an even permutation of the vertices and since every even permutation is the
composition of an even number of transpositions, the Claim is proved.
Definition.
The orientation n of s
i
is called the
orientation
of s
i
induced by the
orientation
m
of
s
.
The Claim implies that ∂
q
is well defined because the “boundary” (q - 1)-chain
q
i
Â
()
[
]
ˆ
1
vvv
◊◊◊
◊◊◊
0
i
q
i
=
0
is well defined for each oriented q-simplex [
v
0
v
1
...
v
q
]. Furthermore, because this
(q - 1)-chain is exactly what the (oriented) boundary of the oriented q-simplex should
be intuitively, we are justified in calling the maps ∂
q
“boundary maps.” For example,
the definition of ∂
2
implies that
[
]
)
=
[
]
-
[
]
+
[
]
(
∂
2
vvv
vv
vv
vv
012
12
02
01
=
[
]
+
[
]
+
[
]
,
vv
vv
vv
12
20
01
which is what we want. Part (1) of the lemma is now proved since to define a homo-
morphism on a free group it suffices to specify it on a basis (Theorem B.5.9).
To prove part (2), we may assume that q ≥ 2. It suffices to show that
(
)
[
(
]
)
=
∂∂
q
o
vv
...
v
0
-
1
q
0
1
q
for every oriented q-simplex [
v
0
v
1
...
v
q
] of K, since these generate C
q
(K). Now
q
Ê
Á
ˆ
˜
i
Â
(
)
[
(
]
)
=
()
[
ˆ
]
∂∂
o
vv
...
v
∂
1
v
◊◊◊
v
◊◊◊
v
q
-
1
q
0
1
q
q
-
1
0
i
q
i
=
0
q
Â
i
=
()
(
[
ˆ
]
)
1
∂
vvv
◊◊◊
◊◊◊
q
-
10
i
q
q
i
-
1
Ê
Á
Â
i
Â
j
=
()
()
[
ˆ
ˆ
]
1
1
vvvv
◊◊◊
◊◊◊
◊◊◊
0
j
i
q
i
=
0
j
=
0
q
]
ˆ
Â
j
-
1
()
[
ˆ
ˆ
+
1
vvvv
◊◊◊
◊◊◊
◊◊◊
.
˜
0
i
j
q
ji
=+
1
