Graphics Reference
In-Depth Information
Let s be a q-simplex in
R
n
. An orientation of s cannot just be an ordering of the
set V of its vertices because there are many orderings. We need an equivalence rela-
tion on the set of orderings. Note that any two orderings of V differ by a permutation
of V.
Definition.
An
orientation
of s is an equivalence class of orderings of the vertices of
s, where two orderings are said to be equivalent if they differ by an even permutation
of the vertices.
The fact that the inverse of an even permutation is an even permutation easily
implies that we are dealing with an equivalence relation. Furthermore, there are only
two equivalence classes. The two possible orientations of a q-simplex s =
v
0
v
1
···
v
q
,
q ≥ 1 are determined by the orderings (
v
0
,
v
1
,...,
v
q
) and (
v
1
,
v
0
,
v
2
,
v
3
,...,
v
q
). If m is one
orientation of s, then it will be convenient to let -m denote the other. we have -(-m)
=m. A 0-simplex has only one orientation. Note the similarity between the definition
of the orientation of a simplex with the definition of the orientation of a vector space.
Definition.
An
oriented q-simplex
[s] is a pair (s,m), where s is a q-simplex and
m is an orientation of s. The notation [
v
0
v
1
···
v
q
] denotes the oriented q-simplex (
v
0
v
1
···
v
q
,m) where m the orientation determined by the ordering (
v
0
,
v
1
,...,
v
q
). If q = 0,
then there is only one orientation, and we shall always write simply
v
0
instead of [
v
0
].
If q ≥ 1 and if [s] is an oriented q-simplex, then -[s] is defined to be the oriented q-
simplex consisting of s together with the opposite orientation, that is, if [s] = [
v
0
v
1
···
v
q
], then -[s] = [
v
1
v
0
v
2
v
3
···
v
q
]. For uniformity of notation, [s] may also be
denoted by +[s].
Now, let K be a simplicial complex and let S
q
denote the set of oriented q-
simplices of K.
Definition.
A
q-chain
of K, 0 £ q £ dim K, is a function f : S
q
Æ
Z
with the additional
property that if q ≥ 1, then
(
-
[]
)
=-
[]
(
)
q
q
f
s
f
s
q
] in S
q
. The set of all q-chains of K is denoted by C
q
(K). Given f, g ΠC
q
(K),
define the sum
for every [s
fgS
q
+
:
Æ
Z
by
)
[]
(
)
=
(
[]
)
+
(
[]
)
(
q
q
q
fg
+
s
f
s
g
s
.
7.2.1.1. Theorem.
(C
q
(K),+) is an abelian group.
Proof.
First of all, it is easy to see that f + g ΠC
q
(K). The additive identity for + is
the zero function, which maps all oriented q-simplices to zero. The additive inverse
of any f in C
q
(K), denoted by -f, is defined by the formula
