Graphics Reference
In-Depth Information
()
[]
(
)
=-
[]
(
(
)
)
q
q
f
s
f
s
q
] ΠS
q
.
for all [s
It is convenient to have C
q
(K) defined for all values of q, including negative values,
even though the only groups that are really interesting to us are the ones with 0 £ q
£ dim K.
Definition.
If q < 0 or q > dim K, define C
q
(K) = 0.
Definition.
For all q, the abelian group (C
q
(K),+) is called the
group of q-chains
of
K. The “vector”
()
=
(
() () ()
)
CK
...,
C KCKCK
,
,
,...
#
-1
0
1
is called the (
oriented
)
chain complex
of K.
The definition of q-chains as functions is neither convenient nor intuitive. We shall
now describe the more common notation that one uses when working with q-chains.
For each oriented q-simplex a=[s
q
] ΠS
q
define a q-chain
()
a
F
Œ
CK
by
()
=
ab
0
1
,
for
b
Œ
S
and
b
π ±
a
,
F
q
()
=
-
()
=-
aa
aa
,
and
F
1
.
F
Such “elementary” q-chains a
F
actually generate C
q
(K). To see this, choose one orienta-
tion for each q-simplex of K and let S
+
be the collection of oriented q-simplices defined
by these choices. (The easiest way to simultaneously pick an orientation for all the sim-
plices of K is to order the vertices of K once and for all and then to take the induced ori-
entation.) The essential property of S
+
is that it is a subset of S
q
satisfying:
(1) If q = 0, then S
+
= S
q
.
(2) If q ≥ 1, then, for any bŒS
q
, either b or -b belongs to S
+
but
not
both.
()
=
Œ
7.2.1.2. Lemma.
CK
Z
.
q
F
+
a
S
q
Proof.
This is easy to prove (Exercise 7.2.1.1). See Appendix B for a clarification of
the notation. A proof can also be found in [AgoM76].
Because the map
ZZ
Æ
Æ
a
a
F
n
n
F
is clearly an isomorphism, it follows from Lemma 7.2.1.2 that if K has n
q
q-simplices,
then C
q
(K) is isomorphic to a free abelian group, which is a direct sum of n
q
copies
