Graphics Reference
In-Depth Information
q
N
=
*
q
+
1
q
+
1
q
+
1
q
+
1
q
+
1
q
+
1
q
+
1
q
+
1
A
◊◊◊
A
B
◊◊◊
B
C
◊◊◊
C
C
◊◊◊
C
1
1
1
g
b
r
r
+
1
g
q
+
1
q
+
1
q
q
q
q
q
A
d
0
ˆ
Ê
1
1
M
0
0
O
0
˜
˜
˜
˜
˜
˜
˜
˜
˜
˜
˜
˜
˜
˜
˜
˜
˜
Á
Á
Á
Á
Á
Á
Á
Á
Á
Á
Á
Á
Á
Á
Á
Á
Á
q
q
A
0
d
p
r
q
q
q
A
1
0
r
+
1
q
M
0
0
0
O
q
A
0
1
g
q
q
B
1
M
0
0
0
0
q
B
b
q
q
C
1
M
0
0
0
0
¯
q
Ë
C
g
q
-
1
The detailed version of the incidence matrices makes computing the homology
groups of K easy because it tells us all we need to know about the groups C
q
(K) and
homomorphisms ∂
q
. In particular, it is easy to see that
{
[]
[
]
[]
[
]
}
q
q
q
q
A
,...,
A
,
B
,...,
B
1
1
r
b
q
q
is a basis for H
q
(K). Also,
o([A
q
]) =d
i
q
, with the d
i
q
's being the torsion coefficients of H
q
(K),
o([B
q
]) =•, and
rk H
q
(K) =b
q
.
Finally, incidence matrices can be defined for CW complexes using their cells. All
the information about homology groups that one could deduce in the simplicial case
remain valid. This greatly simplifies computations because the dimensions of these
matrices will be much smaller.
7.2.6
The Mod 2 Homology Groups
A more precise name for the homology groups of a simplicial complex K as defined
in Section 7.2.1 is to call them the homology groups “with coefficients in
Z
.” The
reason is that chain group C
q
(K) consisted of formal
integer
linear combinations of
oriented q-simplices. It is easy to generalize this.
Let G be an arbitrary abelian group and let S
q
again denote the set of oriented q-
simplices of K. If 0 £ q £ dim K, define the group C
q
(K;G) of
q-chains with coefficients
in G
by
