Graphics Reference
In-Depth Information
(
)
=
{
-
()
=-
()
}
C
K G
;
f
:
S
Æ ≥
G if
q
1, then f
a
f
a
for all in S
q
a
.
q
q
Clearly, C
q
(K) is just C
q
(K;
Z
). The elements of C
q
(K;G) can be thought of as formal
sums
Â
g
a,
(7.4)
a
+
a
Œ
S
q
where g
a
ΠG, motivating the terminology. Elements are added by collecting the coef-
ficients of the a and then adding them using the group addition. Pretty much every-
thing we did earlier carries over without any problem if we simply replace
Z
by G.
There is a boundary homomorphism
(
)
Æ
(
)
∂
q
:
CKG
;
C
KG
;
q
q
-1
and we can define the subgroup Z
q
(K;G) of
q-cycles
and the subgroup B
q
(K;G) of
q-
boundaries with coefficients in G
. Again, B
q
(K;G) Ã Z
q
(K;G). Finally,
Definition.
The group
(
)
ZKG
BKG
;
;
q
q
(
)
=
HKG
;
.
p
(
)
is called the
qth homology group of K with coefficients in G
.
Simplicial maps f : K Æ L induce homomorphisms
(
)
Æ
(
)
f
:
HKGHLG
;
;
,
*
q
q
q
and one can again show that the homology groups H
q
(K;G) are topological invariants
and so we can associate a unique (up to isomorphism) group H
q
(
X
;G) to every poly-
hedron
X
. They are important new invariants associated to a space even though it
turns out that each is completely determined by H
q
(
X
;
Z
) and H
q-1
(
X
;
Z
) by the so-
called “universal coefficient theorem.” A very important special case is the case where
G =
Z
2
and it is worthwhile to look at that case more carefully.
Let K be simplicial complex. Since +1 is the same as -1 in
Z
2
, there is no need to
orient the simplices of K to define C
q
(K;
Z
2
). It follows that if T
q
is the set of all q-sim-
plices in K, then C
q
(K;
Z
2
) can be identified with the set of all maps
gT
q
:
Æ
Z
2
,
that is, an element of C
q
(K;
Z
2
) (usually called a
mod 2 q-chain
) can be thought of as
consisting of a linear sum of q-simplices of K. To add mod 2 q-chains we simply collect
the coefficients of like simplices but must remember that s + s = 0 for every s ΠT
q
because 2 = 0 in
Z
2
. For example, if K is the simplicial complex determined by the
simplex
v
0
v
1
v
2
v
3
, then
