Graphics Reference
In-Depth Information
of
Z
. Since the map aÆa
F
imbeds S
q
in C
q
(K), we shall identify a with a
F
. With this
identification, we now have a rigorous mathematical definition of the notion, referred
to in the motivational part at the beginning of this section, of “formal linear combi-
nations” of oriented q-simplices. Furthermore, by treating the elements of C
q
(K) as
such formal sums, which we shall do in the future, we shall make our computations
more intuitive. The reader should remember, however, that the definition of C
q
(K)
depends only on K and not on any particular choice of orientations.
Now that we know about q-chains, we move on to a definition of the boundary map.
Definition.
The
boundary map
()
Æ
()
∂
q
:
CK
C
K
q
q
-1
is defined as follows:
(1) If 1 £ q £ dim K, then ∂
q
is the unique homomorphism with the property that
q
Â
i
(
[
]
)
=
()
[
ˆ
]
∂
q
vv
◊◊◊
v
1
v
◊◊◊
v
◊◊◊
v
01
q
0
i
q
i
=
0
for each oriented q-simplex [
v
0
v
1
···
v
q
] of K, where “
v
ˆ ” denotes the fact that
the vertex
v
i
has been omitted.
(2) If q £ 0 or q > dim K, then ∂
q
is defined to be the zero homomorphism.
7.2.1.3. Lemma.
(1) The maps ∂
q
are well-defined homomorphisms.
(2) For all q, ∂
q-1
°
∂
q
= 0.
Proof.
Assume that 1 £ q £ dim K, which is the only case where something has to
be proved. Let s be a q-simplex and s
i
a (q - 1)-dimensional face of s. Suppose that
ˆ
s
=
vv
◊◊◊
v
and
s
=
v
◊◊◊
v
◊◊◊
v
.
01
q
i
0
i
q
Let [o] and [o
i
] be the orientations of s and s
i
induced by the orderings
=
(
)
=
(
,...,
ˆ
,...,
)
o
vv
,
,...,
v
and
o
v
v
v
,
01
q
i
0
i
q
respectively.
Claim.
The orientation n=(-1)
i
[o
i
] of s
i
depends only on the orientation m=[o]
and not on the particular ordering o.
First, consider what happens to the orientation [o
i
] when we pass from the order-
ing o to the ordering
¢=
(
)
o
v
0
,...,
v
,...,
v
,...,
v
,
st
<
,
t
s
q
which corresponds to interchanging two vertices
v
s
and
v
t
. If s π i π t, then we have
interchanged two vertices of s
i
, so that [o¢
i
] =-[o
i
] and
