Graphics Reference
In-Depth Information
Proof.
The corollary is an easy consequence of Theorems 7.2.3.5 and 7.2.3.8.
Next, we return to the Euler characteristic as defined in Chapter 6. We are now
in a position to put this invariant in a more general context. What we had in Chapter
6 was a combinatorial concept defined for surfaces that was easy to compute by some
simple counting and yet was claimed to be a topological invariant. We can now define
that topological invariant in a rigorous manner.
Definition.
If K is a simplicial complex, let n
q
(K) denote the number of q-simplices
in K and define the
Euler-Poincaré characteristic
of K, c(K), by
dim
K
q
Â
()
=
()
()
c K
1
n
K
.
q
q
=
0
What makes c(K) a topological invariant is the fact that it is related to the Betti
numbers b
q
(K) of K.
7.2.3.10. Theorem.
(The Euler-Poincaré Formula) Let K be a simplicial complex.
Then
dim
K
Â
q
()
=
()
()
c
K
1
b
K
.
q
q
=
0
Proof.
By definition, the boundary map ∂
q
:C
q
(K) Æ B
q-1
(K) is onto and has kernel
Z
q
(K) and the natural projection Z
q
(K) Æ H
q
(K) is onto and has kernel B
q
(K). There-
fore, Theorem B.5.8 implies that
(
()
)
=
(
()
)
+
(
()
)
rank C
K
rank B
K
rank Z
K
and
q
q
-1
q
(
()
)
=
(
()
)
+
(
()
)
rank Z
K
rank H
K
rank B
K
.
q
q
q
These identities and the fact that rank (C
q
(K)) = n
q
(K) gives us that
dim
K
Â
q
()
=
()
()
c K
1
n
K
q
q
=
0
dim
K
Â
q
()
(
()
)
=
1
rank C
K
q
q
=
0
dim
K
Â
q
()
[
(
()
)
+
(
()
)
+
(
()
)
]
=
1
rank B
K
rank H
K
rank B
K
q
-
1
q
q
q
=
0
dim
K
dim
K
Â
q
Â
q
()
(
()
)
+
()
[
(
(
)
)
+
(
()
)
]
=
1
rank H
K
1
rank B
K
rank B
K
q
q
-
1
q
q
=
0
q
=
0
dim
K
Â
q
()
(
()
)
=
1
rank H
K
q
q
=
0
