Graphics Reference
In-Depth Information
6.5.7. Theorem.
Every surface is homeomorphic either to the sphere, or to a con-
nected sum of tori, or to a connected sum of projective planes.
Proof.
See [AgoM76].
To finish the classification of surfaces we need to show that the surfaces in
Theorem 6.5.7 are nonhomeomorphic.
Definition.
Let
S
be a surface and let (K,j) be any proper triangulation of
S
. Let
n
i
(K) denote the number of i-simplices in K. The
Euler characteristic of
S
,
c
(
S
),
is
defined by
()
=
()
-
()
+
()
c SnKnKnK
.
0
1
2
6.5.8. Proposition.
The Euler characteristic for a surface is a well-defined integer
that does not depend on the triangulation of the surface that is chosen for the defi-
nition. Furthermore,
(1) c(
S
2
) = 2 , c(
S
1
¥
S
1
) = 0, and c(
P
2
) = 1.
(2) If
S
1
and
S
2
are surfaces, then c(
S
1
#
S
2
) =c(
S
1
) +c(
S
2
) - 2.
Proof.
The fact that the Euler characteristic is well defined is a special case of a
much more general result proved later in Section 7.4. Part (1) is easily verified from
triangulations of the spaces in question. (Any reader who has trouble finding a trian-
gulation for the torus or projective plane can find one in Section 7.2.) To prove (2),
let K
1
, K
2
, L, s
1
, and s
2
be as in the definition of the connected sum. The complexes
K
i
and L triangulate the surfaces
S
i
and
S
1
#
S
2
, respectively, and the 2-simplex s
i
belongs to K
i
. Since L has all the 2-simplices of K
1
and K
2
except for s
1
and s
2
, it follows
that
()
=
(
+
(
-
nL nK
nK
2
.
2
2
1
2
2
But, in L, the boundary of s
1
has been identified to the boundary of s
2
. Therefore,
()
=
(
+
(
-
nL nK
nK
3
1
1
1
1
2
and
()
=
(
+
(
-
nL nK
nK
3
,
0
0
1
0
2
because one does not want to count the 0- and 1-simplices in the boundary of s
1
and
s
2
twice. The three equations easily lead to the result in part (2) of the proposition.
It follows from Proposition 6.5.8 and Theorem 6.5.7 that it is easy to compute the
Euler characteristic of any surface.
We are almost ready to prove the second part of the classification theorem for
surfaces. Before we do, we need to bring up orientability again and also define a
commonly used term in connection with surfaces, namely the “genus.” First of all, the
