Graphics Reference
In-Depth Information
sphere and the connected sum of tori are orientable surfaces whereas the connected
sum of projective planes is not. We shall not be able to prove this, however, until the
next chapter and after we have a precise definition of orientability.
Definition.
If
S
is a surface, define the
genus
of
S
to equal 0 if
S
is homeomorphic
to the sphere and equal to n if
S
is homeomorphic to a connected sum of n tori or n
projective planes, n ≥ 1.
Intuitively, if the genus of an orientable surface is n, then the surface is homeo-
morphic to a
sphere with n handles
. See Figure 6.23. Since the projective plane is the
union of a disk and a Moebius strip, it is often referred to as a sphere with a
cross-
cap
in the literature. With this terminology, a nonorientable surface of genus n is called
a
sphere with n crosscaps
.
A simple formula relates the genus g of a surface
S
to its Euler characteristic c:
(
)
g
=-
2
c
c
/,
2
if is orientable
otherwise
S
(6.3)
=-
2
,
.
The next proposition summarizes what we just shown.
6.5.9. Proposition.
Table 6.5.2 shows the Euler characteristic, orientability, and
genus of the listed surfaces.
S
1
¥ S
1
(S
1
¥ S
1
) # (S
1
¥ S
1
)
Genus 2
Figure 6.23.
Two surfaces as spheres with
handles.
Genus 1
Table 6.5.2
The geometric invariants of some surfaces.
Euler
Surface
characteristic
Orientability
Genus
S
2
2
orientable
0
(
11
)
(
11
)
L
12
SS# #SS
¥
¥
2
-
2n
orientable
n
44443
4444
n times
2
644
2
P# #P
L
2
-
n
nonorientable
n
