Graphics Reference
In-Depth Information
Q
8
e
j
= e
j
(8)
w
j
= w
j
(8)
Q
2
w
2
e
1
(2)
w
3
w
1
e
2
e
3
e
4
e
1
= e
9
w
0
(2)
w
4
w
0
= w
8
w
1
(2)
e
5
e
8
e
7
e
6
w
5
w
7
w
6
e
2
(2)
(a)
(b)
Figure 6.18.
Regular k-gons.
the circles
A
1
and
A
2
in Figure 6.17(a) simultaneously, produces Figure 6.17(c). Clearly,
Figures 6.17(b) and (c) can be considered representations of a torus if we understand
the labeling and arrows properly. Notice that we are now labeling the edges and not
the vertices. This will be more intuitive for the cutting and pasting we want to do. It
is easy to pass between the two types of labeling however. The main advantage to
labeling vertices is that the results about geometric realizations of labeled complexes
are easier to state and prove since it is more straightforward to relate this labeling to
abstract complexes. At any rate, our first step in classifying surfaces will be to show
that an arbitrary surface
S
can be represented by a labeled polygon similar to the one
we got for the torus. We will also show that this geometric presentation is equivalent
to an “algebraic” presentation that consists of a formal symbol.
Let k ≥ 3 and let
Q
k
denote the “standard” regular k-gon (k-sided polygon), namely,
the convex hull of the points
()
=
(
)
Œ
1
w
j
k
cos 2 j/k, sin 2
p
p /
j k
S
.
Let
()
=
()
()
e
k
ww
k
k
j
j
-1
j
denote the jth edge of
Q
k
. Figure 6.18(a) shows
Q
8
. Since it will also be convenient to
have a two-sided “polygon,” let
Q
2
=
D
2
be the “polygon” with vertices
()
=
()
()
=-
(
)
w
210
,
and
w
2
10
,
0
1
and “edges”
()
=
1
()
=
1
e
2
S
and
e
2
S
.
1
+
2
-
See Figure 6.18(b).
The next lemma basically proves that any surface can be flattened out into the
plane by cutting it appropriately. More precisely, it shows that a surface can be rep-
