Graphics Reference
In-Depth Information
In the context of the example in Figure 6.15, to define a space
Z
that corresponds
to having “cut”
Y
along the edge
v
1
v
2
, choose a labeled complex (L,m) with ÔLÔ=
Y
as
in Figure 6.15(c). If (L¢,m¢) is the labeled complex shown in Figure 6.15(d), then we
can let
Z
=ÔL¢Ô =
X
(L¢,m¢)
. To define “pasting” we reverse these steps. Assume that we
are given
Z
=ÔL¢Ô and (L¢,m¢) as shown in Figure 6.15(d) and that we want a space
Y
that corresponds to “pasting together” the two arcs from
v
1
to
v
4
in
Z
. Form the labeled
complex (L≤,m≤) as in Figure 6.15(e) by relabeling the vertex
v
i
¢, which is to be iden-
tified with the vertex
v
i
as
v
i
and define
Y
=
X
(L≤,m≤)
. Note that
X
(L≤,m≤)
X
(L,m)
. In prac-
tice, one often omits all but the relevant labels in figures. In fact, one may not even
specify the triangulation since the homeomorphism type of the resulting space is inde-
pendent of the choice. For example, one could easily use Figure 6.15(f) to indicate the
same type of pasting as Figure 6.15(e).
It should be obvious how to extend the definitions above to the situation where
one wants to cut or paste along arbitrary polygonal curves or, more generally, along
curves in a polyhedron (use an appropriate triangulation) and therefore feel free to
use this terminology in what follows.
6.5
The Classification of Surfaces
The classification of surfaces is not only a fascinating chapter in the history of topol-
ogy but it is also a great place to practice what we learned in the last section. His-
torically, the earliest topological invariants were discovered in the study of surfaces.
We have already given one definition of surfaces in Chapter 5. They are two-
dimensional topological manifolds. For the purposes of this section, we restrict our-
selves to
compact
and
connected
surfaces
without boundary
and we now give a
combinatorial definition for these
Definition.
A
combinatorial surface
(without boundary) is a polyhedron
S
together
with a triangulation (K,j) satisfying:
(1) K is a two-dimensional connected simplicial complex.
(2) Each 1-simplex of K is a face of precisely two 2-simplices of K.
(3) For every vertex
v
in K, the distinct 2-simplices s
1
, s
2
,..., s
s
of K to which
v
belongs can be ordered in such a way that s
i
, 1 £ i £ s, meets s
i+1
in precisely
one 1-simplex, where s
s+1
= s
1
.
A triangulation (K,j) satisfying properties (1)-(3) is called a
proper triangulation
.
Note 1.
It is easy to show that every combinatorial surface is in fact a compact con-
nected two-dimensional topological manifold without boundary. The only points
where the property of having a neighborhood that is homeomorphic to a disk is not
obviously true and needs to be checked are the vertices, but this is where condition
(3) is used. The converse is also true:
6.5.1. Theorem.
(Radó) Every two-dimensional topological manifold (without
boundary) in
R
n
admits a (possibly infinite) triangulation. If the manifold is compact
and connected, then every triangulation is proper.
