Computational Topology (Jeff Erickson)
Surfaces
There is nothing “below the surface,” my faithful friend—absolutely
nothing
.
— Letter 52,
The Mahatma Letters to A. P. Sinnett
(1882)
5
Surfaces
For the next several lectures, we will move from the Euclidean plane to a wider class of topological
spaces that locally resemble the plane. A topological space
Σ
is called a
2
manifold
if for every point
x
∈
Σ
, there is an open subset
U
⊆
Σ
such that
x
∈
U
and
U
is homeomorphic to
R
2
. More succinctly, a
2manifold is a space that is
locally homeomorphic
to the plane. 2manifolds are also called
surfaces
.
In this lecture, I’ll describe a classical combinatorial description of 2manifolds that is useful both for
abstract mathematical arguments and as the basis of concrete data structure. The same combinatorial
structure can be described in two equivalent ways:
•
Polygonal schema:
A collection of polygons with edges glued together in pairs.
•
Cellular graph embedding:
An embedding of a graph
G
on a surface
Σ
, such that every face of
the embedding is a topological disk;
5.1
Polygonal Schemata
A
polygonal schema
Π
is a finite collection of polygons with oriented sides identified in pairs. More
explicitly, let
f
1
,
f
2
,...,
f
n
denote a finite set of polygons in the plane, called
faces
, whose total number
of sides is even. (I will always refer to the vertices of these polygons as
corners
, and their edges as
segments
.) Formally, an
orientation
of an edge
e
is a linear map from the unit interval
[
0
,
1
]
to
e
;
however, since there are only two such maps, we can think of an orientation of a side
e
as a permutation
of its endpoints, or a labeling of its
endpoints
as
head
(
e
)
and
tail
(
e
)
. A polygonal schema assigns each
side an orientation and a
label
from some finite set, such that each label is assigned exactly twice. (The
standard choice of labels in illustrations are lowercase letters.)
a
b
c
d
e
a
b
c
e
d
A polygonal schema
Π
with signature
(
a
bc
adb
)(
cde
e
)
; arrows indicate edge orientations.
Each polygonal schema
Π
defines a topological space
Σ(Π)
by identifying pairs of sides with matching
labels and orientations. The labels indicate which sides should be identified; the orientations indicate
how
the sides are to be identified. Each pair of identified sides becomes a single path in
Σ(Π)
, which we
call an
edge
. The identification of sides in
Π
induces an identification of corners. Thus, several corners
may be mapped to the same point in
Σ(Π)
, which we call a
vertex
. These vertices and edges define a
graph
G
(Π)
embedded in
Σ(Π)
.
We can encode any polygonal schema by listing the labels and orientations of sides in cyclic order
around each polygon. Whenever we traverse a side along its orientation, we record its label; when we
traverse an edge against its orientation, we record its label with a bar over it. Thus, in the example
schema on the previous page, traversing each polygon counterclockwise, starting from its bottom left
corner, we obtain the signature
(
a
bc
adb
)(
cde
e
)
. Of course, this is not the only possible encoding of
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 Fall '08
 Staff
 Topology, Algebraic Topology, Graph Theory, Manifold, polygonal schema

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