Computational Topology (Jeff Erickson)
Surface Classification
Q: So you like the grass?
A: Yeah. Well, actually I don’t care what surface I’m playing on.
— Daniela Hantuchová (May 17, 2001)
press conference at the Tennis Masters Series, Rome, Italy
6
Surface Classification
In this lecture, I’ll give a proof of the most fundamental result in 2manifold topology:
Surface Classification Theorem.
Every compact connected 2manifold can be constructed from the
sphere by attaching either a finite number or handles or a finite number of Möbius bands.
A complete proof of this theorem relies on Kerékjártó and Rado’s proof that any compact 2manifold
has a triangulation, but the classification of
triangulated
2manifolds is much older. Different sources
attribute the first proof to Brahana
[
1
]
, Dehn and Heegard
[
2
]
, and Dyck
[
3
]
.
1
Brahana’s proof is the
one that appears in most topology textbooks, thanks to its appearance in an early textbook of Siefert and
Threlfall. Several other proofs are known; we refer in particular to a completely selfcontained proof by
Thomassen
[
10
]
and Conway’s ‘zero irrelevancy’ proof
[
5
]
.
6.1
Attaching Handles and Möbius Bands
To
attach a handle
to a surface
Σ
, find two disjoint closed disks on
Σ
, delete the interiors of the disks,
and glue an annulus to the two boundary circles. The inverse operation—deleting an annulus and gluing
disks onto each boundary circle—is called
detaching a handle
.
Left to right: Attaching a handle.
Right to left: Detaching a handle.
To
attach a Möbius band
to a surface
Σ
, find a single closed disk in
Σ
, delete its interior, and glue a
Möbius band to its boundary circle. The inverse operation—deleting a Möbius band and gluing a disk
onto its boundary circle—is called
detaching a Möbius band
.
Left to right: Attaching a Möbius band.
Right to left: Detaching a Möbius band.
Let
Σ(
g
,
h
)
denote the surface obtained from the sphere by attaching
g
handles and
h
Möbius bands.
(You should convince yourself that it doesn’t matter where the handles and Möbius bands are attached,
or in which order.) For example,
Σ(
0
,
0
)
is the sphere;
Σ(
1
,
0
)
is the torus;
Σ(
0
,
1
)
is the
projective
plane
; and
Σ(
0,2
)
is the
Klein bottle
.
1
However, the classification of orientable surfaces was previously
stated
in various forms by Riemann, Klein, Jordan
[
6
]
,
and Möbius
[
9
]
.
1
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Computational Topology (Jeff Erickson)
Surface Classification
6.2
Edge Surgery
My proof of the classification theorem starts with a polygonal schema for the 2manifold and modifies
its graph through a series of
edge contractions
and
edge deletions
. Both operations remove edges from
the graph, but in different ways.
Let
e
be an edge in the graph
G
. If
e
separates two distinct faces
f
and
f
0
, we can
delete
e
to obtain
a new graph
G
\
e
(pronounced ‘
G
without
e
’). The two faces
f
and
f
0
are merged into a single face in
G
\
e
. If the embedding of
G
is represented by a rotation system, we simply delete both occurrences of
e
.
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 Fall '08
 Staff
 Topology, Graph Theory, Planar graph, Leonhard Euler, Euler characteristic, Mobius band

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