TOPOLOGICAL CLASSIFICATION OF SURFACES
“A topologist is a person who does not know the difference
between a donut and a coffee cup.”
— John Kelly
1. T HE E ULER
CHARACTERISTIC
In this section, we will discuss some elementary aspects of algebraic topology (or combinatorial topology) — with emphasis on the
classification of surfaces. This topological classification of surfaces
was one of the triumphs of 19th mathematics.
We begin with the basic definition of topological equivalence:
Definition 1. A homeomorphism
f : X 1 → X2
between topological spaces is a continuous map for which there exists
a continuous inverse, i.e. a continuous map
g: X2 → X1
such that f ◦ g = 1: X2 → X2 and g ◦ f = 1: X1 → X1 . If a homeomorphism exists between these spaces, they are said to be homeomorphic.
The map f is a local homeomorphism if, for every point x ∈ X1 ,
there exists an open set x ∈ U ⊂ X1 such that f (U ) ⊂ X2 is open and
f : U → f (U ) is a homeomorphism.
Remark. Homeomorphic spaces are regarded as equivalent in topology. This is a very loose form of equivalence and is the reason topology is often described as “rubber sheet” geometry.
Now we can define a surface:
Definition 2. A closed surface, S, is a compact Hausdorff space with
the property that each point p ∈ S has a neighborhood that is homeomorphic to an open disk in R2 . A surface with boundary, S, is a
compact Hausdorff space such that each point has a neighborhood
homeomorphic to an open set in the upper-half-plane R2+ = {(x, y) ∈
R2 |x ≥ 0}.
In order to study surfaces, it is convenient to triangulate them — to
break them into unions of (possibly curved) triangles — see figure 1.4
on page 4 for an example of a triangulated surface.
1
2
TOPOLOGICAL CLASSIFICATION OF SURFACES
F IGURE 1.1.
F IGURE 1.2.
Simplices
A planar graph
We begin by introducing some simple topological concepts:
Definition 3. A 0-simplex is a point. A 1-simplex is a line-segment
with two 0-simplices as its boundary. A 2-simplex is a two dimensional
triangle with three 1-simplices as its boundaries — see figure 1.1.
Remark. One can easily define n-simplices as the set of points
(x1 , . . . , xn+1 ) in Rn+1 satisfying the equations
n+1
X
xi = 1
i=1
xi ≥ 0 for i = 1, . . . , n + 1
but we will only need the definition for n ≤ 2.
Definition 4. A (planar) graph is a union of 1-simplices with the property that two 1-simplices can only intersect at their endpoints — see
figure 1.2. The set of regions associated with a planar graph is the set
of components that remain when the graph is deleted from the plane.
The set of 0-simplices in a graph are called its vertices, and the set
of 1-simplices are called its edges.
Remark. For instance, the graph in figure 1.2 has two regions.
TOPOLOGICAL CLASSIFICATION OF SURFACES
F IGURE 1.3.
3
Property of a triangulation
If a surface is triangulated, we can define:
Definition 5. If G is a planar graph, its Euler characteristic, χ(G) is
defined by
χ(S) = V − E + R
where V is the number of vertices, E is the number of edges, and R is
the number of regions of the graph.
Remark. Euler first defined this in connection with his work on graph
theory — see [2]. The Euler characteristic of the graph in figure 1.2
on the facing page is
χ(G) = 6 − 6 + 2 = 2
Euler proved that all planar connected graphs have an Euler characteristic of 2.
In studying surfaces, it will be advantageous for them to be triangulated:
Definition 6. If S is a surface, a triangulation, T , of S is an expression
S=
t
[
σi2
i=1
where the σi2 are 2-simplices and two such 2-simplices may only meet
in a common 1-simplex on their boundary — see figure 1.3.
Remark. Figure 1.4 on the following page shows a triangulated surface. The proof that all surfaces can be triangulated is surprising
difficult (the paper [5] contains the first published proof and [1] has
the shortest), so we will simply assume that surfaces are triangulated.
We will assume all surfaces are compact — which means that they
can be triangulated with a finite number of triangles.
4
TOPOLOGICAL CLASSIFICATION OF SURFACES
F IGURE 1.4.
F IGURE 1.5.
A triangulated surface of genus 2
Refinement of a triangulation
Although a triangulation of a surface is not a planar graph, it certainly divides a surface into “regions” in a one to one correspondence
with the 2-simplices of the triangulation.
Definition 7. If S is a surface with a triangulation, T , the Euler characteristic, χ(S, T ), is defined by
χ(S, T ) = V − E + F
where F is the number of 2-simplices in the triangulation, E is the
number of edges (i.e. 1-simplices) and V is the number of vertices
(0-simplices).
We will fairly quickly show that this Euler characteristic only depends on the surface, S, and not the triangulation. To that end, we
define:
Definition 8. If T is a triangulation of a surface, S, another triangulation, T 0 , of S is a refinement of T if:
• every vertex of T is in T 0
• every 2-simplex of T 0 is a subset of a corresponding 2- simplex
of T
Remark. Figure 1.5 illustrates refinement.
TOPOLOGICAL CLASSIFICATION OF SURFACES
Case A
Case B
F IGURE 1.6.
5
Case C
Simplex-deletion
Refinement does not change the Euler characteristic:
Proposition 9. If T is a triangulation of a surface, S, and T 0 is a refinement of T then
χ(S, T ) = χ(S, T 0 )
Proof. Let σ be a 2-simplex of T and consider its interior. Although a
triangulation of a surface is not a planar graph, the portion of T 0 that
is interior to σ is. We consider several cases:
A: If s is a 1-simplex interior to σ with vertices of other simplices on
either end (as in figure 1.6, case A), removing s will decrease
E by 1 in the formula χ(S, T 0 ) = V −E +F but will also merge
two distinct regions, making F decrease by 1 as well.
B: If we have a 1-simplex with an isolated vertex, v, at its end (as
in figure 1.6, case B) we can simply delete it and the vertex
without changing the Euler characteristic.
C: If we have an “extra” vertex, e, in an edge of the original triangulation, T , (as in figure 1.6, case C), we can delete it and
“merge” the two 1-simplices incident on it without changing
the Euler characteristic.
The conclusion follows.
Although it appears that the Euler characteristic of a surface depends on its triangulation, it turns out that all triangulations are related:
Theorem 10 (Hauptvermutung). If S is a surface with triangulations
T1 and T2 , then there exists a triangulation, T̄ , that is a refinement of
both. Consequently
χ(S, T1 ) = χ(S, T2 ) = χ(S)
If S1 is homeomorphic to S2 then χ(S1 ) = χ(S2 ).
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TOPOLOGICAL CLASSIFICATION OF SURFACES
A common
refinement
Overlaid
triangulations
F IGURE 1.7.
Two triangulations
Remark. It is interesting (and the reason for the ceremonious title1)
that this result is difficult to prove in three dimensions and not true
in higher dimensions. In two dimensions, it is borderline-trivial.
Proof. A rigorous proof of this is given in [5], but the idea is fairly
simple: simply overlay T1 and T2 . Wherever 1-simplices of T1 intersect those of T2 introduce new vertices. The resulting polygonal
regions can easily be subdivided into triangles. Figure 1.7 illustrates
this construction.
If f : S1 → S2 is a homeomorphism and T is any triangulation of S1 ,
then f (T ) is a triangulation of S2 — and the conclusion follows. Exercise: Using the fact that the Euler characteristic of a 2-sphere is
2, prove that the 5 Platonic solids are “the only ones possible.”
In other words, if a union of polygons that have p edges, with
the property that that q faces meet at each vertex, forms a
surface homeomorphic to a sphere then
(p, q) = (3, 3), (4, 3), (3, 4), (5, 3), (3, 5)
Since each edge lies in exactly 2 faces and q faces meet at each vertex,
we get
pF = 2E = qV
1
Which means “main hypothesis” in German.
TOPOLOGICAL CLASSIFICATION OF SURFACES
Torus: T 2
Sphere: S 2
F IGURE 2.1.
7
Klein bottle: K
Identification spaces
The equation for the Euler characteristic is
V −E+F
2E
2E
−E+
q
p
E E E
− +
q
2
p
1 1
+
q p
= 2
= 2
= 1
=
1
1
+
2 E
so we conclude that
1
1 1
+ >
q p
2
which has the solutions given.
2. C LASSIFICATION
OF SURFACES
Our treatment follows that of Seifert and Threlfall in [6]. Although
it is not the shortest proof (that is Conway’s ZIP proof in [3]), it will
suit our purposes.
We can construct surfaces by identifying edges of a polygon as in
figure 2.1. In the case of a torus, we identify the a-edges to get a
tube whose open ends correspond to the b-edges — then we identify
them to get a “donut.” In the torus and Klein bottle, all four vertices
wind up being identified. The arrows define which of two possible
identifications we should use. If we do the second step with the reversed orientation — as in the third figure — we get a Klein bottle as
in figure 2.2 on the next page.
This turns out to be a universal phenomena:
Proposition 11. Every surface, S, is homeomorphic to a polygon with
edges identified, as in figure 2.3 on the following page. If the surface is
closed, the polygon has an even number of edges. The distinct classes of
vertices and edges can be used to compute the Euler characteristic.
8
TOPOLOGICAL CLASSIFICATION OF SURFACES
F IGURE 2.2.
Klein bottle, K
F IGURE 2.3. Polygon with edges identified
Remark. For instance, counting edges the diagrams in figure 2.1 on
the previous page:
(1) S 2 has vertices p1 , p2 , and p3 , edges a and b, and a single face
so χ(S 2 ) = 3 − 2 + 1 = 2
(2) χ(T 2 ) = χ(K) = 1 − 2 + 1 = 0.
Proof. Let S0 = {T1 , . . . , Tk } be the triangles in the triangulation and
set P0 to (an arbitrary) one of these triangles in the. In each step
(1) Select Ti ∈ Si , a triangle adjacent to the polygon, Pi , and set
(2) set Pi+1 = Pi ∪ Ti (a merge-operation so that the number of
sides of Pi=1 is > than the number of sides of Pi ,
(3) set Si+1 = Si \ {Ti }
This procedure must terminate when Sn = ∅ and all of the triangles
have been merged into Pn . An argument like that used in the proof
of proposition 9 on page 5 shows that these merge-operations do not
change the Euler characteristic.
Given a polygon like that in figure 2.3, we can simplify it and reduce it to a standard form.
TOPOLOGICAL CLASSIFICATION OF SURFACES
9
F IGURE 2.4. Eliminating aa−1 or a−1 a
Claim 12. We begin by introducing a direction to the boundary of the
figure and representing the sequence of edges by a string of characters. These characters have an exponent of ±1 according to whether
their arrows are in the same or the opposite direction from what we
have chosen. For instance, if we choose the clockwise direction in
figure 2.3 on the facing page, starting at p1 at the top left, we get the
symbol
ac−1 bac−1 b
Throughout the rest of this section, we will fix a polygon, P , that
has an orientation or direction as mentioned above. When its edges
are identified, it gives rise to a closed surface S.
There are certain operations we can perform on P and its associated symbol:
• Cyclically permute it. This is because the starting point in our
ordering of the vertices was arbitrary.
• Reverse all arrows with a certain label. This is because the
directions of the arrows only indicate how the corresponding
faces of the polygon are to be identified. In the symbol, it
amounts to replacing all occurrences of ani by a−n
.
i
• Reverse the direction of the boundary — since it was arbitrary to begin with. This replaces the symbol an1 1 · · · ant t by
t
1
a−n
· · · a−n
.
t
1
Proposition 13. We can eliminate edges representing symbols aa−1 or
a−1 a from P .
10
TOPOLOGICAL CLASSIFICATION OF SURFACES
Original P
Cut off triangle
F IGURE 2.5.
F IGURE 2.6.
Eliminating a vertex
Glue onto matching portion of P
Remark. The proof is illustrated in figure 2.4 on the previous page.
We fold the edges marked a inwards, resulting in a single edge and
then delete it (and the vertex at its end).
Henceforth, we assume that P has no edges of the type mentioned
above.
By performing some surgery on our polygon, we can guarantee:
Proposition 14. We can modify P in such a way that the modification
represents the same surface, S, but only has a single vertex.
Remark. In other words, all of the vertices in the modification have
the same label and map to a single vertex in S.
Henceforth, all of the vertices of P will have the same label and will
give rise to the same vertex in S.
Proof. If there is more than one type of vertex, we can perform
“surgery” on the polygon to decrease the number of vertices of a
TOPOLOGICAL CLASSIFICATION OF SURFACES
11
given type. Suppose there are vertices p1 and p2 and we want to
eliminate vertices of type p1 . Figure 2.5 on the facing page shows the
first two steps of the process and figure 2.6 on the preceding page
shows the third.
Afterward, the polygon has one more point of type p2 and one fewer
point of type p1 . A straightforward induction shows that we can eliminate all points of type p1 and eventually have all vertices with the
same label.
We focus on the edges of P : Since S is closed, each such edge is
identified with exactly one other.
Definition 15. If the edges of P are give rise to the symbol
a1 · · · an
then
(1) ai is a symbol of the first kind if there exists a j such that
aj = a−1
i
(2) ai is a symbol of the second kind if there exists a j such that
aj = ai
If all edges of P are of the first kind, the surface S is said to be
orientable.
Remark. For instance a sphere is orientable but a Klein bottle isn’t.
Proposition 16. If c is a symbol of the second kind in P , we can perform
surgery on P to get a polygon, P 0 , where the two copies of c have been
replaced by a · a — two copies of a symbol a that are adjacent — and
all other symbols are unchanged.
Remark. Henceforth, we will assume that all symbols of the second
kind in P are pairs of adjacent edges in the same direction.
Pairs of adjacent edges in the same direction lead to what is called
a crosscap in S: identifying the a-edges in the rightmost diagram in
figure 2.8 on the following page — and assuming that the edges on
the bottom are open leads to a Möbius strip whose circular boundary
lies on the bottom. Figure 2.7 on the next page shows a crosscap
whose bottom has been closed off (by having a disk sewn onto it).
Proof. Figure 2.8 on the following page (left to right) shows what
happens: we join the two copies of c by a new edge, a, cut along this
new edge, and glue the c-edges together. Since the c-edge is interior
12
TOPOLOGICAL CLASSIFICATION OF SURFACES
1
0.5
Z
0
− 0.5
−1
− 0.8
− 0.6
− 0.4
− 0.2
0 0.2
0.4 0.6
Y
0.8
F IGURE 2.7.
Symbol of the
second kind
F IGURE 2.8.
0
0.2
0.4
− 0.4
− 0.2
X
Steiner’s Crosscap
Cut along new edge
a
Glue c-edges
together
Normalizing a symbol of the second kind
to the polygon, we can safely delete it. Note that the remaining edges
are intact (but joined together).
If necessary, we may have to invoke proposition 13 on page 9 to
eliminate any edges of the form e · e−1 that are created.
Now we consider a property of edges of the first kind:
Proposition 17. If a is an edge in P of the first kind then, between a
and a−1 , there exists another edge of the first kind, b such that b lies
between a and a−1 and b−1 lies between a−1 and a, as in
. . . a . . . b . . . a−1 . . . b−1 . . .
TOPOLOGICAL CLASSIFICATION OF SURFACES
13
Proof. Suppose we have a sequence
z = a . . . a−1
with no edges of the first kind sandwiched between them. Then all
of the edges between them will either be of the second kind (which
are of the form c · c) or edges of the first kind entirely contained in
this sequence. In either case, all of the edges in z will be paired with
other edges within z. The same must be true of all edges outside z.
It follows that z and its complement each generate closed surfaces
joined at a single point (namely, the one vertex) — which is not a
valid surface.
We are finally ready to classify orientable surfaces:
Proposition 18. Given a sandwiched pair of edges of the first kind in P
. . . c . . . d . . . c−1 . . . d−1 . . .
we can replace P with a polygon P 0 giving rise to the same surface S
with c- and d-edges replaces by edges a, db of the first kind with a symbol
. . . aba−1 b−1 . . .
It follows that the symbol of P can be assumed to be a concatenation of
substrings of the form
−1
ai · ai and cj dj c−1
j dj
Remark. A sequence like aba−1 b−1 is called a handle because it defines
a shape like that. For instance aba−1 b−1 itself defines a torus — see
the middle image in figure 2.1 on page 7. The sequence
aba−1 b−1 cdc−1 d−1
defines the surface in figure 1.4 on page 4 that has two handles on it.
Proof. As before, this is a matter of surgery. Figures 2.9 on the following page and 2.10 on the next page illustrate the process.
Definition 19. If S is an orientable surface represented by symbol
−1
−1 −1
a1 b1 a−1
1 b 1 · · · at b t at b t
— i.e. a surface with t handles, the number t is called the genus of S.
Remark. If S is an orientable surface of genus t, it is not hard to see
that
χ(S) = 2 − 2 · t
14
TOPOLOGICAL CLASSIFICATION OF SURFACES
Cut along a new edge b
F IGURE 2.9.
Paste along c-edges
Normalizing a handle, part 1
Cut along a new edge a
F IGURE 2.10.
Paste along d-edges
Normalizing a handle, part 2
3. T HE R IEMANN -H URWITZ
FORMULA
We begin with a definition:
Definition 20. If S1 and S2 are surfaces, a degree-d map f : S1 → S2
is a map with the property that there is a finite number of points,
{q1 , . . . qm }, of S1 such that
f |S1 \ {q1 , . . . qm }: S1 \ {q1 , . . . qm } → S2 \ {f (q1 ), . . . , f (qm )}
is a local homeomorphism (see definition 1 on page 1) such that
f −1 (p) has exactly d points for all points p ∈ S2 \ {f (q1 ), . . . , f (qm )}.
The exception qi ∈ S1 is a branch point of degree ei if there exists a
neighborhood qi ∈ Ui ⊂ S1 such that f |Ui is a map of degree ei and
(f |Ui )−1 (f (qi )) = qi .
Remark. Points like qi are also called ramification points of f .
Example 21. If S 1 ⊂ C is the complex unit circle and f : S 1 → S 1 is
the map z 7→ z 3 , then f is of degree 3 and has no branch points. On
the other hand, if we extend f to C then it has degree 3 but z = 0 is
a branch point of degree 3.
TOPOLOGICAL CLASSIFICATION OF SURFACES
15
Here’s a more complex example:
Example 22. Let f : C → C be the map f (z) = z 2 (z −1)3 (z +1)5 . Then
f is a map of degree 10 that has:
• a branch point of degree 2 at z = 0
• a branch point of degree 3 at z = 1 and
• a branch point of degree 5 at z = −1
Suppose f : S1 → S2 is a degree d map and we triangulate S1 with
vertices at the branch points (as well as at other points) and select
vertices of S2 that are the images of vertices of S1 under f . Now select
1- and 2-simplices small enough to fit in the open sets f (U ) ⊂ S2 from
definition 1 on page 1 — that are mapped homeomorphically by f .
Then every 1-simplex (or edge) and 2-simplex of S2 has precisely d
inverse images and induces a compatible triangulation of S1 . A simple
count of simplices gives the Riemann-Hurwitz formula:
Theorem 23 (Riemann-Hurwitz formula). If f : S1 → S2 is a degree-d
map of surfaces with branch-points {q1 , . . . , qt } ∈ S1 that have corresponding degrees {e1 , . . . , et } then
χ(S2 ) = d · χ(S1 ) −
t
X
(ei − 1)
i=1
Remark. Adolf Hurwitz originally proved this in [4], where he used it
to study transformations of Riemann surfaces.
Proof. Suppose S1 and S2 have triangulations compatible with f as
described above. Since f is of degree-d, it follows that the numbers of
1- and 2-simplices of S1 are precisely d× the corresponding numbers
of simplices in S2 . At branch point qi , we will have over-counted the
number of vertices in S1 by precisely ei − 1 since (f |Ui )−1 (f (qi ) =
qj .
In the examples above, regard C as the Riemann sphere, or CP 1 .
Then the map in example 21 on page 14 gives
χ(S 2 ) = 2 = 3 · 2 − 2 − 2
which implies that the map z 7→ z 3 has an additional branch point.
Since has none in the finite plane (other than z = 0) it must have a
branch point of degree 3 at ∞.
R EFERENCES
[1] P. H. Doyle and D. A. Moran. A Short Proof that Compact 2-Manifolds Can Be
Triangulated. Inventiones math., 5:160–162, 1968.
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TOPOLOGICAL CLASSIFICATION OF SURFACES
[2] Leonhard Euler. Demonstratio nonnullarum insignium proprietatum, quibus
solida hedris planis inclusa sunt praedita. Novi Commentarii academiae scientiarum Petropolitanae, 4:140–160, 1758. URL http://math.dartmouth.edu/ euler/pages/E231.html.
[3] George K. Francis and Jeffrey R. Weeks. Conway’s ZIP Proof. American Math
Monthly, 106:393–399, 1999.
[4] Adolf Hurwitz. Ueber algebraische gebilde mit eindeutigen transformationen
insich. Math. Ann., 41:403–442, 1893.
[5] T. Radò. Über den Begriff der Riemannschen Fläche. Acta Litt. Sci. Szeged,
2:101–121, 1925.
[6] Herbert Seifert and William Threlfall. A textbook of topology. Academic Press,
1980.