Math 210 Manifold III, Spring 2017
Euler Characteristics of Surfaces
Hirotaka Tamanoi
1. Euler Characteristic of Surfaces
Leonhard Euler noticed that the number v of vertices, the number e of edges and the
number f of faces of Platonic solids have a curious property that we always have v−e+f = 2,
as shown in the next table.
As we will see, this observation can be generalized to general convex polyhedra for which
v − e + f = 2 is always valid. For arbitrary triangulation of any given surface (these are
general polyhedra not necessarily convex) the identity v − e + f = 2 is no longer valid, but
v − e + f is still independent of the triangulation. This phenomenon is valid in the general
context of cell complexes. Before defining cell complexes, we discuss one example.
Example 1.1. (Soccer Ball) Soccer balls are made with hexagonal faces and pentagonal
faces, with three faces meeting at each vertex. This can be thought of as a convex polyhedron
so that v − e + f = 2 is valid. Suppose here are H hexagonal faces and P pentagonal faces.
Then there are v = (5P + 6H)/3 vertices, e = (5P + 6H)/2 edges, and f = P + H faces.
Then
5P + 6H 5P + 6H
P
2=v−e+f =
−
+ (P + H) = .
3
2
6
Hence P = 12 and such a soccer ball always has 12 pentagonal faces. The number of
hexagonal faces can vary. For example, the regular dodecahedron has no hexagonal faces,
but truncated regular icosahedron (this polyhedron is obtained by chopping off 12 vertices of
the icosahedron, at each vertex of which two hexagonal faces and one pentagonal face meet)
has 20 hexagonal faces. Unfortunately, there does not exist a polyhedron whose faces are
regular pentagons and regular hexagons such that at each vertex, two pentagons and one
hexagon meet. In fact, only convex polyhedron whose faces consist of regular pentagons and
regular hexagons are the above two. See a table of 13 Archimedean solids at the end of this
section. These are semi regular solids. More general solids are called Johnson solids whose
only requirement is that every face is a regular polygon. There are exactly 92 Johnson solids.
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If you relax the condition that every face must be a regular polygon, then it is possible to
have a cell decomposition of S 2 whose 2-cells have 5 or 6 edges around their boundary, and
at each 0-cell, (the closure of) three 2-cells meet.
By definiton, an n-cell is a topological space homeomorphic to an open n-ball for n ≥ 1,
and a 0-cell is a point.
Definition 1.2. A cell complex is a topological space X which is a disjoint union of cells of
various dimensions satisfying the following conditions.
(C) For any cell, its boundary in X is a union of finitely many lower dimensional cells.
(W) A subset A ⊂ X is open (closed) in X if and only if for every cell ek , the intersection
A ∩ ek with the closure of ek in X is open (closed) inside the closure ek of that cell
ek .
The condition (C) is called closure finiteness and the condition (W) says that X has the
weak topology with respect to the cell structure. These conditions (C) and (W) are irrelevant
if X has only finitely many cells. But for cells with infinitely many cells, these conditions
eliminates pathological cases.
Example 1.3. (1) Consider a family of circles in the plane R2 all touching each other at
the origin, and converging to the origin. Namely, let
n
1 2
1o
Cn = (x, y) ∈ R2 x −
+ y2 = 2 .
n
n
S
and Let Y = n≥1 Cn with induced topology from R2 . Y has a natural cell structure
by decomposing each circle Cn as Cn = e0 ∪ e1n , the origin and the rest of the circle. In
Y , the condition (C) is satisfied but the condition (W) is not satisfied. For example, let
yn = (2/n, 0) ∈ Cn be the point in Cn opposite to the origin. The the set A = {yn | n ≥ 1}
is not closed in Y since the origin is a limit point of A but the origin is not in A. However,
A ∩ e1n is closed in the n-th 1-cell e1n in Cn for all n ≥ 1. Thus, Y with this cell structure is
not a cell complex in the above sense.
(2) On the other hand, for each n ≥ 1, let Sn be the standard unit circle with usual base
point xn at (1, 0). Let
a X=
Sn /{x1 , x2 , . . . , xn , . . . }
n≥1
be the quotient space obtained from the topological disjoint union of infinitely many circles
by identifying them at one point. The space X again has a canonical cell decomposition as
before, but this time, X is a cell complex.
Definition 1.4. Let X be a finite cell complex and let an be the number of n-cells in X.
Then the Euler characteristic of X is defined by
X
χ(X) =
(−1)n an .
n≥0
.
For surfaces with finitely many cells, the above definition becomes the following.
Definition 1.5. Given a cell decomposition of a surface M , let ν, e, and f be the number
of 0-cells, 1-cells, and 2-cells. The Euler characteristic of the cell complex M is defined by
χ(M ) = ν − e + f.
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Example 1.6. Some examples of cell decompositions and their Euler characteristic. Here
S 2 denotes the sphere, P denotes the projective plane, and T denotes the torus.
(1) The 2-dimensional sphere S 2 can be decomposed as one point and the rest of the
sphere. This gives S 2 = e0 ∪ e2 . So ν = 1, e = 0, f = 1 and χ(S 2 ) = 2. Another
decomposition will be decompose the equator as one point and the rest of the equator,
together with strictly upper hemisphere and strictly lower hemisphere. This gives
S 2 = e0 ∪ e1 ∪ e2 ∪ e2 . So again χ(S 2 ) = 1 − 1 + 2 = 2.
(2) The five Platonic solidsgive cell decompositions of a sphere. At the beginning of this
section, we saw that in all cases, we have ν − e + f = 2.
(3) We use the standard construction of the projective plane P as 2-dimensional disc
D2 with antipodal boundary points identified. If we decompose the boundary circle
as two antipodal points and two semi arcs without end points, these cells, after
identification, give rise to one 0-cell and one 1-cell. The interior of the disc gives a
2-cell. So we have P = e0 ∪ e1 ∪ e2 . So ν = 1, e = 1, f = 1 and χ(P ) = 1. Another cell
decomposition, which is actually a triangulation, is given below with 10 triangles. In
this cell decomposition, there are six 0-cells, fifteen 1-cells, and ten 2-cells. So again
we have χ(P ) = 6 − 15 + 10 = 1.
(4) Among the cell decompositions on the sphere S 2 coming from Platonic solids, all
but the tetrahedon cell decomposition descend to cell decompositions of the projective plane. In these cell decompositions, the numbers ν, e, f are half of the original
numbers and we get χ(P ) = χ(S 2 )/2 = 1.
(5) T = e0 ∪ e1a ∪ e1b ∪ e2 . So ν = 1, e = 2, f = 1 and χ(T ) = 0.
(6) Here is another cell decomposition of a torus in which all faces are squares. See the
picture below. This is actually a polyhedron homeomorphic to a torus. We have
ν = 24, e = 48, f = 24 and so χ = ν − e + f = 0.
Example 1.7. A triangulation of a compact surface M is a finite collection of triangles
T = {T1 , t2 , . . . , Tn } in M such that
S
(1) these triangles cover M , that is ni=1 Ti = M , and
(2) if two distinct triangles intersect, they do so along a single common vertex or along
a single commom edge. Each triangle does not intersect with itself.
A triangulation naturally gives rise to a finite cell complex structure on the surface. But in
general, one needs many triangles. For example, one can show that on a compact surface
without boundary with Euler characteristic χ, every triangulation on M must satisfy
p
1
2e = 3f,
e = 3(ν − χ),
ν ≥ (7 + 49 − 24χ).
2
3
The last inequality is obtained by using e ≤ ν2 . For the projective plane P = RP 2 with
χ = 1, in every triangulation of P , there must exist at least 10 triangles. Such a triangulation
of P is given below.
0
1
2
2
1
0
Here, along the circle, opposite arcs are identified in the same direction.
A basic property of the Euler characteristic is that it is a topolological invariant of the
surface, and independent of the cell decomposition used to compute it. This is true for any
cell complex, but here we give a proof for surfaces.
Theorem 1.8. The Euler characteristic is independent of the cell complex structure on a
compact surface M , and depends only on the homeomorphism type of M .
Proof. First we observe that the Euler characteristic is invariant under the following three
processes.
(1) Subdivide an edge. This increases the number of vertices by 1 and the number of
edges by 1. Or erasing a degree 2 vertex, combining two edges into one edge.
(2) Subdivide a face into two by inserting an edge connecting two boundary vertices. Or
combining two adjacent faces by removing a common edge.
(3) Introduce a new “dangling” edge in the interior of a face, where the new edge has
one vertex incident to a boundary vertex of the face. Or removing a dangling edge.
Given two cell decomposition D1 and D2 of M , we can use the above processes to convert
these cell decompositions into a finer common cell decomposition D3 obtained by combining
edges of these two cell decompositions and possibly adding additional edges to subdivide
resulting regions into contractible cells using the above three processes. (The argument here
is delicate, since an edge from D1 may intersect with an edge from D2 at infinitely many
points, like the graph of y = sin(1/x) intersects with x-axis. If this happens, we cal always
move edges a little to avoid this situation.) Then χ(D1 ) = χ(D3 ) = χ(D2 ). Hence χ only
depends on the topological type.
Thus in particular, the Euler characteristic of a surface is independent of its cell decomposition. Note that the Euler characteristic does not change under circulation rules. To see
this, we observe that under Circulation Rules (I) and (II), the cell structure remains the
same. Rule (III), sphere rule, corresponds to introducing or eliminating a dangling edge
corresponding to (3) above. The Cylinder Rule (V) and the Möbius Rule (VI) involve introducing an edge which bisects a face corresponding to (2) above. Hence χ(C) is the same as
χ(Cstandard ).
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Since all surfaces are generated by taking connected sum with the torus T or the projective
plane P , to compute Euler characteristic of general surface, we first study its behavior under
connected sums. The following theorem can be proved using convenient cell decomposition
of surfaces M1 and M2 in which interiors of discs removed are 2-cells.
Theorem 1.9. Let M1 , M2 be surfaces. Then
χ(M1 #M2 ) = χ(M1 ) + χ(M2 ) − 2.
Proof. Let M i , i = 1, 2, be a surface obtained by removing an open disc from Mi . Introduce
the same cell decomposition along the boundary circles of M i , say with p 0-cells and p 1cells. Then extend this cell decomposition to the rest of surfaces with boundary. We have
χ(Mi ) = χ(M i ) + 1 for i = 1, 2. Now by counting cells in M1 #M2 , let vi , ei , fi be the
number of 0-cells, 1-cells, and 2-cells in M i for i = 1, 2. Since vertices and edges along the
boundary circles are identified, we have χ(M1 #M2 ) = (v1 +v2 −p)−(e1 +e2 −p)+(f1 +f2 ) =
χ(M 1 ) + χ(M 2 ) = χ(M1 ) − 1 + χ(M2 ) − 1. This proves the formula.
By classification theorem of surfaces, a compact surface without boundary is homeomorphic to either an orientable surface Σg = gT of genus g ≥ 0, or a nonorientable surface Nh
of genus h ≥ 1. By convention, N0 = Σ0 = S 2 is the sphere.
Proposition 1.10. The Euler characteristic of a compact surfaces without boundary are
given as follows.
χ(Σg ) = 2 − 2g,
χ(Nh ) = 2 − h.
Proof. Since χ(T ) = 0 and χ(P ) = 1, the formula above for connected sum gives
χ(M #T ) = χ(M ) + χ(T ) − 2 = χ(M ) − 2,
χ(M #P ) = χ(M ) + χ(P ) − 2 = χ(M ) − 1.
Thus, each time we take a connected sum with a torus, the Euler characteristic decreases by
2, and every time we take a connected sum with the projective plane, the Euler characteristic
decreases by 1. By induction starting with the sphere with χ(S 2 ) = 2, we get the above
formula. In the above formula, the number 2 on the right hand sides is the Euler characteristic
of the sphere.
We can also use standard plane models for the above surfaces to compute the Euler
characteristic. For example, in the standard plane model of genus g orientable surface on
−1
−1 −1
4g-gon with the word a1 b1 a−1
1 b1 · · · ag bg ag bg , the induced cell structure on the surface has
one 0-cell, 2g 1-cells, and pne 2-cells. Hence χ(Σg ) = 1 − 2g + 1 = 2 − 2g.
Similar computation works for the genus h nonorientable surface Nh .
Corollary 1.11. Homeomorphism type of closed surface is classified by its orientability and
its Euler characteristic. More precisely, let X be a closed surface.
(1) If X is orientable and χ(X) = 2 − 2g, then X is an orientable surface Σg of genus
g ≥ 0. Here, genus 0 surface is a sphere.
(2) If X is nonorientable and χ(X) = 2 − h, then X is a nonorientable surface Nh of
genus h ≥ 1.
Note that due to Dyck’s identity, we have
P #Σg ∼
= (2g + 1)P,
K#Σg ∼
= (2g + 2)P
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for g ≥ 0. Thus, all closed surfaces are generated from a sphere S 2 , a torus T , a projective
plane P , and a Klein bottle K by connected sum, where P or K only have to be used at
most once.
2. Applications of Euler Characteristics: Identification of surfaces
Surfaces can be classified by orientability and the Euler characteristic. So, we can use
Euler characteristic to identify surfaces. We illustrate the method by examples.
2.1. Application 1. Plane models. Euler characteristic provides a quick way to identify
the surface presented by a plane model. What we use is a fact that a closed surface is
determined, up to a homeomorphism, by orientability and the Euler characteristic. As
an example, let M be a surface whose plane model has a word dac−1 bca−1 b−1 d−1 . Since
each label appears exactly twice with opposite exponent, this word represents an orientable
surface. Plane model induces a cell decomposition on the surface M with one big 2 cell
coming from the interior of the plane model, so f = 1. The number of 1-cells is equal to the
number of labels. By examining vertex identification induced by edge identification (draw a
vertex identification graph), we have ν = 3, e = 4, f = 1. Since the word is orientable and
χ(M ) = 3 − 4 + 1 = 0, the surface M must be a torus.
To draw a vertex identification diagram, label all vertices of the plane model as 1, 2, . . . , n
(if there are n vertices), and arrange all vertices in a circular fashion. Suppose edge labels
are a, b, c, . . . . When we glue two edges carrying the label a in the same direction, the two
head vertices are identified, and two tail vertices are identified. Draw an edge in the vertex
diagram between two vertices which are identified. Do this for all edges. The number of
connected components of the resulting graph is the number ν of distinct vertices on the glued
surface M .
Let M be a surface having a plane model with a word ca−1 b−1 cdab−1 d. This is a nonorientable surface word with ν = 1 after examining the vertex identification diagram. So,
χ(M ) = 1 − 4 + 1 = −2. Hence M = 4P = N4 .
2.2. Application 2. Surfaces spanning knots and links. Given a knot (or a link) in
R3 , we can find a surface whose boundary is the given knot (or a link), called a spanning
surface. To identify this surface, follow the following procedure.
(1) Decide orientability of a given surface. If there exists a loop on the surface which goes
through odd number of crossings in the knot’s plane drawing, then it is an orientation
reversing loop, and the surface is non-orientable. If you cannot find any orientation
reversing loops, then your spanning surface is orientable.
(2) Count the number of boundary circles. This is the number of components of a link.
If we start with a knot, then the number is 1.
(3) Introduce a cell decomposition of a surface and compute the Euler characteristic
of M . Basically we introduce new edges and vertices on the surface to “cut” the
surface into contractible regions, and the interiors of contractible regions are 2-cells,
and the cuts used are part of 1-cells and 0-cells of the new cell decomposition. For
example, if the knot (or link) diagram has c crossings, we can introduce one “cut”
for each crossing. This is an edge close to the crossing which you can use to cut
your surface. Introduction of this edge uses one 1-cell and two 0-cells (end vertices).
We introduce a cut for each crossing. Then, we have 2c vertices (=0-cells), and c
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short 1-cells (cuts) and the original knot (or link) is divided by the above 2c vertices
into 2c arcs, which are 1-cells. Hence altogether, we have ν = 2c many 0-cells
and e = c + 2c = 3c many 1-cells. You notice that each region of the surface
separated by cuts is a contractible region, so its interior is a 2-cell. Suppose there
are f regions. Then χ(M ) = ν − e + f = f − c. This gives a quick way to compute
the Euler characteristic of the spanning surface simply by counting the number c of
knot crosings and the number of regions from knot diagram. Of course we can use
a smaller cell decomposition without using as many cells as above. We just have to
make sure that after cuts are made, each region is contractible.
f be the closed surface obtained by capping bound(4) For a spanning surface M , let M
aries, that is, by gluing a disc along each boundary component. If there are k boundaries, this capping process adds k 2-cells to the cell decomposition of M , so we have
f) = χ(M ) + k. If M is a spanning surface of a knot, then k = 1. If M is a
χ(M
spanning surface of a link, then k is the number of components of a link. Use the
f as gT or
orientability and the Euler chracteristic to identify the closed surface M
hP .
f with k open discs removed.
(5) The original surface M can be described as M
We can span a surface for the following knots, and apply the above method to identify the
surfaces. In general, given a knot, there are several possible spanning surfaces. The easiest
one is to imagine a soap film spanning the wire in the shape of a knot. Less obvious one
is a surface which extends outward and curls up to close. These spanning surfaces of the
same knot (or link) may not be homeomorphic. Soem of them may be orientable, and some
of them are not. However, one can always find an orientable spanning surface called Seifert
surface. This is the one which extends out and curls up to close.
The Euler characteristic is characterized by the following property. This often gives quick
computation.
Proposition 2.1. For compact topological spaces posessing a cell decomposition, the Euler
characteristic is characterized by the following properties.
(1)
(2)
(3)
(4)
χ(X) is a topological invariant of X.
χ(X) = 0 is X is empty.
χ(X) = 1 is X is contractible.
Let X1 and X2 be subcomplexes of X. Then
χ(X1 ∪ X2 ) = χ(X1 ) + χ(X2 ) − χ(X1 ∩ X2 ).
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The proof is basically the Inclusion-Exclusion Principle in set theory. That is, for two
finite sets A, B, the number of elements of sets behave as follows.
|A ∪ B| = |A| + |B| − |A ∩ B|.
For the proof of the above formula of the Euler characteristic, consider the set of k-dimensional
cells for k = 0, 1, 2, · · · . For each dimension, write down the above Inclusion-Exclusion Principle, then take alternating sum.
3. Regular Complexes on Surfaces
Definition 3.1. A regular complex on a surface M is a cell decomposition on M such that
(1)
(2)
(3)
(4)
each face has the same number of edges a ≥ 3,
each vertex has the same valency b ≥ 3,
two faecs meet along a single edge, at a single vertex, or none at all,
no two faces meet with itself.
A regular complex above on M is denoted by (a, b)M.
Example 3.2. Platonic solids are regular complexes on the sphere S.
(1)
(2)
(3)
(4)
(5)
Tetrahedron = (3, 3)S with ν = 4, e = 6, f = 4.
Cube = (4, 3)S with ν = 8, e = 12, f = 6.
Octahedron = (3, 4)S with ν = 6, e = 12, f = 8.
Dedecahedron = (5, 3)S with ν = 20, e = 30, f = 12.
Icosahedron = (3, 5)S with ν = 12, e = 30, f = 20.
Lemma 3.3. Consider a regular complex of type (a, b) on a surface M . Then
af = 2e,
νb = 2e.
The Euler characteristic of M is given by
χ(M ) = 2e
1 1 1
+ −
a b 2
.
Proposition 3.4. Platonic solids are the only regular convex polytopes.
Proof. Since χ(S 2 ) = 2, we have e( a1 + 1b − 21 ) = 1. From this, only possibilities are (a, b) =
(3, 3), (3, 4), (3, 5), (4, 3), (5, 3). The numbers ν, e, f can be computed from a, b, and these are
Platonic solids.
Definition 3.5. Given a cell complex C on a surface M , its dual cell complex C 0 can be
constructed as follows. Place a veretx in the middle of each face of C. There is an edge
between two new vertices if and only if the corresponding faces of C share a common edge.
These new vertices and edges divide the surface M into 2-cells which are the faces of C 0 .
Suppose cell complex is an (a, b)M. In the dualization process, an a-gon face in C becomes
a vertex of valence b in the dual complex C 0 . A vertex of valence b in C becomes a b-gon
face. Thus, C 0 = (b, a)M.
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4. Regular complex structure on P, T, K
Proposition 4.1. (1) On the projective plane P, all the possible regular complexes are of
type (3, 5) and (5, 3).
(2) On the torus T, all the possible regular complexes are of type (3, 6), (4, 4), (6, 3). The
number of edges, vertices and faces are not fixed, and there are infinitely many for each type.
(3) On the Klein bottle K, all the possible regular complexes are of type (3, 6), (4, 4), (6, 3).
The number of edges, vertices and faces are not fixed, and there are infinitely many for each
type.
5. b-valent complexes
A b-valent complex is a cell complex defined by the requirement of the regular coimplex
without the condition that all faces have the same number of edges.
Lemma 5.1. Let fn be the number of faces with n edges. on a trivalent (b = 3) complex on
S 2 , we have
12 = 3f3 + 2f4 + f5 − f7 − 2f8 − · · · .
In particular, there must be a face with fewer than 6 sides.
By chopping off corners of regular solids, we get trivalent complexes on S 2 .
Example 5.2. A convex polyhedron with only triangle faces and square faces has a property
that at each vertex, 2 triangles and 2 squares meet. Compute the number of vertices, edges,
triangle faces, and square faces. (This solid is called cuboctahedron. See the table at the
end of this section.)
Solution Being a convex polyhedron means that the polyhedron is homeomorphic to a
sphere. So v − e + f = χ(S) = 2. Let f3 and f4 be the number of triangles and the number
of squares. By counting the number of pairs (edge, vertex on the edge) in two different ways,
we get 4v = 2e. By counting the number of pairs (triangle, a vertex on the triangle) in two
ways, we get 3f3 = 2v. This is because at each vertex there are two triangles, so there are 2v
such pairs. On the other hand, for each triangle, there are three possible vertices we can pair
the triangle with. So the number of such pairs are 3f3 . Hence we have 3f3 = 2v. Similarly,
we get 4f4 = 2v. Using these identities, we get v = 12, e = 24, f3 = 8, f4 = 6. We can obtain
this convex polyhedron by cutting off vertices from a cube by cuts through middle of edges.
The above solid is an example of an Archimedean solid.
Definition 5.3. (Archimedean solids) An Archimedean solid is a convex polyhedron
whose faces consist of two or more types of regular polygons and whose vertices are identical.
That is, for any choice of two vertices, there is an isometry of the polygon sending one vertex
to the other.
Here are some examples of Archimedian solids.
There are 13 Archimedean solids, and they are listed below.
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