Math 210 Manifold III, Spring 2018 Euler Characteristics of Surfaces Hirotaka Tamanoi

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1 Math 210 Manifold III, Spring 2018 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triangulation of any given surface (these are general polyhedra not necessarily convex) the identity v e + f = 2 is no longer valid, but v e + f is still independent of the triangulation.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 2 = v e + f = + (P + H) = P 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. 1

2 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. The following statement is not quite precise definition of a cell complex. But it will give you a good idea of it. For a precise definition, wee Hatcher Chaopter 0 on cell complexes. 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 e k, the intersection A e k with the closure of e k in X is open (closed) inside the closure e k of that cell e k. 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 R 2 all touching each other at the origin, and converging to the origin. Namely, let ( C n = {(x, y) R 2 x 1 ) 2 + y 2 = 1 }. n n 2 and Let Y = n 1 C n with induced topology from R 2. Y has a natural cell structure by decomposing each circle C n as C n = e 0 e 1 n, 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 y n = (2/n, 0) C n be the point in C n opposite to the origin. The the set A = {y n 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 e 1 n is closed in the n-th 1-cell e 1 n in C n 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 S n be the standard unit circle with usual base point x n at (1, 0). Let ( X = S n )/{x 1, x 2,..., x n,... } 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 a n be the number of n-cells in X. Then the Euler characteristic of X is defined by χ(x) = n 0( 1) n a n.. For surfaces with finitely many cells, the above definition becomes the following. 2

3 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. 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 = e 0 e 2. 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 = e 0 e 1 e 2 e 2. So again χ(s 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 D 2 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 = e 0 e 1 e 2. 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 ) = = 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 = e 0 e 1 a e 1 b 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 = {T 1, t 2,..., T n } in M such that (1) these triangles cover M, that is n i=1 T i = 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 3

4 without boundary with Euler characteristic χ, every triangulation on M must satisfy 2e = 3f, e = 3(ν χ), ν 1 2 ( χ). The last inequality is obtained by using e ( ν 2). 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 Here, along the circle, opposite arcs are identified in the same direction. 0 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 D 1 and D 2 of M, we can use the above processes to convert these cell decompositions into a finer common cell decomposition D 3 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 D 1 may intersect with an edge from D 2 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(d 1 ) = χ(d 3 ) = χ(d 2 ). 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 4

5 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(c standard ). 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 M 1 and M 2 in which interiors of discs removed are 2-cells. Theorem 1.9. Let M 1, M 2 be surfaces. Then χ(m 1 #M 2 ) = χ(m 1 ) + χ(m 2 ) 2. Proof. Let M i, i = 1, 2, be a surface obtained by removing an open disc from M i. Introduce the same cell decomposition along the boundary circles of M i, say with p 0-cells and p 1- cells. Then extend this cell decomposition to the rest of surfaces with boundary. We have χ(m i ) = χ(m i ) + 1 for i = 1, 2. Now by counting cells in M 1 #M 2, let v i, e i, f i 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(m 1 #M 2 ) = (v 1 +v 2 p) (e 1 +e 2 p)+(f 1 +f 2 ) = χ(m 1 ) + χ(m 2 ) = χ(m 1 ) 1 + χ(m 2 ) 1. This proves the formula. By classification theorem of surfaces, a compact surface without boundary is homeomorphic to either an orientable surface Σ g = gt of genus g 0, or a nonorientable surface N h of genus h 1. By convention, N 0 = Σ 0 = S 2 is the sphere. Proposition The Euler characteristic of a compact surfaces without boundary are given as follows. χ(σ g ) = 2 2g, χ(n h ) = 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 4g-gon with the word a 1 b 1 a 1 1 b 1 1 a g b g a 1 g b 1 g, the induced cell structure on the surface has one 0-cell, 2g 1-cells, and pne 2-cells. Hence χ(σ g ) = 1 2g + 1 = 2 2g. Similar computation works for the genus h nonorientable surface N h. Corollary 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 Σ g of genus g 0. Here, genus 0 surface is a sphere. 5

6 (2) If X is nonorientable and χ(x) = 2 h, then X is a nonorientable surface N h of genus h 1. Note that due to Dyck s identity, we have P #Σ g = (2g + 1)P, K#Σg = (2g + 2)P 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 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) = = 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) = = 2. Hence M = 4P = N Application 2. Surfaces spanning knots and links. Given a knot (or a link) in R 3, 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, 6

7 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 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. (4) For a spanning surface M, let M be the closed surface obtained by capping boundaries, 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 χ( M) = χ(m) + k. If M is a spanning surface of a knot, then k = 1. If M is a spanning surface of a link, then k is the number of components of a link. Use the orientability and the Euler chracteristic to identify the closed surface M as gt or hp. (5) The original surface M can be described as M with k open discs removed. 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) χ(x) is a topological invariant of X. (2) χ(x) = 0 is X is empty. (3) χ(x) = 1 is X is contractible. 7

8 (4) Let X 1 and X 2 be subcomplexes of X. Then χ(x 1 X 2 ) = χ(x 1 ) + χ(x 2 ) χ(x 1 X 2 ). 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) each face has the same number of edges a 3, (2) each vertex has the same valency b 3, (3) two faecs meet along a single edge, at a single vertex, or none at all, (4) 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) Tetrahedron = (3, 3)S with ν = 4, e = 6, f = 4. (2) Cube = (4, 3)S with ν = 8, e = 12, f = 6. (3) Octahedron = (3, 4)S with ν = 6, e = 12, f = 8. (4) Dedecahedron = (5, 3)S with ν = 20, e = 30, f = 12. (5) 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 The Euler characteristic of M is given by af = 2e, νb = 2e. χ(m) = 2e ( 1 a + 1 b 1 ). 2 Proposition 3.4. Platonic solids are the only regular convex polytopes. Proof. Since χ(s 2 ) = 2, we have e( ) = 1. From this, only possibilities are (a, b) = a b 2 (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 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. 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. A vertex of valence b in C becomes a b-gon face. Thus, C = (b, a)m. 8

9 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 f n be the number of faces with n edges. on a trivalent (b = 3) complex on S 2, we have 12 = 3f 3 + 2f 4 + f 5 f 7 2f 8. 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 f 3 and f 4 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 3f 3 = 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 3f 3. Hence we have 3f 3 = 2v. Similarly, we get 4f 4 = 2v. Using these identities, we get v = 12, e = 24, f 3 = 8, f 4 = 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. 9

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