Classification of Surfaces

Transcription

1 Classification of Surfaces Corrin Clarkson REU Project August 17, 2006 One of the simplest topological spaces is that of the surface. There are many reasons why surfaces are nice objects to study. Our natural intuition about space can easily be adopted to this study. Many surfaces can be modeled in three space and so are things we can literally get our hands on. For the most part the surfaces that we can understand intuitively are compact, meaning they don t go off to infinity. Probably the nicest property of compact surfaces is the fact that they can be classified. This is very useful, because it means we know exactly how many different types of compact surfaces there are. The goal of this paper is to present the proof of the classification of compact connected surfaces without boundary as well as some of the key underlying proofs and definitions. 1 A Few Definitions Definition 1.1 (Homeomorphism). A homeomorphism is a continuous invertible function mapping one topological space to another. The inverse of a homeomorphism is also continuous. Two Spaces are said to be homeomorphic, topologically equivalent, if there exists a homeomorphism mapping one to the other. We write A B, if A is homeomorphic to B. Definition 1.2 (Surface). A surface or 2-manifold is a topological space X with the following characteristics: 1. X is Hausdorff 2. x X a neighborhood of x that is homeomorphic to R 2 1

2 Figure 1: The torus After the first pair of edges is associated the square looks like a cylinder. When the second pair is associated we get the torus. Hence the torus can be thought of as the surface of a doughnut. 3. X has a countable basis The following three surfaces are very important. Our goal will be to prove that they are in fact the building blocks of all compact surfaces. Sphere S 2 = {x R 3 x = 1} with the topology induced by the standard topology of R 3 Torus A torus or T 2 is the quotient space of the unit square obtained by the equivalence relations (x, 0) (x, 1) and (0, y) (1, y). See Figure 1. It is also R 2 /Z 2. It s topology comes from the standard topology of R 2 Projective plane A projective plane or RP 2 is the quotient space obtained by associating each point on the sphere with its polar opposite. It can also be thought of as a 2-gon with its edges associated with opposite orientation. See Figure 2 Definition 1.3 (Connected Sum). The connected sum of two surfaces is the space obtained when an open disk is removed from each surface, and the resulting boundaries are mapped together via a continuous function. We write A#B for the connected sum A and B. See Figures 3. The connected sum is a well defined operation. Definition 1.4 (Triangulation). A triangulation of a surface is a finite set of subsets that cover the surface with the following properties. Each subset is homeomorphic to a triangle in the Euclidean plane. Any two subsets are 2

3 Figure 2: The projective plane A sphere with points associated to their polar opposites is homeomorphic to a hemisphere with points on the boundary associated to their polar opposites. This in turn is homeomorphic to a 2-gon with its edges associated with opposite orientation. Figure 3: The connected sum of two tori disjoint or have a single vertex or a single edge in common. (Here the terms vertex and edge refer to those points that map respectively to vertices and edges of a triangle under the homeomorphism.) See Figure 4 for examples for illegal intersections. Definition 1.5 (Euler Characteristic). The Euler Characteristic of a triangulation is the defined by the equation X(M) = V E + F, where V is the number of vertices in a triangulation of M, E the number of edges, and F the number of faces or triangles. The Euler Characteristic is well defined for all surfaces that can be triangulated. This can be rigorously proven, but due to time constraints the reader is asked to take it as a matter of faith. Figure 4: Illegal intersections 3

4 Figure 5: A oriented pair of triangles Definition 1.6 (Orientable). A surface is said to be orientable if all triangulations can be oriented. A triangulation is said to be oriented if an ordering of vertices of each face can be chosen such that the orderings of any two triangles with a common edge induce opposite orderings on the shared edge. See Figure 5 2 Some Theorems For the purposes of this paper let M be an arbitrary, compact, connected 2-manifold without boundary. Theorem 2.1. M can be triangulated. 1 Proof. Let {B i } be an irreducible finite cover of M consisting of closed balls, B 1, B 2,..., B n. Such a cover exists due to the fact that M is compact. Let C = n B n i=1 Also let A be the set of all points in C that do not have Euclidean neighborhoods. If A is empty, then M is the sphere. The proof of this goes as follows. A = implies one of three things: {B i } consists of only a single closed ball, none of the boundaries of the balls intersect, or the only boundary intersections involve balls having their entire boundary in common. The first case is impossible due to the fact that M has no boundary and so can not be covered by a single closed ball. 1 This theorem was originally proven by Rado [6]. A shorter version is given by Doyle and Moran [1] 4

5 In general the fact that M has no boundary implies B m B 1 m n B {B i }\B m This means that in case two the boundary of any B i is completely contained in the interior of some B j where i j. Because the cover is irreducible, we know that B i B j. We have that B i int(b j ) B j int(b i ) This in turn gives us B i B j homeomorphic to S 2. The sphere can be covered by two disks, one slightly larger than the upper hemisphere and the other slightly larger than the lower hemisphere. The boundary of each is completely contained in the other. This covering is homeomorphic to B i B j. This gives us a subset of M homeomorphic to S 2. M is a 2-manifold so B m st B m (B i B j ) we have either B m (B i B j ). There are however no such B m, because the cover is irreducible. M is connected B m st B m (B i B j ) =. It follows that M is homeomorphic to S 2. Similarly in the third case, two balls with common boundary are homeomorphic to the covering of the sphere that consists simply of the upper and lower hemispheres. This leads to M being homeomorphic to the sphere in this case as well. Therefore A = M S 2. The sphere is easily triangulated. An example of a triangulation of the sphere is the tetrahedron. Therefore if A is empty, M can be triangulated. If A is nonempty, we cover it with a closed 2-cell, D. We can do this, because A is compact and totally disconnected. M = M/D M, because modding out by a 2-cell doesn t change the topology. Let R be the image of C under the quotient map. This implies that M \ R M \ (C D), because C D by definition of D. R is the one-point union of countably many simple closed curves. We also have that all points in R except for the one that corresponds to the image of D have Euclidean neighborhoods. Let p be the point corresponding to the image of D. Now let V be a 2-cell neighborhood of p. V covers all but finitely many simple closed curves in R, by the compactness of R. Therefore we can assume R is the one point union of finitely many simple closed curves, because we can always mod out by V if necessary. This in turn implies that there exists a 2-cell neighborhood of p, V that meets each curve in R in exactly two points. By the Jordan-Schoenflies Theorem the complement of R V is composed of finitely many open 2-cells. If we thicken each of the arcs of R V we obtain a surface which we can triangulate. This triangulation is then extended to the open 2-cells of the complement. Thus M can be triangulated. 5

6 Figure 6: Triangulation proof on the torus (A) A torus covered by finitely many closed balls. (B) A two cell that covers the points with non-euclidean neighborhoods. (C) Modding out by the 2-cell (D) The neighborhood of p (E) The union of the boundaries with the neighbor hood of p (F) The set on a torus, the open 2-cells of the complement Corollary 2.2. M is homeomorphic to a polygon, P with edges glued together in pairs. Proof. By Theorem 2.1 we have shown that M can be triangulated. Let {T i } be a triangulation of M consisting of the triangles T 1, T 2,..., T n. Let the triangles be ordered such that T i shares an edge e i with some T j where 1 j < i i. We can do this, because M is connected. Each edge in {T i } belongs to exactly two triangles. This is due to M being a 2-manifold without boundary. If there were and edge that belonged to only one triangle, it would be a boundary component of M. If there were an edge that belonged to three 6

7 Figure 7: The normal form for the sphere or more triangles, the points belonging to it would not have neighborhoods that were homeomorphic to an open disk. Therefore if an edge belonged to three or more triangles M would not be a manifold. This implies that the space obtained by associating all the e i is a polygon with its edges paired. This can be easily seen if you look at the process step by step. First start with T 1 and then add T 2 at edge e 2. The resulting space is homeomorphic to a disk. The space remains homeomorphic to the disk as each triangle is added, because there are never more than two triangles meeting at a single edge. The space that results when all the e i have been associated is homeomorphic to a disk and so must be a polygon. Each edge of this polygon is an edge of a triangle in the triangulation and so must be associated with exactly one other edge. Therefore M is homeomorphic to a polygon with its edges associated in pairs. Remark 2.3. From this point on when referring to pair of edges will mean only a set of two edges that are identified with one another. For the classification theorem we will need a normal form for the sphere, the projective plane and the torus. We will also need a normal form for the connected sum of these surfaces. The normal form for the torus and the projective plane have already been mentioned. That of the torus is a square with opposite sides associated with the same orientation. The normal form of the projective plane is the 2-gon with its edges associated with opposite orientation, as shown in Figure 2. The normal form for the sphere is similar to that of the projective plane. It is a 2-gon with its edges associated with the same orientation. See Figure 7. The normal form for the connected sums is obtained by summing the normal forms as shown in Figures 8 and 9. We assign a letter to each pair of edges. If an edge is oriented counterclockwise we write the letter as its inverse. Thus we can now represent each normal form with a sequence of letters, for example the normal form for a torus would be written aba 1 b 1. 7

8 Figure 8: The normal form for the connected sum of two tori Figure 9: The normal form for the connected sum of two RP 2 Similarly the normal forms for the sphere and the projective plane are written aa 1 and aa respectively. It is fairly easy to see that there is a connection between the normal forms of surfaces and that of their connected sums. We would write aabbcc for the connected sum of three RP 2 and aba 1 b 1 cdc 1 d 1 for the connected sum of two tori. One would like to infer from this that we could write aabcb 1 c 1 for the connected sum of a torus and a projective plane. In turns out however that this is the same as the connected sum of three RP 2 and so should have the same normal form. In general we will call a polygon of normal form if it is the normal form for one of the following figures, S 2, T 2, RP 2, the connected sum of n tori, or the connected sum of n projective planes. Our goal is to show that these are in fact all connected compact surfaces without boundary. Lemma 2.4. The connected sum of three RP 2 is homeomorphic to the connected sum of RP 2 and T 2. Proof. The proof of this lemma is easily accomplished by taking the normal form of the connected sum of three RP 2 and cutting/gluing it so that it is in the normal form for RP 2 #T 2. The process is laid out in Figure 10. 8

9 Figure 10: The connected sum of three RP 2 (RP 2 #T 2 ) The dotted lies show where cuts are to occur and the grey lines show where edges have been glued. Theorem 2.5. M is homeomorphic to a polyhedron of normal form. 2 Proof. By Corollary 2.2 M is homeomorphic to a polyhedron P with edges identified in pairs. Below is a step by step method for transforming P into a normal form while preserving its topological properties. 1. If P has 4 edges, then eliminate any pair that consists of adjacent edges of opposite orientation. 2. We will call two vertices equivalent if they are identified with the same point. Our goal is to make all the vertices of P equivalent. The method to accomplish this is the following. Assume step 1 has been completed. If P has 2 edges it is either the projective plane or the sphere, both of which are normal forms. If it has 4 edges, then we can assume there are no pairs of consisting of adjacent edges of opposite orientation. We will now further examine the case where P has 4 edges. If all the vertices are not equivalent, then there must be a pair A, B such that A and B are not equivalent, and they are separated by exactly one edge, x. Let y be the edge on the other side of A. If x and y form a pair and have the same orientation, then A must be equivalent to B. If they were a pair of opposite orientation, then they would have been eliminated by step 1. Therefore the two edges x and y do not form a pair. There must then be some other edge y of P that is identified with y. 2 This proof can be found in both Massy [3] and Seifert-Threlfall [5] 9

10 Now cut along a line from B to the vertex of y that is not A, and then glue y to y. The resulting polygon is still homeomorphic to M and has all its edges paired. It also has one less vertex in the equivalence class of A and one more in that of B. We can continue this process, returning to step 1 whenever possible, until all the vertices are in the same equivalence class or P has been reduced to a sphere. 3. As the normal form requires that pairs of the same orientation be adjacent, we are in need of a way to bring such a pair into this position. The following procedure is an example of how this can be done. If P has a pair of nonadjacent edges a, a of the same orientation, then cut along the line, d, connecting the first point of a to the first point of a. Now all that remains is to glue a and a together. The resulting polygon has a pair d and d that are adjacent. As this proceeder does not separate any adjacent pairs, it can be repeated until all pairs of the same orientation are adjacent. As before repeat step 1 when possible. If at this point there are no pairs of opposite orientation, P is in a normal form. 4. We will now work through the case where there are pairs of opposite orientation. Given one pair of opposite orientation a, a 1, there must be a second pair of opposite orientation b, b 1 such that these four edges are in the following arrangement, a... b... a 1... b. The proof of this statement follows from the simplification accomplished by the previous steps. From step 1 we have that there must be at least one edge between a and a 1. As a and a 1 are of opposite orientation, the edges of P can be divided into two groups, those that lie between the fist points of a and a 1 and those that lie between the last. From step 3 we have that any pair of the same orientation can not be split between these two groups, but must be fully contained in one. Therefore if there is no pair of opposite orientation that is split between the two groups, all pairs must be fully contained in one group or the other. This implies however that the first point of a can not be identified with the last point of a, a contradiction of step 2. Therefore there must be a pair b, b 1 of the given arrangement. In order to achieve the normal form aba 1 b 1 cut along the line, c, connecting the first points of a and a 1 and then glue b and b 1 together. 10

11 After this we cut along the line, d, connecting the first points of c and c and glue a and a 1 together. The resulting polygon is homeomorphic to M and has edges in the following arrangement cdc d. As this procedure doesn t disrupt the arrangements of other such pairs it can be repeated until all pairs of opposite orientation are in this arrangement. Once again we repeat step 1 whenever possible. If at this point there are no pairs of the same orientation P is in a normal form. 5. After completion of step 4 P is reduced to the connect sum of projective planes and tori. From Theorem 2.4 we have that the connect sum of three RP 2 is homeomorphic to the connect sum of a torus and a projective plane. Therefore if P has pairs of the same orientation, it can be simplified to a normal form by converting the tori into pairs of projective planes. After completion of the above procedure P is in the normal form. As each step preserves the topology of P, it follows that M is homeomorphic to a polyhedron in normal form. Lemma 2.6. The Euler Characteristic of the connect sum of two compact connected surfaces without boundary, L and N, is two less than the sum of the Euler Characteristics of L and N, ie X(L#N) = X(L) + X(N) 2 Proof. Let L and N be triangulated. Let T L and T N be triangles in the triangulations of L and N respectively. Both T L and T N are homeomorphic to triangles in the plane by definition. This implies that they are both homeomorphic to closed disks. We now remove the interior of T L and that of T N from L and N respectively. Lastly associate the edges of the two triangles in pairs. The resulting space is L#N. Its Euler Characteristic is X(L#N) = (V (L) + V (N) 3) (E(L) + E(N) 3) + (F (L) + F (N) 2) = V (L) + V (N) E(L) E(N) + F (L) + F (N) 2 = X(L) + X(N) 2 Because both the Euler Characteristic and the connected sum are well defined, this is enough to show that the lemma always holds. 11

12 Lemma 2.7. Each orientable normal form has a different Euler Characteristic. Also each non-orientable normal form has a different Euler Characteristic. Proof. The connected sum of tori is always orientable, because the torus is orientable. By Lemma 2.6 the Euler Characteristic of the connected sum of n tori is n X(T ) 2n, and hence is different for all n. Similarly the connected sum of projective planes is never orientable, because the projective plane is not orientable. We have that the Euler Characteristic of the connected sum of n projective planes is n X(RP 2 ) 2n, by Lemma 2.6. Therefore the Euler Characteristic of each non-orientable normal form is different. Theorem 2.8. If two compact connected surfaces without boundary are homeomorphic, then they have the same Euler Characteristic. Proof. Let L and N be homeomorphic compact connected surfaces without boundary. By Theorem 2.1 L can be triangulated. Let ψ : L N be the homeomorphism form L to N. ψ is subjective by the definition of a homeomorphism. This implies that the image of the triangulation of L under ψ covers N. The images of the triangles must be homeomorphic to triangles in R 2, because is a transitive relation. Similarly ψ must map vertices to vertices and edges to edges. ψ is also injective by definition. This implies that the number of triangles in the image is the same as the number in the triangulation. Similarly the number of edges and faces must be the same. Therefore L and N must have the same Euler Characteristic. Corollary 2.9. M is homeomorphic to exactly one surface of normal form. Proof. An orientable surface can not be homeomorphic to a non-orientable surface. This is do to the fact that for any triangulation of a surface there exists a corresponding triangulation for any homeomorphic surface, as shown in the proof of Theorem 2.8. Therefore if you can orient every triangulation of a surface you can do the same for any homeomorphic surface. By Lemma 2.7 all the orientable normal forms have different Euler Characteristic. By Theorem 2.8 homeomorphic surfaces have the same Euler Characteristic. Therefore each orientable normal form is unique. Similarly each non-orientable normal form is unique. By Theorem 2.5 we have that M is homeomorphic to a surface of normal form. This implies that M is homeomorphic to exactly 12

13 one surface of normal form, because we have just shown that all such surfaces are unique. We have obtained a full classification of compact, connected surfaces without boundary. By Corollary 2.9 we have that all such surfaces are either the sphere or the connected sum of n tori or the connected sum of n projective planes. This classification can easily be extended to compact surfaces without boundary that are not connected, as such surfaces must simply be unions of finitely many compact connected surfaces without boundary. It turns out that compact surfaces with boundary also lone themselves to classification. The ease with which 2-manifolds can be classified might lead one to hope that such a classification is possible for all n-manifolds. This is however not the case, with higher dimension the classification theorem get increasingly difficult. Advances in the classification of 3-manifolds and 4-manifolds aare still in progress.. References [1] P. H. Doyle and D. A. Moran A short proof that compact 2-manifolds can be triangulated Inventiones Mathematicae 5: , Springer, Berlin [2] Klaus Jänich Topology trans. Silvio Levy, Springer-Verlag, Berlin [3] William S. Massey Algebraic Topology: An Introduction Springer- Verlag, Berlin [4] James R. Munkres Topology 2 nd Ed., Prentice Hall, Upper Saddle River, NJ [5] H. Seifert and W. Threlfall Lehrbuch der Topologie Chelsea Publishing Company, New York [6] Radó, T Über den Begriff der Riemannschen Fläche Acts. Litt. Sci. Szeged. 2, (1925) 13