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Classification of Surfaces

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Classification of Surfaces 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 in- vertible function mapping one topological space to another. The inverse of a homeomorphism is also continuous. Two Spaces are said to be homeomor- phic, 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 R2 1 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 S2 = {x ∈ R3 | |x| = 1} with the topology induced by the standard topology of R3 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 R2/Z2. It’s topology comes from the standard topology of R2 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 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 trian- gulation 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 Figure 5: A oriented pair of triangles Definition 1.6 (Orientable). A surface is said to be orientable if all triangu- lations 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 {Bi} be an irreducible finite cover of M consisting of closed balls, B1,B2,...,Bn. Such a cover exists due to the fact that M is compact. Let n [ C = ∂Bn i=1 Also let A be the set of all points in C that do not have Euclidean neighbor- hoods. If A is empty, then M is the sphere. The proof of this goes as follows. A = ∅ implies one of three things: {Bi} 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. 1This theorem was originally proven by Rado [6] . A shorter version is given by Doyle and Moran [1] 4 In general the fact that M has no boundary implies [ ∂Bm ⊂ B ∀ 1 ≤ m ≤ n B∈{Bi}\Bm This means that in case two the boundary of any Bi is completely contained in the interior of some Bj where i 6= j. Because the cover is irreducible, we know that Bi 6= Bj. We have that ∂Bi ⊂ int(Bj) ⇒ ∂Bj ⊂ int(Bi) 2 This in turn gives us Bi ∪ Bj homeomorphic to S . 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 Bi ∪Bj. 2 This gives us a subset of M homeomorphic to S . M is a 2-manifold so ∀ Bm st Bm ∩ (Bi ∪ Bj) 6= ∅ we have either Bm ⊂ (Bi ∪ Bj). There are however no such Bm, because the cover is irreducible. M is connected ⇒ @ Bm st 2 Bm ∩ (Bi ∪ Bj) = ∅. It follows that M is homeomorphic to S . Similarly in the third case, two balls with common boundary are homeo- morphic 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 ∼ S2. 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 0 = 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 0 \ R ∼ M \ (C ∪ D), because C ∩ D 6= ∅ 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 0 that meets each curve in R in exactly two points. By the Jordan-Schoenflies Theorem the complement of R∪V 0 is composed of finitely many open 2-cells. If we thicken each of the arcs of R ∩ V 0 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 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 {Ti} be a triangulation of M consisting of the triangles T1,T2,...,Tn. Let the triangles be ordered such that Ti shares an edge ei with some Tj where 1 ≤ j < i ∀ i. We can do this, because M is connected. Each edge in {Ti} 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.
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